libstdc++-v3/doc/xml/manual/test_policy_data_structures.xml

   1 <section xmlns="http://docbook.org/ns/docbook" version="5.0"
   2          xml:id="pbds.test" xreflabel="Test">
   3   <info><title>Testing</title></info>
   4   <?dbhtml filename="policy_based_data_structures_test.html"?>
   5
   6   <!-- S01 regression -->
   7   <section xml:id="pbds.test.regression">
   8     <info><title>Regression</title></info>
   9
  10     <para>The library contains a single comprehensive regression test.
  11     For a given container type in this library, the test creates
  12     an object of the container type and an object of the
  13     corresponding standard type (e.g., <classname>std::set</classname>). It
  14     then performs a random sequence of methods with random
  15     arguments (e.g., inserts, erases, and so forth) on both
  16     objects. At each operation, the test checks the return value of
  17     the method, and optionally both compares this library's
  18     object with the standard's object as well as performing other
  19     consistency checks on this library's object (e.g.,
  20     order preservation, when applicable, or node invariants, when
  21     applicable).</para>
  22
  23     <para>Additionally, the test integrally checks exception safety
  24     and resource leaks. This is done as follows. A special
  25     allocator type, written for the purpose of the test, both
  26     randomly throws an exceptions when allocations are performed,
  27     and tracks allocations and de-allocations. The exceptions thrown
  28     at allocations simulate memory-allocation failures; the
  29     tracking mechanism checks for memory-related bugs (e.g.,
  30     resource leaks and multiple de-allocations). Both
  31     this library's containers and the containers' value-types are
  32     configured to use this allocator.</para>
  33
  34     <para>For granularity, the test is split into the
  35     several sources, each checking only some containers.</para>
  36
  37     <para>For more details, consult the files in
  38     <filename class="directory">testsuite/ext/pb_ds/regression</filename>.
  39     </para>
  40   </section>
  41
  42   <!-- S02 performance -->
  43   <section xml:id="pbds.test.performance">
  44     <info><title>Performance</title></info>
  45
  46     <section xml:id="performance.hash">
  47       <info><title>Hash-Based</title></info>
  48       <para></para>
  49
  50       <!-- 01 <a href="hash_text_find_find_timing_test"> -->
  51       <section xml:id="performance.hash.text_find">
  52         <info><title>
  53           Text <function>find</function>
  54         </title></info>
  55         <para></para>
  56
  57         <section xml:id="hash.text_find.info">
  58           <info><title>
  59             Description
  60           </title></info>
  61
  62           <para>
  63             This test inserts a number of values with keys from an
  64             arbitrary text (<xref
  65             linkend="biblio.wickland96thirty"/>) into a container,
  66             then performs a series of finds using
  67             <function>find</function> . It measures the average
  68             time for <function>find</function> as a function of
  69           the number of values inserted.</para>
  70           <para>
  71             It uses the test file:
  72             <filename>performance/ext/pb_ds/text_find_timing_test.cc</filename>
  73           </para>
  74
  75           <para>
  76             And uses the data file:
  77             <filename>filethirty_years_among_the_dead_preproc.txt</filename>
  78           </para>
  79
  80           <para>The test checks the effect of different range-hashing
  81           functions, trigger policies, and cache-hashing policies.
  82           </para>
  83
  84         </section>
  85
  86         <section xml:id="hash.text_find.results">
  87           <info><title>
  88             Results
  89           </title></info>
  90
  91           <para>The graphic below show the results for the native
  92           and collision-chaining hash types the function
  93           applied being a text find timing test using
  94           <function>find</function>.
  95           </para>
  96
  97           <!-- results graphic -->
  98           <informalfigure>
  99             <mediaobject>
 100               <imageobject>
 101                 <imagedata align="center" format="PDF" scale="75"
 102                            fileref="../images/pbds_hash_text_find.pdf"/>
 103               </imageobject>
 104               <imageobject>
 105                 <imagedata align="center" format="PNG" scale="100"
 106                            fileref="../images/pbds_hash_text_find.png"/>
 107               </imageobject>
 108             </mediaobject>
 109           </informalfigure>
 110
 111           <para>
 112             The abbreviated names in the legend of the graphic above are
 113             instantiated with the types in the following table.
 114           </para>
 115
 116
 117           <informaltable frame="all">
 118
 119             <tgroup cols="5" align="left" colsep="1" rowsep="1">
 120               <colspec colname="c1"/>
 121               <colspec colname="c2"/>
 122               <colspec colname="c3"/>
 123               <colspec colname="c4"/>
 124               <colspec colname="c5"/>
 125               <thead>
 126                 <row>
 127                   <entry><emphasis>Name/Instantiating Type</emphasis></entry>
 128                   <entry><emphasis>Parameter</emphasis></entry>
 129                   <entry><emphasis>Details</emphasis></entry>
 130                   <entry><emphasis>Parameter</emphasis></entry>
 131                   <entry><emphasis>Details</emphasis></entry>
 132                 </row>
 133               </thead>
 134
 135               <tbody>
 136
 137                 <!-- native -->
 138                 <row>
 139                   <?dbhtml bgcolor="#B0B0B0" ?>
 140                   <entry namest="c1" nameend="c5">
 141                     n_hash_map_ncah
 142                   </entry>
 143                 </row>
 144
 145                 <row>
 146                   <entry>
 147                     <classname>std::tr1::unordered_map</classname>
 148                   </entry>
 149                   <entry>
 150                     <classname>cache_hash_code</classname>
 151                   </entry>
 152                   <entry>
 153                     <constant>false</constant>
 154                   </entry>
 155                   <entry namest="c4" nameend="c5"></entry>
 156                 </row>
 157
 158                 <!-- hash 01 -->
 159                 <row>
 160                   <?dbhtml bgcolor="#B0B0B0" ?>
 161                   <entry namest="c1" nameend="c5">
 162                     cc_hash_mod_prime_1div1_nsth_map
 163                   </entry>
 164                 </row>
 165
 166                 <row>
 167                   <entry morerows="2" valign="top">
 168                     <classname>cc_hash_table</classname>
 169                   </entry>
 170                   <entry>
 171                     <classname>Comb_Hash_Fn</classname>
 172                   </entry>
 173                   <entry>
 174                     <classname>direct_mod_range_hashing</classname>
 175                   </entry>
 176                   <entry namest="c4" nameend="c5"></entry>
 177                 </row>
 178
 179                 <row>
 180                   <entry morerows="1" valign="top">
 181                     <classname>Resize_Policy</classname>
 182                   </entry>
 183                   <entry morerows="1" valign="top">
 184                     <classname>hash_standard_resize_policy</classname>
 185                   </entry>
 186                   <entry>
 187                     <classname>Size_Policy</classname>
 188                   </entry>
 189                   <entry>
 190                     <classname>hash_prime_size_policy</classname>
 191                   </entry>
 192                 </row>
 193
 194                 <row>
 195                   <entry valign="top">
 196                     <classname>Trigger_Policy</classname>
 197                   </entry>
 198                   <entry>
 199                     <classname>hash_load_check_resize_trigger</classname> with
 200                     α<subscript>min</subscript> = 1/8 and α<subscript>max</subscript> = 1/1
 201                   </entry>
 202                 </row>
 203
 204                 <!-- hash 02 -->
 205                 <row>
 206                   <?dbhtml bgcolor="#B0B0B0" ?>
 207                   <entry namest="c1" nameend="c5">
 208                     cc_hash_mask_exp_1div2_sth_map
 209                   </entry>
 210                 </row>
 211
 212                 <row>
 213                   <entry morerows="2" valign="top">
 214                     <classname>
 215                       cc_hash_table
 216                     </classname>
 217                   </entry>
 218                   <entry>
 219                     <classname>Comb_Hash_Fn</classname>
 220                   </entry>
 221                   <entry>
 222                     <classname>direct_mask_range_hashing</classname>
 223                   </entry>
 224                   <entry namest="c4" nameend="c5"></entry>
 225                 </row>
 226
 227                 <row>
 228                   <entry morerows="1" valign="top">
 229                     <classname>Resize_Policy</classname>
 230                   </entry>
 231                   <entry morerows="1" valign="top">
 232                     <classname>hash_standard_resize_policy</classname>
 233                   </entry>
 234                   <entry>
 235                     <classname>Size_Policy</classname>
 236                   </entry>
 237                   <entry>
 238                     <classname>hash_exponential_size_policy</classname>
 239                   </entry>
 240                 </row>
 241
 242                 <row>
 243                   <entry valign="top">
 244                     <classname>Trigger_Policy</classname>
 245                   </entry>
 246                   <entry>
 247                     <classname>hash_load_check_resize_trigger</classname> with
 248                     α<subscript>min</subscript> = 1/8 and α<subscript>max</subscript> = 1/2
 249                   </entry>
 250                 </row>
 251
 252                 <!-- hash 03 -->
 253                 <row>
 254                   <?dbhtml bgcolor="#B0B0B0" ?>
 255                   <entry namest="c1" nameend="c5">
 256                     cc_hash_mask_exp_1div1_nsth_map
 257                   </entry>
 258                 </row>
 259
 260                 <row>
 261                   <entry morerows="2" valign="top">
 262                     <classname>cc_hash_table</classname>
 263                   </entry>
 264                   <entry>
 265                     <classname>Comb_Hash_Fn</classname>
 266                   </entry>
 267                   <entry>
 268                     <classname>direct_mask_range_hashing</classname>
 269                   </entry>
 270                   <entry namest="c4" nameend="c5"></entry>
 271                 </row>
 272
 273                 <row>
 274                   <entry morerows="1" valign="top">
 275                     <classname>Resize_Policy</classname>
 276                   </entry>
 277                   <entry morerows="1" valign="top">
 278                     <classname>hash_standard_resize_policy</classname>
 279                   </entry>
 280                   <entry>
 281                     <classname>Size_Policy</classname>
 282                   </entry>
 283                   <entry>
 284                     <classname>hash_exponential_size_policy</classname>
 285                   </entry>
 286                 </row>
 287
 288                 <row>
 289                   <entry valign="top">
 290                     <classname>Trigger_Policy</classname>
 291                   </entry>
 292                   <entry>
 293                     <classname>hash_load_check_resize_trigger</classname> with
 294                     α<subscript>min</subscript> = 1/8 and α<subscript>max</subscript> = 1/1
 295                   </entry>
 296                 </row>
 297
 298                 <!-- hash 04 -->
 299                 <row>
 300                   <?dbhtml bgcolor="#B0B0B0" ?>
 301                   <entry namest="c1" nameend="c5">
 302                     cc_hash_mask_exp_1div2_nsth_map
 303                   </entry>
 304                 </row>
 305                 <row>
 306                   <entry morerows="2" valign="top">
 307                     <classname>cc_hash_table</classname>
 308                   </entry>
 309                   <entry>
 310                     <classname>Comb_Hash_Fn</classname>
 311                   </entry>
 312                   <entry>
 313                     <classname>direct_mask_range_hashing</classname>
 314                   </entry>
 315                   <entry namest="c4" nameend="c5"></entry>
 316                 </row>
 317
 318                 <row>
 319                   <entry morerows="1" valign="top">
 320                     <classname>Resize_Policy</classname>
 321                   </entry>
 322                   <entry morerows="1" valign="top">
 323                     <classname>hash_standard_resize_policy</classname>
 324                   </entry>
 325                   <entry>
 326                     <classname>Size_Policy</classname>
 327                   </entry>
 328                   <entry>
 329                     <classname>hash_exponential_size_policy</classname>
 330                   </entry>
 331                 </row>
 332
 333                 <row>
 334                   <entry valign="top">
 335                     <classname>Trigger_Policy</classname>
 336                   </entry>
 337                   <entry>
 338                     <classname>hash_load_check_resize_trigger</classname> with
 339                     α<subscript>min</subscript> = 1/8 and α<subscript>max</subscript> = 1/2
 340                   </entry>
 341                 </row>
 342
 343               </tbody>
 344             </tgroup>
 345           </informaltable>
 346
 347         </section>
 348
 349         <section xml:id="hash.text_find.observations">
 350           <info><title>
 351             Observations
 352           </title></info>
 353
 354           <para>In this setting, the range-hashing scheme affects performance
 355           more than other policies. As the results show, containers using
 356           mod-based range-hashing (including the native hash-based container,
 357           which is currently hard-wired to this scheme) have lower performance
 358           than those using mask-based range-hashing. A modulo-based
 359           range-hashing scheme's main benefit is that it takes into account
 360           all hash-value bits. Standard string hash-functions are designed to
 361           create hash values that are nearly-uniform as is (<xref
 362           linkend="biblio.knuth98sorting"/>).</para>
 363
 364           <para>Trigger policies, i.e. the load-checks constants, affect
 365           performance to a lesser extent.</para>
 366
 367           <para>Perhaps surprisingly, storing the hash value alongside each
 368           entry affects performance only marginally, at least in this
 369           library's implementation. (Unfortunately, it was not possible to run
 370           the tests with <classname>std::tr1::unordered_map</classname> 's
 371           <classname>cache_hash_code = true</classname> , as it appeared to
 372           malfuntion.)</para>
 373
 374         </section>
 375
 376       </section>
 377
 378       <!-- 02 <a href="hash_int_find_timing_test"> -->
 379       <section xml:id="performance.hash.int_find">
 380         <info><title>
 381           Integer <function>find</function>
 382         </title></info>
 383         <para></para>
 384
 385         <section xml:id="hash.int_find.info">
 386           <info><title>
 387             Description
 388           </title></info>
 389
 390           <para>This test inserts a number of values with uniform
 391           integer keys into a container, then performs a series of finds
 392           using <function>find</function>. It measures the average time
 393           for <function>find</function> as a function of the number of values
 394           inserted.</para>
 395
 396           <para>
 397             It uses the test file:
 398             <filename>performance/ext/pb_ds/random_int_find_timing.cc</filename>
 399           </para>
 400
 401           <para>The test checks the effect of different underlying
 402           hash-tables,
 403           range-hashing functions, and trigger policies.</para>
 404
 405         </section>
 406
 407         <section xml:id="hash.int_find.results">
 408           <info><title>
 409             Results
 410           </title></info>
 411
 412           <para>
 413             There are two sets of results for this type, one for
 414             collision-chaining hashes, and one for general-probe hashes.
 415           </para>
 416
 417           <para>The first graphic below shows the results for the native and
 418           collision-chaining hash types. The function applied being a random
 419           integer timing test using <function>find</function>.
 420           </para>
 421
 422           <!-- results graphic 01 -->
 423           <informalfigure>
 424             <mediaobject>
 425               <imageobject>
 426                 <imagedata align="center" format="PDF" scale="75"
 427                            fileref="../images/pbds_cc_hash_int_find.pdf"/>
 428               </imageobject>
 429               <imageobject>
 430                 <imagedata align="center" format="PNG" scale="100"
 431                            fileref="../images/pbds_cc_hash_int_find.png"/>
 432               </imageobject>
 433             </mediaobject>
 434           </informalfigure>
 435
 436           <para>
 437             The abbreviated names in the legend of the graphic above are
 438             instantiated with the types in the following table.
 439           </para>
 440
 441
 442           <informaltable frame="all">
 443
 444             <tgroup cols="5" align="left" colsep="1" rowsep="1">
 445               <colspec colname="c1"/>
 446               <colspec colname="c2"/>
 447               <colspec colname="c3"/>
 448               <colspec colname="c4"/>
 449               <colspec colname="c5"/>
 450               <thead>
 451                 <row>
 452                   <entry><emphasis>Name/Instantiating Type</emphasis></entry>
 453                   <entry><emphasis>Parameter</emphasis></entry>
 454                   <entry><emphasis>Details</emphasis></entry>
 455                   <entry><emphasis>Parameter</emphasis></entry>
 456                   <entry><emphasis>Details</emphasis></entry>
 457                 </row>
 458               </thead>
 459
 460               <tbody>
 461
 462                 <!-- native -->
 463                 <row>
 464                   <?dbhtml bgcolor="#B0B0B0" ?>
 465                   <entry namest="c1" nameend="c5">
 466                     n_hash_map_ncah
 467                   </entry>
 468                 </row>
 469
 470                 <row>
 471                   <entry>
 472                     <classname>std::tr1::unordered_map</classname>
 473                   </entry>
 474                   <entry>
 475                     <classname>cache_hash_code</classname>
 476                   </entry>
 477                   <entry>
 478                     <constant>false</constant>
 479                   </entry>
 480                   <entry namest="c4" nameend="c5"></entry>
 481                 </row>
 482
 483                 <!-- hash 01 -->
 484                 <row>
 485                   <?dbhtml bgcolor="#B0B0B0" ?>
 486                   <entry namest="c1" nameend="c5">
 487                     cc_hash_mod_prime_1div1_nsth_map
 488                   </entry>
 489                 </row>
 490
 491                 <row>
 492                   <entry morerows="2" valign="top">
 493                     <classname>cc_hash_table</classname>
 494                   </entry>
 495                   <entry>
 496                     <classname>Comb_Hash_Fn</classname>
 497                   </entry>
 498                   <entry>
 499                     <classname>direct_mod_range_hashing</classname>
 500                   </entry>
 501                   <entry namest="c4" nameend="c5"></entry>
 502                 </row>
 503
 504                 <row>
 505                   <entry morerows="1" valign="top">
 506                     <classname>Resize_Policy</classname>
 507                   </entry>
 508                   <entry morerows="1" valign="top">
 509                     <classname>hash_standard_resize_policy</classname>
 510                   </entry>
 511                   <entry>
 512                     <classname>Size_Policy</classname>
 513                   </entry>
 514                   <entry>
 515                     <classname>hash_prime_size_policy</classname>
 516                   </entry>
 517                 </row>
 518
 519                 <row>
 520                   <entry valign="top">
 521                     <classname>Trigger_Policy</classname>
 522                   </entry>
 523                   <entry>
 524                     <classname>hash_load_check_resize_trigger</classname> with
 525                     α<subscript>min</subscript> = 1/8 and α<subscript>max</subscript> = 1/1
 526                   </entry>
 527                 </row>
 528
 529                 <!-- hash 02 -->
 530                 <row>
 531                   <?dbhtml bgcolor="#B0B0B0" ?>
 532                   <entry namest="c1" nameend="c5">
 533                     cc_hash_mod_prime_1div2_nsth_map
 534                   </entry>
 535                 </row>
 536
 537                 <row>
 538                   <entry morerows="2" valign="top">
 539                     <classname>
 540                       cc_hash_table
 541                     </classname>
 542                   </entry>
 543                   <entry>
 544                     <classname>Comb_Hash_Fn</classname>
 545                   </entry>
 546                   <entry>
 547                     <classname>direct_mod_range_hashing</classname>
 548                   </entry>
 549                   <entry namest="c4" nameend="c5"></entry>
 550                 </row>
 551
 552                 <row>
 553                   <entry morerows="1" valign="top">
 554                     <classname>Resize_Policy</classname>
 555                   </entry>
 556                   <entry morerows="1" valign="top">
 557                     <classname>hash_standard_resize_policy</classname>
 558                   </entry>
 559                   <entry>
 560                     <classname>Size_Policy</classname>
 561                   </entry>
 562                   <entry>
 563                     <classname>hash_prime_size_policy</classname>
 564                   </entry>
 565                 </row>
 566
 567                 <row>
 568                   <entry valign="top">
 569                     <classname>Trigger_Policy</classname>
 570                   </entry>
 571                   <entry>
 572                     <classname>hash_load_check_resize_trigger</classname> with
 573                     α<subscript>min</subscript> = 1/8 and α<subscript>max</subscript> = 1/2
 574                   </entry>
 575                 </row>
 576
 577                 <!-- hash 03 -->
 578                 <row>
 579                   <?dbhtml bgcolor="#B0B0B0" ?>
 580                   <entry namest="c1" nameend="c5">
 581                     cc_hash_mask_exp_1div1_nsth_map
 582                   </entry>
 583                 </row>
 584
 585                 <row>
 586                   <entry morerows="2" valign="top">
 587                     <classname>cc_hash_table</classname>
 588                   </entry>
 589                   <entry>
 590                     <classname>Comb_Hash_Fn</classname>
 591                   </entry>
 592                   <entry>
 593                     <classname>direct_mask_range_hashing</classname>
 594                   </entry>
 595                   <entry namest="c4" nameend="c5"></entry>
 596                 </row>
 597
 598                 <row>
 599                   <entry morerows="1" valign="top">
 600                     <classname>Resize_Policy</classname>
 601                   </entry>
 602                   <entry morerows="1" valign="top">
 603                     <classname>hash_standard_resize_policy</classname>
 604                   </entry>
 605                   <entry>
 606                     <classname>Size_Policy</classname>
 607                   </entry>
 608                   <entry>
 609                     <classname>hash_exponential_size_policy</classname>
 610                   </entry>
 611                 </row>
 612
 613                 <row>
 614                   <entry valign="top">
 615                     <classname>Trigger_Policy</classname>
 616                   </entry>
 617                   <entry>
 618                     <classname>hash_load_check_resize_trigger</classname> with
 619                     α<subscript>min</subscript> = 1/8 and α<subscript>max</subscript> = 1/1
 620                   </entry>
 621                 </row>
 622
 623                 <!-- hash 04 -->
 624                 <row>
 625                   <?dbhtml bgcolor="#B0B0B0" ?>
 626                   <entry namest="c1" nameend="c5">
 627                     cc_hash_mask_exp_1div2_nsth_map
 628                   </entry>
 629                 </row>
 630                 <row>
 631                   <entry morerows="2" valign="top">
 632                     <classname>cc_hash_table</classname>
 633                   </entry>
 634                   <entry>
 635                     <classname>Comb_Hash_Fn</classname>
 636                   </entry>
 637                   <entry>
 638                     <classname>direct_mask_range_hashing</classname>
 639                   </entry>
 640                   <entry namest="c4" nameend="c5"></entry>
 641                 </row>
 642
 643                 <row>
 644                   <entry morerows="1" valign="top">
 645                     <classname>Resize_Policy</classname>
 646                   </entry>
 647                   <entry morerows="1" valign="top">
 648                     <classname>hash_standard_resize_policy</classname>
 649                   </entry>
 650                   <entry>
 651                     <classname>Size_Policy</classname>
 652                   </entry>
 653                   <entry>
 654                     <classname>hash_exponential_size_policy</classname>
 655                   </entry>
 656                 </row>
 657
 658                 <row>
 659                   <entry valign="top">
 660                     <classname>Trigger_Policy</classname>
 661                   </entry>
 662                   <entry>
 663                     <classname>hash_load_check_resize_trigger</classname> with
 664                     α<subscript>min</subscript> = 1/8 and α<subscript>max</subscript> = 1/2
 665                   </entry>
 666                 </row>
 667
 668               </tbody>
 669             </tgroup>
 670           </informaltable>
 671
 672           <para>
 673           </para>
 674
 675           <para>
 676           </para>
 677
 678           <para>And the second graphic shows the results for the native and
 679           general-probe hash types. The function applied being a random
 680           integer timing test using <function>find</function>.
 681           </para>
 682
 683           <!-- results graphic 02 -->
 684           <informalfigure>
 685             <mediaobject>
 686               <imageobject>
 687                 <imagedata align="center" format="PDF" scale="75"
 688                            fileref="../images/pbds_gp_hash_int_find.pdf"/>
 689               </imageobject>
 690               <imageobject>
 691                 <imagedata align="center" format="PNG" scale="100"
 692                            fileref="../images/pbds_gp_hash_int_find.png"/>
 693               </imageobject>
 694             </mediaobject>
 695           </informalfigure>
 696
 697           <para>
 698             The abbreviated names in the legend of the graphic above are
 699             instantiated with the types in the following table.
 700           </para>
 701
 702
 703           <informaltable frame="all">
 704
 705             <tgroup cols="5" align="left" colsep="1" rowsep="1">
 706               <colspec colname="c1"/>
 707               <colspec colname="c2"/>
 708               <colspec colname="c3"/>
 709               <colspec colname="c4"/>
 710               <colspec colname="c5"/>
 711               <thead>
 712                 <row>
 713                   <entry><emphasis>Name/Instantiating Type</emphasis></entry>
 714                   <entry><emphasis>Parameter</emphasis></entry>
 715                   <entry><emphasis>Details</emphasis></entry>
 716                   <entry><emphasis>Parameter</emphasis></entry>
 717                   <entry><emphasis>Details</emphasis></entry>
 718                 </row>
 719               </thead>
 720
 721               <tbody>
 722
 723                 <!-- native -->
 724                 <row>
 725                   <?dbhtml bgcolor="#B0B0B0" ?>
 726                   <entry namest="c1" nameend="c5">
 727                     n_hash_map_ncah
 728                   </entry>
 729                 </row>
 730
 731                 <row>
 732                   <entry>
 733                     <classname>std::tr1::unordered_map</classname>
 734                   </entry>
 735                   <entry>
 736                     <classname>cache_hash_code</classname>
 737                   </entry>
 738                   <entry>
 739                     <constant>false</constant>
 740                   </entry>
 741                   <entry namest="c4" nameend="c5"></entry>
 742                 </row>
 743
 744                 <!-- hash 01 -->
 745                 <row>
 746                   <?dbhtml bgcolor="#B0B0B0" ?>
 747                   <entry namest="c1" nameend="c5">
 748                     gp_hash_mod_quadp_prime_1div2_nsth_map
 749                   </entry>
 750                 </row>
 751
 752                 <row>
 753                   <entry morerows="3" valign="top">
 754                     <classname>gp_hash_table</classname>
 755                   </entry>
 756                   <entry>
 757                     <classname>Comb_Hash_Fn</classname>
 758                   </entry>
 759                   <entry>
 760                     <classname>direct_mod_range_hashing</classname>
 761                   </entry>
 762                   <entry namest="c4" nameend="c5"></entry>
 763                 </row>
 764
 765                 <row>
 766                   <entry>
 767                     <classname>Probe_Fn</classname>
 768                   </entry>
 769                   <entry>
 770                     <classname>quadratic_probe_fn</classname>
 771                   </entry>
 772                   <entry namest="c4" nameend="c5"></entry>
 773                 </row>
 774
 775                 <row>
 776                   <entry morerows="1" valign="top">
 777                     <classname>Resize_Policy</classname>
 778                   </entry>
 779                   <entry morerows="1" valign="top">
 780                     <classname>hash_standard_resize_policy</classname>
 781                   </entry>
 782                   <entry>
 783                     <classname>Size_Policy</classname>
 784                   </entry>
 785                   <entry>
 786                     <classname>hash_prime_size_policy</classname>
 787                   </entry>
 788                 </row>
 789
 790                 <row>
 791                   <entry valign="top">
 792                     <classname>Trigger_Policy</classname>
 793                   </entry>
 794                   <entry>
 795                     <classname>hash_load_check_resize_trigger</classname> with
 796                     α<subscript>min</subscript> = 1/8 and α<subscript>max</subscript> = 1/2
 797                   </entry>
 798                 </row>
 799
 800                 <!-- hash 02 -->
 801                 <row>
 802                   <?dbhtml bgcolor="#B0B0B0" ?>
 803                   <entry namest="c1" nameend="c5">
 804                     gp_hash_mask_linp_exp_1div2_nsth_map
 805                   </entry>
 806                 </row>
 807
 808                 <row>
 809                   <entry morerows="3" valign="top">
 810                     <classname>
 811                       gp_hash_table
 812                     </classname>
 813                   </entry>
 814                   <entry>
 815                     <classname>Comb_Hash_Fn</classname>
 816                   </entry>
 817                   <entry>
 818                     <classname>direct_mask_range_hashing</classname>
 819                   </entry>
 820                   <entry namest="c4" nameend="c5"></entry>
 821                 </row>
 822
 823                 <row>
 824                   <entry>
 825                     <classname>Probe_Fn</classname>
 826                   </entry>
 827                   <entry>
 828                     <classname>linear_probe_fn</classname>
 829                   </entry>
 830                   <entry namest="c4" nameend="c5"></entry>
 831                 </row>
 832
 833                 <row>
 834                   <entry morerows="1" valign="top">
 835                     <classname>Resize_Policy</classname>
 836                   </entry>
 837                   <entry morerows="1" valign="top">
 838                     <classname>hash_standard_resize_policy</classname>
 839                   </entry>
 840                   <entry>
 841                     <classname>Size_Policy</classname>
 842                   </entry>
 843                   <entry>
 844                     <classname>hash_exponential_size_policy</classname>
 845                   </entry>
 846                 </row>
 847
 848                 <row>
 849                   <entry valign="top">
 850                     <classname>Trigger_Policy</classname>
 851                   </entry>
 852                   <entry>
 853                     <classname>hash_load_check_resize_trigger</classname> with
 854                     α<subscript>min</subscript> = 1/8 and α<subscript>max</subscript> = 1/2
 855                   </entry>
 856                 </row>
 857
 858               </tbody>
 859             </tgroup>
 860           </informaltable>
 861
 862         </section>
 863
 864         <section xml:id="hash.int_find.observations">
 865           <info><title>
 866             Observations
 867           </title></info>
 868
 869           <para>In this setting, the choice of underlying hash-table affects
 870           performance most, then the range-hashing scheme and, only finally,
 871           other policies.</para>
 872
 873           <para>When comparing probing and chaining containers, it is
 874           apparent that the probing containers are less efficient than the
 875           collision-chaining containers (
 876           <classname>std::tr1::unordered_map</classname> uses
 877           collision-chaining) in this case.</para>
 878
 879           <para>Hash-Based Integer Subscript Insert Timing Test shows
 880           a different case, where the situation is reversed;
 881           </para>
 882
 883           <para>Within each type of hash-table, the range-hashing scheme
 884           affects performance more than other policies; Hash-Based Text
 885           <function>find</function> Find Timing Test also shows this. In the
 886           above graphics should be noted that
 887           <classname>std::tr1::unordered_map</classname> are hard-wired
 888           currently to mod-based schemes.
 889           </para>
 890
 891         </section>
 892
 893       </section>
 894
 895       <!-- 03 <a href="hash_int_subscript_find_test"> -->
 896       <section xml:id="performance.hash.int_subscript_find">
 897         <info><title>
 898           Integer Subscript <function>find</function>
 899         </title></info>
 900         <para></para>
 901
 902         <section xml:id="hash.int_subscript_find.info">
 903           <info><title>
 904             Description
 905           </title></info>
 906
 907           <para>This test inserts a number of values with uniform
 908           integer keys into a container, then performs a series of finds
 909           using <function>operator[]</function>. It measures the average time
 910           for <function>operator[]</function> as a function of the number of
 911           values inserted.</para>
 912
 913           <para>
 914             It uses the test file:
 915             <filename>performance/ext/pb_ds/random_int_subscript_find_timing.cc</filename>
 916           </para>
 917
 918           <para>The test checks the effect of different underlying
 919           hash-tables, range-hashing functions, and trigger policies.</para>
 920
 921
 922         </section>
 923
 924         <section xml:id="hash.int_subscript_find.results">
 925           <info><title>
 926             Results
 927           </title></info>
 928
 929           <para>
 930             There are two sets of results for this type, one for
 931             collision-chaining hashes, and one for general-probe hashes.
 932           </para>
 933
 934           <para>The first graphic below shows the results for the native
 935           and collision-chaining hash types, using as the function
 936           applied an integer subscript timing test with
 937           <function>find</function>.
 938           </para>
 939
 940           <!-- results graphic -->
 941           <informalfigure>
 942             <mediaobject>
 943               <imageobject>
 944                 <imagedata align="center" format="PDF" scale="75"
 945                            fileref="../images/pbds_cc_hash_int_subscript_find.pdf"/>
 946               </imageobject>
 947               <imageobject>
 948                 <imagedata align="center" format="PNG" scale="100"
 949                            fileref="../images/pbds_cc_hash_int_subscript_find.png"/>
 950               </imageobject>
 951             </mediaobject>
 952           </informalfigure>
 953
 954           <para>
 955             The abbreviated names in the legend of the graphic above are
 956             instantiated with the types in the following table.
 957           </para>
 958
 959           <informaltable frame="all">
 960
 961             <tgroup cols="5" align="left" colsep="1" rowsep="1">
 962               <colspec colname="c1"/>
 963               <colspec colname="c2"/>
 964               <colspec colname="c3"/>
 965               <colspec colname="c4"/>
 966               <colspec colname="c5"/>
 967               <thead>
 968                 <row>
 969                   <entry><emphasis>Name/Instantiating Type</emphasis></entry>
 970                   <entry><emphasis>Parameter</emphasis></entry>
 971                   <entry><emphasis>Details</emphasis></entry>
 972                   <entry><emphasis>Parameter</emphasis></entry>
 973                   <entry><emphasis>Details</emphasis></entry>
 974                 </row>
 975               </thead>
 976
 977               <tbody>
 978
 979                 <!-- native -->
 980                 <row>
 981                   <?dbhtml bgcolor="#B0B0B0" ?>
 982                   <entry namest="c1" nameend="c5">
 983                     n_hash_map_ncah
 984                   </entry>
 985                 </row>
 986
 987                 <row>
 988                   <entry>
 989                     <classname>std::tr1::unordered_map</classname>
 990                   </entry>
 991                   <entry>
 992                     <classname>cache_hash_code</classname>
 993                   </entry>
 994                   <entry>
 995                     <constant>false</constant>
 996                   </entry>
 997                   <entry namest="c4" nameend="c5"></entry>
 998                 </row>
 999
1000                 <!-- hash 01 -->
1001                 <row>
1002                   <?dbhtml bgcolor="#B0B0B0" ?>
1003                   <entry namest="c1" nameend="c5">
1004                     cc_hash_mod_prime_1div1_nsth_map
1005                   </entry>
1006                 </row>
1007
1008                 <row>
1009                   <entry morerows="2" valign="top">
1010                     <classname>cc_hash_table</classname>
1011                   </entry>
1012                   <entry>
1013                     <classname>Comb_Hash_Fn</classname>
1014                   </entry>
1015                   <entry>
1016                     <classname>direct_mod_range_hashing</classname>
1017                   </entry>
1018                   <entry namest="c4" nameend="c5"></entry>
1019                 </row>
1020
1021                 <row>
1022                   <entry morerows="1" valign="top">
1023                     <classname>Resize_Policy</classname>
1024                   </entry>
1025                   <entry morerows="1" valign="top">
1026                     <classname>hash_standard_resize_policy</classname>
1027                   </entry>
1028                   <entry>
1029                     <classname>Size_Policy</classname>
1030                   </entry>
1031                   <entry>
1032                     <classname>hash_prime_size_policy</classname>
1033                   </entry>
1034                 </row>
1035
1036                 <row>
1037                   <entry valign="top">
1038                     <classname>Trigger_Policy</classname>
1039                   </entry>
1040                   <entry>
1041                     <classname>hash_load_check_resize_trigger</classname> with
1042                     α<subscript>min</subscript> = 1/8 and α<subscript>max</subscript> = 1/1
1043                   </entry>
1044                 </row>
1045
1046                 <!-- hash 02 -->
1047                 <row>
1048                   <?dbhtml bgcolor="#B0B0B0" ?>
1049                   <entry namest="c1" nameend="c5">
1050                     cc_hash_mod_prime_1div2_nsth_map
1051                   </entry>
1052                 </row>
1053
1054                 <row>
1055                   <entry morerows="2" valign="top">
1056                     <classname>cc_hash_table</classname>
1057                   </entry>
1058                   <entry>
1059                     <classname>Comb_Hash_Fn</classname>
1060                   </entry>
1061                   <entry>
1062                     <classname>direct_mod_range_hashing</classname>
1063                   </entry>
1064                   <entry namest="c4" nameend="c5"></entry>
1065                 </row>
1066
1067                 <row>
1068                   <entry morerows="1" valign="top">
1069                     <classname>Resize_Policy</classname>
1070                   </entry>
1071                   <entry morerows="1" valign="top">
1072                     <classname>hash_standard_resize_policy</classname>
1073                   </entry>
1074                   <entry>
1075                     <classname>Size_Policy</classname>
1076                   </entry>
1077                   <entry>
1078                     <classname>hash_prime_size_policy</classname>
1079                   </entry>
1080                 </row>
1081
1082                 <row>
1083                   <entry valign="top">
1084                     <classname>Trigger_Policy</classname>
1085                   </entry>
1086                   <entry>
1087                     <classname>hash_load_check_resize_trigger</classname> with
1088                     α<subscript>min</subscript> = 1/8 and α<subscript>max</subscript> = 1/2
1089                   </entry>
1090                 </row>
1091
1092                 <!-- hash 03 -->
1093                 <row>
1094                   <?dbhtml bgcolor="#B0B0B0" ?>
1095                   <entry namest="c1" nameend="c5">
1096                     cc_hash_mask_exp_1div1_nsth_map
1097                   </entry>
1098                 </row>
1099
1100                 <row>
1101                   <entry morerows="2" valign="top">
1102                     <classname>cc_hash_table</classname>
1103                   </entry>
1104                   <entry>
1105                     <classname>Comb_Hash_Fn</classname>
1106                   </entry>
1107                   <entry>
1108                     <classname>direct_mask_range_hashing</classname>
1109                   </entry>
1110                   <entry namest="c4" nameend="c5"></entry>
1111                 </row>
1112
1113                 <row>
1114                   <entry morerows="1" valign="top">
1115                     <classname>Resize_Policy</classname>
1116                   </entry>
1117                   <entry morerows="1" valign="top">
1118                     <classname>hash_standard_resize_policy</classname>
1119                   </entry>
1120                   <entry>
1121                     <classname>Size_Policy</classname>
1122                   </entry>
1123                   <entry>
1124                     <classname>hash_exponential_size_policy</classname>
1125                   </entry>
1126                 </row>
1127
1128                 <row>
1129                   <entry valign="top">
1130                     <classname>Trigger_Policy</classname>
1131                   </entry>
1132                   <entry>
1133                     <classname>hash_load_check_resize_trigger</classname> with
1134                     α<subscript>min</subscript> = 1/8 and α<subscript>max</subscript> = 1/1
1135                   </entry>
1136                 </row>
1137
1138                 <!-- hash 04 -->
1139                 <row>
1140                   <?dbhtml bgcolor="#B0B0B0" ?>
1141                   <entry namest="c1" nameend="c5">
1142                     cc_hash_mask_exp_1div2_nsth_map
1143                   </entry>
1144                 </row>
1145                 <row>
1146                   <entry morerows="2" valign="top">
1147                     <classname>cc_hash_table</classname>
1148                   </entry>
1149                   <entry>
1150                     <classname>Comb_Hash_Fn</classname>
1151                   </entry>
1152                   <entry>
1153                     <classname>direct_mask_range_hashing</classname>
1154                   </entry>
1155                   <entry namest="c4" nameend="c5"></entry>
1156                 </row>
1157
1158                 <row>
1159                   <entry morerows="1" valign="top">
1160                     <classname>Resize_Policy</classname>
1161                   </entry>
1162                   <entry morerows="1" valign="top">
1163                     <classname>hash_standard_resize_policy</classname>
1164                   </entry>
1165                   <entry>
1166                     <classname>Size_Policy</classname>
1167                   </entry>
1168                   <entry>
1169                     <classname>hash_exponential_size_policy</classname>
1170                   </entry>
1171                 </row>
1172
1173                 <row>
1174                   <entry valign="top">
1175                     <classname>Trigger_Policy</classname>
1176                   </entry>
1177                   <entry>
1178                     <classname>hash_load_check_resize_trigger</classname> with
1179                     α<subscript>min</subscript> = 1/8 and α<subscript>max</subscript> = 1/2
1180                   </entry>
1181                 </row>
1182
1183               </tbody>
1184             </tgroup>
1185           </informaltable>
1186
1187
1188           <para>
1189           </para>
1190
1191           <para>
1192           </para>
1193
1194           <para>And the second graphic shows the results for the native and
1195           general-probe hash types. The function applied being a random
1196           integer timing test using <function>find</function>.
1197           </para>
1198
1199           <!-- results graphic 02 -->
1200           <informalfigure>
1201             <mediaobject>
1202               <imageobject>
1203                 <imagedata align="center" format="PDF" scale="75"
1204                            fileref="../images/pbds_gp_hash_int_subscript_find.pdf"/>
1205               </imageobject>
1206               <imageobject>
1207                 <imagedata align="center" format="PNG" scale="100"
1208                            fileref="../images/pbds_gp_hash_int_subscript_find.png"/>
1209               </imageobject>
1210             </mediaobject>
1211           </informalfigure>
1212
1213           <para>
1214             The abbreviated names in the legend of the graphic above are
1215             instantiated with the types in the following table.
1216           </para>
1217
1218
1219           <informaltable frame="all">
1220
1221             <tgroup cols="5" align="left" colsep="1" rowsep="1">
1222               <colspec colname="c1"/>
1223               <colspec colname="c2"/>
1224               <colspec colname="c3"/>
1225               <colspec colname="c4"/>
1226               <colspec colname="c5"/>
1227               <thead>
1228                 <row>
1229                   <entry><emphasis>Name/Instantiating Type</emphasis></entry>
1230                   <entry><emphasis>Parameter</emphasis></entry>
1231                   <entry><emphasis>Details</emphasis></entry>
1232                   <entry><emphasis>Parameter</emphasis></entry>
1233                   <entry><emphasis>Details</emphasis></entry>
1234                 </row>
1235               </thead>
1236
1237               <tbody>
1238
1239                 <!-- native -->
1240                 <row>
1241                   <?dbhtml bgcolor="#B0B0B0" ?>
1242                   <entry namest="c1" nameend="c5">
1243                     n_hash_map_ncah
1244                   </entry>
1245                 </row>
1246
1247                 <row>
1248                   <entry>
1249                     <classname>std::tr1::unordered_map</classname>
1250                   </entry>
1251                   <entry>
1252                     <classname>cache_hash_code</classname>
1253                   </entry>
1254                   <entry>
1255                     <constant>false</constant>
1256                   </entry>
1257                   <entry namest="c4" nameend="c5"></entry>
1258                 </row>
1259
1260                 <!-- hash 01 -->
1261                 <row>
1262                   <?dbhtml bgcolor="#B0B0B0" ?>
1263                   <entry namest="c1" nameend="c5">
1264                     gp_hash_mod_quadp_prime_1div2_nsth_map
1265                   </entry>
1266                 </row>
1267
1268                 <row>
1269                   <entry morerows="3" valign="top">
1270                     <classname>gp_hash_table</classname>
1271                   </entry>
1272                   <entry>
1273                     <classname>Comb_Hash_Fn</classname>
1274                   </entry>
1275                   <entry>
1276                     <classname>direct_mod_range_hashing</classname>
1277                   </entry>
1278                   <entry namest="c4" nameend="c5"></entry>
1279                 </row>
1280
1281                 <row>
1282                   <entry>
1283                     <classname>Probe_Fn</classname>
1284                   </entry>
1285                   <entry>
1286                     <classname>quadratic_probe_fn</classname>
1287                   </entry>
1288                   <entry namest="c4" nameend="c5"></entry>
1289                 </row>
1290
1291                 <row>
1292                   <entry morerows="1" valign="top">
1293                     <classname>Resize_Policy</classname>
1294                   </entry>
1295                   <entry morerows="1" valign="top">
1296                     <classname>hash_standard_resize_policy</classname>
1297                   </entry>
1298                   <entry>
1299                     <classname>Size_Policy</classname>
1300                   </entry>
1301                   <entry>
1302                     <classname>hash_prime_size_policy</classname>
1303                   </entry>
1304                 </row>
1305
1306                 <row>
1307                   <entry valign="top">
1308                     <classname>Trigger_Policy</classname>
1309                   </entry>
1310                   <entry>
1311                     <classname>hash_load_check_resize_trigger</classname> with
1312                     α<subscript>min</subscript> = 1/8 and α<subscript>max</subscript> = 1/2
1313                   </entry>
1314                 </row>
1315
1316                 <!-- hash 02 -->
1317                 <row>
1318                   <?dbhtml bgcolor="#B0B0B0" ?>
1319                   <entry namest="c1" nameend="c5">
1320                     gp_hash_mask_linp_exp_1div2_nsth_map
1321                   </entry>
1322                 </row>
1323
1324                 <row>
1325                   <entry morerows="3" valign="top">
1326                     <classname>
1327                       gp_hash_table
1328                     </classname>
1329                   </entry>
1330                   <entry>
1331                     <classname>Comb_Hash_Fn</classname>
1332                   </entry>
1333                   <entry>
1334                     <classname>direct_mask_range_hashing</classname>
1335                   </entry>
1336                   <entry namest="c4" nameend="c5"></entry>
1337                 </row>
1338
1339                 <row>
1340                   <entry>
1341                     <classname>Probe_Fn</classname>
1342                   </entry>
1343                   <entry>
1344                     <classname>linear_probe_fn</classname>
1345                   </entry>
1346                   <entry namest="c4" nameend="c5"></entry>
1347                 </row>
1348
1349                 <row>
1350                   <entry morerows="1" valign="top">
1351                     <classname>Resize_Policy</classname>
1352                   </entry>
1353                   <entry morerows="1" valign="top">
1354                     <classname>hash_standard_resize_policy</classname>
1355                   </entry>
1356                   <entry>
1357                     <classname>Size_Policy</classname>
1358                   </entry>
1359                   <entry>
1360                     <classname>hash_exponential_size_policy</classname>
1361                   </entry>
1362                 </row>
1363
1364                 <row>
1365                   <entry valign="top">
1366                     <classname>Trigger_Policy</classname>
1367                   </entry>
1368                   <entry>
1369                     <classname>hash_load_check_resize_trigger</classname> with
1370                     α<subscript>min</subscript> = 1/8 and α<subscript>max</subscript> = 1/2
1371                   </entry>
1372                 </row>
1373
1374               </tbody>
1375             </tgroup>
1376           </informaltable>
1377
1378
1379         </section>
1380
1381         <section xml:id="hash.int_subscript_find.observations">
1382           <info><title>
1383             Observations
1384           </title></info>
1385           <para>This test shows similar results to Hash-Based
1386           Integer <classname>find</classname> Find Timing test.</para>
1387
1388         </section>
1389
1390       </section>
1391
1392       <!-- 04 <a href="hash_random_int_subscript_insert_timing_test"> -->
1393       <section xml:id="performance.hash.int_subscript_insert">
1394         <info><title>
1395           Integer Subscript <function>insert</function>
1396         </title></info>
1397         <para></para>
1398
1399         <section xml:id="hash.int_subscript_insert.info">
1400           <info><title>
1401             Description
1402           </title></info>
1403
1404           <para>This test inserts a number of values with uniform i.i.d.
1405           integer keys into a container, using
1406           <function>operator[]</function>. It measures the average time for
1407           <function>operator[]</function> as a function of the number of
1408           values inserted.</para>
1409
1410           <para>
1411             It uses the test file:
1412             <filename>performance/ext/pb_ds/random_int_subscript_insert_timing.cc</filename>
1413           </para>
1414
1415           <para>The test checks the effect of different underlying
1416           hash-tables.</para>
1417
1418
1419         </section>
1420
1421         <section xml:id="hash.int_subscript_insert.results">
1422           <info><title>
1423             Results
1424           </title></info>
1425
1426           <para>
1427             There are two sets of results for this type, one for
1428             collision-chaining hashes, and one for general-probe hashes.
1429           </para>
1430
1431           <para>The first graphic below shows the results for the native
1432           and collision-chaining hash types, using as the function
1433           applied an integer subscript timing test with
1434           <function>insert</function>.
1435           </para>
1436
1437           <!-- results graphic -->
1438           <informalfigure>
1439             <mediaobject>
1440               <imageobject>
1441                 <imagedata align="center" format="PDF" scale="75"
1442                            fileref="../images/pbds_cc_hash_int_subscript_insert.pdf"/>
1443               </imageobject>
1444               <imageobject>
1445                 <imagedata align="center" format="PNG" scale="100"
1446                            fileref="../images/pbds_cc_hash_int_subscript_insert.png"/>
1447               </imageobject>
1448             </mediaobject>
1449           </informalfigure>
1450
1451           <para>
1452             The abbreviated names in the legend of the graphic above are
1453             instantiated with the types in the following table.
1454           </para>
1455
1456           <informaltable frame="all">
1457
1458             <tgroup cols="5" align="left" colsep="1" rowsep="1">
1459               <colspec colname="c1"/>
1460               <colspec colname="c2"/>
1461               <colspec colname="c3"/>
1462               <colspec colname="c4"/>
1463               <colspec colname="c5"/>
1464               <thead>
1465                 <row>
1466                   <entry><emphasis>Name/Instantiating Type</emphasis></entry>
1467                   <entry><emphasis>Parameter</emphasis></entry>
1468                   <entry><emphasis>Details</emphasis></entry>
1469                   <entry><emphasis>Parameter</emphasis></entry>
1470                   <entry><emphasis>Details</emphasis></entry>
1471                 </row>
1472               </thead>
1473
1474               <tbody>
1475
1476                 <!-- native -->
1477                 <row>
1478                   <?dbhtml bgcolor="#B0B0B0" ?>
1479                   <entry namest="c1" nameend="c5">
1480                     n_hash_map_ncah
1481                   </entry>
1482                 </row>
1483
1484                 <row>
1485                   <entry>
1486                     <classname>std::tr1::unordered_map</classname>
1487                   </entry>
1488                   <entry>
1489                     <classname>cache_hash_code</classname>
1490                   </entry>
1491                   <entry>
1492                     <constant>false</constant>
1493                   </entry>
1494                   <entry namest="c4" nameend="c5"></entry>
1495                 </row>
1496
1497                 <!-- hash 01 -->
1498                 <row>
1499                   <?dbhtml bgcolor="#B0B0B0" ?>
1500                   <entry namest="c1" nameend="c5">
1501                     cc_hash_mod_prime_1div1_nsth_map
1502                   </entry>
1503                 </row>
1504
1505                 <row>
1506                   <entry morerows="2" valign="top">
1507                     <classname>cc_hash_table</classname>
1508                   </entry>
1509                   <entry>
1510                     <classname>Comb_Hash_Fn</classname>
1511                   </entry>
1512                   <entry>
1513                     <classname>direct_mod_range_hashing</classname>
1514                   </entry>
1515                   <entry namest="c4" nameend="c5"></entry>
1516                 </row>
1517
1518                 <row>
1519                   <entry morerows="1" valign="top">
1520                     <classname>Resize_Policy</classname>
1521                   </entry>
1522                   <entry morerows="1" valign="top">
1523                     <classname>hash_standard_resize_policy</classname>
1524                   </entry>
1525                   <entry>
1526                     <classname>Size_Policy</classname>
1527                   </entry>
1528                   <entry>
1529                     <classname>hash_prime_size_policy</classname>
1530                   </entry>
1531                 </row>
1532
1533                 <row>
1534                   <entry valign="top">
1535                     <classname>Trigger_Policy</classname>
1536                   </entry>
1537                   <entry>
1538                     <classname>hash_load_check_resize_trigger</classname> with
1539                     α<subscript>min</subscript> = 1/8 and α<subscript>max</subscript> = 1/1
1540                   </entry>
1541                 </row>
1542
1543                 <!-- hash 02 -->
1544                 <row>
1545                   <?dbhtml bgcolor="#B0B0B0" ?>
1546                   <entry namest="c1" nameend="c5">
1547                     cc_hash_mod_prime_1div2_nsth_map
1548                   </entry>
1549                 </row>
1550
1551                 <row>
1552                   <entry morerows="2" valign="top">
1553                     <classname>cc_hash_table</classname>
1554                   </entry>
1555                   <entry>
1556                     <classname>Comb_Hash_Fn</classname>
1557                   </entry>
1558                   <entry>
1559                     <classname>direct_mod_range_hashing</classname>
1560                   </entry>
1561                   <entry namest="c4" nameend="c5"></entry>
1562                 </row>
1563
1564                 <row>
1565                   <entry morerows="1" valign="top">
1566                     <classname>Resize_Policy</classname>
1567                   </entry>
1568                   <entry morerows="1" valign="top">
1569                     <classname>hash_standard_resize_policy</classname>
1570                   </entry>
1571                   <entry>
1572                     <classname>Size_Policy</classname>
1573                   </entry>
1574                   <entry>
1575                     <classname>hash_prime_size_policy</classname>
1576                   </entry>
1577                 </row>
1578
1579                 <row>
1580                   <entry valign="top">
1581                     <classname>Trigger_Policy</classname>
1582                   </entry>
1583                   <entry>
1584                     <classname>hash_load_check_resize_trigger</classname> with
1585                     α<subscript>min</subscript> = 1/8 and α<subscript>max</subscript> = 1/2
1586                   </entry>
1587                 </row>
1588
1589                 <!-- hash 03 -->
1590                 <row>
1591                   <?dbhtml bgcolor="#B0B0B0" ?>
1592                   <entry namest="c1" nameend="c5">
1593                     cc_hash_mask_exp_1div1_nsth_map
1594                   </entry>
1595                 </row>
1596
1597                 <row>
1598                   <entry morerows="2" valign="top">
1599                     <classname>cc_hash_table</classname>
1600                   </entry>
1601                   <entry>
1602                     <classname>Comb_Hash_Fn</classname>
1603                   </entry>
1604                   <entry>
1605                     <classname>direct_mask_range_hashing</classname>
1606                   </entry>
1607                   <entry namest="c4" nameend="c5"></entry>
1608                 </row>
1609
1610                 <row>
1611                   <entry morerows="1" valign="top">
1612                     <classname>Resize_Policy</classname>
1613                   </entry>
1614                   <entry morerows="1" valign="top">
1615                     <classname>hash_standard_resize_policy</classname>
1616                   </entry>
1617                   <entry>
1618                     <classname>Size_Policy</classname>
1619                   </entry>
1620                   <entry>
1621                     <classname>hash_exponential_size_policy</classname>
1622                   </entry>
1623                 </row>
1624
1625                 <row>
1626                   <entry valign="top">
1627                     <classname>Trigger_Policy</classname>
1628                   </entry>
1629                   <entry>
1630                     <classname>hash_load_check_resize_trigger</classname> with
1631                     α<subscript>min</subscript> = 1/8 and α<subscript>max</subscript> = 1/1
1632                   </entry>
1633                 </row>
1634
1635                 <!-- hash 04 -->
1636                 <row>
1637                   <?dbhtml bgcolor="#B0B0B0" ?>
1638                   <entry namest="c1" nameend="c5">
1639                     cc_hash_mask_exp_1div2_nsth_map
1640                   </entry>
1641                 </row>
1642                 <row>
1643                   <entry morerows="2" valign="top">
1644                     <classname>cc_hash_table</classname>
1645                   </entry>
1646                   <entry>
1647                     <classname>Comb_Hash_Fn</classname>
1648                   </entry>
1649                   <entry>
1650                     <classname>direct_mask_range_hashing</classname>
1651                   </entry>
1652                   <entry namest="c4" nameend="c5"></entry>
1653                 </row>
1654
1655                 <row>
1656                   <entry morerows="1" valign="top">
1657                     <classname>Resize_Policy</classname>
1658                   </entry>
1659                   <entry morerows="1" valign="top">
1660                     <classname>hash_standard_resize_policy</classname>
1661                   </entry>
1662                   <entry>
1663                     <classname>Size_Policy</classname>
1664                   </entry>
1665                   <entry>
1666                     <classname>hash_exponential_size_policy</classname>
1667                   </entry>
1668                 </row>
1669
1670                 <row>
1671                   <entry valign="top">
1672                     <classname>Trigger_Policy</classname>
1673                   </entry>
1674                   <entry>
1675                     <classname>hash_load_check_resize_trigger</classname> with
1676                     α<subscript>min</subscript> = 1/8 and α<subscript>max</subscript> = 1/2
1677                   </entry>
1678                 </row>
1679
1680               </tbody>
1681             </tgroup>
1682           </informaltable>
1683
1684
1685           <para>
1686           </para>
1687
1688           <para>
1689           </para>
1690
1691           <para>And the second graphic shows the results for the native and
1692           general-probe hash types. The function applied being a random
1693           integer timing test using <function>find</function>.
1694           </para>
1695
1696           <!-- results graphic 02 -->
1697           <informalfigure>
1698             <mediaobject>
1699               <imageobject>
1700                 <imagedata align="center" format="PDF" scale="75"
1701                            fileref="../images/pbds_gp_hash_int_subscript_insert.pdf"/>
1702               </imageobject>
1703               <imageobject>
1704                 <imagedata align="center" format="PNG" scale="100"
1705                            fileref="../images/pbds_gp_hash_int_subscript_insert.png"/>
1706               </imageobject>
1707             </mediaobject>
1708           </informalfigure>
1709
1710           <para>
1711             The abbreviated names in the legend of the graphic above are
1712             instantiated with the types in the following table.
1713           </para>
1714
1715
1716           <informaltable frame="all">
1717
1718             <tgroup cols="5" align="left" colsep="1" rowsep="1">
1719               <colspec colname="c1"/>
1720               <colspec colname="c2"/>
1721               <colspec colname="c3"/>
1722               <colspec colname="c4"/>
1723               <colspec colname="c5"/>
1724               <thead>
1725                 <row>
1726                   <entry><emphasis>Name/Instantiating Type</emphasis></entry>
1727                   <entry><emphasis>Parameter</emphasis></entry>
1728                   <entry><emphasis>Details</emphasis></entry>
1729                   <entry><emphasis>Parameter</emphasis></entry>
1730                   <entry><emphasis>Details</emphasis></entry>
1731                 </row>
1732               </thead>
1733
1734               <tbody>
1735
1736                 <!-- native -->
1737                 <row>
1738                   <?dbhtml bgcolor="#B0B0B0" ?>
1739                   <entry namest="c1" nameend="c5">
1740                     n_hash_map_ncah
1741                   </entry>
1742                 </row>
1743
1744                 <row>
1745                   <entry>
1746                     <classname>std::tr1::unordered_map</classname>
1747                   </entry>
1748                   <entry>
1749                     <classname>cache_hash_code</classname>
1750                   </entry>
1751                   <entry>
1752                     <constant>false</constant>
1753                   </entry>
1754                   <entry namest="c4" nameend="c5"></entry>
1755                 </row>
1756
1757                 <!-- hash 01 -->
1758                 <row>
1759                   <?dbhtml bgcolor="#B0B0B0" ?>
1760                   <entry namest="c1" nameend="c5">
1761                     gp_hash_mod_quadp_prime_1div2_nsth_map
1762                   </entry>
1763                 </row>
1764
1765                 <row>
1766                   <entry morerows="3" valign="top">
1767                     <classname>gp_hash_table</classname>
1768                   </entry>
1769                   <entry>
1770                     <classname>Comb_Hash_Fn</classname>
1771                   </entry>
1772                   <entry>
1773                     <classname>direct_mod_range_hashing</classname>
1774                   </entry>
1775                   <entry namest="c4" nameend="c5"></entry>
1776                 </row>
1777
1778                 <row>
1779                   <entry>
1780                     <classname>Probe_Fn</classname>
1781                   </entry>
1782                   <entry>
1783                     <classname>quadratic_probe_fn</classname>
1784                   </entry>
1785                   <entry namest="c4" nameend="c5"></entry>
1786                 </row>
1787
1788                 <row>
1789                   <entry morerows="1" valign="top">
1790                     <classname>Resize_Policy</classname>
1791                   </entry>
1792                   <entry morerows="1" valign="top">
1793                     <classname>hash_standard_resize_policy</classname>
1794                   </entry>
1795                   <entry>
1796                     <classname>Size_Policy</classname>
1797                   </entry>
1798                   <entry>
1799                     <classname>hash_prime_size_policy</classname>
1800                   </entry>
1801                 </row>
1802
1803                 <row>
1804                   <entry valign="top">
1805                     <classname>Trigger_Policy</classname>
1806                   </entry>
1807                   <entry>
1808                     <classname>hash_load_check_resize_trigger</classname> with
1809                     α<subscript>min</subscript> = 1/8 and α<subscript>max</subscript> = 1/2
1810                   </entry>
1811                 </row>
1812
1813                 <!-- hash 02 -->
1814                 <row>
1815                   <?dbhtml bgcolor="#B0B0B0" ?>
1816                   <entry namest="c1" nameend="c5">
1817                     gp_hash_mask_linp_exp_1div2_nsth_map
1818                   </entry>
1819                 </row>
1820
1821                 <row>
1822                   <entry morerows="3" valign="top">
1823                     <classname>
1824                       gp_hash_table
1825                     </classname>
1826                   </entry>
1827                   <entry>
1828                     <classname>Comb_Hash_Fn</classname>
1829                   </entry>
1830                   <entry>
1831                     <classname>direct_mask_range_hashing</classname>
1832                   </entry>
1833                   <entry namest="c4" nameend="c5"></entry>
1834                 </row>
1835
1836                 <row>
1837                   <entry>
1838                     <classname>Probe_Fn</classname>
1839                   </entry>
1840                   <entry>
1841                     <classname>linear_probe_fn</classname>
1842                   </entry>
1843                   <entry namest="c4" nameend="c5"></entry>
1844                 </row>
1845
1846                 <row>
1847                   <entry morerows="1" valign="top">
1848                     <classname>Resize_Policy</classname>
1849                   </entry>
1850                   <entry morerows="1" valign="top">
1851                     <classname>hash_standard_resize_policy</classname>
1852                   </entry>
1853                   <entry>
1854                     <classname>Size_Policy</classname>
1855                   </entry>
1856                   <entry>
1857                     <classname>hash_exponential_size_policy</classname>
1858                   </entry>
1859                 </row>
1860
1861                 <row>
1862                   <entry valign="top">
1863                     <classname>Trigger_Policy</classname>
1864                   </entry>
1865                   <entry>
1866                     <classname>hash_load_check_resize_trigger</classname> with
1867                     α<subscript>min</subscript> = 1/8 and α<subscript>max</subscript> = 1/2
1868                   </entry>
1869                 </row>
1870
1871               </tbody>
1872             </tgroup>
1873           </informaltable>
1874
1875
1876         </section>
1877
1878         <section xml:id="hash.int_subscript_insert.observations">
1879           <info><title>
1880             Observations
1881           </title></info>
1882
1883           <para>In this setting, as in Hash-Based Text
1884           <function>find</function> Find Timing test and Hash-Based
1885           Integer <function>find</function> Find Timing test , the choice
1886           of underlying hash-table underlying hash-table affects performance
1887           most, then the range-hashing scheme, and
1888           finally any other policies.</para>
1889           <para>There are some differences, however:</para>
1890           <orderedlist>
1891             <listitem><para>In this setting, probing tables function sometimes more
1892             efficiently than collision-chaining tables.
1893             This is explained shortly.</para></listitem>
1894             <listitem><para>The performance graphs have a "saw-tooth" shape. The
1895             average insert time rises and falls. As values are inserted
1896             into the container, the load factor grows larger. Eventually,
1897             a resize occurs. The reallocations and rehashing are
1898             relatively expensive. After this, the load factor is smaller
1899             than before.</para></listitem>
1900           </orderedlist>
1901
1902           <para>Collision-chaining containers use indirection for greater
1903           flexibility; probing containers store values contiguously, in
1904           an array (see Figure Motivation::Different
1905           underlying data structures A and B, respectively). It
1906           follows that for simple data types, probing containers access
1907           their allocator less frequently than collision-chaining
1908           containers, (although they still have less efficient probing
1909           sequences). This explains why some probing containers fare
1910           better than collision-chaining containers in this case.</para>
1911
1912           <para>
1913             Within each type of hash-table, the range-hashing scheme affects
1914             performance more than other policies. This is similar to the
1915             situation in Hash-Based Text
1916             <function>find</function> Find Timing Test and Hash-Based
1917             Integer <function>find</function> Find Timing Test.
1918             Unsurprisingly, however, containers with lower α<subscript>max</subscript> perform worse in this case,
1919           since more re-hashes are performed.</para>
1920
1921         </section>
1922
1923       </section>
1924
1925
1926       <!-- 05 <a href="hash_zlob_random_int_find_find_timing_test"> -->
1927
1928       <!-- 05 <a href="hash_zlob_random_int_find_find_timing_test"> -->
1929       <section xml:id="performance.hash.zlob_int_find">
1930         <info><title>
1931           Integer <function>find</function> with Skewed-Distribution
1932         </title></info>
1933         <para></para>
1934
1935         <section xml:id="hash.zlob_int_find.info">
1936           <info><title>
1937             Description
1938           </title></info>
1939
1940           <para>This test inserts a number of values with a markedly
1941           non-uniform integer keys into a container, then performs
1942           a series of finds using <function>find</function>. It measures the average
1943           time for <function>find</function> as a function of the number of values in
1944           the containers. The keys are generated as follows. First, a
1945           uniform integer is created. Then it is then shifted left 8 bits.</para>
1946
1947           <para>
1948             It uses the test file:
1949             <filename>performance/ext/pb_ds/hash_zlob_random_int_find_timing.cc</filename>
1950           </para>
1951
1952           <para>The test checks the effect of different range-hashing
1953           functions and trigger policies.</para>
1954
1955
1956         </section>
1957
1958         <section xml:id="hash.zlob_int_find.results">
1959           <info><title>
1960             Results
1961           </title></info>
1962
1963           <para>The graphic below show the results for the native, collision-chaining, and general-probing hash types.
1964           </para>
1965
1966           <!-- results graphic -->
1967           <informalfigure>
1968             <mediaobject>
1969               <imageobject>
1970                 <imagedata align="center" format="PDF" scale="75"
1971                            fileref="../images/pbds_hash_zlob_int_find.pdf"/>
1972               </imageobject>
1973               <imageobject>
1974                 <imagedata align="center" format="PNG" scale="100"
1975                            fileref="../images/pbds_hash_zlob_int_find.png"/>
1976               </imageobject>
1977             </mediaobject>
1978           </informalfigure>
1979
1980           <para>
1981             The abbreviated names in the legend of the graphic above are
1982             instantiated with the types in the following table.
1983           </para>
1984
1985
1986           <informaltable frame="all">
1987
1988             <tgroup cols="5" align="left" colsep="1" rowsep="1">
1989               <colspec colname="c1"/>
1990               <colspec colname="c2"/>
1991               <colspec colname="c3"/>
1992               <colspec colname="c4"/>
1993               <colspec colname="c5"/>
1994               <thead>
1995                 <row>
1996                   <entry><emphasis>Name/Instantiating Type</emphasis></entry>
1997                   <entry><emphasis>Parameter</emphasis></entry>
1998                   <entry><emphasis>Details</emphasis></entry>
1999                   <entry><emphasis>Parameter</emphasis></entry>
2000                   <entry><emphasis>Details</emphasis></entry>
2001                 </row>
2002               </thead>
2003
2004               <tbody>
2005
2006                 <!-- native -->
2007                 <row>
2008                   <?dbhtml bgcolor="#B0B0B0" ?>
2009                   <entry namest="c1" nameend="c5">
2010                     n_hash_map_ncah
2011                   </entry>
2012                 </row>
2013
2014                 <row>
2015                   <entry>
2016                     <classname>std::tr1::unordered_map</classname>
2017                   </entry>
2018                   <entry>
2019                     <classname>cache_hash_code</classname>
2020                   </entry>
2021                   <entry>
2022                     <constant>false</constant>
2023                   </entry>
2024                   <entry namest="c4" nameend="c5"></entry>
2025                 </row>
2026
2027                 <!-- hash 01 -->
2028                 <row>
2029                   <?dbhtml bgcolor="#B0B0B0" ?>
2030                   <entry namest="c1" nameend="c5">
2031                     cc_hash_mod_prime_1div1_nsth_map
2032                   </entry>
2033                 </row>
2034
2035                 <row>
2036                   <entry morerows="2" valign="top">
2037                     <classname>cc_hash_table</classname>
2038                   </entry>
2039                   <entry>
2040                     <classname>Comb_Hash_Fn</classname>
2041                   </entry>
2042                   <entry>
2043                     <classname>direct_mod_range_hashing</classname>
2044                   </entry>
2045                   <entry namest="c4" nameend="c5"></entry>
2046                 </row>
2047
2048                 <row>
2049                   <entry morerows="1" valign="top">
2050                     <classname>Resize_Policy</classname>
2051                   </entry>
2052                   <entry morerows="1" valign="top">
2053                     <classname>hash_standard_resize_policy</classname>
2054                   </entry>
2055                   <entry>
2056                     <classname>Size_Policy</classname>
2057                   </entry>
2058                   <entry>
2059                     <classname>hash_prime_size_policy</classname>
2060                   </entry>
2061                 </row>
2062
2063                 <row>
2064                   <entry valign="top">
2065                     <classname>Trigger_Policy</classname>
2066                   </entry>
2067                   <entry>
2068                     <classname>hash_load_check_resize_trigger</classname> with
2069                     α<subscript>min</subscript> = 1/8 and α<subscript>max</subscript> = 1/1
2070                   </entry>
2071                 </row>
2072
2073                 <!-- hash 02 -->
2074                 <row>
2075                   <?dbhtml bgcolor="#B0B0B0" ?>
2076                   <entry namest="c1" nameend="c5">
2077                     cc_hash_mask_exp_1div1_nsth_map
2078                   </entry>
2079                 </row>
2080
2081                 <row>
2082                   <entry morerows="2" valign="top">
2083                     <classname>
2084                       cc_hash_table
2085                     </classname>
2086                   </entry>
2087                   <entry>
2088                     <classname>Comb_Hash_Fn</classname>
2089                   </entry>
2090                   <entry>
2091                     <classname>direct_mask_range_hashing</classname>
2092                   </entry>
2093                   <entry namest="c4" nameend="c5"></entry>
2094                 </row>
2095
2096                 <row>
2097                   <entry morerows="1" valign="top">
2098                     <classname>Resize_Policy</classname>
2099                   </entry>
2100                   <entry morerows="1" valign="top">
2101                     <classname>hash_standard_resize_policy</classname>
2102                   </entry>
2103                   <entry>
2104                     <classname>Size_Policy</classname>
2105                   </entry>
2106                   <entry>
2107                     <classname>hash_exponential_size_policy</classname>
2108                   </entry>
2109                 </row>
2110
2111                 <row>
2112                   <entry valign="top">
2113                     <classname>Trigger_Policy</classname>
2114                   </entry>
2115                   <entry>
2116                     <classname>hash_load_check_resize_trigger</classname> with
2117                     α<subscript>min</subscript> = 1/8 and α<subscript>max</subscript> = 1/1
2118                   </entry>
2119                 </row>
2120
2121                 <!-- hash 01 -->
2122                 <row>
2123                   <?dbhtml bgcolor="#B0B0B0" ?>
2124                   <entry namest="c1" nameend="c5">
2125                     gp_hash_mod_quadp_prime_1div2_nsth_map
2126                   </entry>
2127                 </row>
2128
2129                 <row>
2130                   <entry morerows="3" valign="top">
2131                     <classname>gp_hash_table</classname>
2132                   </entry>
2133                   <entry>
2134                     <classname>Comb_Hash_Fn</classname>
2135                   </entry>
2136                   <entry>
2137                     <classname>direct_mod_range_hashing</classname>
2138                   </entry>
2139                   <entry namest="c4" nameend="c5"></entry>
2140                 </row>
2141
2142                 <row>
2143                   <entry>
2144                     <classname>Probe_Fn</classname>
2145                   </entry>
2146                   <entry>
2147                     <classname>quadratic_probe_fn</classname>
2148                   </entry>
2149                   <entry namest="c4" nameend="c5"></entry>
2150                 </row>
2151
2152                 <row>
2153                   <entry morerows="1" valign="top">
2154                     <classname>Resize_Policy</classname>
2155                   </entry>
2156                   <entry morerows="1" valign="top">
2157                     <classname>hash_standard_resize_policy</classname>
2158                   </entry>
2159                   <entry>
2160                     <classname>Size_Policy</classname>
2161                   </entry>
2162                   <entry>
2163                     <classname>hash_prime_size_policy</classname>
2164                   </entry>
2165                 </row>
2166
2167                 <row>
2168                   <entry valign="top">
2169                     <classname>Trigger_Policy</classname>
2170                   </entry>
2171                   <entry>
2172                     <classname>hash_load_check_resize_trigger</classname> with
2173                     α<subscript>min</subscript> = 1/8 and α<subscript>max</subscript> = 1/2
2174                   </entry>
2175                 </row>
2176
2177
2178               </tbody>
2179             </tgroup>
2180           </informaltable>
2181
2182         </section>
2183
2184         <section xml:id="hash.zlob_int_find.observations">
2185           <info><title>
2186             Observations
2187           </title></info>
2188
2189           <para>In this setting, the distribution of keys is so skewed that
2190           the underlying hash-table type affects performance marginally.
2191           (This is in contrast with Hash-Based Text
2192           <function>find</function> Find Timing Test, Hash-Based
2193           Integer <function>find</function> Find Timing Test, Hash-Based
2194           Integer Subscript Find Timing Test and Hash-Based
2195           Integer Subscript Insert Timing Test.)</para>
2196
2197           <para>The range-hashing scheme affects performance dramatically. A
2198           mask-based range-hashing scheme effectively maps all values
2199           into the same bucket. Access degenerates into a search within
2200           an unordered linked-list. In the graphic above, it should be noted that
2201           <classname>std::tr1::unordered_map</classname> is hard-wired currently to mod-based and mask-based schemes,
2202           respectively.</para>
2203
2204           <para>When observing the settings of this test, it is apparent
2205           that the keys' distribution is far from natural. One might ask
2206           if the test is not contrived to show that, in some cases,
2207           mod-based range hashing does better than mask-based range
2208           hashing. This is, in fact just the case. A
2209           more natural case in which mod-based range hashing is better was not encountered.
2210           Thus the inescapable conclusion: real-life key distributions are handled better
2211           with an appropriate hash function and a mask-based
2212           range-hashing function. (<filename>pb_ds/example/hash_shift_mask.cc</filename>
2213           shows an example of handling this a-priori known skewed
2214           distribution with a mask-based range-hashing function). If hash
2215           performance is bad, a χ<superscript>2</superscript> test can be used
2216           to check how to transform it into a more uniform
2217           distribution.</para>
2218           <para>For this reason, this library's default range-hashing
2219           function is mask-based.</para>
2220
2221         </section>
2222
2223       </section>
2224
2225
2226       <!-- 06 <a href="hash_random_int_erase_mem_usage_test"> -->
2227
2228       <!-- 06 <a href="hash_random_int_erase_mem_usage_test"> -->
2229       <section xml:id="performance.hash.erase_mem">
2230         <info><title>
2231           Erase Memory Use
2232         </title></info>
2233         <para></para>
2234
2235         <section xml:id="hash.erase_mem.info">
2236           <info><title>
2237             Description
2238           </title></info>
2239
2240           <para>This test inserts a number of uniform integer keys
2241           into a container, then erases all keys except one. It measures
2242           the final size of the container.</para>
2243
2244           <para>
2245             It uses the test file:
2246             <filename>performance/ext/pb_ds/hash_random_int_erase_mem_usage.cc</filename>
2247           </para>
2248
2249
2250           <para>The test checks how containers adjust internally as their
2251           logical size decreases.</para>
2252
2253         </section>
2254
2255         <section xml:id="hash.erase_mem.results">
2256           <info><title>
2257             Results
2258           </title></info>
2259
2260           <para>The graphic below show the results for the native, collision-chaining, and general-probing hash types.
2261           </para>
2262
2263           <!-- results graphic -->
2264           <informalfigure>
2265             <mediaobject>
2266               <imageobject>
2267                 <imagedata align="center" format="PDF" scale="75"
2268                            fileref="../images/pbds_hash_int_erase_mem.pdf"/>
2269               </imageobject>
2270               <imageobject>
2271                 <imagedata align="center" format="PNG" scale="100"
2272                            fileref="../images/pbds_hash_int_erase_mem.png"/>
2273               </imageobject>
2274             </mediaobject>
2275           </informalfigure>
2276
2277           <para>
2278             The abbreviated names in the legend of the graphic above are
2279             instantiated with the types in the following table.
2280           </para>
2281
2282
2283           <informaltable frame="all">
2284
2285             <tgroup cols="5" align="left" colsep="1" rowsep="1">
2286               <colspec colname="c1"/>
2287               <colspec colname="c2"/>
2288               <colspec colname="c3"/>
2289               <colspec colname="c4"/>
2290               <colspec colname="c5"/>
2291               <thead>
2292                 <row>
2293                   <entry><emphasis>Name/Instantiating Type</emphasis></entry>
2294                   <entry><emphasis>Parameter</emphasis></entry>
2295                   <entry><emphasis>Details</emphasis></entry>
2296                   <entry><emphasis>Parameter</emphasis></entry>
2297                   <entry><emphasis>Details</emphasis></entry>
2298                 </row>
2299               </thead>
2300
2301               <tbody>
2302
2303                 <!-- native -->
2304                 <row>
2305                   <?dbhtml bgcolor="#B0B0B0" ?>
2306                   <entry namest="c1" nameend="c5">
2307                     n_hash_map_ncah
2308                   </entry>
2309                 </row>
2310
2311                 <row>
2312                   <entry>
2313                     <classname>std::tr1::unordered_map</classname>
2314                   </entry>
2315                   <entry>
2316                     <classname>cache_hash_code</classname>
2317                   </entry>
2318                   <entry>
2319                     <constant>false</constant>
2320                   </entry>
2321                   <entry namest="c4" nameend="c5"></entry>
2322                 </row>
2323
2324                 <!-- hash 01 -->
2325                 <row>
2326                   <?dbhtml bgcolor="#B0B0B0" ?>
2327                   <entry namest="c1" nameend="c5">
2328                     cc_hash_mod_prime_1div1_nsth_map
2329                   </entry>
2330                 </row>
2331
2332                 <row>
2333                   <entry morerows="2" valign="top">
2334                     <classname>cc_hash_table</classname>
2335                   </entry>
2336                   <entry>
2337                     <classname>Comb_Hash_Fn</classname>
2338                   </entry>
2339                   <entry>
2340                     <classname>direct_mod_range_hashing</classname>
2341                   </entry>
2342                   <entry namest="c4" nameend="c5"></entry>
2343                 </row>
2344
2345                 <row>
2346                   <entry morerows="1" valign="top">
2347                     <classname>Resize_Policy</classname>
2348                   </entry>
2349                   <entry morerows="1" valign="top">
2350                     <classname>hash_standard_resize_policy</classname>
2351                   </entry>
2352                   <entry>
2353                     <classname>Size_Policy</classname>
2354                   </entry>
2355                   <entry>
2356                     <classname>hash_prime_size_policy</classname>
2357                   </entry>
2358                 </row>
2359
2360                 <row>
2361                   <entry valign="top">
2362                     <classname>Trigger_Policy</classname>
2363                   </entry>
2364                   <entry>
2365                     <classname>hash_load_check_resize_trigger</classname> with
2366                     α<subscript>min</subscript> = 1/8 and α<subscript>max</subscript> = 1/1
2367                   </entry>
2368                 </row>
2369
2370                 <!-- hash 02 -->
2371                 <row>
2372                   <?dbhtml bgcolor="#B0B0B0" ?>
2373                   <entry namest="c1" nameend="c5">
2374                     cc_hash_mask_exp_1div2_nsth_map
2375                   </entry>
2376                 </row>
2377
2378                 <row>
2379                   <entry morerows="2" valign="top">
2380                     <classname>
2381                       cc_hash_table
2382                     </classname>
2383                   </entry>
2384                   <entry>
2385                     <classname>Comb_Hash_Fn</classname>
2386                   </entry>
2387                   <entry>
2388                     <classname>direct_mask_range_hashing</classname>
2389                   </entry>
2390                   <entry namest="c4" nameend="c5"></entry>
2391                 </row>
2392
2393                 <row>
2394                   <entry morerows="1" valign="top">
2395                     <classname>Resize_Policy</classname>
2396                   </entry>
2397                   <entry morerows="1" valign="top">
2398                     <classname>hash_standard_resize_policy</classname>
2399                   </entry>
2400                   <entry>
2401                     <classname>Size_Policy</classname>
2402                   </entry>
2403                   <entry>
2404                     <classname>hash_exponential_size_policy</classname>
2405                   </entry>
2406                 </row>
2407
2408                 <row>
2409                   <entry valign="top">
2410                     <classname>Trigger_Policy</classname>
2411                   </entry>
2412                   <entry>
2413                     <classname>hash_load_check_resize_trigger</classname> with
2414                     α<subscript>min</subscript> = 1/8 and α<subscript>max</subscript> = 1/2
2415                   </entry>
2416                 </row>
2417
2418                 <!-- hash 03 -->
2419                 <row>
2420                   <?dbhtml bgcolor="#B0B0B0" ?>
2421                   <entry namest="c1" nameend="c5">
2422                     gp_hash_mask_linp_exp_1div2_nsth_set
2423                   </entry>
2424                 </row>
2425
2426                 <row>
2427                   <entry morerows="3" valign="top">
2428                     <classname>gp_hash_table</classname>
2429                   </entry>
2430                   <entry>
2431                     <classname>Comb_Hash_Fn</classname>
2432                   </entry>
2433                   <entry>
2434                     <classname>direct_mask_range_hashing</classname>
2435                   </entry>
2436                   <entry namest="c4" nameend="c5"></entry>
2437                 </row>
2438
2439                 <row>
2440                   <entry>
2441                     <classname>Probe_Fn</classname>
2442                   </entry>
2443                   <entry>
2444                     <classname>linear_probe_fn</classname>
2445                   </entry>
2446                   <entry namest="c4" nameend="c5"></entry>
2447                 </row>
2448
2449                 <row>
2450                   <entry morerows="1" valign="top">
2451                     <classname>Resize_Policy</classname>
2452                   </entry>
2453                   <entry morerows="1" valign="top">
2454                     <classname>hash_standard_resize_policy</classname>
2455                   </entry>
2456                   <entry>
2457                     <classname>Size_Policy</classname>
2458                   </entry>
2459                   <entry>
2460                     <classname>hash_exponential_size_policy</classname>
2461                   </entry>
2462                 </row>
2463
2464                 <row>
2465                   <entry valign="top">
2466                     <classname>Trigger_Policy</classname>
2467                   </entry>
2468                   <entry>
2469                     <classname>hash_load_check_resize_trigger</classname> with
2470                     α<subscript>min</subscript> = 1/8 and α<subscript>max</subscript> = 1/2
2471                   </entry>
2472                 </row>
2473
2474
2475               </tbody>
2476             </tgroup>
2477           </informaltable>
2478
2479         </section>
2480
2481         <section xml:id="hash.erase_mem.observations">
2482           <info><title>
2483             Observations
2484           </title></info>
2485           <para>The standard's hash-based containers act very differently than trees in
2486           this respect. When erasing numerous keys from an standard
2487           associative-container, the resulting memory user varies greatly
2488           depending on whether the container is tree-based or hash-based.
2489           This is a fundamental consequence of the standard's interface for
2490           associative containers, and it is not due to a specific
2491           implementation.</para>
2492         </section>
2493
2494       </section>
2495     </section>
2496
2497
2498     <section xml:id="performance.branch">
2499       <info><title>Branch-Based</title></info>
2500       <para></para>
2501
2502       <!-- 01 <a href="tree_text_insert_timing_test"> -->
2503       <section xml:id="performance.branch.text_insert">
2504         <info><title>
2505           Text <function>insert</function>
2506         </title></info>
2507         <para></para>
2508
2509         <section xml:id="branch.text_insert.info">
2510           <info><title>
2511             Description
2512           </title></info>
2513
2514
2515           <para>This test inserts a number of values with keys from an arbitrary
2516           text ([ wickland96thirty ]) into a container
2517           using <function>insert</function> . It measures the average time
2518           for <function>insert</function> as a function of the number of
2519           values inserted.</para>
2520
2521           <para>The test checks the effect of different underlying
2522           data structures.</para>
2523
2524           <para>
2525             It uses the test file:
2526             <filename>performance/ext/pb_ds/tree_text_insert_timing.cc</filename>
2527           </para>
2528
2529
2530         </section>
2531
2532         <section xml:id="branch.text_insert.results">
2533           <info><title>
2534             Results
2535           </title></info>
2536
2537           <para>The three graphics below show the results for the native
2538           tree and this library's node-based trees, the native tree and
2539           this library's vector-based trees, and the native tree
2540           and this library's PATRICIA-trie, respectively.
2541           </para>
2542
2543           <para>The graphic immediately below shows the results for the
2544           native tree type and several node-based tree types.
2545           </para>
2546
2547           <!-- results graphic -->
2548           <informalfigure>
2549             <mediaobject>
2550               <imageobject>
2551                 <imagedata align="center" format="PDF" scale="75"
2552                            fileref="../images/pbds_tree_text_insert_node.pdf"/>
2553               </imageobject>
2554               <imageobject>
2555                 <imagedata align="center" format="PNG" scale="100"
2556                            fileref="../images/pbds_tree_text_insert_node.png"/>
2557               </imageobject>
2558             </mediaobject>
2559
2560
2561             <para>
2562               The abbreviated names in the legend of the graphic above are
2563               instantiated with the types in the following table.
2564             </para>
2565           </informalfigure>
2566
2567           <informaltable frame="all">
2568             <tgroup cols="3" align="left" colsep="1" rowsep="1">
2569               <colspec colname="c1"/>
2570               <colspec colname="c2"/>
2571               <colspec colname="c3"/>
2572               <thead>
2573                 <row>
2574                   <entry><emphasis>Name/Instantiating Type</emphasis></entry>
2575                   <entry><emphasis>Parameter</emphasis></entry>
2576                   <entry><emphasis>Details</emphasis></entry>
2577                 </row>
2578               </thead>
2579
2580               <tbody>
2581
2582                 <!-- native -->
2583                 <row>
2584                   <?dbhtml bgcolor="#B0B0B0" ?>
2585                   <entry namest="c1" nameend="c3">
2586                     n_map
2587                   </entry>
2588                 </row>
2589
2590                 <row>
2591                   <entry>
2592                     <classname>std::map</classname>
2593                   </entry>
2594                   <entry namest="c2" nameend="c3"></entry>
2595                 </row>
2596
2597                 <!-- branch 01 -->
2598                 <row>
2599                   <?dbhtml bgcolor="#B0B0B0" ?>
2600                   <entry namest="c1" nameend="c3">
2601                     splay_tree_map
2602                   </entry>
2603                 </row>
2604
2605                 <row>
2606                   <entry morerows="1" valign="top">
2607                     <classname>tree</classname>
2608                   </entry>
2609                   <entry>
2610                     <classname>Tag</classname>
2611                   </entry>
2612                   <entry>
2613                     <classname>splay_tree_tag</classname>
2614                   </entry>
2615                 </row>
2616
2617                 <row>
2618                   <entry>
2619                     <classname>Node_update</classname>
2620                   </entry>
2621                   <entry>
2622                     <classname>null_node_update</classname>
2623                   </entry>
2624                 </row>
2625
2626
2627                 <!-- branch 02 -->
2628                 <row>
2629                   <?dbhtml bgcolor="#B0B0B0" ?>
2630                   <entry namest="c1" nameend="c3">
2631                     rb_tree_map
2632                   </entry>
2633                 </row>
2634
2635                 <row>
2636                   <entry morerows="1" valign="top">
2637                     <classname>tree</classname>
2638                   </entry>
2639                   <entry>
2640                     <classname>Tag</classname>
2641                   </entry>
2642                   <entry>
2643                     <classname>rb_tree_tag</classname>
2644                   </entry>
2645                 </row>
2646
2647                 <row>
2648                   <entry>
2649                     <classname>Node_update</classname>
2650                   </entry>
2651                   <entry>
2652                     <classname>null_node_update</classname>
2653                   </entry>
2654                 </row>
2655
2656               </tbody>
2657             </tgroup>
2658           </informaltable>
2659
2660
2661
2662           <para>The graphic below shows the results for the
2663           native tree type and a vector-based tree type.
2664           </para>
2665
2666           <!-- results graphic -->
2667           <informalfigure>
2668             <mediaobject>
2669               <imageobject>
2670                 <imagedata align="center" format="PDF" scale="75"
2671                            fileref="../images/pbds_tree_text_insert_vector.pdf"/>
2672               </imageobject>
2673               <imageobject>
2674                 <imagedata align="center" format="PNG" scale="100"
2675                            fileref="../images/pbds_tree_text_insert_vector.png"/>
2676               </imageobject>
2677             </mediaobject>
2678           </informalfigure>
2679
2680           <para>
2681             The abbreviated names in the legend of the graphic above are
2682             instantiated with the types in the following table.
2683           </para>
2684
2685
2686           <informaltable frame="all">
2687
2688             <tgroup cols="3" align="left" colsep="1" rowsep="1">
2689               <colspec colname="c1"/>
2690               <colspec colname="c2"/>
2691               <colspec colname="c3"/>
2692               <thead>
2693                 <row>
2694                   <entry><emphasis>Name/Instantiating Type</emphasis></entry>
2695                   <entry><emphasis>Parameter</emphasis></entry>
2696                   <entry><emphasis>Details</emphasis></entry>
2697                 </row>
2698               </thead>
2699
2700               <tbody>
2701
2702                 <!-- native -->
2703                 <row>
2704                   <?dbhtml bgcolor="#B0B0B0" ?>
2705                   <entry namest="c1" nameend="c3">
2706                     n_map
2707                   </entry>
2708                 </row>
2709
2710                 <row>
2711                   <entry>
2712                     <classname>std::map</classname>
2713                   </entry>
2714                   <entry namest="c2" nameend="c3"></entry>
2715                 </row>
2716
2717                 <!-- branch 01 -->
2718                 <row>
2719                   <?dbhtml bgcolor="#B0B0B0" ?>
2720                   <entry namest="c1" nameend="c3">
2721                     ov_tree_map
2722                   </entry>
2723                 </row>
2724
2725                 <row>
2726                   <entry morerows="1" valign="top">
2727                     <classname>tree</classname>
2728                   </entry>
2729                   <entry>
2730                     <classname>Tag</classname>
2731                   </entry>
2732                   <entry>
2733                     <classname>ov_tree_tag</classname>
2734                   </entry>
2735                 </row>
2736
2737                 <row>
2738                   <entry>
2739                     <classname>Node_update</classname>
2740                   </entry>
2741                   <entry>
2742                     <classname>null_node_update</classname>
2743                   </entry>
2744                 </row>
2745
2746
2747               </tbody>
2748             </tgroup>
2749           </informaltable>
2750
2751
2752
2753
2754           <para>The graphic below shows the results for the
2755           native tree type and a PATRICIA trie type.
2756           </para>
2757
2758           <!-- results graphic -->
2759           <informalfigure>
2760             <mediaobject>
2761               <imageobject>
2762                 <imagedata align="center" format="PDF" scale="75"
2763                            fileref="../images/pbds_tree_text_insert_trie.pdf"/>
2764               </imageobject>
2765               <imageobject>
2766                 <imagedata align="center" format="PNG" scale="100"
2767                            fileref="../images/pbds_tree_text_insert_trie.png"/>
2768               </imageobject>
2769             </mediaobject>
2770           </informalfigure>
2771
2772           <para>
2773             The abbreviated names in the legend of the graphic above are
2774             instantiated with the types in the following table.
2775           </para>
2776
2777
2778           <informaltable frame="all">
2779
2780             <tgroup cols="3" align="left" colsep="1" rowsep="1">
2781               <colspec colname="c1"/>
2782               <colspec colname="c2"/>
2783               <colspec colname="c3"/>
2784               <thead>
2785                 <row>
2786                   <entry><emphasis>Name/Instantiating Type</emphasis></entry>
2787                   <entry><emphasis>Parameter</emphasis></entry>
2788                   <entry><emphasis>Details</emphasis></entry>
2789                 </row>
2790               </thead>
2791
2792               <tbody>
2793
2794                 <!-- native -->
2795                 <row>
2796                   <?dbhtml bgcolor="#B0B0B0" ?>
2797                   <entry namest="c1" nameend="c3">
2798                     n_map
2799                   </entry>
2800                 </row>
2801
2802                 <row>
2803                   <entry>
2804                     <classname>std::map</classname>
2805                   </entry>
2806                   <entry namest="c2" nameend="c3"></entry>
2807                 </row>
2808
2809                 <!-- branch 01 -->
2810                 <row>
2811                   <?dbhtml bgcolor="#B0B0B0" ?>
2812                   <entry namest="c1" nameend="c3">
2813                     pat_trie_map
2814                   </entry>
2815                 </row>
2816
2817                 <row>
2818                   <entry morerows="1" valign="top">
2819                     <classname>tree</classname>
2820                   </entry>
2821                   <entry>
2822                     <classname>Tag</classname>
2823                   </entry>
2824                   <entry>
2825                     <classname>pat_trie_tag</classname>
2826                   </entry>
2827                 </row>
2828
2829                 <row>
2830                   <entry>
2831                     <classname>Node_update</classname>
2832                   </entry>
2833                   <entry>
2834                     <classname>null_node_update</classname>
2835                   </entry>
2836                 </row>
2837
2838
2839               </tbody>
2840             </tgroup>
2841           </informaltable>
2842
2843         </section>
2844
2845         <section xml:id="branch.text_insert.observations">
2846           <info><title>
2847             Observations
2848           </title></info>
2849
2850           <para>Observing the first graphic implies that for this setting, a splay tree
2851           (<classname>tree</classname> with <classname>Tag
2852           </classname> = <classname>splay_tree_tag</classname>) does not do
2853           well. See also the Branch-Based
2854           Text <function>find</function> Find Timing Test. The two
2855           red-black trees perform better.</para>
2856
2857           <para>Observing the second graphic, an ordered-vector tree
2858           (<classname>tree</classname> with <classname>Tag
2859           </classname> = <classname>ov_tree_tag</classname>) performs
2860           abysmally. Inserting into this type of tree has linear complexity
2861           [ austern00noset].</para>
2862
2863           <para>Observing the third and last graphic, A PATRICIA trie
2864           (<classname>trie</classname> with <classname>Tag
2865           </classname> = <classname>pat_trie_tag</classname>) has abysmal
2866           performance, as well. This is not that surprising, since a
2867           large-fan-out PATRICIA trie works like a hash table with
2868           collisions resolved by a sub-trie. Each time a collision is
2869           encountered, a new "hash-table" is built A large fan-out PATRICIA
2870           trie, however, doe does well in look-ups (see Branch-Based
2871           Text <function>find</function> Find Timing Test). It may be
2872           beneficial in semi-static settings.</para>
2873         </section>
2874
2875       </section>
2876
2877
2878       <!-- 02 <a href="tree_text_find_find_timing_test"> -->
2879       <section xml:id="performance.branch.text_find">
2880         <info><title>
2881           Text <function>find</function>
2882         </title></info>
2883         <para></para>
2884
2885         <section xml:id="branch.text_find.info">
2886           <info><title>
2887             Description
2888           </title></info>
2889
2890
2891           <para>This test inserts a number of values with keys from an
2892           arbitrary text ([wickland96thirty]) into
2893           a container, then performs a series of finds using
2894           <function>find</function>. It measures the average time
2895           for <function>find</function> as a function of the number of
2896           values inserted.</para>
2897
2898           <para>
2899             It uses the test file:
2900             <filename>performance/ext/pb_ds/text_find_timing.cc</filename>
2901           </para>
2902
2903
2904           <para>The test checks the effect of different underlying
2905           data structures.</para>
2906
2907         </section>
2908
2909         <section xml:id="branch.text_find.results">
2910           <info><title>
2911             Results
2912           </title></info>
2913
2914           <para>The graphic immediately below shows the results for the
2915           native tree type and several other tree types.
2916           </para>
2917
2918           <!-- results graphic -->
2919           <informalfigure>
2920             <mediaobject>
2921               <imageobject>
2922                 <imagedata align="center" format="PDF" scale="75"
2923                            fileref="../images/pbds_tree_text_find.pdf"/>
2924               </imageobject>
2925               <imageobject>
2926                 <imagedata align="center" format="PNG" scale="100"
2927                            fileref="../images/pbds_tree_text_find.png"/>
2928               </imageobject>
2929             </mediaobject>
2930           </informalfigure>
2931
2932           <para>
2933             The abbreviated names in the legend of the graphic above are
2934             instantiated with the types in the following table.
2935           </para>
2936
2937
2938           <informaltable frame="all">
2939
2940             <tgroup cols="3" align="left" colsep="1" rowsep="1">
2941               <colspec colname="c1"/>
2942               <colspec colname="c2"/>
2943               <colspec colname="c3"/>
2944               <thead>
2945                 <row>
2946                   <entry><emphasis>Name/Instantiating Type</emphasis></entry>
2947                   <entry><emphasis>Parameter</emphasis></entry>
2948                   <entry><emphasis>Details</emphasis></entry>
2949                 </row>
2950               </thead>
2951
2952               <tbody>
2953
2954                 <!-- native -->
2955                 <row>
2956                   <?dbhtml bgcolor="#B0B0B0" ?>
2957                   <entry namest="c1" nameend="c3">
2958                     n_map
2959                   </entry>
2960                 </row>
2961
2962                 <row>
2963                   <entry>
2964                     <classname>std::map</classname>
2965                   </entry>
2966                   <entry namest="c2" nameend="c3"></entry>
2967                 </row>
2968
2969                 <!-- branch 01 -->
2970                 <row>
2971                   <?dbhtml bgcolor="#B0B0B0" ?>
2972                   <entry namest="c1" nameend="c3">
2973                     splay_tree_map
2974                   </entry>
2975                 </row>
2976
2977                 <row>
2978                   <entry morerows="1" valign="top">
2979                     <classname>tree</classname>
2980                   </entry>
2981                   <entry>
2982                     <classname>Tag</classname>
2983                   </entry>
2984                   <entry>
2985                     <classname>splay_tree_tag</classname>
2986                   </entry>
2987                 </row>
2988
2989                 <row>
2990                   <entry>
2991                     <classname>Node_Update</classname>
2992                   </entry>
2993                   <entry>
2994                     <classname>null_node_update</classname>
2995                   </entry>
2996                 </row>
2997
2998
2999                 <!-- branch 02 -->
3000                 <row>
3001                   <?dbhtml bgcolor="#B0B0B0" ?>
3002                   <entry namest="c1" nameend="c3">
3003                     rb_tree_map
3004                   </entry>
3005                 </row>
3006
3007                 <row>
3008                   <entry morerows="1" valign="top">
3009                     <classname>tree</classname>
3010                   </entry>
3011                   <entry>
3012                     <classname>Tag</classname>
3013                   </entry>
3014                   <entry>
3015                     <classname>rb_tree_tag</classname>
3016                   </entry>
3017                 </row>
3018
3019                 <row>
3020                   <entry>
3021                     <classname>Node_Update</classname>
3022                   </entry>
3023                   <entry>
3024                     <classname>null_node_update</classname>
3025                   </entry>
3026                 </row>
3027
3028
3029                 <!-- branch 03 -->
3030                 <row>
3031                   <?dbhtml bgcolor="#B0B0B0" ?>
3032                   <entry namest="c1" nameend="c3">
3033                     ov_tree_map
3034                   </entry>
3035                 </row>
3036
3037                 <row>
3038                   <entry morerows="1" valign="top">
3039                     <classname>tree</classname>
3040                   </entry>
3041                   <entry>
3042                     <classname>Tag</classname>
3043                   </entry>
3044                   <entry>
3045                     <classname>ov_tree_tag</classname>
3046                   </entry>
3047                 </row>
3048
3049                 <row>
3050                   <entry>
3051                     <classname>Node_Update</classname>
3052                   </entry>
3053                   <entry>
3054                     <classname>null_node_update</classname>
3055                   </entry>
3056                 </row>
3057
3058
3059                 <!-- branch 05 -->
3060                 <row>
3061                   <?dbhtml bgcolor="#B0B0B0" ?>
3062                   <entry namest="c1" nameend="c3">
3063                     pat_trie_map
3064                   </entry>
3065                 </row>
3066
3067                 <row>
3068                   <entry morerows="1" valign="top">
3069                     <classname>tree</classname>
3070                   </entry>
3071                   <entry>
3072                     <classname>Tag</classname>
3073                   </entry>
3074                   <entry>
3075                     <classname>pat_trie_tag</classname>
3076                   </entry>
3077                 </row>
3078
3079                 <row>
3080                   <entry>
3081                     <classname>Node_Update</classname>
3082                   </entry>
3083                   <entry>
3084                     <classname>null_node_update</classname>
3085                   </entry>
3086                 </row>
3087
3088               </tbody>
3089             </tgroup>
3090           </informaltable>
3091
3092         </section>
3093
3094         <section xml:id="branch.text_find.observations">
3095           <info><title>
3096             Observations
3097           </title></info>
3098
3099           <para>For this setting, a splay tree (<classname>tree</classname>
3100           with <classname>Tag
3101           </classname> = <classname>splay_tree_tag</classname>) does not do
3102           well. This is possibly due to two reasons:</para>
3103
3104           <orderedlist>
3105             <listitem><para>A splay tree is not guaranteed to be balanced [motwani95random]. If a
3106             splay tree contains n nodes, its average root-leaf
3107             path can be m &gt;&gt; log(n).</para></listitem>
3108             <listitem><para>Assume a specific root-leaf search path has length
3109             m, and the search-target node has distance m'
3110             from the root. A red-black tree will require m + 1
3111             comparisons to find the required node; a splay tree will
3112             require 2 m' comparisons. A splay tree, consequently,
3113             can perform many more comparisons than a red-black tree.</para></listitem>
3114           </orderedlist>
3115           <para>An ordered-vector tree (<classname>tree</classname>
3116           with <classname>Tag</classname> =  <classname>ov_tree_tag</classname>), a red-black
3117           tree (<classname>tree</classname>
3118           with <classname>Tag</classname>  = <classname>rb_tree_tag</classname>), and the
3119           native red-black tree all share approximately the same
3120           performance.</para>
3121           <para>An ordered-vector tree is slightly slower than red-black
3122           trees, since it requires, in order to find a key, more math
3123           operations than they do. Conversely, an ordered-vector tree
3124           requires far lower space than the others. ([austern00noset], however,
3125           seems to have an implementation that is also faster than a
3126           red-black tree).</para>
3127           <para>A PATRICIA trie (<classname>trie</classname>
3128           with <classname>Tag</classname> = <classname>pat_trie_tag</classname>) has good
3129           look-up performance, due to its large fan-out in this case. In
3130           this setting, a PATRICIA trie has look-up performance comparable
3131           to a hash table (see Hash-Based Text
3132           <classname>find</classname> Timing Test), but it is order
3133           preserving. This is not that surprising, since a large-fan-out
3134           PATRICIA trie works like a hash table with collisions resolved
3135           by a sub-trie. A large-fan-out PATRICIA trie does not do well on
3136           modifications (see Tree-Based and Trie-Based
3137           Text Insert Timing Test). Therefore, it is possibly beneficial in
3138           semi-static settings.</para>
3139         </section>
3140       </section>
3141
3142
3143       <!-- 03 <a href="tree_text_lor_find_find_timing_test"> -->
3144       <section xml:id="performance.branch.text_lor_find">
3145
3146         <info><title>
3147           Text <function>find</function> with Locality-of-Reference
3148         </title></info>
3149         <para></para>
3150
3151         <section xml:id="branch.text_lor_find.info">
3152           <info><title>
3153             Description
3154           </title></info>
3155
3156
3157
3158           <para>This test inserts a number of values with keys from an
3159           arbitrary text ([ wickland96thirty ]) into
3160           a container, then performs a series of finds using
3161           <function>find</function>. It is different than Tree-Based and
3162           Trie-Based Text <function>find</function> Find Timing Test in the
3163           sequence of finds it performs: this test performs multiple
3164           <function>find</function>s on the same key before moving on to the next
3165           key. It measures the average time for <function>find</function> as a
3166           function of the number of values inserted.</para>
3167
3168
3169           <para>
3170             It uses the test file:
3171             <filename>performance/ext/pb_ds/tree_text_lor_find_timing.cc</filename>
3172           </para>
3173
3174           <para>The test checks the effect of different underlying
3175           data structures in a locality-of-reference setting.</para>
3176
3177         </section>
3178
3179         <section xml:id="branch.text_lor_find.results">
3180           <info><title>
3181             Results
3182           </title></info>
3183
3184           <para>The graphic immediately below shows the results for the
3185           native tree type and several other tree types.
3186           </para>
3187
3188           <!-- results graphic -->
3189           <informalfigure>
3190             <mediaobject>
3191               <imageobject>
3192                 <imagedata align="center" format="PDF" scale="75"
3193                            fileref="../images/pbds_tree_text_lor_find.pdf"/>
3194               </imageobject>
3195               <imageobject>
3196                 <imagedata align="center" format="PNG" scale="100"
3197                            fileref="../images/pbds_tree_text_lor_find.png"/>
3198               </imageobject>
3199             </mediaobject>
3200           </informalfigure>
3201
3202           <para>
3203             The abbreviated names in the legend of the graphic above are
3204             instantiated with the types in the following table.
3205           </para>
3206
3207
3208           <informaltable frame="all">
3209
3210             <tgroup cols="3" align="left" colsep="1" rowsep="1">
3211               <colspec colname="c1"/>
3212               <colspec colname="c2"/>
3213               <colspec colname="c3"/>
3214               <thead>
3215                 <row>
3216                   <entry><emphasis>Name/Instantiating Type</emphasis></entry>
3217                   <entry><emphasis>Parameter</emphasis></entry>
3218                   <entry><emphasis>Details</emphasis></entry>
3219                 </row>
3220               </thead>
3221
3222               <tbody>
3223
3224                 <!-- native -->
3225                 <row>
3226                   <?dbhtml bgcolor="#B0B0B0" ?>
3227                   <entry namest="c1" nameend="c3">
3228                     n_map
3229                   </entry>
3230                 </row>
3231
3232                 <row>
3233                   <entry>
3234                     <classname>std::map</classname>
3235                   </entry>
3236                   <entry namest="c2" nameend="c3"></entry>
3237                 </row>
3238
3239                 <!-- branch 01 -->
3240                 <row>
3241                   <?dbhtml bgcolor="#B0B0B0" ?>
3242                   <entry namest="c1" nameend="c3">
3243                     splay_tree_map
3244                   </entry>
3245                 </row>
3246
3247                 <row>
3248                   <entry morerows="1" valign="top">
3249                     <classname>tree</classname>
3250                   </entry>
3251                   <entry>
3252                     <classname>Tag</classname>
3253                   </entry>
3254                   <entry>
3255                     <classname>splay_tree_tag</classname>
3256                   </entry>
3257                 </row>
3258
3259                 <row>
3260                   <entry>
3261                     <classname>Node_Update</classname>
3262                   </entry>
3263                   <entry>
3264                     <classname>null_node_update</classname>
3265                   </entry>
3266                 </row>
3267
3268
3269                 <!-- branch 02 -->
3270                 <row>
3271                   <?dbhtml bgcolor="#B0B0B0" ?>
3272                   <entry namest="c1" nameend="c3">
3273                     rb_tree_map
3274                   </entry>
3275                 </row>
3276
3277                 <row>
3278                   <entry morerows="1" valign="top">
3279                     <classname>tree</classname>
3280                   </entry>
3281                   <entry>
3282                     <classname>Tag</classname>
3283                   </entry>
3284                   <entry>
3285                     <classname>rb_tree_tag</classname>
3286                   </entry>
3287                 </row>
3288
3289                 <row>
3290                   <entry>
3291                     <classname>Node_Update</classname>
3292                   </entry>
3293                   <entry>
3294                     <classname>null_node_update</classname>
3295                   </entry>
3296                 </row>
3297
3298
3299                 <!-- branch 03 -->
3300                 <row>
3301                   <?dbhtml bgcolor="#B0B0B0" ?>
3302                   <entry namest="c1" nameend="c3">
3303                     ov_tree_map
3304                   </entry>
3305                 </row>
3306
3307                 <row>
3308                   <entry morerows="1" valign="top">
3309                     <classname>tree</classname>
3310                   </entry>
3311                   <entry>
3312                     <classname>Tag</classname>
3313                   </entry>
3314                   <entry>
3315                     <classname>ov_tree_tag</classname>
3316                   </entry>
3317                 </row>
3318
3319                 <row>
3320                   <entry>
3321                     <classname>Node_Update</classname>
3322                   </entry>
3323                   <entry>
3324                     <classname>null_node_update</classname>
3325                   </entry>
3326                 </row>
3327
3328
3329                 <!-- branch 05 -->
3330                 <row>
3331                   <?dbhtml bgcolor="#B0B0B0" ?>
3332                   <entry namest="c1" nameend="c3">
3333                     pat_trie_map
3334                   </entry>
3335                 </row>
3336
3337                 <row>
3338                   <entry morerows="1" valign="top">
3339                     <classname>tree</classname>
3340                   </entry>
3341                   <entry>
3342                     <classname>Tag</classname>
3343                   </entry>
3344                   <entry>
3345                     <classname>pat_trie_tag</classname>
3346                   </entry>
3347                 </row>
3348
3349                 <row>
3350                   <entry>
3351                     <classname>Node_Update</classname>
3352                   </entry>
3353                   <entry>
3354                     <classname>null_node_update</classname>
3355                   </entry>
3356                 </row>
3357
3358               </tbody>
3359             </tgroup>
3360           </informaltable>
3361
3362         </section>
3363
3364         <section xml:id="branch.text_lor_find.observations">
3365           <info><title>
3366             Observations
3367           </title></info>
3368
3369           <para>For this setting, an ordered-vector tree
3370           (<classname>tree</classname> with <classname>Tag</classname>
3371           = <classname>ov_tree_tag</classname>), a red-black tree
3372           (<classname>tree</classname> with <classname>Tag</classname>
3373           = <classname>rb_tree_tag</classname>), and the native red-black
3374           tree all share approximately the same performance.</para>
3375           <para>A splay tree (<classname>tree</classname>
3376           with <classname>Tag</classname> = <classname>splay_tree_tag</classname>) does
3377           much better, since each (successful) find "bubbles" the
3378           corresponding node to the root of the tree.</para>
3379
3380         </section>
3381       </section>
3382
3383       <!-- 04 <a href="tree_split_join_timing_test"> -->
3384       <section xml:id="performance.branch.split_join">
3385
3386         <info><title>
3387           <function>split</function> and <function>join</function>
3388         </title></info>
3389         <para></para>
3390
3391         <section xml:id="branch.split_join.info">
3392           <info><title>
3393             Description
3394           </title></info>
3395
3396
3397           <para>This test a container, inserts into a number of values, splits
3398           the container at the median, and joins the two containers. (If the
3399           containers are one of this library's trees,
3400           it splits and joins with the <function>split</function> and
3401           <function>join</function> method; otherwise, it uses the <function>erase</function> and
3402           <function>insert</function> methods.) It measures the time for splitting
3403           and joining the containers as a function of the number of
3404           values inserted.</para>
3405
3406           <para>
3407             It uses the test file:
3408             <filename>performance/ext/pb_ds/tree_split_join_timing.cc</filename>
3409           </para>
3410
3411
3412           <para>The test checks the performance difference of <function>join</function>
3413           as opposed to a sequence of <function>insert</function> operations; by
3414           implication, this test checks the most efficient way to erase a
3415           sub-sequence from a tree-like-based container, since this can
3416           always be performed by a small sequence of splits and joins.
3417           </para>
3418         </section>
3419
3420         <section xml:id="branch.split_join.results">
3421           <info><title>
3422             Results
3423           </title></info>
3424
3425           <para>The graphic immediately below shows the results for the
3426           native tree type and several other tree types.
3427           </para>
3428
3429           <!-- results graphic -->
3430           <informalfigure>
3431             <mediaobject>
3432               <imageobject>
3433                 <imagedata align="center" format="PDF" scale="75"
3434                            fileref="../images/pbds_tree_split_join.pdf"/>
3435               </imageobject>
3436               <imageobject>
3437                 <imagedata align="center" format="PNG" scale="100"
3438                            fileref="../images/pbds_tree_split_join.png"/>
3439               </imageobject>
3440             </mediaobject>
3441           </informalfigure>
3442
3443           <para>
3444             The abbreviated names in the legend of the graphic above are
3445             instantiated with the types in the following table.
3446           </para>
3447
3448
3449           <informaltable frame="all">
3450
3451             <tgroup cols="3" align="left" colsep="1" rowsep="1">
3452               <colspec colname="c1"/>
3453               <colspec colname="c2"/>
3454               <colspec colname="c3"/>
3455               <thead>
3456                 <row>
3457                   <entry><emphasis>Name/Instantiating Type</emphasis></entry>
3458                   <entry><emphasis>Parameter</emphasis></entry>
3459                   <entry><emphasis>Details</emphasis></entry>
3460                 </row>
3461               </thead>
3462
3463               <tbody>
3464
3465                 <!-- native -->
3466                 <row>
3467                   <?dbhtml bgcolor="#B0B0B0" ?>
3468                   <entry namest="c1" nameend="c3">
3469                     n_set
3470                   </entry>
3471                 </row>
3472
3473                 <row>
3474                   <entry>
3475                     <classname>std::set</classname>
3476                   </entry>
3477                   <entry namest="c2" nameend="c3"></entry>
3478                 </row>
3479
3480                 <!-- branch 01 -->
3481                 <row>
3482                   <?dbhtml bgcolor="#B0B0B0" ?>
3483                   <entry namest="c1" nameend="c3">
3484                     splay_tree_set
3485                   </entry>
3486                 </row>
3487
3488                 <row>
3489                   <entry morerows="1" valign="top">
3490                     <classname>tree</classname>
3491                   </entry>
3492                   <entry>
3493                     <classname>Tag</classname>
3494                   </entry>
3495                   <entry>
3496                     <classname>splay_tree_tag</classname>
3497                   </entry>
3498                 </row>
3499
3500                 <row>
3501                   <entry>
3502                     <classname>Node_Update</classname>
3503                   </entry>
3504                   <entry>
3505                     <classname>null_node_update</classname>
3506                   </entry>
3507                 </row>
3508
3509
3510                 <!-- branch 02 -->
3511                 <row>
3512                   <?dbhtml bgcolor="#B0B0B0" ?>
3513                   <entry namest="c1" nameend="c3">
3514                     rb_tree_set
3515                   </entry>
3516                 </row>
3517
3518                 <row>
3519                   <entry morerows="1" valign="top">
3520                     <classname>tree</classname>
3521                   </entry>
3522                   <entry>
3523                     <classname>Tag</classname>
3524                   </entry>
3525                   <entry>
3526                     <classname>rb_tree_tag</classname>
3527                   </entry>
3528                 </row>
3529
3530                 <row>
3531                   <entry>
3532                     <classname>Node_Update</classname>
3533                   </entry>
3534                   <entry>
3535                     <classname>null_node_update</classname>
3536                   </entry>
3537                 </row>
3538
3539
3540                 <!-- branch 03 -->
3541                 <row>
3542                   <?dbhtml bgcolor="#B0B0B0" ?>
3543                   <entry namest="c1" nameend="c3">
3544                     ov_tree_set
3545                   </entry>
3546                 </row>
3547
3548                 <row>
3549                   <entry morerows="1" valign="top">
3550                     <classname>tree</classname>
3551                   </entry>
3552                   <entry>
3553                     <classname>Tag</classname>
3554                   </entry>
3555                   <entry>
3556                     <classname>ov_tree_tag</classname>
3557                   </entry>
3558                 </row>
3559
3560                 <row>
3561                   <entry>
3562                     <classname>Node_Update</classname>
3563                   </entry>
3564                   <entry>
3565                     <classname>null_node_update</classname>
3566                   </entry>
3567                 </row>
3568
3569
3570                 <!-- branch 05 -->
3571                 <row>
3572                   <?dbhtml bgcolor="#B0B0B0" ?>
3573                   <entry namest="c1" nameend="c3">
3574                     pat_trie_map
3575                   </entry>
3576                 </row>
3577
3578                 <row>
3579                   <entry morerows="1" valign="top">
3580                     <classname>tree</classname>
3581                   </entry>
3582                   <entry>
3583                     <classname>Tag</classname>
3584                   </entry>
3585                   <entry>
3586                     <classname>pat_trie_tag</classname>
3587                   </entry>
3588                 </row>
3589
3590                 <row>
3591                   <entry>
3592                     <classname>Node_Update</classname>
3593                   </entry>
3594                   <entry>
3595                     <classname>null_node_update</classname>
3596                   </entry>
3597                 </row>
3598
3599               </tbody>
3600             </tgroup>
3601           </informaltable>
3602
3603         </section>
3604
3605         <section xml:id="branch.split_join.observations">
3606           <info><title>
3607             Observations
3608           </title></info>
3609
3610           <para>In this test, the native red-black trees must be split and
3611           joined externally, through a sequence of <function>erase</function> and
3612           <function>insert</function> operations. This is clearly
3613           super-linear, and it is not that surprising that the cost is
3614           high.</para>
3615           <para>This library's tree-based containers use in this test the
3616           <function>split</function> and <function>join</function> methods,
3617           which have lower complexity: the <function>join</function> method
3618           of a splay tree (<classname>tree</classname>
3619           with <classname>Tag </classname>
3620           = <classname>splay_tree_tag</classname>) is quadratic in the
3621           length of the longest root-leaf path, and linear in the total
3622           number of elements; the <function>join</function> method of a
3623           red-black tree (<classname>tree</classname>
3624           with <classname>Tag </classname>
3625           = <classname>rb_tree_tag</classname>) or an ordered-vector tree
3626           (<classname>tree</classname> with <classname>Tag </classname>
3627           = <classname>ov_tree_tag</classname>) is linear in the number of
3628           elements.</para>
3629
3630           <para>Asides from orders of growth, this library's trees access their
3631           allocator very little in these operations, and some of them do not
3632           access it at all. This leads to lower constants in their
3633           complexity, and, for some containers, to exception-free splits and
3634           joins (which can be determined
3635           via <classname>container_traits</classname>).</para>
3636
3637           <para>It is important to note that <function>split</function> and
3638           <function>join</function> are not esoteric methods - they are the most
3639           efficient means of erasing a contiguous range of values from a
3640           tree based container.</para>
3641         </section>
3642       </section>
3643
3644       <!-- 05 <a href="tree_order_statistics_timing_test"> -->
3645       <section xml:id="performance.branch.order_statistics">
3646
3647         <info><title>
3648           Order-Statistics
3649         </title></info>
3650         <para></para>
3651
3652         <section xml:id="branch.order_statistics.info">
3653           <info><title>
3654             Description
3655           </title></info>
3656
3657           <para>This test creates a container, inserts random integers into the
3658           the container, and then checks the order-statistics of the
3659           container's values. (If the container is one of this
3660           library's trees, it does this with
3661           the <function>order_of_key</function> method of
3662           <classname>tree_order_statistics_node_update</classname>
3663           ; otherwise, it uses the <function>find</function> method and
3664           <function>std::distance</function>.) It measures the average
3665           time for such queries as a function of the number of values
3666           inserted.</para>
3667
3668           <para>
3669             It uses the test file:
3670             <filename>performance/ext/pb_ds/tree_order_statistics_timing.cc</filename>
3671           </para>
3672
3673           <para>The test checks the performance difference of policies based
3674           on node-invariant as opposed to a external functions.</para>
3675
3676         </section>
3677
3678         <section xml:id="branch.order_statistics.results">
3679           <info><title>
3680             Results
3681           </title></info>
3682
3683           <para>The graphic immediately below shows the results for the
3684           native tree type and several other tree types.
3685           </para>
3686
3687           <!-- results graphic -->
3688           <informalfigure>
3689             <mediaobject>
3690               <imageobject>
3691                 <imagedata align="center" format="PDF" scale="75"
3692                            fileref="../images/pbds_tree_order_statistics.pdf"/>
3693               </imageobject>
3694               <imageobject>
3695                 <imagedata align="center" format="PNG" scale="100"
3696                            fileref="../images/pbds_tree_order_statistics.png"/>
3697               </imageobject>
3698             </mediaobject>
3699           </informalfigure>
3700
3701           <para>
3702             The abbreviated names in the legend of the graphic above are
3703             instantiated with the types in the following table.
3704           </para>
3705
3706
3707           <informaltable frame="all">
3708
3709             <tgroup cols="3" align="left" colsep="1" rowsep="1">
3710               <colspec colname="c1"/>
3711               <colspec colname="c2"/>
3712               <colspec colname="c3"/>
3713               <thead>
3714                 <row>
3715                   <entry><emphasis>Name/Instantiating Type</emphasis></entry>
3716                   <entry><emphasis>Parameter</emphasis></entry>
3717                   <entry><emphasis>Details</emphasis></entry>
3718                 </row>
3719               </thead>
3720
3721               <tbody>
3722
3723                 <!-- native -->
3724                 <row>
3725                   <?dbhtml bgcolor="#B0B0B0" ?>
3726                   <entry namest="c1" nameend="c3">
3727                     n_set
3728                   </entry>
3729                 </row>
3730
3731                 <row>
3732                   <entry>
3733                     <classname>std::set</classname>
3734                   </entry>
3735                   <entry namest="c2" nameend="c3"></entry>
3736                 </row>
3737
3738                 <!-- branch 01 -->
3739                 <row>
3740                   <?dbhtml bgcolor="#B0B0B0" ?>
3741                   <entry namest="c1" nameend="c3">
3742                     splay_tree_ost_set
3743                   </entry>
3744                 </row>
3745
3746                 <row>
3747                   <entry morerows="1" valign="top">
3748                     <classname>tree</classname>
3749                   </entry>
3750                   <entry>
3751                     <classname>Tag</classname>
3752                   </entry>
3753                   <entry>
3754                     <classname>splay_tree_tag</classname>
3755                   </entry>
3756                 </row>
3757
3758                 <row>
3759                   <entry>
3760                     <classname>Node_Update</classname>
3761                   </entry>
3762                   <entry>
3763                     <classname>tree_order_statistics_node_update</classname>
3764                   </entry>
3765                 </row>
3766
3767
3768                 <!-- branch 02 -->
3769                 <row>
3770                   <?dbhtml bgcolor="#B0B0B0" ?>
3771                   <entry namest="c1" nameend="c3">
3772                     rb_tree_ost_set
3773                   </entry>
3774                 </row>
3775
3776                 <row>
3777                   <entry morerows="1" valign="top">
3778                     <classname>tree</classname>
3779                   </entry>
3780                   <entry>
3781                     <classname>Tag</classname>
3782                   </entry>
3783                   <entry>
3784                     <classname>rb_tree_tag</classname>
3785                   </entry>
3786                 </row>
3787
3788                 <row>
3789                   <entry>
3790                     <classname>Node_Update</classname>
3791                   </entry>
3792                   <entry>
3793                     <classname>tree_order_statistics_node_update</classname>
3794                   </entry>
3795                 </row>
3796
3797
3798               </tbody>
3799             </tgroup>
3800           </informaltable>
3801
3802         </section>
3803
3804         <section xml:id="branch.order_statistics.observations">
3805           <info><title>
3806             Observations
3807           </title></info>
3808           <para>In this test, the native red-black tree can support
3809           order-statistics queries only externally, by performing a
3810           <classname>find</classname> (alternatively, <classname>lower_bound</classname> or
3811           <classname>upper_bound</classname> ) and then using <classname>std::distance</classname> .
3812           This is clearly linear, and it is not that surprising that the
3813           cost is high.</para>
3814           <para>This library's tree-based containers use in this test the
3815           <classname>order_of_key</classname> method of <classname>tree_order_statistics_node_update</classname>.
3816           This method has only linear complexity in the length of the
3817           root-node path. Unfortunately, the average path of a splay tree
3818           (<classname>tree</classname>
3819           with <classname>Tag =</classname> <classname>splay_tree_tag</classname> ) can
3820           be higher than logarithmic; the longest path of a red-black
3821           tree (<classname>tree</classname>
3822           with <classname>Tag =</classname> <classname>rb_tree_tag</classname> ) is
3823           logarithmic in the number of elements. Consequently, the splay
3824           tree has worse performance than the red-black tree.</para>
3825         </section>
3826       </section>
3827
3828     </section> <!-- branch -->
3829
3830     <section xml:id="performance.multimap">
3831       <info><title>Multimap</title></info>
3832       <para></para>
3833
3834
3835       <!-- 01 <a href="multimap_text_find_timing_test_small"> -->
3836       <section xml:id="performance.multimap.text_find_small">
3837         <info><title>
3838           Text <function>find</function> with Small Secondary-to-Primary Key Ratios
3839         </title></info>
3840         <para></para>
3841
3842         <section xml:id="multimap.text_find_small.info">
3843           <info><title>
3844             Description
3845           </title></info>
3846
3847           <para>This test inserts a number of pairs into a container. The
3848           first item of each pair is a string from an arbitrary text
3849           [wickland96thirty], and
3850           the second is a uniform i.i.d.integer. The container is a
3851           "multimap" - it considers the first member of each pair as a
3852           primary key, and the second member of each pair as a secondary
3853           key (see Motivation::Associative
3854           Containers::Alternative to Multiple Equivalent Keys). There
3855           are 400 distinct primary keys, and the ratio of secondary keys
3856           to primary keys ranges from 1 to 5.</para>
3857           <para>The test measures the average find-time as a function of the
3858           number of values inserted. For this library's containers, it
3859           finds the secondary key from a container obtained from finding
3860           a primary key. For the native multimaps, it searches a range
3861           obtained using <classname>std::equal_range</classname> on a primary key.</para>
3862
3863           <para>
3864             It uses the test file:
3865             <filename>performance/ext/pb_ds/multimap_text_find_timing_small.cc</filename>
3866           </para>
3867
3868           <para>The test checks the find-time scalability of different
3869           "multimap" designs.</para>
3870
3871         </section>
3872
3873         <section xml:id="multimap.text_find_small.results">
3874           <info><title>
3875             Results
3876           </title></info>
3877
3878           <para>The graphic below show the results for "multimaps" which
3879           use a tree-based container for primary keys.
3880           </para>
3881
3882           <!-- results graphic -->
3883           <informalfigure>
3884             <mediaobject>
3885               <imageobject>
3886                 <imagedata align="center" format="PNG" scale="100"
3887                            fileref="../images/pbds_multimap_text_find_small_s2p_tree.png"/>
3888               </imageobject>
3889               <imageobject>
3890                 <imagedata align="center" format="PDF" scale="33"
3891                            fileref="../images/pbds_multimap_text_find_small_s2p_tree.pdf"/>
3892               </imageobject>
3893             </mediaobject>
3894           </informalfigure>
3895
3896           <para>
3897             The abbreviated names in the legend of the graphic above are
3898             instantiated with the types in the following table.
3899           </para>
3900
3901
3902           <informaltable frame="all">
3903
3904             <tgroup cols="7" align="left" colsep="1" rowsep="1">
3905               <colspec colname="c1"/>
3906               <colspec colname="c2"/>
3907               <colspec colname="c3"/>
3908               <colspec colname="c4"/>
3909               <colspec colname="c5"/>
3910               <colspec colname="c6"/>
3911               <colspec colname="c7"/>
3912               <thead>
3913                 <row>
3914                   <entry><emphasis>Name/Instantiating Type</emphasis></entry>
3915                   <entry><emphasis>Parameter</emphasis></entry>
3916                   <entry><emphasis>Details</emphasis></entry>
3917                   <entry><emphasis>Parameter</emphasis></entry>
3918                   <entry><emphasis>Details</emphasis></entry>
3919                   <entry><emphasis>Parameter</emphasis></entry>
3920                   <entry><emphasis>Details</emphasis></entry>
3921                 </row>
3922               </thead>
3923
3924               <tbody>
3925
3926                 <!-- native -->
3927                 <row>
3928                   <?dbhtml bgcolor="#B0B0B0" ?>
3929                   <entry namest="c1" nameend="c7">
3930                     n_mmap
3931                   </entry>
3932                 </row>
3933
3934                 <row>
3935                   <entry>
3936                     <classname>std::multimap</classname>
3937                   </entry>
3938                   <entry namest="c2" nameend="c7"></entry>
3939                 </row>
3940
3941                 <!-- multimap 01 -->
3942                 <row>
3943                   <?dbhtml bgcolor="#B0B0B0" ?>
3944                   <entry namest="c1" nameend="c7">
3945                     rb_tree_mmap_lu_mtf_set
3946                   </entry>
3947                 </row>
3948
3949                 <row>
3950                   <entry morerows="2" valign="top">
3951                     <classname>tree</classname>
3952                   </entry>
3953                   <entry>
3954                     <classname>Tag</classname>
3955                   </entry>
3956                   <entry>
3957                     <classname>rb_tree_tag</classname>
3958                   </entry>
3959                   <entry namest="c4" nameend="c7"></entry>
3960                 </row>
3961
3962                 <row>
3963                   <entry>
3964                     <classname>Node_Update</classname>
3965                   </entry>
3966                   <entry>
3967                     <classname>null_node_update</classname>
3968                   </entry>
3969                   <entry namest="c4" nameend="c7"></entry>
3970                 </row>
3971                 <row>
3972                   <entry>
3973                     <classname>Mapped</classname>
3974                   </entry>
3975                   <entry>
3976                     <classname>list_update</classname>
3977                   </entry>
3978                   <entry>
3979                     <classname>Update_Policy</classname>
3980                   </entry>
3981                   <entry>
3982                     <classname>lu_move_to_front_policy</classname>
3983                   </entry>
3984                   <entry namest="c6" nameend="c7"></entry>
3985                 </row>
3986
3987                 <!-- multimap 02 -->
3988                 <row>
3989                   <?dbhtml bgcolor="#B0B0B0" ?>
3990                   <entry namest="c1" nameend="c7">
3991                     rb_tree_mmap_cc_hash_mask_exp_1div2_nsth_set
3992                   </entry>
3993                 </row>
3994
3995                 <row>
3996                   <entry morerows="4" valign="top">
3997                     <classname>tree</classname>
3998                   </entry>
3999                   <entry>
4000                     <classname>Tag</classname>
4001                   </entry>
4002                   <entry>
4003                     <classname>rb_tree_tag</classname>
4004                   </entry>
4005                   <entry namest="c4" nameend="c7"></entry>
4006                 </row>
4007
4008                 <row>
4009                   <entry>
4010                     <classname>Node_Update</classname>
4011                   </entry>
4012                   <entry>
4013                     <classname>null_node_update</classname>
4014                   </entry>
4015                   <entry namest="c4" nameend="c7"></entry>
4016                 </row>
4017
4018                 <row>
4019                   <entry morerows="2" valign="top">
4020                     <classname>Mapped</classname>
4021                   </entry>
4022                   <entry morerows="2" valign="top">
4023                     <classname>cc_hash_table</classname>
4024                   </entry>
4025                   <entry>
4026                     <classname>Comb_Hash_Fn</classname>
4027                   </entry>
4028                   <entry>
4029                     <classname>direct_mask_range_hashing</classname>
4030                   </entry>
4031                   <entry namest="c6" nameend="c7"></entry>
4032                 </row>
4033
4034                 <row>
4035                   <entry morerows="1" valign="top">
4036                     <classname>Resize_Policy</classname>
4037                   </entry>
4038                   <entry morerows="1" valign="top">
4039                     <classname>hash_standard_resize_policy</classname>
4040                   </entry>
4041                   <entry>
4042                     <classname>Size_Policy</classname>
4043                   </entry>
4044                   <entry>
4045                     <classname>hash_exponential_size_policy</classname>
4046                   </entry>
4047                 </row>
4048
4049                 <row>
4050                   <entry valign="top">
4051                     <classname>Trigger_Policy</classname>
4052                   </entry>
4053                   <entry>
4054                     <classname>hash_load_check_resize_trigger</classname> with
4055                     α<subscript>min</subscript> = 1/8 and α<subscript>max</subscript> = 1/2
4056                   </entry>
4057                 </row>
4058
4059               </tbody>
4060             </tgroup>
4061           </informaltable>
4062
4063
4064           <para>The graphic below show the results for "multimaps" which
4065           use a hash-based container for primary keys.
4066           </para>
4067
4068           <!-- results graphic -->
4069           <informalfigure>
4070             <mediaobject>
4071               <imageobject>
4072                 <imagedata align="center" format="PNG" scale="100"
4073                            fileref="../images/pbds_multimap_text_find_small_s2p_hash.png"/>
4074               </imageobject>
4075               <imageobject>
4076                 <imagedata align="center" format="PDF" scale="33"
4077                            fileref="../images/pbds_multimap_text_find_small_s2p_hash.pdf"/>
4078               </imageobject>
4079             </mediaobject>
4080           </informalfigure>
4081
4082           <para>
4083             The abbreviated names in the legend of the graphic above are
4084             instantiated with the types in the following table.
4085           </para>
4086
4087           <informaltable frame="all">
4088
4089             <tgroup cols="7" align="left" colsep="1" rowsep="1">
4090               <colspec colname="c1"/>
4091               <colspec colname="c2"/>
4092               <colspec colname="c3"/>
4093               <colspec colname="c4"/>
4094               <colspec colname="c5"/>
4095               <colspec colname="c6"/>
4096               <colspec colname="c7"/>
4097               <thead>
4098                 <row>
4099                   <entry><emphasis>Name/Instantiating Type</emphasis></entry>
4100                   <entry><emphasis>Parameter</emphasis></entry>
4101                   <entry><emphasis>Details</emphasis></entry>
4102                   <entry><emphasis>Parameter</emphasis></entry>
4103                   <entry><emphasis>Details</emphasis></entry>
4104                   <entry><emphasis>Parameter</emphasis></entry>
4105                   <entry><emphasis>Details</emphasis></entry>
4106                 </row>
4107               </thead>
4108
4109               <tbody>
4110
4111                 <!-- native -->
4112                 <row>
4113                   <?dbhtml bgcolor="#B0B0B0" ?>
4114                   <entry namest="c1" nameend="c7">
4115                     n_hash_mmap
4116                   </entry>
4117                 </row>
4118
4119                 <row>
4120                   <entry>
4121                     <classname>std::tr1::unordered_multimap</classname>
4122                   </entry>
4123                   <entry namest="c2" nameend="c7"></entry>
4124                 </row>
4125
4126                 <!-- multimap 01 -->
4127                 <row>
4128                   <?dbhtml bgcolor="#B0B0B0" ?>
4129                   <entry namest="c1" nameend="c7">
4130                     rb_tree_mmap_lu_mtf_set
4131                   </entry>
4132                 </row>
4133
4134                 <row>
4135                   <entry morerows="3" valign="top">
4136                     <classname>
4137                       cc_hash_table
4138                     </classname>
4139                   </entry>
4140                   <entry>
4141                     <classname>Comb_Hash_Fn</classname>
4142                   </entry>
4143                   <entry>
4144                     <classname>direct_mask_range_hashing</classname>
4145                   </entry>
4146                   <entry namest="c4" nameend="c7"></entry>
4147                 </row>
4148                 <row>
4149                   <entry morerows="1" valign="top">
4150                     <classname>Resize_Policy</classname>
4151                   </entry>
4152                   <entry morerows="1" valign="top">
4153                     <classname>hash_standard_resize_policy</classname>
4154                   </entry>
4155                   <entry>
4156                     <classname>Size_Policy</classname>
4157                   </entry>
4158                   <entry>
4159                     <classname>hash_exponential_size_policy</classname>
4160                   </entry>
4161                   <entry namest="c6" nameend="c7"></entry>
4162                 </row>
4163
4164                 <row>
4165                   <entry valign="top">
4166                     <classname>Trigger_Policy</classname>
4167                   </entry>
4168                   <entry>
4169                     <classname>hash_load_check_resize_trigger</classname> with
4170                     α<subscript>min</subscript> = 1/8 and α<subscript>max</subscript> = 1/2
4171                   </entry>
4172                   <entry namest="c6" nameend="c7"></entry>
4173                 </row>
4174
4175                 <row>
4176                   <entry>
4177                     <classname>Mapped</classname>
4178                   </entry>
4179                   <entry>
4180                     <classname>list_update</classname>
4181                   </entry>
4182                   <entry>
4183                     <classname>Update_Policy</classname>
4184                   </entry>
4185                   <entry>
4186                     <classname>lu_move_to_front_policy</classname>
4187                   </entry>
4188                   <entry namest="c6" nameend="c7"></entry>
4189                 </row>
4190
4191                 <!-- multimap 02 -->
4192                 <row>
4193                   <?dbhtml bgcolor="#B0B0B0" ?>
4194                   <entry namest="c1" nameend="c7">
4195                     rb_tree_mmap_cc_hash_mask_exp_1div2_nsth_set
4196                   </entry>
4197                 </row>
4198
4199                 <row>
4200                   <entry morerows="5" valign="top">
4201                     <classname>
4202                       cc_hash_table
4203                     </classname>
4204                   </entry>
4205                   <entry>
4206                     <classname>Comb_Hash_Fn</classname>
4207                   </entry>
4208                   <entry>
4209                     <classname>direct_mask_range_hashing</classname>
4210                   </entry>
4211                   <entry namest="c4" nameend="c7"></entry>
4212                 </row>
4213                 <row>
4214                   <entry morerows="1" valign="top">
4215                     <classname>Resize_Policy</classname>
4216                   </entry>
4217                   <entry morerows="1" valign="top">
4218                     <classname>hash_standard_resize_policy</classname>
4219                   </entry>
4220                   <entry>
4221                     <classname>Size_Policy</classname>
4222                   </entry>
4223                   <entry>
4224                     <classname>hash_exponential_size_policy</classname>
4225                   </entry>
4226                   <entry namest="c6" nameend="c7"></entry>
4227                 </row>
4228
4229                 <row>
4230                   <entry valign="top">
4231                     <classname>Trigger_Policy</classname>
4232                   </entry>
4233                   <entry>
4234                     <classname>hash_load_check_resize_trigger</classname> with
4235                     α<subscript>min</subscript> = 1/8 and α<subscript>max</subscript> = 1/2
4236                   </entry>
4237                   <entry namest="c6" nameend="c7"></entry>
4238                 </row>
4239
4240                 <row>
4241                   <entry morerows="2" valign="top">
4242                     <classname>Mapped</classname>
4243                   </entry>
4244                   <entry morerows="2" valign="top">
4245                     <classname>cc_hash_table</classname>
4246                   </entry>
4247                   <entry>
4248                     <classname>Comb_Hash_Fn</classname>
4249                   </entry>
4250                   <entry>
4251                     <classname>direct_mask_range_hashing</classname>
4252                   </entry>
4253                   <entry namest="c6" nameend="c7"></entry>
4254                 </row>
4255
4256                 <row>
4257                   <entry morerows="1" valign="top">
4258                     <classname>Resize_Policy</classname>
4259                   </entry>
4260                   <entry morerows="1" valign="top">
4261                     <classname>hash_standard_resize_policy</classname>
4262                   </entry>
4263                   <entry>
4264                     <classname>Size_Policy</classname>
4265                   </entry>
4266                   <entry>
4267                     <classname>hash_exponential_size_policy</classname>
4268                   </entry>
4269                 </row>
4270
4271                 <row>
4272                   <entry valign="top">
4273                     <classname>Trigger_Policy</classname>
4274                   </entry>
4275                   <entry>
4276                     <classname>hash_load_check_resize_trigger</classname> with
4277                     α<subscript>min</subscript> = 1/8 and α<subscript>max</subscript> = 1/2
4278                   </entry>
4279                 </row>
4280
4281               </tbody>
4282             </tgroup>
4283           </informaltable>
4284
4285         </section>
4286
4287         <section xml:id="multimap.text_find_small.observations">
4288           <info><title>
4289             Observations
4290           </title></info>
4291
4292           <para>See Observations::Mapping-Semantics
4293           Considerations.</para>
4294
4295         </section>
4296
4297       </section>
4298
4299       <!-- 02 <a href="multimap_text_find_timing_test_large"> -->
4300       <section xml:id="performance.multimap.text_find_large">
4301         <info><title>
4302           Text <function>find</function> with Large Secondary-to-Primary Key Ratios
4303         </title></info>
4304         <para></para>
4305
4306         <section xml:id="multimap.text_find_large.info">
4307           <info><title>
4308             Description
4309           </title></info>
4310
4311           <para>This test inserts a number of pairs into a container. The
4312           first item of each pair is a string from an arbitrary text
4313           [wickland96thirty], and
4314           the second is a uniform integer. The container is a
4315           "multimap" - it considers the first member of each pair as a
4316           primary key, and the second member of each pair as a secondary
4317           key. There
4318           are 400 distinct primary keys, and the ratio of secondary keys
4319           to primary keys ranges from 1 to 5.</para>
4320           <para>The test measures the average find-time as a function of the
4321           number of values inserted. For this library's containers, it
4322           finds the secondary key from a container obtained from finding
4323           a primary key. For the native multimaps, it searches a range
4324           obtained using <classname>std::equal_range</classname> on a primary key.</para>
4325
4326           <para>
4327             It uses the test file:
4328             <filename>performance/ext/pb_ds/multimap_text_find_timing_large.cc</filename>
4329           </para>
4330
4331           <para>The test checks the find-time scalability of different
4332           "multimap" designs.</para>
4333
4334         </section>
4335
4336         <section xml:id="multimap.text_find_large.results">
4337           <info><title>
4338             Results
4339           </title></info>
4340
4341           <para>The graphic below show the results for "multimaps" which
4342           use a tree-based container for primary keys.
4343           </para>
4344
4345           <!-- results graphic -->
4346           <informalfigure>
4347             <mediaobject>
4348               <imageobject>
4349                 <imagedata align="center" format="PNG" scale="100"
4350                            fileref="../images/pbds_multimap_text_find_large_s2p_tree.png"/>
4351               </imageobject>
4352               <imageobject>
4353                 <imagedata align="center" format="PDF" scale="33"
4354                            fileref="../images/pbds_multimap_text_find_large_s2p_tree.pdf"/>
4355               </imageobject>
4356             </mediaobject>
4357           </informalfigure>
4358
4359           <para>
4360             The abbreviated names in the legend of the graphic above are
4361             instantiated with the types in the following table.
4362           </para>
4363
4364
4365           <informaltable frame="all">
4366
4367             <tgroup cols="7" align="left" colsep="1" rowsep="1">
4368               <colspec colname="c1"/>
4369               <colspec colname="c2"/>
4370               <colspec colname="c3"/>
4371               <colspec colname="c4"/>
4372               <colspec colname="c5"/>
4373               <colspec colname="c6"/>
4374               <colspec colname="c7"/>
4375               <thead>
4376                 <row>
4377                   <entry><emphasis>Name/Instantiating Type</emphasis></entry>
4378                   <entry><emphasis>Parameter</emphasis></entry>
4379                   <entry><emphasis>Details</emphasis></entry>
4380                   <entry><emphasis>Parameter</emphasis></entry>
4381                   <entry><emphasis>Details</emphasis></entry>
4382                   <entry><emphasis>Parameter</emphasis></entry>
4383                   <entry><emphasis>Details</emphasis></entry>
4384                 </row>
4385               </thead>
4386
4387               <tbody>
4388
4389                 <!-- native -->
4390                 <row>
4391                   <?dbhtml bgcolor="#B0B0B0" ?>
4392                   <entry namest="c1" nameend="c7">
4393                     n_mmap
4394                   </entry>
4395                 </row>
4396
4397                 <row>
4398                   <entry>
4399                     <classname>std::multimap</classname>
4400                   </entry>
4401                   <entry namest="c2" nameend="c7"></entry>
4402                 </row>
4403
4404                 <!-- multimap 01 -->
4405                 <row>
4406                   <?dbhtml bgcolor="#B0B0B0" ?>
4407                   <entry namest="c1" nameend="c7">
4408                     rb_tree_mmap_lu_mtf_set
4409                   </entry>
4410                 </row>
4411
4412                 <row>
4413                   <entry morerows="2" valign="top">
4414                     <classname>tree</classname>
4415                   </entry>
4416                   <entry>
4417                     <classname>Tag</classname>
4418                   </entry>
4419                   <entry>
4420                     <classname>rb_tree_tag</classname>
4421                   </entry>
4422                   <entry namest="c4" nameend="c7"></entry>
4423                 </row>
4424
4425                 <row>
4426                   <entry>
4427                     <classname>Node_Update</classname>
4428                   </entry>
4429                   <entry>
4430                     <classname>null_node_update</classname>
4431                   </entry>
4432                   <entry namest="c4" nameend="c7"></entry>
4433                 </row>
4434                 <row>
4435                   <entry>
4436                     <classname>Mapped</classname>
4437                   </entry>
4438                   <entry>
4439                     <classname>list_update</classname>
4440                   </entry>
4441                   <entry>
4442                     <classname>Update_Policy</classname>
4443                   </entry>
4444                   <entry>
4445                     <classname>lu_move_to_front_policy</classname>
4446                   </entry>
4447                   <entry namest="c6" nameend="c7"></entry>
4448                 </row>
4449
4450                 <!-- multimap 02 -->
4451                 <row>
4452                   <?dbhtml bgcolor="#B0B0B0" ?>
4453                   <entry namest="c1" nameend="c7">
4454                     rb_tree_mmap_cc_hash_mask_exp_1div2_nsth_set
4455                   </entry>
4456                 </row>
4457
4458                 <row>
4459                   <entry morerows="4" valign="top">
4460                     <classname>tree</classname>
4461                   </entry>
4462                   <entry>
4463                     <classname>Tag</classname>
4464                   </entry>
4465                   <entry>
4466                     <classname>rb_tree_tag</classname>
4467                   </entry>
4468                   <entry namest="c4" nameend="c7"></entry>
4469                 </row>
4470
4471                 <row>
4472                   <entry>
4473                     <classname>Node_Update</classname>
4474                   </entry>
4475                   <entry>
4476                     <classname>null_node_update</classname>
4477                   </entry>
4478                   <entry namest="c4" nameend="c7"></entry>
4479                 </row>
4480
4481                 <row>
4482                   <entry morerows="2" valign="top">
4483                     <classname>Mapped</classname>
4484                   </entry>
4485                   <entry morerows="2" valign="top">
4486                     <classname>cc_hash_table</classname>
4487                   </entry>
4488                   <entry>
4489                     <classname>Comb_Hash_Fn</classname>
4490                   </entry>
4491                   <entry>
4492                     <classname>direct_mask_range_hashing</classname>
4493                   </entry>
4494                   <entry namest="c6" nameend="c7"></entry>
4495                 </row>
4496
4497                 <row>
4498                   <entry morerows="1" valign="top">
4499                     <classname>Resize_Policy</classname>
4500                   </entry>
4501                   <entry morerows="1" valign="top">
4502                     <classname>hash_standard_resize_policy</classname>
4503                   </entry>
4504                   <entry>
4505                     <classname>Size_Policy</classname>
4506                   </entry>
4507                   <entry>
4508                     <classname>hash_exponential_size_policy</classname>
4509                   </entry>
4510                 </row>
4511
4512                 <row>
4513                   <entry valign="top">
4514                     <classname>Trigger_Policy</classname>
4515                   </entry>
4516                   <entry>
4517                     <classname>hash_load_check_resize_trigger</classname> with
4518                     α<subscript>min</subscript> = 1/8 and α<subscript>max</subscript> = 1/2
4519                   </entry>
4520                 </row>
4521
4522               </tbody>
4523             </tgroup>
4524           </informaltable>
4525
4526
4527           <para>The graphic below show the results for "multimaps" which
4528           use a hash-based container for primary keys.
4529           </para>
4530
4531           <!-- results graphic -->
4532           <informalfigure>
4533             <mediaobject>
4534               <imageobject>
4535                 <imagedata align="center" format="PNG" scale="100"
4536                            fileref="../images/pbds_multimap_text_find_large_s2p_hash.png"/>
4537               </imageobject>
4538               <imageobject>
4539                 <imagedata align="center" format="PDF" scale="33"
4540                            fileref="../images/pbds_multimap_text_find_large_s2p_hash.pdf"/>
4541               </imageobject>
4542             </mediaobject>
4543           </informalfigure>
4544
4545           <para>
4546             The abbreviated names in the legend of the graphic above are
4547             instantiated with the types in the following table.
4548           </para>
4549
4550           <informaltable frame="all">
4551
4552             <tgroup cols="7" align="left" colsep="1" rowsep="1">
4553               <colspec colname="c1"/>
4554               <colspec colname="c2"/>
4555               <colspec colname="c3"/>
4556               <colspec colname="c4"/>
4557               <colspec colname="c5"/>
4558               <colspec colname="c6"/>
4559               <colspec colname="c7"/>
4560               <thead>
4561                 <row>
4562                   <entry><emphasis>Name/Instantiating Type</emphasis></entry>
4563                   <entry><emphasis>Parameter</emphasis></entry>
4564                   <entry><emphasis>Details</emphasis></entry>
4565                   <entry><emphasis>Parameter</emphasis></entry>
4566                   <entry><emphasis>Details</emphasis></entry>
4567                   <entry><emphasis>Parameter</emphasis></entry>
4568                   <entry><emphasis>Details</emphasis></entry>
4569                 </row>
4570               </thead>
4571
4572               <tbody>
4573
4574                 <!-- native -->
4575                 <row>
4576                   <?dbhtml bgcolor="#B0B0B0" ?>
4577                   <entry namest="c1" nameend="c7">
4578                     n_hash_mmap
4579                   </entry>
4580                 </row>
4581
4582                 <row>
4583                   <entry>
4584                     <classname>std::tr1::unordered_multimap</classname>
4585                   </entry>
4586                   <entry namest="c2" nameend="c7"></entry>
4587                 </row>
4588
4589                 <!-- multimap 01 -->
4590                 <row>
4591                   <?dbhtml bgcolor="#B0B0B0" ?>
4592                   <entry namest="c1" nameend="c7">
4593                     rb_tree_mmap_lu_mtf_set
4594                   </entry>
4595                 </row>
4596
4597                 <row>
4598                   <entry morerows="3" valign="top">
4599                     <classname>
4600                       cc_hash_table
4601                     </classname>
4602                   </entry>
4603                   <entry>
4604                     <classname>Comb_Hash_Fn</classname>
4605                   </entry>
4606                   <entry>
4607                     <classname>direct_mask_range_hashing</classname>
4608                   </entry>
4609                   <entry namest="c4" nameend="c7"></entry>
4610                 </row>
4611                 <row>
4612                   <entry morerows="1" valign="top">
4613                     <classname>Resize_Policy</classname>
4614                   </entry>
4615                   <entry morerows="1" valign="top">
4616                     <classname>hash_standard_resize_policy</classname>
4617                   </entry>
4618                   <entry>
4619                     <classname>Size_Policy</classname>
4620                   </entry>
4621                   <entry>
4622                     <classname>hash_exponential_size_policy</classname>
4623                   </entry>
4624                   <entry namest="c6" nameend="c7"></entry>
4625                 </row>
4626
4627                 <row>
4628                   <entry valign="top">
4629                     <classname>Trigger_Policy</classname>
4630                   </entry>
4631                   <entry>
4632                     <classname>hash_load_check_resize_trigger</classname> with
4633                     α<subscript>min</subscript> = 1/8 and α<subscript>max</subscript> = 1/2
4634                   </entry>
4635                   <entry namest="c6" nameend="c7"></entry>
4636                 </row>
4637
4638                 <row>
4639                   <entry>
4640                     <classname>Mapped</classname>
4641                   </entry>
4642                   <entry>
4643                     <classname>list_update</classname>
4644                   </entry>
4645                   <entry>
4646                     <classname>Update_Policy</classname>
4647                   </entry>
4648                   <entry>
4649                     <classname>lu_move_to_front_policy</classname>
4650                   </entry>
4651                   <entry namest="c6" nameend="c7"></entry>
4652                 </row>
4653
4654                 <!-- multimap 02 -->
4655                 <row>
4656                   <?dbhtml bgcolor="#B0B0B0" ?>
4657                   <entry namest="c1" nameend="c7">
4658                     rb_tree_mmap_cc_hash_mask_exp_1div2_nsth_set
4659                   </entry>
4660                 </row>
4661
4662                 <row>
4663                   <entry morerows="5" valign="top">
4664                     <classname>
4665                       cc_hash_table
4666                     </classname>
4667                   </entry>
4668                   <entry>
4669                     <classname>Comb_Hash_Fn</classname>
4670                   </entry>
4671                   <entry>
4672                     <classname>direct_mask_range_hashing</classname>
4673                   </entry>
4674                   <entry namest="c4" nameend="c7"></entry>
4675                 </row>
4676                 <row>
4677                   <entry morerows="1" valign="top">
4678                     <classname>Resize_Policy</classname>
4679                   </entry>
4680                   <entry morerows="1" valign="top">
4681                     <classname>hash_standard_resize_policy</classname>
4682                   </entry>
4683                   <entry>
4684                     <classname>Size_Policy</classname>
4685                   </entry>
4686                   <entry>
4687                     <classname>hash_exponential_size_policy</classname>
4688                   </entry>
4689                   <entry namest="c6" nameend="c7"></entry>
4690                 </row>
4691
4692                 <row>
4693                   <entry valign="top">
4694                     <classname>Trigger_Policy</classname>
4695                   </entry>
4696                   <entry>
4697                     <classname>hash_load_check_resize_trigger</classname> with
4698                     α<subscript>min</subscript> = 1/8 and α<subscript>max</subscript> = 1/2
4699                   </entry>
4700                   <entry namest="c6" nameend="c7"></entry>
4701                 </row>
4702
4703                 <row>
4704                   <entry morerows="2" valign="top">
4705                     <classname>Mapped</classname>
4706                   </entry>
4707                   <entry morerows="2" valign="top">
4708                     <classname>cc_hash_table</classname>
4709                   </entry>
4710                   <entry>
4711                     <classname>Comb_Hash_Fn</classname>
4712                   </entry>
4713                   <entry>
4714                     <classname>direct_mask_range_hashing</classname>
4715                   </entry>
4716                   <entry namest="c6" nameend="c7"></entry>
4717                 </row>
4718
4719                 <row>
4720                   <entry morerows="1" valign="top">
4721                     <classname>Resize_Policy</classname>
4722                   </entry>
4723                   <entry morerows="1" valign="top">
4724                     <classname>hash_standard_resize_policy</classname>
4725                   </entry>
4726                   <entry>
4727                     <classname>Size_Policy</classname>
4728                   </entry>
4729                   <entry>
4730                     <classname>hash_exponential_size_policy</classname>
4731                   </entry>
4732                 </row>
4733
4734                 <row>
4735                   <entry valign="top">
4736                     <classname>Trigger_Policy</classname>
4737                   </entry>
4738                   <entry>
4739                     <classname>hash_load_check_resize_trigger</classname> with
4740                     α<subscript>min</subscript> = 1/8 and α<subscript>max</subscript> = 1/2
4741                   </entry>
4742                 </row>
4743
4744               </tbody>
4745             </tgroup>
4746           </informaltable>
4747
4748         </section>
4749
4750         <section xml:id="multimap.text_find_large.observations">
4751           <info><title>
4752             Observations
4753           </title></info>
4754
4755           <para>See Observations::Mapping-Semantics
4756           Considerations.</para>
4757
4758         </section>
4759
4760       </section>
4761
4762
4763       <!-- 03 <a href="multimap_text_insert_timing_test_small"> -->
4764       <section xml:id="performance.multimap.text_insert_small">
4765         <info><title>
4766           Text <function>insert</function> with Small
4767           Secondary-to-Primary Key Ratios
4768         </title></info>
4769         <para></para>
4770
4771         <section xml:id="multimap.text_insert_small.info">
4772           <info><title>
4773             Description
4774           </title></info>
4775
4776           <para>This test inserts a number of pairs into a container. The
4777           first item of each pair is a string from an arbitrary text
4778           [wickland96thirty], and
4779           the second is a uniform integer. The container is a
4780           "multimap" - it considers the first member of each pair as a
4781           primary key, and the second member of each pair as a secondary
4782           key. There
4783           are 400 distinct primary keys, and the ratio of secondary keys
4784           to primary keys ranges from 1 to 5.</para>
4785           <para>The test measures the average insert-time as a function of
4786           the number of values inserted. For this library's containers,
4787           it inserts a primary key into the primary associative
4788           container, then a secondary key into the secondary associative
4789           container. For the native multimaps, it obtains a range using
4790           <classname>std::equal_range</classname>, and inserts a value only if it was
4791           not contained already.</para>
4792
4793           <para>
4794             It uses the test file:
4795             <filename>performance/ext/pb_ds/multimap_text_insert_timing_small.cc</filename>
4796           </para>
4797
4798           <para>The test checks the insert-time scalability of different
4799           "multimap" designs.</para>
4800
4801         </section>
4802
4803         <section xml:id="multimap.text_insert_small.results">
4804           <info><title>
4805             Results
4806           </title></info>
4807
4808           <para>The graphic below show the results for "multimaps" which
4809           use a tree-based container for primary keys.
4810           </para>
4811
4812           <!-- results graphic -->
4813           <informalfigure>
4814             <mediaobject>
4815               <imageobject>
4816                 <imagedata align="center" format="PNG" scale="100"
4817                            fileref="../images/pbds_multimap_text_insert_small_s2p_tree.png"/>
4818               </imageobject>
4819               <imageobject>
4820                 <imagedata align="center" format="PDF" scale="33"
4821                            fileref="../images/pbds_multimap_text_insert_small_s2p_tree.pdf"/>
4822               </imageobject>
4823             </mediaobject>
4824           </informalfigure>
4825
4826           <para>
4827             The abbreviated names in the legend of the graphic above are
4828             instantiated with the types in the following table.
4829           </para>
4830
4831
4832           <informaltable frame="all">
4833
4834             <tgroup cols="7" align="left" colsep="1" rowsep="1">
4835               <colspec colname="c1"/>
4836               <colspec colname="c2"/>
4837               <colspec colname="c3"/>
4838               <colspec colname="c4"/>
4839               <colspec colname="c5"/>
4840               <colspec colname="c6"/>
4841               <colspec colname="c7"/>
4842               <thead>
4843                 <row>
4844                   <entry><emphasis>Name/Instantiating Type</emphasis></entry>
4845                   <entry><emphasis>Parameter</emphasis></entry>
4846                   <entry><emphasis>Details</emphasis></entry>
4847                   <entry><emphasis>Parameter</emphasis></entry>
4848                   <entry><emphasis>Details</emphasis></entry>
4849                   <entry><emphasis>Parameter</emphasis></entry>
4850                   <entry><emphasis>Details</emphasis></entry>
4851                 </row>
4852               </thead>
4853
4854               <tbody>
4855
4856                 <!-- native -->
4857                 <row>
4858                   <?dbhtml bgcolor="#B0B0B0" ?>
4859                   <entry namest="c1" nameend="c7">
4860                     n_mmap
4861                   </entry>
4862                 </row>
4863
4864                 <row>
4865                   <entry>
4866                     <classname>std::multimap</classname>
4867                   </entry>
4868                   <entry namest="c2" nameend="c7"></entry>
4869                 </row>
4870
4871                 <!-- multimap 01 -->
4872                 <row>
4873                   <?dbhtml bgcolor="#B0B0B0" ?>
4874                   <entry namest="c1" nameend="c7">
4875                     rb_tree_mmap_lu_mtf_set
4876                   </entry>
4877                 </row>
4878
4879                 <row>
4880                   <entry morerows="2" valign="top">
4881                     <classname>tree</classname>
4882                   </entry>
4883                   <entry>
4884                     <classname>Tag</classname>
4885                   </entry>
4886                   <entry>
4887                     <classname>rb_tree_tag</classname>
4888                   </entry>
4889                   <entry namest="c4" nameend="c7"></entry>
4890                 </row>
4891
4892                 <row>
4893                   <entry>
4894                     <classname>Node_Update</classname>
4895                   </entry>
4896                   <entry>
4897                     <classname>null_node_update</classname>
4898                   </entry>
4899                   <entry namest="c4" nameend="c7"></entry>
4900                 </row>
4901                 <row>
4902                   <entry>
4903                     <classname>Mapped</classname>
4904                   </entry>
4905                   <entry>
4906                     <classname>list_update</classname>
4907                   </entry>
4908                   <entry>
4909                     <classname>Update_Policy</classname>
4910                   </entry>
4911                   <entry>
4912                     <classname>lu_move_to_front_policy</classname>
4913                   </entry>
4914                   <entry namest="c6" nameend="c7"></entry>
4915                 </row>
4916
4917                 <!-- multimap 02 -->
4918                 <row>
4919                   <?dbhtml bgcolor="#B0B0B0" ?>
4920                   <entry namest="c1" nameend="c7">
4921                     rb_tree_mmap_cc_hash_mask_exp_1div2_nsth_set
4922                   </entry>
4923                 </row>
4924
4925                 <row>
4926                   <entry morerows="4" valign="top">
4927                     <classname>tree</classname>
4928                   </entry>
4929                   <entry>
4930                     <classname>Tag</classname>
4931                   </entry>
4932                   <entry>
4933                     <classname>rb_tree_tag</classname>
4934                   </entry>
4935                   <entry namest="c4" nameend="c7"></entry>
4936                 </row>
4937
4938                 <row>
4939                   <entry>
4940                     <classname>Node_Update</classname>
4941                   </entry>
4942                   <entry>
4943                     <classname>null_node_update</classname>
4944                   </entry>
4945                   <entry namest="c4" nameend="c7"></entry>
4946                 </row>
4947
4948                 <row>
4949                   <entry morerows="2" valign="top">
4950                     <classname>Mapped</classname>
4951                   </entry>
4952                   <entry morerows="2" valign="top">
4953                     <classname>cc_hash_table</classname>
4954                   </entry>
4955                   <entry>
4956                     <classname>Comb_Hash_Fn</classname>
4957                   </entry>
4958                   <entry>
4959                     <classname>direct_mask_range_hashing</classname>
4960                   </entry>
4961                   <entry namest="c6" nameend="c7"></entry>
4962                 </row>
4963
4964                 <row>
4965                   <entry morerows="1" valign="top">
4966                     <classname>Resize_Policy</classname>
4967                   </entry>
4968                   <entry morerows="1" valign="top">
4969                     <classname>hash_standard_resize_policy</classname>
4970                   </entry>
4971                   <entry>
4972                     <classname>Size_Policy</classname>
4973                   </entry>
4974                   <entry>
4975                     <classname>hash_exponential_size_policy</classname>
4976                   </entry>
4977                 </row>
4978
4979                 <row>
4980                   <entry valign="top">
4981                     <classname>Trigger_Policy</classname>
4982                   </entry>
4983                   <entry>
4984                     <classname>hash_load_check_resize_trigger</classname> with
4985                     α<subscript>min</subscript> = 1/8 and α<subscript>max</subscript> = 1/2
4986                   </entry>
4987                 </row>
4988
4989               </tbody>
4990             </tgroup>
4991           </informaltable>
4992
4993
4994           <para>The graphic below show the results for "multimaps" which
4995           use a hash-based container for primary keys.
4996           </para>
4997
4998           <!-- results graphic -->
4999           <informalfigure>
5000             <mediaobject>
5001               <imageobject>
5002                 <imagedata align="center" format="PNG" scale="100"
5003                            fileref="../images/pbds_multimap_text_find_small_s2p_hash.png"/>
5004               </imageobject>
5005               <imageobject>
5006                 <imagedata align="center" format="PDF" scale="33"
5007                            fileref="../images/pbds_multimap_text_find_small_s2p_hash.pdf"/>
5008               </imageobject>
5009             </mediaobject>
5010           </informalfigure>
5011
5012           <para>
5013             The abbreviated names in the legend of the graphic above are
5014             instantiated with the types in the following table.
5015           </para>
5016
5017           <informaltable frame="all">
5018
5019             <tgroup cols="7" align="left" colsep="1" rowsep="1">
5020               <colspec colname="c1"/>
5021               <colspec colname="c2"/>
5022               <colspec colname="c3"/>
5023               <colspec colname="c4"/>
5024               <colspec colname="c5"/>
5025               <colspec colname="c6"/>
5026               <colspec colname="c7"/>
5027               <thead>
5028                 <row>
5029                   <entry><emphasis>Name/Instantiating Type</emphasis></entry>
5030                   <entry><emphasis>Parameter</emphasis></entry>
5031                   <entry><emphasis>Details</emphasis></entry>
5032                   <entry><emphasis>Parameter</emphasis></entry>
5033                   <entry><emphasis>Details</emphasis></entry>
5034                   <entry><emphasis>Parameter</emphasis></entry>
5035                   <entry><emphasis>Details</emphasis></entry>
5036                 </row>
5037               </thead>
5038
5039               <tbody>
5040
5041                 <!-- native -->
5042                 <row>
5043                   <?dbhtml bgcolor="#B0B0B0" ?>
5044                   <entry namest="c1" nameend="c7">
5045                     n_hash_mmap
5046                   </entry>
5047                 </row>
5048
5049                 <row>
5050                   <entry>
5051                     <classname>std::tr1::unordered_multimap</classname>
5052                   </entry>
5053                   <entry namest="c2" nameend="c7"></entry>
5054                 </row>
5055
5056                 <!-- multimap 01 -->
5057                 <row>
5058                   <?dbhtml bgcolor="#B0B0B0" ?>
5059                   <entry namest="c1" nameend="c7">
5060                     rb_tree_mmap_lu_mtf_set
5061                   </entry>
5062                 </row>
5063
5064                 <row>
5065                   <entry morerows="3" valign="top">
5066                     <classname>
5067                       cc_hash_table
5068                     </classname>
5069                   </entry>
5070                   <entry>
5071                     <classname>Comb_Hash_Fn</classname>
5072                   </entry>
5073                   <entry>
5074                     <classname>direct_mask_range_hashing</classname>
5075                   </entry>
5076                   <entry namest="c4" nameend="c7"></entry>
5077                 </row>
5078                 <row>
5079                   <entry morerows="1" valign="top">
5080                     <classname>Resize_Policy</classname>
5081                   </entry>
5082                   <entry morerows="1" valign="top">
5083                     <classname>hash_standard_resize_policy</classname>
5084                   </entry>
5085                   <entry>
5086                     <classname>Size_Policy</classname>
5087                   </entry>
5088                   <entry>
5089                     <classname>hash_exponential_size_policy</classname>
5090                   </entry>
5091                   <entry namest="c6" nameend="c7"></entry>
5092                 </row>
5093
5094                 <row>
5095                   <entry valign="top">
5096                     <classname>Trigger_Policy</classname>
5097                   </entry>
5098                   <entry>
5099                     <classname>hash_load_check_resize_trigger</classname> with
5100                     α<subscript>min</subscript> = 1/8 and α<subscript>max</subscript> = 1/2
5101                   </entry>
5102                   <entry namest="c6" nameend="c7"></entry>
5103                 </row>
5104
5105                 <row>
5106                   <entry>
5107                     <classname>Mapped</classname>
5108                   </entry>
5109                   <entry>
5110                     <classname>list_update</classname>
5111                   </entry>
5112                   <entry>
5113                     <classname>Update_Policy</classname>
5114                   </entry>
5115                   <entry>
5116                     <classname>lu_move_to_front_policy</classname>
5117                   </entry>
5118                   <entry namest="c6" nameend="c7"></entry>
5119                 </row>
5120
5121                 <!-- multimap 02 -->
5122                 <row>
5123                   <?dbhtml bgcolor="#B0B0B0" ?>
5124                   <entry namest="c1" nameend="c7">
5125                     rb_tree_mmap_cc_hash_mask_exp_1div2_nsth_set
5126                   </entry>
5127                 </row>
5128
5129                 <row>
5130                   <entry morerows="5" valign="top">
5131                     <classname>
5132                       cc_hash_table
5133                     </classname>
5134                   </entry>
5135                   <entry>
5136                     <classname>Comb_Hash_Fn</classname>
5137                   </entry>
5138                   <entry>
5139                     <classname>direct_mask_range_hashing</classname>
5140                   </entry>
5141                   <entry namest="c4" nameend="c7"></entry>
5142                 </row>
5143                 <row>
5144                   <entry morerows="1" valign="top">
5145                     <classname>Resize_Policy</classname>
5146                   </entry>
5147                   <entry morerows="1" valign="top">
5148                     <classname>hash_standard_resize_policy</classname>
5149                   </entry>
5150                   <entry>
5151                     <classname>Size_Policy</classname>
5152                   </entry>
5153                   <entry>
5154                     <classname>hash_exponential_size_policy</classname>
5155                   </entry>
5156                   <entry namest="c6" nameend="c7"></entry>
5157                 </row>
5158
5159                 <row>
5160                   <entry valign="top">
5161                     <classname>Trigger_Policy</classname>
5162                   </entry>
5163                   <entry>
5164                     <classname>hash_load_check_resize_trigger</classname> with
5165                     α<subscript>min</subscript> = 1/8 and α<subscript>max</subscript> = 1/2
5166                   </entry>
5167                   <entry namest="c6" nameend="c7"></entry>
5168                 </row>
5169
5170                 <row>
5171                   <entry morerows="2" valign="top">
5172                     <classname>Mapped</classname>
5173                   </entry>
5174                   <entry morerows="2" valign="top">
5175                     <classname>cc_hash_table</classname>
5176                   </entry>
5177                   <entry>
5178                     <classname>Comb_Hash_Fn</classname>
5179                   </entry>
5180                   <entry>
5181                     <classname>direct_mask_range_hashing</classname>
5182                   </entry>
5183                   <entry namest="c6" nameend="c7"></entry>
5184                 </row>
5185
5186                 <row>
5187                   <entry morerows="1" valign="top">
5188                     <classname>Resize_Policy</classname>
5189                   </entry>
5190                   <entry morerows="1" valign="top">
5191                     <classname>hash_standard_resize_policy</classname>
5192                   </entry>
5193                   <entry>
5194                     <classname>Size_Policy</classname>
5195                   </entry>
5196                   <entry>
5197                     <classname>hash_exponential_size_policy</classname>
5198                   </entry>
5199                 </row>
5200
5201                 <row>
5202                   <entry valign="top">
5203                     <classname>Trigger_Policy</classname>
5204                   </entry>
5205                   <entry>
5206                     <classname>hash_load_check_resize_trigger</classname> with
5207                     α<subscript>min</subscript> = 1/8 and α<subscript>max</subscript> = 1/2
5208                   </entry>
5209                 </row>
5210
5211               </tbody>
5212             </tgroup>
5213           </informaltable>
5214
5215         </section>
5216
5217         <section xml:id="multimap.text_insert_small.observations">
5218           <info><title>
5219             Observations
5220           </title></info>
5221
5222           <para>See Observations::Mapping-Semantics
5223           Considerations.</para>
5224
5225         </section>
5226
5227       </section>
5228
5229
5230       <!-- 04 <a href="multimap_text_insert_timing_test_large"> -->
5231       <section xml:id="performance.multimap.text_insert_large">
5232         <info><title>
5233           Text <function>insert</function> with Small
5234           Secondary-to-Primary Key Ratios
5235         </title></info>
5236         <para></para>
5237
5238         <section xml:id="multimap.text_insert_large.info">
5239           <info><title>
5240             Description
5241           </title></info>
5242
5243           <para>This test inserts a number of pairs into a container. The
5244           first item of each pair is a string from an arbitrary text
5245           [wickland96thirty], and
5246           the second is a uniform integer. The container is a
5247           "multimap" - it considers the first member of each pair as a
5248           primary key, and the second member of each pair as a secondary
5249           key. There
5250           are 400 distinct primary keys, and the ratio of secondary keys
5251           to primary keys ranges from 1 to 5.</para>
5252           <para>The test measures the average insert-time as a function of
5253           the number of values inserted. For this library's containers,
5254           it inserts a primary key into the primary associative
5255           container, then a secondary key into the secondary associative
5256           container. For the native multimaps, it obtains a range using
5257           <classname>std::equal_range</classname>, and inserts a value only if it was
5258           not contained already.</para>
5259
5260           <para>
5261             It uses the test file:
5262             <filename>performance/ext/pb_ds/multimap_text_insert_timing_large.cc</filename>
5263           </para>
5264
5265           <para>The test checks the insert-time scalability of different
5266           "multimap" designs.</para>
5267
5268         </section>
5269
5270         <section xml:id="multimap.text_insert_large.results">
5271           <info><title>
5272             Results
5273           </title></info>
5274
5275           <para>The graphic below show the results for "multimaps" which
5276           use a tree-based container for primary keys.
5277           </para>
5278
5279           <!-- results graphic -->
5280           <informalfigure>
5281             <mediaobject>
5282               <imageobject>
5283                 <imagedata align="center" format="PNG" scale="100"
5284                            fileref="../images/pbds_multimap_text_insert_large_s2p_tree.png"/>
5285               </imageobject>
5286               <imageobject>
5287                 <imagedata align="center" format="PDF" scale="33"
5288                            fileref="../images/pbds_multimap_text_insert_large_s2p_tree.pdf"/>
5289               </imageobject>
5290             </mediaobject>
5291           </informalfigure>
5292
5293           <para>
5294             The abbreviated names in the legend of the graphic above are
5295             instantiated with the types in the following table.
5296           </para>
5297
5298
5299           <informaltable frame="all">
5300
5301             <tgroup cols="7" align="left" colsep="1" rowsep="1">
5302               <colspec colname="c1"/>
5303               <colspec colname="c2"/>
5304               <colspec colname="c3"/>
5305               <colspec colname="c4"/>
5306               <colspec colname="c5"/>
5307               <colspec colname="c6"/>
5308               <colspec colname="c7"/>
5309               <thead>
5310                 <row>
5311                   <entry><emphasis>Name/Instantiating Type</emphasis></entry>
5312                   <entry><emphasis>Parameter</emphasis></entry>
5313                   <entry><emphasis>Details</emphasis></entry>
5314                   <entry><emphasis>Parameter</emphasis></entry>
5315                   <entry><emphasis>Details</emphasis></entry>
5316                   <entry><emphasis>Parameter</emphasis></entry>
5317                   <entry><emphasis>Details</emphasis></entry>
5318                 </row>
5319               </thead>
5320
5321               <tbody>
5322
5323                 <!-- native -->
5324                 <row>
5325                   <?dbhtml bgcolor="#B0B0B0" ?>
5326                   <entry namest="c1" nameend="c7">
5327                     n_mmap
5328                   </entry>
5329                 </row>
5330
5331                 <row>
5332                   <entry>
5333                     <classname>std::multimap</classname>
5334                   </entry>
5335                   <entry namest="c2" nameend="c7"></entry>
5336                 </row>
5337
5338                 <!-- multimap 01 -->
5339                 <row>
5340                   <?dbhtml bgcolor="#B0B0B0" ?>
5341                   <entry namest="c1" nameend="c7">
5342                     rb_tree_mmap_lu_mtf_set
5343                   </entry>
5344                 </row>
5345
5346                 <row>
5347                   <entry morerows="2" valign="top">
5348                     <classname>tree</classname>
5349                   </entry>
5350                   <entry>
5351                     <classname>Tag</classname>
5352                   </entry>
5353                   <entry>
5354                     <classname>rb_tree_tag</classname>
5355                   </entry>
5356                   <entry namest="c4" nameend="c7"></entry>
5357                 </row>
5358
5359                 <row>
5360                   <entry>
5361                     <classname>Node_Update</classname>
5362                   </entry>
5363                   <entry>
5364                     <classname>null_node_update</classname>
5365                   </entry>
5366                   <entry namest="c4" nameend="c7"></entry>
5367                 </row>
5368                 <row>
5369                   <entry>
5370                     <classname>Mapped</classname>
5371                   </entry>
5372                   <entry>
5373                     <classname>list_update</classname>
5374                   </entry>
5375                   <entry>
5376                     <classname>Update_Policy</classname>
5377                   </entry>
5378                   <entry>
5379                     <classname>lu_move_to_front_policy</classname>
5380                   </entry>
5381                   <entry namest="c6" nameend="c7"></entry>
5382                 </row>
5383
5384                 <!-- multimap 02 -->
5385                 <row>
5386                   <?dbhtml bgcolor="#B0B0B0" ?>
5387                   <entry namest="c1" nameend="c7">
5388                     rb_tree_mmap_cc_hash_mask_exp_1div2_nsth_set
5389                   </entry>
5390                 </row>
5391
5392                 <row>
5393                   <entry morerows="4" valign="top">
5394                     <classname>tree</classname>
5395                   </entry>
5396                   <entry>
5397                     <classname>Tag</classname>
5398                   </entry>
5399                   <entry>
5400                     <classname>rb_tree_tag</classname>
5401                   </entry>
5402                   <entry namest="c4" nameend="c7"></entry>
5403                 </row>
5404
5405                 <row>
5406                   <entry>
5407                     <classname>Node_Update</classname>
5408                   </entry>
5409                   <entry>
5410                     <classname>null_node_update</classname>
5411                   </entry>
5412                   <entry namest="c4" nameend="c7"></entry>
5413                 </row>
5414
5415                 <row>
5416                   <entry morerows="2" valign="top">
5417                     <classname>Mapped</classname>
5418                   </entry>
5419                   <entry morerows="2" valign="top">
5420                     <classname>cc_hash_table</classname>
5421                   </entry>
5422                   <entry>
5423                     <classname>Comb_Hash_Fn</classname>
5424                   </entry>
5425                   <entry>
5426                     <classname>direct_mask_range_hashing</classname>
5427                   </entry>
5428                   <entry namest="c6" nameend="c7"></entry>
5429                 </row>
5430
5431                 <row>
5432                   <entry morerows="1" valign="top">
5433                     <classname>Resize_Policy</classname>
5434                   </entry>
5435                   <entry morerows="1" valign="top">
5436                     <classname>hash_standard_resize_policy</classname>
5437                   </entry>
5438                   <entry>
5439                     <classname>Size_Policy</classname>
5440                   </entry>
5441                   <entry>
5442                     <classname>hash_exponential_size_policy</classname>
5443                   </entry>
5444                 </row>
5445
5446                 <row>
5447                   <entry valign="top">
5448                     <classname>Trigger_Policy</classname>
5449                   </entry>
5450                   <entry>
5451                     <classname>hash_load_check_resize_trigger</classname> with
5452                     α<subscript>min</subscript> = 1/8 and α<subscript>max</subscript> = 1/2
5453                   </entry>
5454                 </row>
5455
5456               </tbody>
5457             </tgroup>
5458           </informaltable>
5459
5460
5461           <para>The graphic below show the results for "multimaps" which
5462           use a hash-based container for primary keys.
5463           </para>
5464
5465           <!-- results graphic -->
5466           <informalfigure>
5467             <mediaobject>
5468               <imageobject>
5469                 <imagedata align="center" format="PNG" scale="100"
5470                            fileref="../images/pbds_multimap_text_find_large_s2p_hash.png"/>
5471               </imageobject>
5472               <imageobject>
5473                 <imagedata align="center" format="PDF" scale="33"
5474                            fileref="../images/pbds_multimap_text_find_large_s2p_hash.pdf"/>
5475               </imageobject>
5476             </mediaobject>
5477           </informalfigure>
5478
5479           <para>
5480             The abbreviated names in the legend of the graphic above are
5481             instantiated with the types in the following table.
5482           </para>
5483
5484           <informaltable frame="all">
5485
5486             <tgroup cols="7" align="left" colsep="1" rowsep="1">
5487               <colspec colname="c1"/>
5488               <colspec colname="c2"/>
5489               <colspec colname="c3"/>
5490               <colspec colname="c4"/>
5491               <colspec colname="c5"/>
5492               <colspec colname="c6"/>
5493               <colspec colname="c7"/>
5494               <thead>
5495                 <row>
5496                   <entry><emphasis>Name/Instantiating Type</emphasis></entry>
5497                   <entry><emphasis>Parameter</emphasis></entry>
5498                   <entry><emphasis>Details</emphasis></entry>
5499                   <entry><emphasis>Parameter</emphasis></entry>
5500                   <entry><emphasis>Details</emphasis></entry>
5501                   <entry><emphasis>Parameter</emphasis></entry>
5502                   <entry><emphasis>Details</emphasis></entry>
5503                 </row>
5504               </thead>
5505
5506               <tbody>
5507
5508                 <!-- native -->
5509                 <row>
5510                   <?dbhtml bgcolor="#B0B0B0" ?>
5511                   <entry namest="c1" nameend="c7">
5512                     n_hash_mmap
5513                   </entry>
5514                 </row>
5515
5516                 <row>
5517                   <entry>
5518                     <classname>std::tr1::unordered_multimap</classname>
5519                   </entry>
5520                   <entry namest="c2" nameend="c7"></entry>
5521                 </row>
5522
5523                 <!-- multimap 01 -->
5524                 <row>
5525                   <?dbhtml bgcolor="#B0B0B0" ?>
5526                   <entry namest="c1" nameend="c7">
5527                     rb_tree_mmap_lu_mtf_set
5528                   </entry>
5529                 </row>
5530
5531                 <row>
5532                   <entry morerows="3" valign="top">
5533                     <classname>
5534                       cc_hash_table
5535                     </classname>
5536                   </entry>
5537                   <entry>
5538                     <classname>Comb_Hash_Fn</classname>
5539                   </entry>
5540                   <entry>
5541                     <classname>direct_mask_range_hashing</classname>
5542                   </entry>
5543                   <entry namest="c4" nameend="c7"></entry>
5544                 </row>
5545                 <row>
5546                   <entry morerows="1" valign="top">
5547                     <classname>Resize_Policy</classname>
5548                   </entry>
5549                   <entry morerows="1" valign="top">
5550                     <classname>hash_standard_resize_policy</classname>
5551                   </entry>
5552                   <entry>
5553                     <classname>Size_Policy</classname>
5554                   </entry>
5555                   <entry>
5556                     <classname>hash_exponential_size_policy</classname>
5557                   </entry>
5558                   <entry namest="c6" nameend="c7"></entry>
5559                 </row>
5560
5561                 <row>
5562                   <entry valign="top">
5563                     <classname>Trigger_Policy</classname>
5564                   </entry>
5565                   <entry>
5566                     <classname>hash_load_check_resize_trigger</classname> with
5567                     α<subscript>min</subscript> = 1/8 and α<subscript>max</subscript> = 1/2
5568                   </entry>
5569                   <entry namest="c6" nameend="c7"></entry>
5570                 </row>
5571
5572                 <row>
5573                   <entry>
5574                     <classname>Mapped</classname>
5575                   </entry>
5576                   <entry>
5577                     <classname>list_update</classname>
5578                   </entry>
5579                   <entry>
5580                     <classname>Update_Policy</classname>
5581                   </entry>
5582                   <entry>
5583                     <classname>lu_move_to_front_policy</classname>
5584                   </entry>
5585                   <entry namest="c6" nameend="c7"></entry>
5586                 </row>
5587
5588                 <!-- multimap 02 -->
5589                 <row>
5590                   <?dbhtml bgcolor="#B0B0B0" ?>
5591                   <entry namest="c1" nameend="c7">
5592                     rb_tree_mmap_cc_hash_mask_exp_1div2_nsth_set
5593                   </entry>
5594                 </row>
5595
5596                 <row>
5597                   <entry morerows="5" valign="top">
5598                     <classname>
5599                       cc_hash_table
5600                     </classname>
5601                   </entry>
5602                   <entry>
5603                     <classname>Comb_Hash_Fn</classname>
5604                   </entry>
5605                   <entry>
5606                     <classname>direct_mask_range_hashing</classname>
5607                   </entry>
5608                   <entry namest="c4" nameend="c7"></entry>
5609                 </row>
5610                 <row>
5611                   <entry morerows="1" valign="top">
5612                     <classname>Resize_Policy</classname>
5613                   </entry>
5614                   <entry morerows="1" valign="top">
5615                     <classname>hash_standard_resize_policy</classname>
5616                   </entry>
5617                   <entry>
5618                     <classname>Size_Policy</classname>
5619                   </entry>
5620                   <entry>
5621                     <classname>hash_exponential_size_policy</classname>
5622                   </entry>
5623                   <entry namest="c6" nameend="c7"></entry>
5624                 </row>
5625
5626                 <row>
5627                   <entry valign="top">
5628                     <classname>Trigger_Policy</classname>
5629                   </entry>
5630                   <entry>
5631                     <classname>hash_load_check_resize_trigger</classname> with
5632                     α<subscript>min</subscript> = 1/8 and α<subscript>max</subscript> = 1/2
5633                   </entry>
5634                   <entry namest="c6" nameend="c7"></entry>
5635                 </row>
5636
5637                 <row>
5638                   <entry morerows="2" valign="top">
5639                     <classname>Mapped</classname>
5640                   </entry>
5641                   <entry morerows="2" valign="top">
5642                     <classname>cc_hash_table</classname>
5643                   </entry>
5644                   <entry>
5645                     <classname>Comb_Hash_Fn</classname>
5646                   </entry>
5647                   <entry>
5648                     <classname>direct_mask_range_hashing</classname>
5649                   </entry>
5650                   <entry namest="c6" nameend="c7"></entry>
5651                 </row>
5652
5653                 <row>
5654                   <entry morerows="1" valign="top">
5655                     <classname>Resize_Policy</classname>
5656                   </entry>
5657                   <entry morerows="1" valign="top">
5658                     <classname>hash_standard_resize_policy</classname>
5659                   </entry>
5660                   <entry>
5661                     <classname>Size_Policy</classname>
5662                   </entry>
5663                   <entry>
5664                     <classname>hash_exponential_size_policy</classname>
5665                   </entry>
5666                 </row>
5667
5668                 <row>
5669                   <entry valign="top">
5670                     <classname>Trigger_Policy</classname>
5671                   </entry>
5672                   <entry>
5673                     <classname>hash_load_check_resize_trigger</classname> with
5674                     α<subscript>min</subscript> = 1/8 and α<subscript>max</subscript> = 1/2
5675                   </entry>
5676                 </row>
5677
5678               </tbody>
5679             </tgroup>
5680           </informaltable>
5681
5682         </section>
5683
5684         <section xml:id="multimap.text_insert_large.observations">
5685           <info><title>
5686             Observations
5687           </title></info>
5688
5689           <para>See Observations::Mapping-Semantics
5690           Considerations.</para>
5691
5692         </section>
5693
5694       </section>
5695
5696
5697       <!-- 05 <a href="multimap_text_insert_mem_usage_test_small"> -->
5698       <section xml:id="performance.multimap.text_insert_mem_small">
5699         <info><title>
5700           Text <function>insert</function> with Small
5701           Secondary-to-Primary Key Ratios Memory Use
5702         </title></info>
5703         <para></para>
5704
5705         <section xml:id="multimap.text_insert_mem_small.info">
5706           <info><title>
5707             Description
5708           </title></info>
5709           <para>This test inserts a number of pairs into a container. The
5710           first item of each pair is a string from an arbitrary text
5711           [wickland96thirty], and
5712           the second is a uniform integer. The container is a
5713           "multimap" - it considers the first member of each pair as a
5714           primary key, and the second member of each pair as a secondary
5715           key. There
5716           are 100 distinct primary keys, and the ratio of secondary keys
5717           to primary keys ranges to about 20.</para>
5718           <para>The test measures the memory use as a function of the number
5719           of values inserted.</para>
5720
5721           <para>
5722             It uses the test file:
5723             <filename>performance/ext/pb_ds/multimap_text_insert_mem_usage_small.cc</filename>
5724           </para>
5725
5726           <para>The test checks the memory scalability of different
5727           "multimap" designs.</para>
5728
5729         </section>
5730
5731         <section xml:id="multimap.text_insert_mem_small.results">
5732           <info><title>
5733             Results
5734           </title></info>
5735
5736           <para>The graphic below show the results for "multimaps" which
5737           use a tree-based container for primary keys.
5738           </para>
5739
5740           <!-- results graphic -->
5741           <informalfigure>
5742             <mediaobject>
5743               <imageobject>
5744                 <imagedata align="center" format="PNG" scale="100"
5745                            fileref="../images/pbds_multimap_text_insert_mem_small_s2p_tree.png"/>
5746               </imageobject>
5747               <imageobject>
5748                 <imagedata align="center" format="PDF" scale="33"
5749                            fileref="../images/pbds_multimap_text_insert_mem_small_s2p_tree.pdf"/>
5750               </imageobject>
5751             </mediaobject>
5752           </informalfigure>
5753
5754           <para>
5755             The abbreviated names in the legend of the graphic above are
5756             instantiated with the types in the following table.
5757           </para>
5758
5759
5760           <informaltable frame="all">
5761
5762             <tgroup cols="7" align="left" colsep="1" rowsep="1">
5763               <colspec colname="c1"/>
5764               <colspec colname="c2"/>
5765               <colspec colname="c3"/>
5766               <colspec colname="c4"/>
5767               <colspec colname="c5"/>
5768               <colspec colname="c6"/>
5769               <colspec colname="c7"/>
5770               <thead>
5771                 <row>
5772                   <entry><emphasis>Name/Instantiating Type</emphasis></entry>
5773                   <entry><emphasis>Parameter</emphasis></entry>
5774                   <entry><emphasis>Details</emphasis></entry>
5775                   <entry><emphasis>Parameter</emphasis></entry>
5776                   <entry><emphasis>Details</emphasis></entry>
5777                   <entry><emphasis>Parameter</emphasis></entry>
5778                   <entry><emphasis>Details</emphasis></entry>
5779                 </row>
5780               </thead>
5781
5782               <tbody>
5783
5784                 <!-- native -->
5785                 <row>
5786                   <?dbhtml bgcolor="#B0B0B0" ?>
5787                   <entry namest="c1" nameend="c7">
5788                     n_mmap
5789                   </entry>
5790                 </row>
5791
5792                 <row>
5793                   <entry>
5794                     <classname>std::multimap</classname>
5795                   </entry>
5796                   <entry namest="c2" nameend="c7"></entry>
5797                 </row>
5798
5799                 <!-- multimap 01 -->
5800                 <row>
5801                   <?dbhtml bgcolor="#B0B0B0" ?>
5802                   <entry namest="c1" nameend="c7">
5803                     rb_tree_mmap_lu_mtf_set
5804                   </entry>
5805                 </row>
5806
5807                 <row>
5808                   <entry morerows="2" valign="top">
5809                     <classname>tree</classname>
5810                   </entry>
5811                   <entry>
5812                     <classname>Tag</classname>
5813                   </entry>
5814                   <entry>
5815                     <classname>rb_tree_tag</classname>
5816                   </entry>
5817                   <entry namest="c4" nameend="c7"></entry>
5818                 </row>
5819
5820                 <row>
5821                   <entry>
5822                     <classname>Node_Update</classname>
5823                   </entry>
5824                   <entry>
5825                     <classname>null_node_update</classname>
5826                   </entry>
5827                   <entry namest="c4" nameend="c7"></entry>
5828                 </row>
5829                 <row>
5830                   <entry>
5831                     <classname>Mapped</classname>
5832                   </entry>
5833                   <entry>
5834                     <classname>list_update</classname>
5835                   </entry>
5836                   <entry>
5837                     <classname>Update_Policy</classname>
5838                   </entry>
5839                   <entry>
5840                     <classname>lu_move_to_front_policy</classname>
5841                   </entry>
5842                   <entry namest="c6" nameend="c7"></entry>
5843                 </row>
5844
5845                 <!-- multimap 02 -->
5846                 <row>
5847                   <?dbhtml bgcolor="#B0B0B0" ?>
5848                   <entry namest="c1" nameend="c7">
5849                     rb_tree_mmap_cc_hash_mask_exp_1div2_nsth_set
5850                   </entry>
5851                 </row>
5852
5853                 <row>
5854                   <entry morerows="4" valign="top">
5855                     <classname>tree</classname>
5856                   </entry>
5857                   <entry>
5858                     <classname>Tag</classname>
5859                   </entry>
5860                   <entry>
5861                     <classname>rb_tree_tag</classname>
5862                   </entry>
5863                   <entry namest="c4" nameend="c7"></entry>
5864                 </row>
5865
5866                 <row>
5867                   <entry>
5868                     <classname>Node_Update</classname>
5869                   </entry>
5870                   <entry>
5871                     <classname>null_node_update</classname>
5872                   </entry>
5873                   <entry namest="c4" nameend="c7"></entry>
5874                 </row>
5875
5876                 <row>
5877                   <entry morerows="2" valign="top">
5878                     <classname>Mapped</classname>
5879                   </entry>
5880                   <entry morerows="2" valign="top">
5881                     <classname>cc_hash_table</classname>
5882                   </entry>
5883                   <entry>
5884                     <classname>Comb_Hash_Fn</classname>
5885                   </entry>
5886                   <entry>
5887                     <classname>direct_mask_range_hashing</classname>
5888                   </entry>
5889                   <entry namest="c6" nameend="c7"></entry>
5890                 </row>
5891
5892                 <row>
5893                   <entry morerows="1" valign="top">
5894                     <classname>Resize_Policy</classname>
5895                   </entry>
5896                   <entry morerows="1" valign="top">
5897                     <classname>hash_standard_resize_policy</classname>
5898                   </entry>
5899                   <entry>
5900                     <classname>Size_Policy</classname>
5901                   </entry>
5902                   <entry>
5903                     <classname>hash_exponential_size_policy</classname>
5904                   </entry>
5905                 </row>
5906
5907                 <row>
5908                   <entry valign="top">
5909                     <classname>Trigger_Policy</classname>
5910                   </entry>
5911                   <entry>
5912                     <classname>hash_load_check_resize_trigger</classname> with
5913                     α<subscript>min</subscript> = 1/8 and α<subscript>max</subscript> = 1/2
5914                   </entry>
5915                 </row>
5916
5917               </tbody>
5918             </tgroup>
5919           </informaltable>
5920
5921
5922           <para>The graphic below show the results for "multimaps" which
5923           use a hash-based container for primary keys.
5924           </para>
5925
5926           <!-- results graphic -->
5927           <informalfigure>
5928             <mediaobject>
5929               <imageobject>
5930                 <imagedata align="center" format="PNG" scale="100"
5931                            fileref="../images/pbds_multimap_text_find_large_s2p_hash.png"/>
5932               </imageobject>
5933               <imageobject>
5934                 <imagedata align="center" format="PDF" scale="33"
5935                            fileref="../images/pbds_multimap_text_find_large_s2p_hash.pdf"/>
5936               </imageobject>
5937             </mediaobject>
5938           </informalfigure>
5939
5940           <para>
5941             The abbreviated names in the legend of the graphic above are
5942             instantiated with the types in the following table.
5943           </para>
5944
5945           <informaltable frame="all">
5946
5947             <tgroup cols="7" align="left" colsep="1" rowsep="1">
5948               <colspec colname="c1"/>
5949               <colspec colname="c2"/>
5950               <colspec colname="c3"/>
5951               <colspec colname="c4"/>
5952               <colspec colname="c5"/>
5953               <colspec colname="c6"/>
5954               <colspec colname="c7"/>
5955               <thead>
5956                 <row>
5957                   <entry><emphasis>Name/Instantiating Type</emphasis></entry>
5958                   <entry><emphasis>Parameter</emphasis></entry>
5959                   <entry><emphasis>Details</emphasis></entry>
5960                   <entry><emphasis>Parameter</emphasis></entry>
5961                   <entry><emphasis>Details</emphasis></entry>
5962                   <entry><emphasis>Parameter</emphasis></entry>
5963                   <entry><emphasis>Details</emphasis></entry>
5964                 </row>
5965               </thead>
5966
5967               <tbody>
5968
5969                 <!-- native -->
5970                 <row>
5971                   <?dbhtml bgcolor="#B0B0B0" ?>
5972                   <entry namest="c1" nameend="c7">
5973                     n_hash_mmap
5974                   </entry>
5975                 </row>
5976
5977                 <row>
5978                   <entry>
5979                     <classname>std::tr1::unordered_multimap</classname>
5980                   </entry>
5981                   <entry namest="c2" nameend="c7"></entry>
5982                 </row>
5983
5984                 <!-- multimap 01 -->
5985                 <row>
5986                   <?dbhtml bgcolor="#B0B0B0" ?>
5987                   <entry namest="c1" nameend="c7">
5988                     rb_tree_mmap_lu_mtf_set
5989                   </entry>
5990                 </row>
5991
5992                 <row>
5993                   <entry morerows="3" valign="top">
5994                     <classname>
5995                       cc_hash_table
5996                     </classname>
5997                   </entry>
5998                   <entry>
5999                     <classname>Comb_Hash_Fn</classname>
6000                   </entry>
6001                   <entry>
6002                     <classname>direct_mask_range_hashing</classname>
6003                   </entry>
6004                   <entry namest="c4" nameend="c7"></entry>
6005                 </row>
6006                 <row>
6007                   <entry morerows="1" valign="top">
6008                     <classname>Resize_Policy</classname>
6009                   </entry>
6010                   <entry morerows="1" valign="top">
6011                     <classname>hash_standard_resize_policy</classname>
6012                   </entry>
6013                   <entry>
6014                     <classname>Size_Policy</classname>
6015                   </entry>
6016                   <entry>
6017                     <classname>hash_exponential_size_policy</classname>
6018                   </entry>
6019                   <entry namest="c6" nameend="c7"></entry>
6020                 </row>
6021
6022                 <row>
6023                   <entry valign="top">
6024                     <classname>Trigger_Policy</classname>
6025                   </entry>
6026                   <entry>
6027                     <classname>hash_load_check_resize_trigger</classname> with
6028                     α<subscript>min</subscript> = 1/8 and α<subscript>max</subscript> = 1/2
6029                   </entry>
6030                   <entry namest="c6" nameend="c7"></entry>
6031                 </row>
6032
6033                 <row>
6034                   <entry>
6035                     <classname>Mapped</classname>
6036                   </entry>
6037                   <entry>
6038                     <classname>list_update</classname>
6039                   </entry>
6040                   <entry>
6041                     <classname>Update_Policy</classname>
6042                   </entry>
6043                   <entry>
6044                     <classname>lu_move_to_front_policy</classname>
6045                   </entry>
6046                   <entry namest="c6" nameend="c7"></entry>
6047                 </row>
6048
6049                 <!-- multimap 02 -->
6050                 <row>
6051                   <?dbhtml bgcolor="#B0B0B0" ?>
6052                   <entry namest="c1" nameend="c7">
6053                     rb_tree_mmap_cc_hash_mask_exp_1div2_nsth_set
6054                   </entry>
6055                 </row>
6056
6057                 <row>
6058                   <entry morerows="5" valign="top">
6059                     <classname>
6060                       cc_hash_table
6061                     </classname>
6062                   </entry>
6063                   <entry>
6064                     <classname>Comb_Hash_Fn</classname>
6065                   </entry>
6066                   <entry>
6067                     <classname>direct_mask_range_hashing</classname>
6068                   </entry>
6069                   <entry namest="c4" nameend="c7"></entry>
6070                 </row>
6071                 <row>
6072                   <entry morerows="1" valign="top">
6073                     <classname>Resize_Policy</classname>
6074                   </entry>
6075                   <entry morerows="1" valign="top">
6076                     <classname>hash_standard_resize_policy</classname>
6077                   </entry>
6078                   <entry>
6079                     <classname>Size_Policy</classname>
6080                   </entry>
6081                   <entry>
6082                     <classname>hash_exponential_size_policy</classname>
6083                   </entry>
6084                   <entry namest="c6" nameend="c7"></entry>
6085                 </row>
6086
6087                 <row>
6088                   <entry valign="top">
6089                     <classname>Trigger_Policy</classname>
6090                   </entry>
6091                   <entry>
6092                     <classname>hash_load_check_resize_trigger</classname> with
6093                     α<subscript>min</subscript> = 1/8 and α<subscript>max</subscript> = 1/2
6094                   </entry>
6095                   <entry namest="c6" nameend="c7"></entry>
6096                 </row>
6097
6098                 <row>
6099                   <entry morerows="2" valign="top">
6100                     <classname>Mapped</classname>
6101                   </entry>
6102                   <entry morerows="2" valign="top">
6103                     <classname>cc_hash_table</classname>
6104                   </entry>
6105                   <entry>
6106                     <classname>Comb_Hash_Fn</classname>
6107                   </entry>
6108                   <entry>
6109                     <classname>direct_mask_range_hashing</classname>
6110                   </entry>
6111                   <entry namest="c6" nameend="c7"></entry>
6112                 </row>
6113
6114                 <row>
6115                   <entry morerows="1" valign="top">
6116                     <classname>Resize_Policy</classname>
6117                   </entry>
6118                   <entry morerows="1" valign="top">
6119                     <classname>hash_standard_resize_policy</classname>
6120                   </entry>
6121                   <entry>
6122                     <classname>Size_Policy</classname>
6123                   </entry>
6124                   <entry>
6125                     <classname>hash_exponential_size_policy</classname>
6126                   </entry>
6127                 </row>
6128
6129                 <row>
6130                   <entry valign="top">
6131                     <classname>Trigger_Policy</classname>
6132                   </entry>
6133                   <entry>
6134                     <classname>hash_load_check_resize_trigger</classname> with
6135                     α<subscript>min</subscript> = 1/8 and α<subscript>max</subscript> = 1/2
6136                   </entry>
6137                 </row>
6138
6139               </tbody>
6140             </tgroup>
6141           </informaltable>
6142
6143         </section>
6144
6145         <section xml:id="multimap.text_insert_mem_small.observations">
6146           <info><title>
6147             Observations
6148           </title></info>
6149
6150           <para>See Observations::Mapping-Semantics
6151           Considerations.</para>
6152
6153         </section>
6154
6155       </section>
6156
6157       <!-- 06 <a href="multimap_text_insert_mem_usage_test_large"> -->
6158       <section xml:id="performance.multimap.text_insert_mem_large">
6159         <info><title>
6160           Text <function>insert</function> with Small
6161           Secondary-to-Primary Key Ratios Memory Use
6162         </title></info>
6163         <para></para>
6164
6165         <section xml:id="multimap.text_insert_mem_large.info">
6166           <info><title>
6167             Description
6168           </title></info>
6169           <para>This test inserts a number of pairs into a container. The
6170           first item of each pair is a string from an arbitrary text
6171           [wickland96thirty], and
6172           the second is a uniform integer. The container is a
6173           "multimap" - it considers the first member of each pair as a
6174           primary key, and the second member of each pair as a secondary
6175           key. There
6176           are 100 distinct primary keys, and the ratio of secondary keys
6177           to primary keys ranges to about 20.</para>
6178           <para>The test measures the memory use as a function of the number
6179           of values inserted.</para>
6180
6181           <para>
6182             It uses the test file:
6183             <filename>performance/ext/pb_ds/multimap_text_insert_mem_usage_large.cc</filename>
6184           </para>
6185
6186           <para>The test checks the memory scalability of different
6187           "multimap" designs.</para>
6188
6189         </section>
6190
6191         <section xml:id="multimap.text_insert_mem_large.results">
6192           <info><title>
6193             Results
6194           </title></info>
6195
6196           <para>The graphic below show the results for "multimaps" which
6197           use a tree-based container for primary keys.
6198           </para>
6199
6200           <!-- results graphic -->
6201           <informalfigure>
6202             <mediaobject>
6203               <imageobject>
6204                 <imagedata align="center" format="PNG" scale="100"
6205                            fileref="../images/pbds_multimap_text_insert_mem_large_s2p_tree.png"/>
6206               </imageobject>
6207               <imageobject>
6208                 <imagedata align="center" format="PDF" scale="33"
6209                            fileref="../images/pbds_multimap_text_insert_mem_large_s2p_tree.pdf"/>
6210               </imageobject>
6211             </mediaobject>
6212           </informalfigure>
6213
6214           <para>
6215             The abbreviated names in the legend of the graphic above are
6216             instantiated with the types in the following table.
6217           </para>
6218
6219
6220           <informaltable frame="all">
6221
6222             <tgroup cols="7" align="left" colsep="1" rowsep="1">
6223               <colspec colname="c1"/>
6224               <colspec colname="c2"/>
6225               <colspec colname="c3"/>
6226               <colspec colname="c4"/>
6227               <colspec colname="c5"/>
6228               <colspec colname="c6"/>
6229               <colspec colname="c7"/>
6230               <thead>
6231                 <row>
6232                   <entry><emphasis>Name/Instantiating Type</emphasis></entry>
6233                   <entry><emphasis>Parameter</emphasis></entry>
6234                   <entry><emphasis>Details</emphasis></entry>
6235                   <entry><emphasis>Parameter</emphasis></entry>
6236                   <entry><emphasis>Details</emphasis></entry>
6237                   <entry><emphasis>Parameter</emphasis></entry>
6238                   <entry><emphasis>Details</emphasis></entry>
6239                 </row>
6240               </thead>
6241
6242               <tbody>
6243
6244                 <!-- native -->
6245                 <row>
6246                   <?dbhtml bgcolor="#B0B0B0" ?>
6247                   <entry namest="c1" nameend="c7">
6248                     n_mmap
6249                   </entry>
6250                 </row>
6251
6252                 <row>
6253                   <entry>
6254                     <classname>std::multimap</classname>
6255                   </entry>
6256                   <entry namest="c2" nameend="c7"></entry>
6257                 </row>
6258
6259                 <!-- multimap 01 -->
6260                 <row>
6261                   <?dbhtml bgcolor="#B0B0B0" ?>
6262                   <entry namest="c1" nameend="c7">
6263                     rb_tree_mmap_lu_mtf_set
6264                   </entry>
6265                 </row>
6266
6267                 <row>
6268                   <entry morerows="2" valign="top">
6269                     <classname>tree</classname>
6270                   </entry>
6271                   <entry>
6272                     <classname>Tag</classname>
6273                   </entry>
6274                   <entry>
6275                     <classname>rb_tree_tag</classname>
6276                   </entry>
6277                   <entry namest="c4" nameend="c7"></entry>
6278                 </row>
6279
6280                 <row>
6281                   <entry>
6282                     <classname>Node_Update</classname>
6283                   </entry>
6284                   <entry>
6285                     <classname>null_node_update</classname>
6286                   </entry>
6287                   <entry namest="c4" nameend="c7"></entry>
6288                 </row>
6289                 <row>
6290                   <entry>
6291                     <classname>Mapped</classname>
6292                   </entry>
6293                   <entry>
6294                     <classname>list_update</classname>
6295                   </entry>
6296                   <entry>
6297                     <classname>Update_Policy</classname>
6298                   </entry>
6299                   <entry>
6300                     <classname>lu_move_to_front_policy</classname>
6301                   </entry>
6302                   <entry namest="c6" nameend="c7"></entry>
6303                 </row>
6304
6305                 <!-- multimap 02 -->
6306                 <row>
6307                   <?dbhtml bgcolor="#B0B0B0" ?>
6308                   <entry namest="c1" nameend="c7">
6309                     rb_tree_mmap_cc_hash_mask_exp_1div2_nsth_set
6310                   </entry>
6311                 </row>
6312
6313                 <row>
6314                   <entry morerows="4" valign="top">
6315                     <classname>tree</classname>
6316                   </entry>
6317                   <entry>
6318                     <classname>Tag</classname>
6319                   </entry>
6320                   <entry>
6321                     <classname>rb_tree_tag</classname>
6322                   </entry>
6323                   <entry namest="c4" nameend="c7"></entry>
6324                 </row>
6325
6326                 <row>
6327                   <entry>
6328                     <classname>Node_Update</classname>
6329                   </entry>
6330                   <entry>
6331                     <classname>null_node_update</classname>
6332                   </entry>
6333                   <entry namest="c4" nameend="c7"></entry>
6334                 </row>
6335
6336                 <row>
6337                   <entry morerows="2" valign="top">
6338                     <classname>Mapped</classname>
6339                   </entry>
6340                   <entry morerows="2" valign="top">
6341                     <classname>cc_hash_table</classname>
6342                   </entry>
6343                   <entry>
6344                     <classname>Comb_Hash_Fn</classname>
6345                   </entry>
6346                   <entry>
6347                     <classname>direct_mask_range_hashing</classname>
6348                   </entry>
6349                   <entry namest="c6" nameend="c7"></entry>
6350                 </row>
6351
6352                 <row>
6353                   <entry morerows="1" valign="top">
6354                     <classname>Resize_Policy</classname>
6355                   </entry>
6356                   <entry morerows="1" valign="top">
6357                     <classname>hash_standard_resize_policy</classname>
6358                   </entry>
6359                   <entry>
6360                     <classname>Size_Policy</classname>
6361                   </entry>
6362                   <entry>
6363                     <classname>hash_exponential_size_policy</classname>
6364                   </entry>
6365                 </row>
6366
6367                 <row>
6368                   <entry valign="top">
6369                     <classname>Trigger_Policy</classname>
6370                   </entry>
6371                   <entry>
6372                     <classname>hash_load_check_resize_trigger</classname> with
6373                     α<subscript>min</subscript> = 1/8 and α<subscript>max</subscript> = 1/2
6374                   </entry>
6375                 </row>
6376
6377               </tbody>
6378             </tgroup>
6379           </informaltable>
6380
6381
6382           <para>The graphic below show the results for "multimaps" which
6383           use a hash-based container for primary keys.
6384           </para>
6385
6386           <!-- results graphic -->
6387           <informalfigure>
6388             <mediaobject>
6389               <imageobject>
6390                 <imagedata align="center" format="PNG" scale="100"
6391                            fileref="../images/pbds_multimap_text_find_large_s2p_hash.png"/>
6392               </imageobject>
6393               <imageobject>
6394                 <imagedata align="center" format="PDF" scale="33"
6395                            fileref="../images/pbds_multimap_text_find_large_s2p_hash.pdf"/>
6396               </imageobject>
6397             </mediaobject>
6398           </informalfigure>
6399
6400           <para>
6401             The abbreviated names in the legend of the graphic above are
6402             instantiated with the types in the following table.
6403           </para>
6404
6405           <informaltable frame="all">
6406
6407             <tgroup cols="7" align="left" colsep="1" rowsep="1">
6408               <colspec colname="c1"/>
6409               <colspec colname="c2"/>
6410               <colspec colname="c3"/>
6411               <colspec colname="c4"/>
6412               <colspec colname="c5"/>
6413               <colspec colname="c6"/>
6414               <colspec colname="c7"/>
6415               <thead>
6416                 <row>
6417                   <entry><emphasis>Name/Instantiating Type</emphasis></entry>
6418                   <entry><emphasis>Parameter</emphasis></entry>
6419                   <entry><emphasis>Details</emphasis></entry>
6420                   <entry><emphasis>Parameter</emphasis></entry>
6421                   <entry><emphasis>Details</emphasis></entry>
6422                   <entry><emphasis>Parameter</emphasis></entry>
6423                   <entry><emphasis>Details</emphasis></entry>
6424                 </row>
6425               </thead>
6426
6427               <tbody>
6428
6429                 <!-- native -->
6430                 <row>
6431                   <?dbhtml bgcolor="#B0B0B0" ?>
6432                   <entry namest="c1" nameend="c7">
6433                     n_hash_mmap
6434                   </entry>
6435                 </row>
6436
6437                 <row>
6438                   <entry>
6439                     <classname>std::tr1::unordered_multimap</classname>
6440                   </entry>
6441                   <entry namest="c2" nameend="c7"></entry>
6442                 </row>
6443
6444                 <!-- multimap 01 -->
6445                 <row>
6446                   <?dbhtml bgcolor="#B0B0B0" ?>
6447                   <entry namest="c1" nameend="c7">
6448                     rb_tree_mmap_lu_mtf_set
6449                   </entry>
6450                 </row>
6451
6452                 <row>
6453                   <entry morerows="3" valign="top">
6454                     <classname>
6455                       cc_hash_table
6456                     </classname>
6457                   </entry>
6458                   <entry>
6459                     <classname>Comb_Hash_Fn</classname>
6460                   </entry>
6461                   <entry>
6462                     <classname>direct_mask_range_hashing</classname>
6463                   </entry>
6464                   <entry namest="c4" nameend="c7"></entry>
6465                 </row>
6466                 <row>
6467                   <entry morerows="1" valign="top">
6468                     <classname>Resize_Policy</classname>
6469                   </entry>
6470                   <entry morerows="1" valign="top">
6471                     <classname>hash_standard_resize_policy</classname>
6472                   </entry>
6473                   <entry>
6474                     <classname>Size_Policy</classname>
6475                   </entry>
6476                   <entry>
6477                     <classname>hash_exponential_size_policy</classname>
6478                   </entry>
6479                   <entry namest="c6" nameend="c7"></entry>
6480                 </row>
6481
6482                 <row>
6483                   <entry valign="top">
6484                     <classname>Trigger_Policy</classname>
6485                   </entry>
6486                   <entry>
6487                     <classname>hash_load_check_resize_trigger</classname> with
6488                     α<subscript>min</subscript> = 1/8 and α<subscript>max</subscript> = 1/2
6489                   </entry>
6490                   <entry namest="c6" nameend="c7"></entry>
6491                 </row>
6492
6493                 <row>
6494                   <entry>
6495                     <classname>Mapped</classname>
6496                   </entry>
6497                   <entry>
6498                     <classname>list_update</classname>
6499                   </entry>
6500                   <entry>
6501                     <classname>Update_Policy</classname>
6502                   </entry>
6503                   <entry>
6504                     <classname>lu_move_to_front_policy</classname>
6505                   </entry>
6506                   <entry namest="c6" nameend="c7"></entry>
6507                 </row>
6508
6509                 <!-- multimap 02 -->
6510                 <row>
6511                   <?dbhtml bgcolor="#B0B0B0" ?>
6512                   <entry namest="c1" nameend="c7">
6513                     rb_tree_mmap_cc_hash_mask_exp_1div2_nsth_set
6514                   </entry>
6515                 </row>
6516
6517                 <row>
6518                   <entry morerows="5" valign="top">
6519                     <classname>
6520                       cc_hash_table
6521                     </classname>
6522                   </entry>
6523                   <entry>
6524                     <classname>Comb_Hash_Fn</classname>
6525                   </entry>
6526                   <entry>
6527                     <classname>direct_mask_range_hashing</classname>
6528                   </entry>
6529                   <entry namest="c4" nameend="c7"></entry>
6530                 </row>
6531                 <row>
6532                   <entry morerows="1" valign="top">
6533                     <classname>Resize_Policy</classname>
6534                   </entry>
6535                   <entry morerows="1" valign="top">
6536                     <classname>hash_standard_resize_policy</classname>
6537                   </entry>
6538                   <entry>
6539                     <classname>Size_Policy</classname>
6540                   </entry>
6541                   <entry>
6542                     <classname>hash_exponential_size_policy</classname>
6543                   </entry>
6544                   <entry namest="c6" nameend="c7"></entry>
6545                 </row>
6546
6547                 <row>
6548                   <entry valign="top">
6549                     <classname>Trigger_Policy</classname>
6550                   </entry>
6551                   <entry>
6552                     <classname>hash_load_check_resize_trigger</classname> with
6553                     α<subscript>min</subscript> = 1/8 and α<subscript>max</subscript> = 1/2
6554                   </entry>
6555                   <entry namest="c6" nameend="c7"></entry>
6556                 </row>
6557
6558                 <row>
6559                   <entry morerows="2" valign="top">
6560                     <classname>Mapped</classname>
6561                   </entry>
6562                   <entry morerows="2" valign="top">
6563                     <classname>cc_hash_table</classname>
6564                   </entry>
6565                   <entry>
6566                     <classname>Comb_Hash_Fn</classname>
6567                   </entry>
6568                   <entry>
6569                     <classname>direct_mask_range_hashing</classname>
6570                   </entry>
6571                   <entry namest="c6" nameend="c7"></entry>
6572                 </row>
6573
6574                 <row>
6575                   <entry morerows="1" valign="top">
6576                     <classname>Resize_Policy</classname>
6577                   </entry>
6578                   <entry morerows="1" valign="top">
6579                     <classname>hash_standard_resize_policy</classname>
6580                   </entry>
6581                   <entry>
6582                     <classname>Size_Policy</classname>
6583                   </entry>
6584                   <entry>
6585                     <classname>hash_exponential_size_policy</classname>
6586                   </entry>
6587                 </row>
6588
6589                 <row>
6590                   <entry valign="top">
6591                     <classname>Trigger_Policy</classname>
6592                   </entry>
6593                   <entry>
6594                     <classname>hash_load_check_resize_trigger</classname> with
6595                     α<subscript>min</subscript> = 1/8 and α<subscript>max</subscript> = 1/2
6596                   </entry>
6597                 </row>
6598
6599               </tbody>
6600             </tgroup>
6601           </informaltable>
6602
6603         </section>
6604
6605         <section xml:id="multimap.text_insert_mem_large.observations">
6606           <info><title>
6607             Observations
6608           </title></info>
6609
6610           <para>See Observations::Mapping-Semantics
6611           Considerations.</para>
6612
6613         </section>
6614
6615       </section>
6616
6617     </section> <!-- multimap -->
6618
6619     <section xml:id="performance.priority_queue">
6620       <info><title>Priority Queue</title></info>
6621
6622       <!-- 01 <a href="priority_queue_text_push_timing_test"> -->
6623       <section xml:id="performance.priority_queue.text_push">
6624         <info><title>
6625           Text <function>push</function>
6626         </title></info>
6627         <para></para>
6628
6629         <section xml:id="priority_queue.text_push.info">
6630           <info><title>
6631             Description
6632           </title></info>
6633
6634
6635           <para>This test inserts a number of values with keys from an
6636           arbitrary text ([ wickland96thirty ]) into
6637           a container using <function>push</function>. It measures the average time
6638           for <function>push</function> as a function of the number of values
6639           pushed.</para>
6640
6641           <para>
6642             It uses the test file:
6643             <filename>performance/ext/pb_ds/priority_queue_text_push_timing.cc</filename>
6644           </para>
6645
6646           <para>The test checks the effect of different underlying data
6647           structures.
6648           </para>
6649
6650         </section>
6651
6652         <section xml:id="priority_queue.text_push.results">
6653           <info><title>
6654             Results
6655           </title></info>
6656
6657           <para>The two graphics below show the results for the native
6658           priority_queues and this library's priority_queues.
6659           </para>
6660
6661           <para>The graphic immediately below shows the results for the
6662           native priority_queue type instantiated with different underlying
6663           container types versus several different versions of library's
6664           priority_queues.
6665           </para>
6666
6667           <!-- results graphic -->
6668           <informalfigure>
6669             <mediaobject>
6670               <imageobject>
6671                 <imagedata align="center" format="PDF" scale="75"
6672                            fileref="../images/pbds_priority_queue_text_push.pdf"/>
6673               </imageobject>
6674               <imageobject>
6675                 <imagedata align="center" format="PNG" scale="100"
6676                            fileref="../images/pbds_priority_queue_text_push.png"/>
6677               </imageobject>
6678             </mediaobject>
6679           </informalfigure>
6680
6681           <para>
6682             The abbreviated names in the legend of the graphic above are
6683             instantiated with the types in the following table.
6684           </para>
6685
6686
6687           <informaltable frame="all">
6688
6689             <tgroup cols="3" align="left" colsep="1" rowsep="1">
6690               <colspec colname="c1"/>
6691               <colspec colname="c2"/>
6692               <colspec colname="c3"/>
6693               <thead>
6694                 <row>
6695                   <entry><emphasis>Name/Instantiating Type</emphasis></entry>
6696                   <entry><emphasis>Parameter</emphasis></entry>
6697                   <entry><emphasis>Details</emphasis></entry>
6698                 </row>
6699               </thead>
6700
6701               <tbody>
6702
6703                 <!-- native 01 -->
6704                 <row>
6705                   <?dbhtml bgcolor="#B0B0B0" ?>
6706                   <entry namest="c1" nameend="c3">
6707                     n_pq_vector
6708                   </entry>
6709                 </row>
6710
6711                 <row>
6712                   <entry>
6713                     <classname>std::priority_queue</classname>
6714                   </entry>
6715                   <entry>
6716                     <classname>Sequence</classname>
6717                   </entry>
6718                   <entry>
6719                     <classname>std::vector</classname>
6720                   </entry>
6721                 </row>
6722
6723                 <!-- native 02 -->
6724                 <row>
6725                   <?dbhtml bgcolor="#B0B0B0" ?>
6726                   <entry namest="c1" nameend="c3">
6727                     n_pq_deque
6728                   </entry>
6729                 </row>
6730
6731                 <row>
6732                   <entry>
6733                     <classname>std::priority_queue</classname>
6734                   </entry>
6735                   <entry>
6736                     <classname>Sequence</classname>
6737                   </entry>
6738                   <entry>
6739                     <classname>std::deque</classname>
6740                   </entry>
6741                 </row>
6742
6743                 <!-- priority_queue 01 -->
6744                 <row>
6745                   <?dbhtml bgcolor="#B0B0B0" ?>
6746                   <entry namest="c1" nameend="c3">
6747                     binary_heap
6748                   </entry>
6749                 </row>
6750
6751                 <row>
6752                   <entry>
6753                     <classname>priority_queue</classname>
6754                   </entry>
6755                   <entry>
6756                     <classname>Tag</classname>
6757                   </entry>
6758                   <entry>
6759                     <classname>binary_heap_tag</classname>
6760                   </entry>
6761                 </row>
6762
6763                 <!-- priority_queue 02 -->
6764                 <row>
6765                   <?dbhtml bgcolor="#B0B0B0" ?>
6766                   <entry namest="c1" nameend="c3">
6767                     binomial_heap
6768                   </entry>
6769                 </row>
6770
6771                 <row>
6772                   <entry>
6773                     <classname>priority_queue</classname>
6774                   </entry>
6775                   <entry>
6776                     <classname>Tag</classname>
6777                   </entry>
6778                   <entry>
6779                     <classname>binomial_heap_tag</classname>
6780                   </entry>
6781                 </row>
6782
6783                 <!-- priority_queue 03 -->
6784                 <row>
6785                   <?dbhtml bgcolor="#B0B0B0" ?>
6786                   <entry namest="c1" nameend="c3">
6787                     rc_binomial_heap
6788                   </entry>
6789                 </row>
6790
6791                 <row>
6792                   <entry>
6793                     <classname>priority_queue</classname>
6794                   </entry>
6795                   <entry>
6796                     <classname>Tag</classname>
6797                   </entry>
6798                   <entry>
6799                     <classname>rc_binomial_heap_tag</classname>
6800                   </entry>
6801                 </row>
6802
6803                 <!-- priority_queue 04 -->
6804                 <row>
6805                   <?dbhtml bgcolor="#B0B0B0" ?>
6806                   <entry namest="c1" nameend="c3">
6807                     thin_heap
6808                   </entry>
6809                 </row>
6810
6811                 <row>
6812                   <entry>
6813                     <classname>priority_queue</classname>
6814                   </entry>
6815                   <entry>
6816                     <classname>Tag</classname>
6817                   </entry>
6818                   <entry>
6819                     <classname>thin_heap_tag</classname>
6820                   </entry>
6821                 </row>
6822
6823
6824                 <!-- priority_queue 05 -->
6825                 <row>
6826                   <?dbhtml bgcolor="#B0B0B0" ?>
6827                   <entry namest="c1" nameend="c3">
6828                     pairing_heap
6829                   </entry>
6830                 </row>
6831
6832                 <row>
6833                   <entry>
6834                     <classname>priority_queue</classname>
6835                   </entry>
6836                   <entry>
6837                     <classname>Tag</classname>
6838                   </entry>
6839                   <entry>
6840                     <classname>pairing_heap_tag</classname>
6841                   </entry>
6842                 </row>
6843
6844               </tbody>
6845             </tgroup>
6846           </informaltable>
6847
6848
6849
6850           <para>The graphic below shows the results for the binary-heap
6851           based native priority queues and this library's pairing-heap
6852           priority_queue data structures.
6853           </para>
6854
6855           <!-- results graphic -->
6856           <informalfigure>
6857             <mediaobject>
6858               <imageobject>
6859                 <imagedata align="center" format="PDF" scale="75"
6860                            fileref="../images/pbds_pairing_priority_queue_text_push.pdf"/>
6861               </imageobject>
6862               <imageobject>
6863                 <imagedata align="center" format="PNG" scale="100"
6864                            fileref="../images/pbds_pairing_priority_queue_text_push.png"/>
6865               </imageobject>
6866             </mediaobject>
6867           </informalfigure>
6868
6869           <para>
6870             The abbreviated names in the legend of the graphic above are
6871             instantiated with the types in the following table.
6872           </para>
6873
6874
6875           <informaltable frame="all">
6876
6877             <tgroup cols="3" align="left" colsep="1" rowsep="1">
6878               <colspec colname="c1"/>
6879               <colspec colname="c2"/>
6880               <colspec colname="c3"/>
6881               <thead>
6882                 <row>
6883                   <entry><emphasis>Name/Instantiating Type</emphasis></entry>
6884                   <entry><emphasis>Parameter</emphasis></entry>
6885                   <entry><emphasis>Details</emphasis></entry>
6886                 </row>
6887               </thead>
6888
6889               <tbody>
6890                 <!-- native 01 -->
6891                 <row>
6892                   <?dbhtml bgcolor="#B0B0B0" ?>
6893                   <entry namest="c1" nameend="c3">
6894                     n_pq_vector
6895                   </entry>
6896                 </row>
6897
6898                 <row>
6899                   <entry>
6900                     <classname>std::priority_queue</classname>
6901                   </entry>
6902                   <entry>
6903                     <classname>Sequence</classname>
6904                   </entry>
6905                   <entry>
6906                     <classname>std::vector</classname>
6907                   </entry>
6908                 </row>
6909
6910                 <!-- native 02 -->
6911                 <row>
6912                   <?dbhtml bgcolor="#B0B0B0" ?>
6913                   <entry namest="c1" nameend="c3">
6914                     n_pq_deque
6915                   </entry>
6916                 </row>
6917
6918                 <row>
6919                   <entry>
6920                     <classname>std::priority_queue</classname>
6921                   </entry>
6922                   <entry>
6923                     <classname>Sequence</classname>
6924                   </entry>
6925                   <entry>
6926                     <classname>std::deque</classname>
6927                   </entry>
6928
6929                 </row>
6930
6931                 <!-- priority_queue 01 -->
6932                 <row>
6933                   <?dbhtml bgcolor="#B0B0B0" ?>
6934                   <entry namest="c1" nameend="c3">
6935                     thin_heap
6936                   </entry>
6937                 </row>
6938
6939                 <row>
6940                   <entry>
6941                     <classname>priority_queue</classname>
6942                   </entry>
6943                   <entry>
6944                     <classname>Tag</classname>
6945                   </entry>
6946                   <entry>
6947                     <classname>thin_heap_tag</classname>
6948                   </entry>
6949                 </row>
6950
6951
6952                 <!-- priority_queue 02 -->
6953                 <row>
6954                   <?dbhtml bgcolor="#B0B0B0" ?>
6955                   <entry namest="c1" nameend="c3">
6956                     pairing_heap
6957                   </entry>
6958                 </row>
6959
6960                 <row>
6961                   <entry>
6962                     <classname>priority_queue</classname>
6963                   </entry>
6964                   <entry>
6965                     <classname>Tag</classname>
6966                   </entry>
6967                   <entry>
6968                     <classname>pairing_heap_tag</classname>
6969                   </entry>
6970                 </row>
6971
6972               </tbody>
6973             </tgroup>
6974           </informaltable>
6975
6976
6977         </section>
6978
6979         <section xml:id="priority_queue.text_push.observations">
6980           <info><title>
6981             Observations
6982           </title></info>
6983
6984           <para>Pairing heaps (<classname>priority_queue</classname> with
6985           <classname>Tag</classname> = <classname>pairing_heap_tag</classname>)
6986           are the most suited for sequences of <function>push</function> and
6987           <function>pop</function> operations of non-primitive types (e.g.
6988           <classname>std::string</classname>s). (See Priority Queue
6989           Text <function>push</function> and <function>pop</function> Timing Test.) They are
6990           less constrained than binomial heaps, e.g., and since
6991           they are node-based, they outperform binary heaps. (See
6992           Priority
6993           Queue Random Integer <function>push</function> Timing Test for the case
6994           of primitive types.)</para>
6995
6996           <para>The standard's priority queues do not seem to perform well in
6997           this case: the <classname>std::vector</classname> implementation needs to
6998           perform a logarithmic sequence of string operations for each
6999           operation, and the deque implementation is possibly hampered by
7000           its need to manipulate a relatively-complex type (deques
7001           support a O(1) <function>push_front</function>, even though it is
7002           not used by <classname>std::priority_queue</classname>.)</para>
7003
7004         </section>
7005       </section>
7006
7007       <!-- 02 <a href="priority_queue_text_push_pop_timing_test"> -->
7008       <section xml:id="performance.priority_queue.text_push_pop">
7009         <info><title>
7010           Text <function>push</function> and <function>pop</function>
7011         </title></info>
7012         <para></para>
7013
7014         <section xml:id="priority_queue.text_push_pop.info">
7015           <info><title>
7016             Description
7017           </title></info>
7018
7019           <para>This test inserts a number of values with keys from an
7020           arbitrary text ([ wickland96thirty ]) into
7021           a container using <classname>push</classname> , then removes them using
7022           <classname>pop</classname> . It measures the average time for <classname>push</classname>
7023           as a function of the number of values.</para>
7024
7025
7026           <para>
7027             It uses the test file:
7028             <filename>performance/ext/pb_ds/priority_queue_text_push_pop_timing.cc</filename>
7029           </para>
7030
7031           <para>The test checks the effect of different underlying data
7032           structures.
7033           </para>
7034
7035         </section>
7036
7037         <section xml:id="priority_queue.text_push_pop.results">
7038           <info><title>
7039             Results
7040           </title></info>
7041
7042           <para>The two graphics below show the results for the native
7043           priority_queues and this library's priority_queues.
7044           </para>
7045
7046           <para>The graphic immediately below shows the results for the
7047           native priority_queue type instantiated with different underlying
7048           container types versus several different versions of library's
7049           priority_queues.
7050           </para>
7051
7052           <!-- results graphic -->
7053           <informalfigure>
7054             <mediaobject>
7055               <imageobject>
7056                 <imagedata align="center" format="PDF" scale="75"
7057                            fileref="../images/pbds_priority_queue_text_push_pop.pdf"/>
7058               </imageobject>
7059               <imageobject>
7060                 <imagedata align="center" format="PNG" scale="100"
7061                            fileref="../images/pbds_priority_queue_text_push_pop.png"/>
7062               </imageobject>
7063             </mediaobject>
7064           </informalfigure>
7065
7066           <para>
7067             The abbreviated names in the legend of the graphic above are
7068             instantiated with the types in the following table.
7069           </para>
7070
7071
7072           <informaltable frame="all">
7073
7074             <tgroup cols="3" align="left" colsep="1" rowsep="1">
7075               <colspec colname="c1"/>
7076               <colspec colname="c2"/>
7077               <colspec colname="c3"/>
7078               <thead>
7079                 <row>
7080                   <entry><emphasis>Name/Instantiating Type</emphasis></entry>
7081                   <entry><emphasis>Parameter</emphasis></entry>
7082                   <entry><emphasis>Details</emphasis></entry>
7083                 </row>
7084               </thead>
7085
7086               <tbody>
7087
7088                 <!-- native 01 -->
7089                 <row>
7090                   <?dbhtml bgcolor="#B0B0B0" ?>
7091                   <entry namest="c1" nameend="c3">
7092                     n_pq_vector
7093                   </entry>
7094                 </row>
7095
7096                 <row>
7097                   <entry>
7098                     <classname>std::priority_queue</classname>
7099                   </entry>
7100                   <entry>
7101                     <classname>Sequence</classname>
7102                   </entry>
7103                   <entry>
7104                     <classname>std::vector</classname>
7105                   </entry>
7106                 </row>
7107
7108                 <!-- native 02 -->
7109                 <row>
7110                   <?dbhtml bgcolor="#B0B0B0" ?>
7111                   <entry namest="c1" nameend="c3">
7112                     n_pq_deque
7113                   </entry>
7114                 </row>
7115
7116                 <row>
7117                   <entry>
7118                     <classname>std::priority_queue</classname>
7119                   </entry>
7120                   <entry>
7121                     <classname>Sequence</classname>
7122                   </entry>
7123                   <entry>
7124                     <classname>std::deque</classname>
7125                   </entry>
7126
7127                 </row>
7128
7129                 <!-- priority_queue 01 -->
7130                 <row>
7131                   <?dbhtml bgcolor="#B0B0B0" ?>
7132                   <entry namest="c1" nameend="c3">
7133                     binary_heap
7134                   </entry>
7135                 </row>
7136
7137                 <row>
7138                   <entry>
7139                     <classname>priority_queue</classname>
7140                   </entry>
7141                   <entry>
7142                     <classname>Tag</classname>
7143                   </entry>
7144                   <entry>
7145                     <classname>binary_heap_tag</classname>
7146                   </entry>
7147                 </row>
7148
7149                 <!-- priority_queue 02 -->
7150                 <row>
7151                   <?dbhtml bgcolor="#B0B0B0" ?>
7152                   <entry namest="c1" nameend="c3">
7153                     binomial_heap
7154                   </entry>
7155                 </row>
7156
7157                 <row>
7158                   <entry>
7159                     <classname>priority_queue</classname>
7160                   </entry>
7161                   <entry>
7162                     <classname>Tag</classname>
7163                   </entry>
7164                   <entry>
7165                     <classname>binomial_heap_tag</classname>
7166                   </entry>
7167                 </row>
7168
7169                 <!-- priority_queue 03 -->
7170                 <row>
7171                   <?dbhtml bgcolor="#B0B0B0" ?>
7172                   <entry namest="c1" nameend="c3">
7173                     rc_binomial_heap
7174                   </entry>
7175                 </row>
7176
7177                 <row>
7178                   <entry>
7179                     <classname>priority_queue</classname>
7180                   </entry>
7181                   <entry>
7182                     <classname>Tag</classname>
7183                   </entry>
7184                   <entry>
7185                     <classname>rc_binomial_heap_tag</classname>
7186                   </entry>
7187                 </row>
7188
7189                 <!-- priority_queue 04 -->
7190                 <row>
7191                   <?dbhtml bgcolor="#B0B0B0" ?>
7192                   <entry namest="c1" nameend="c3">
7193                     thin_heap
7194                   </entry>
7195                 </row>
7196
7197                 <row>
7198                   <entry>
7199                     <classname>priority_queue</classname>
7200                   </entry>
7201                   <entry>
7202                     <classname>Tag</classname>
7203                   </entry>
7204                   <entry>
7205                     <classname>thin_heap_tag</classname>
7206                   </entry>
7207                 </row>
7208
7209                 <!-- priority_queue 05 -->
7210                 <row>
7211                   <?dbhtml bgcolor="#B0B0B0" ?>
7212                   <entry namest="c1" nameend="c3">
7213                     pairing_heap
7214                   </entry>
7215                 </row>
7216
7217                 <row>
7218                   <entry>
7219                     <classname>priority_queue</classname>
7220                   </entry>
7221                   <entry>
7222                     <classname>Tag</classname>
7223                   </entry>
7224                   <entry>
7225                     <classname>pairing_heap_tag</classname>
7226                   </entry>
7227                 </row>
7228
7229               </tbody>
7230             </tgroup>
7231           </informaltable>
7232
7233
7234
7235           <para>The graphic below shows the results for the native priority
7236           queues and this library's pairing-heap priority_queue data
7237           structures.
7238           </para>
7239
7240           <!-- results graphic -->
7241           <informalfigure>
7242             <mediaobject>
7243               <imageobject>
7244                 <imagedata align="center" format="PDF" scale="75"
7245                            fileref="../images/pbds_pairing_priority_queue_text_push_pop.pdf"/>
7246               </imageobject>
7247               <imageobject>
7248                 <imagedata align="center" format="PNG" scale="100"
7249                            fileref="../images/pbds_pairing_priority_queue_text_push_pop.png"/>
7250               </imageobject>
7251             </mediaobject>
7252           </informalfigure>
7253
7254           <para>
7255             The abbreviated names in the legend of the graphic above are
7256             instantiated with the types in the following table.
7257           </para>
7258
7259
7260           <informaltable frame="all">
7261
7262             <tgroup cols="3" align="left" colsep="1" rowsep="1">
7263               <colspec colname="c1"/>
7264               <colspec colname="c2"/>
7265               <colspec colname="c3"/>
7266               <thead>
7267                 <row>
7268                   <entry><emphasis>Name/Instantiating Type</emphasis></entry>
7269                   <entry><emphasis>Parameter</emphasis></entry>
7270                   <entry><emphasis>Details</emphasis></entry>
7271                 </row>
7272               </thead>
7273
7274               <tbody>
7275
7276                 <!-- native 01 -->
7277                 <row>
7278                   <?dbhtml bgcolor="#B0B0B0" ?>
7279                   <entry namest="c1" nameend="c3">
7280                     n_pq_vector
7281                   </entry>
7282                 </row>
7283
7284                 <row>
7285                   <entry>
7286                     <classname>std::priority_queue</classname> adapting <classname>std::vector</classname>
7287                   </entry>
7288                   <entry>
7289                     <classname>Sequence</classname>
7290                   </entry>
7291                   <entry>
7292                     <classname>std::vector</classname>
7293                   </entry>
7294
7295                 </row>
7296
7297                 <!-- native 02 -->
7298                 <row>
7299                   <?dbhtml bgcolor="#B0B0B0" ?>
7300                   <entry namest="c1" nameend="c3">
7301                     n_pq_deque
7302                   </entry>
7303                 </row>
7304
7305                 <row>
7306                   <entry>
7307                     <classname>std::priority_queue</classname>
7308                   </entry>
7309                   <entry>
7310                     <classname>Sequence</classname>
7311                   </entry>
7312                   <entry>
7313                     <classname>std::deque</classname>
7314                   </entry>
7315
7316                 </row>
7317
7318                 <!-- priority_queue 01 -->
7319                 <row>
7320                   <?dbhtml bgcolor="#B0B0B0" ?>
7321                   <entry namest="c1" nameend="c3">
7322                     pairing_heap
7323                   </entry>
7324                 </row>
7325
7326                 <row>
7327                   <entry>
7328                     <classname>priority_queue</classname>
7329                   </entry>
7330                   <entry>
7331                     <classname>Tag</classname>
7332                   </entry>
7333                   <entry>
7334                     <classname>pairing_heap_tag</classname>
7335                   </entry>
7336                 </row>
7337
7338               </tbody>
7339             </tgroup>
7340           </informaltable>
7341
7342
7343         </section>
7344
7345         <section xml:id="priority_queue.text_push_pop.observations">
7346           <info><title>
7347             Observations
7348           </title></info>
7349
7350           <para>These results are very similar to Priority Queue Text
7351           <function>push</function> Timing Test. As stated there, pairing heaps
7352           (<classname>priority_queue</classname> with
7353           <classname>Tag</classname>
7354           = <classname>pairing_heap_tag</classname>) are most suited
7355           for <function>push</function> and <function>pop</function>
7356           sequences of non-primitive types such as strings. Observing these
7357           two tests, one can note that a pairing heap outperforms the others
7358           in terms of <function>push</function> operations, but equals
7359           binary heaps (<classname>priority_queue</classname> with
7360           <classname>Tag</classname>
7361           = <classname>binary_heap_tag</classname>) if the number
7362           of <function>push</function> and <function>pop</function>
7363           operations is equal. As the number of <function>pop</function>
7364           operations is at most equal to the number
7365           of <function>push</function> operations, pairing heaps are better
7366           in this case. See Priority Queue Random
7367           Integer <function>push</function> and <function>pop</function>
7368           Timing Test for a case which is different.</para>
7369
7370         </section>
7371       </section>
7372
7373
7374       <!-- 03 <a href="priority_queue_random_int_push_timing_test"> -->
7375       <section xml:id="performance.priority_queue.int_push">
7376         <info><title>
7377           Integer <function>push</function>
7378         </title></info>
7379         <para></para>
7380
7381         <section xml:id="priority_queue.int_push.info">
7382           <info><title>
7383             Description
7384           </title></info>
7385
7386           <para>This test inserts a number of values with integer keys
7387           into a container using <function>push</function>. It
7388           measures the average time for <function>push</function> as a
7389           function of the number of values.</para>
7390
7391           <para>
7392             It uses the test file:
7393             <filename>performance/ext/pb_ds/priority_queue_random_int_push_timing.cc</filename>
7394           </para>
7395
7396           <para>The test checks the effect of different underlying data
7397           structures.
7398           </para>
7399
7400         </section>
7401
7402         <section xml:id="priority_queue.int_push.results">
7403           <info><title>
7404             Results
7405           </title></info>
7406
7407           <para>The two graphics below show the results for the native
7408           priority_queues and this library's priority_queues.
7409           </para>
7410
7411           <para>The graphic immediately below shows the results for the
7412           native priority_queue type instantiated with different underlying
7413           container types versus several different versions of library's
7414           priority_queues.
7415           </para>
7416
7417           <!-- results graphic -->
7418           <informalfigure>
7419             <mediaobject>
7420               <imageobject>
7421                 <imagedata align="center" format="PDF" scale="75"
7422                            fileref="../images/pbds_priority_queue_int_push.pdf"/>
7423               </imageobject>
7424               <imageobject>
7425                 <imagedata align="center" format="PNG" scale="100"
7426                            fileref="../images/pbds_priority_queue_int_push.png"/>
7427               </imageobject>
7428             </mediaobject>
7429           </informalfigure>
7430
7431           <para>
7432             The abbreviated names in the legend of the graphic above are
7433             instantiated with the types in the following table.
7434           </para>
7435
7436
7437           <informaltable frame="all">
7438
7439             <tgroup cols="3" align="left" colsep="1" rowsep="1">
7440               <colspec colname="c1"/>
7441               <colspec colname="c2"/>
7442               <colspec colname="c3"/>
7443               <thead>
7444                 <row>
7445                   <entry><emphasis>Name/Instantiating Type</emphasis></entry>
7446                   <entry><emphasis>Parameter</emphasis></entry>
7447                   <entry><emphasis>Details</emphasis></entry>
7448                 </row>
7449               </thead>
7450
7451               <tbody>
7452
7453                 <!-- native 01 -->
7454                 <row>
7455                   <?dbhtml bgcolor="#B0B0B0" ?>
7456                   <entry namest="c1" nameend="c3">
7457                     n_pq_vector
7458                   </entry>
7459                 </row>
7460
7461                 <row>
7462                   <entry>
7463                     <classname>std::priority_queue</classname>
7464                   </entry>
7465                   <entry>
7466                     <classname>Sequence</classname>
7467                   </entry>
7468                   <entry>
7469                     <classname>std::vector</classname>
7470                   </entry>
7471                 </row>
7472
7473                 <!-- native 02 -->
7474                 <row>
7475                   <?dbhtml bgcolor="#B0B0B0" ?>
7476                   <entry namest="c1" nameend="c3">
7477                     n_pq_deque
7478                   </entry>
7479                 </row>
7480
7481                 <row>
7482                   <entry>
7483                     <classname>std::priority_queue</classname>
7484                   </entry>
7485                   <entry>
7486                     <classname>Sequence</classname>
7487                   </entry>
7488                   <entry>
7489                     <classname>std::deque</classname>
7490                   </entry>
7491
7492                 </row>
7493
7494                 <!-- priority_queue 01 -->
7495                 <row>
7496                   <?dbhtml bgcolor="#B0B0B0" ?>
7497                   <entry namest="c1" nameend="c3">
7498                     binary_heap
7499                   </entry>
7500                 </row>
7501
7502                 <row>
7503                   <entry>
7504                     <classname>priority_queue</classname>
7505                   </entry>
7506                   <entry>
7507                     <classname>Tag</classname>
7508                   </entry>
7509                   <entry>
7510                     <classname>binary_heap_tag</classname>
7511                   </entry>
7512                 </row>
7513
7514                 <!-- priority_queue 02 -->
7515                 <row>
7516                   <?dbhtml bgcolor="#B0B0B0" ?>
7517                   <entry namest="c1" nameend="c3">
7518                     binomial_heap
7519                   </entry>
7520                 </row>
7521
7522                 <row>
7523                   <entry>
7524                     <classname>priority_queue</classname>
7525                   </entry>
7526                   <entry>
7527                     <classname>Tag</classname>
7528                   </entry>
7529                   <entry>
7530                     <classname>binomial_heap_tag</classname>
7531                   </entry>
7532                 </row>
7533
7534                 <!-- priority_queue 03 -->
7535                 <row>
7536                   <?dbhtml bgcolor="#B0B0B0" ?>
7537                   <entry namest="c1" nameend="c3">
7538                     rc_binomial_heap
7539                   </entry>
7540                 </row>
7541
7542                 <row>
7543                   <entry>
7544                     <classname>priority_queue</classname>
7545                   </entry>
7546                   <entry>
7547                     <classname>Tag</classname>
7548                   </entry>
7549                   <entry>
7550                     <classname>rc_binomial_heap_tag</classname>
7551                   </entry>
7552                 </row>
7553
7554                 <!-- priority_queue 04 -->
7555                 <row>
7556                   <?dbhtml bgcolor="#B0B0B0" ?>
7557                   <entry namest="c1" nameend="c3">
7558                     thin_heap
7559                   </entry>
7560                 </row>
7561
7562                 <row>
7563                   <entry>
7564                     <classname>priority_queue</classname>
7565                   </entry>
7566                   <entry>
7567                     <classname>Tag</classname>
7568                   </entry>
7569                   <entry>
7570                     <classname>thin_heap_tag</classname>
7571                   </entry>
7572                 </row>
7573
7574                 <!-- priority_queue 05 -->
7575                 <row>
7576                   <?dbhtml bgcolor="#B0B0B0" ?>
7577                   <entry namest="c1" nameend="c3">
7578                     pairing_heap
7579                   </entry>
7580                 </row>
7581
7582                 <row>
7583                   <entry>
7584                     <classname>priority_queue</classname>
7585                   </entry>
7586                   <entry>
7587                     <classname>Tag</classname>
7588                   </entry>
7589                   <entry>
7590                     <classname>pairing_heap_tag</classname>
7591                   </entry>
7592                 </row>
7593
7594               </tbody>
7595             </tgroup>
7596           </informaltable>
7597
7598
7599
7600           <para>The graphic below shows the results for the binary-heap
7601           based native priority queues and this library's
7602           priority_queue data structures.
7603           </para>
7604
7605           <!-- results graphic -->
7606           <informalfigure>
7607             <mediaobject>
7608               <imageobject>
7609                 <imagedata align="center" format="PDF" scale="75"
7610                            fileref="../images/pbds_binary_priority_queue_int_push.pdf"/>
7611               </imageobject>
7612               <imageobject>
7613                 <imagedata align="center" format="PNG" scale="100"
7614                            fileref="../images/pbds_binary_priority_queue_int_push.png"/>
7615               </imageobject>
7616             </mediaobject>
7617           </informalfigure>
7618
7619           <para>
7620             The abbreviated names in the legend of the graphic above are
7621             instantiated with the types in the following table.
7622           </para>
7623
7624
7625           <informaltable frame="all">
7626
7627             <tgroup cols="3" align="left" colsep="1" rowsep="1">
7628               <colspec colname="c1"/>
7629               <colspec colname="c2"/>
7630               <colspec colname="c3"/>
7631               <thead>
7632                 <row>
7633                   <entry><emphasis>Name/Instantiating Type</emphasis></entry>
7634                   <entry><emphasis>Parameter</emphasis></entry>
7635                   <entry><emphasis>Details</emphasis></entry>
7636                 </row>
7637               </thead>
7638
7639               <tbody>
7640
7641                 <!-- native 01 -->
7642                 <row>
7643                   <?dbhtml bgcolor="#B0B0B0" ?>
7644                   <entry namest="c1" nameend="c3">
7645                     n_pq_vector
7646                   </entry>
7647                 </row>
7648
7649                 <row>
7650                   <entry>
7651                     <classname>std::priority_queue</classname> adapting <classname>std::vector</classname>
7652                   </entry>
7653                   <entry>
7654                     <classname>Sequence</classname>
7655                   </entry>
7656                   <entry>
7657                     <classname>std::vector</classname>
7658                   </entry>
7659
7660                 </row>
7661
7662                 <!-- native 02 -->
7663                 <row>
7664                   <?dbhtml bgcolor="#B0B0B0" ?>
7665                   <entry namest="c1" nameend="c3">
7666                     n_pq_deque
7667                   </entry>
7668                 </row>
7669
7670                 <row>
7671                   <entry>
7672                     <classname>std::priority_queue</classname>
7673                   </entry>
7674                   <entry>
7675                     <classname>Sequence</classname>
7676                   </entry>
7677                   <entry>
7678                     <classname>std::deque</classname>
7679                   </entry>
7680
7681                 </row>
7682
7683                 <!-- priority_queue 01 -->
7684                 <row>
7685                   <?dbhtml bgcolor="#B0B0B0" ?>
7686                   <entry namest="c1" nameend="c3">
7687                     binary_heap
7688                   </entry>
7689                 </row>
7690
7691                 <row>
7692                   <entry>
7693                     <classname>priority_queue</classname>
7694                   </entry>
7695                   <entry>
7696                     <classname>Tag</classname>
7697                   </entry>
7698                   <entry>
7699                     <classname>binary_heap_tag</classname>
7700                   </entry>
7701                 </row>
7702
7703               </tbody>
7704             </tgroup>
7705           </informaltable>
7706
7707
7708         </section>
7709
7710         <section xml:id="priority_queue.int_push.observations">
7711           <info><title>
7712             Observations
7713           </title></info>
7714
7715
7716           <para>Binary heaps are the most suited for sequences of
7717           <function>push</function> and <function>pop</function> operations of primitive types
7718           (e.g. <type>int</type>s). They are less constrained
7719           than any other type, and since it is very efficient to store
7720           such types in arrays, they outperform even pairing heaps. (See
7721           Priority
7722           Queue Text <function>push</function> Timing Test for the case of
7723           non-primitive types.)</para>
7724         </section>
7725       </section>
7726
7727       <!-- 04 "priority_queue_random_int_push_pop_timing_test" -->
7728       <section xml:id="performance.priority_queue.int_push_pop">
7729         <info><title>
7730           Integer <function>push</function>
7731         </title></info>
7732         <para></para>
7733
7734         <section xml:id="priority_queue.int_push_pop.info">
7735           <info><title>
7736             Description
7737           </title></info>
7738
7739           <para>This test inserts a number of values with integer keys
7740           into a container using <function>push</function> , then removes them
7741           using <function>pop</function> . It measures the average time for
7742           <function>push</function> and <function>pop</function> as a function
7743           of the number of values.</para>
7744
7745           <para>
7746             It uses the test file:
7747             <filename>performance/ext/pb_ds/priority_queue_random_int_push_pop_timing.cc</filename>
7748           </para>
7749
7750           <para>The test checks the effect of different underlying data
7751           structures.
7752           </para>
7753
7754         </section>
7755
7756         <section xml:id="priority_queue.int_push_pop.results">
7757           <info><title>
7758             Results
7759           </title></info>
7760
7761           <para>The graphic immediately below shows the results for the
7762           native priority_queue type instantiated with different underlying
7763           container types versus several different versions of library's
7764           priority_queues.
7765           </para>
7766
7767           <!-- results graphic -->
7768           <informalfigure>
7769             <mediaobject>
7770               <imageobject>
7771                 <imagedata align="center" format="PDF" scale="75"
7772                            fileref="../images/pbds_priority_queue_int_push_pop.pdf"/>
7773               </imageobject>
7774               <imageobject>
7775                 <imagedata align="center" format="PNG" scale="100"
7776                            fileref="../images/pbds_priority_queue_int_push_pop.png"/>
7777               </imageobject>
7778             </mediaobject>
7779           </informalfigure>
7780
7781           <para>
7782             The abbreviated names in the legend of the graphic above are
7783             instantiated with the types in the following table.
7784           </para>
7785
7786
7787           <informaltable frame="all">
7788
7789             <tgroup cols="3" align="left" colsep="1" rowsep="1">
7790               <colspec colname="c1"/>
7791               <colspec colname="c2"/>
7792               <colspec colname="c3"/>
7793               <thead>
7794                 <row>
7795                   <entry><emphasis>Name/Instantiating Type</emphasis></entry>
7796                   <entry><emphasis>Parameter</emphasis></entry>
7797                   <entry><emphasis>Details</emphasis></entry>
7798                 </row>
7799               </thead>
7800
7801               <tbody>
7802
7803                 <!-- native 01 -->
7804                 <row>
7805                   <?dbhtml bgcolor="#B0B0B0" ?>
7806                   <entry namest="c1" nameend="c3">
7807                     n_pq_vector
7808                   </entry>
7809                 </row>
7810
7811                 <row>
7812                   <entry>
7813                     <classname>std::priority_queue</classname>
7814                   </entry>
7815                   <entry>
7816                     <classname>Sequence</classname>
7817                   </entry>
7818                   <entry>
7819                     <classname>std::vector</classname>
7820                   </entry>
7821                 </row>
7822
7823                 <!-- native 02 -->
7824                 <row>
7825                   <?dbhtml bgcolor="#B0B0B0" ?>
7826                   <entry namest="c1" nameend="c3">
7827                     n_pq_deque
7828                   </entry>
7829                 </row>
7830
7831                 <row>
7832                   <entry>
7833                     <classname>std::priority_queue</classname>
7834                   </entry>
7835                   <entry>
7836                     <classname>Sequence</classname>
7837                   </entry>
7838                   <entry>
7839                     <classname>std::deque</classname>
7840                   </entry>
7841
7842                 </row>
7843
7844                 <!-- priority_queue 01 -->
7845                 <row>
7846                   <?dbhtml bgcolor="#B0B0B0" ?>
7847                   <entry namest="c1" nameend="c3">
7848                     binary_heap
7849                   </entry>
7850                 </row>
7851
7852                 <row>
7853                   <entry>
7854                     <classname>priority_queue</classname>
7855                   </entry>
7856                   <entry>
7857                     <classname>Tag</classname>
7858                   </entry>
7859                   <entry>
7860                     <classname>binary_heap_tag</classname>
7861                   </entry>
7862                 </row>
7863
7864                 <!-- priority_queue 02 -->
7865                 <row>
7866                   <?dbhtml bgcolor="#B0B0B0" ?>
7867                   <entry namest="c1" nameend="c3">
7868                     binomial_heap
7869                   </entry>
7870                 </row>
7871
7872                 <row>
7873                   <entry>
7874                     <classname>priority_queue</classname>
7875                   </entry>
7876                   <entry>
7877                     <classname>Tag</classname>
7878                   </entry>
7879                   <entry>
7880                     <classname>binomial_heap_tag</classname>
7881                   </entry>
7882                 </row>
7883
7884                 <!-- priority_queue 03 -->
7885                 <row>
7886                   <?dbhtml bgcolor="#B0B0B0" ?>
7887                   <entry namest="c1" nameend="c3">
7888                     rc_binomial_heap
7889                   </entry>
7890                 </row>
7891
7892                 <row>
7893                   <entry>
7894                     <classname>priority_queue</classname>
7895                   </entry>
7896                   <entry>
7897                     <classname>Tag</classname>
7898                   </entry>
7899                   <entry>
7900                     <classname>rc_binomial_heap_tag</classname>
7901                   </entry>
7902                 </row>
7903
7904                 <!-- priority_queue 04 -->
7905                 <row>
7906                   <?dbhtml bgcolor="#B0B0B0" ?>
7907                   <entry namest="c1" nameend="c3">
7908                     thin_heap
7909                   </entry>
7910                 </row>
7911
7912                 <row>
7913                   <entry>
7914                     <classname>priority_queue</classname>
7915                   </entry>
7916                   <entry>
7917                     <classname>Tag</classname>
7918                   </entry>
7919                   <entry>
7920                     <classname>thin_heap_tag</classname>
7921                   </entry>
7922                 </row>
7923
7924                 <!-- priority_queue 05 -->
7925                 <row>
7926                   <?dbhtml bgcolor="#B0B0B0" ?>
7927                   <entry namest="c1" nameend="c3">
7928                     pairing_heap
7929                   </entry>
7930                 </row>
7931
7932                 <row>
7933                   <entry>
7934                     <classname>priority_queue</classname>
7935                   </entry>
7936                   <entry>
7937                     <classname>Tag</classname>
7938                   </entry>
7939                   <entry>
7940                     <classname>pairing_heap_tag</classname>
7941                   </entry>
7942                 </row>
7943
7944               </tbody>
7945             </tgroup>
7946           </informaltable>
7947
7948
7949
7950         </section>
7951
7952         <section xml:id="priority_queue.int_push_pop.observations">
7953           <info><title>
7954             Observations
7955           </title></info>
7956
7957           <para>Binary heaps are the most suited for sequences of
7958           <function>push</function> and <function>pop</function> operations of primitive types
7959           (e.g. <type>int</type>s). This is explained in
7960           Priority
7961           Queue Random Int <function>push</function> Timing Test. (See Priority Queue
7962           Text <function>push</function> Timing Test for the case of primitive
7963           types.)</para>
7964
7965           <para>At first glance it seems that the standard's vector-based
7966           priority queue is approximately on par with this
7967           library's corresponding priority queue. There are two
7968           differences however:</para>
7969           <orderedlist>
7970             <listitem><para>The standard's priority queue does not downsize the underlying
7971             vector (or deque) as the priority queue becomes smaller
7972             (see Priority Queue
7973             Text <function>pop</function> Memory Use Test). It is therefore
7974             gaining some speed at the expense of space.</para></listitem>
7975             <listitem><para>From Priority Queue Random
7976             Integer <function>push</function> and <function>pop</function>
7977             Timing Test, it seems that the standard's priority queue is
7978             slower in terms of <function>push</function> operations. Since
7979             the number of
7980             <function>pop</function> operations is at most that of <function>push</function>
7981             operations, the test here is the "best" for the standard's
7982             priority queue.</para></listitem>
7983           </orderedlist>
7984
7985
7986         </section>
7987       </section>
7988
7989
7990       <!-- 05 <a href="priority_queue_text_pop_mem_usage_test"> -->
7991       <section xml:id="performance.priority_queue.text_pop">
7992         <info><title>
7993           Text <function>pop</function> Memory Use
7994         </title></info>
7995         <para></para>
7996
7997         <section xml:id="priority_queue.text_pop.info">
7998           <info><title>
7999             Description
8000           </title></info>
8001
8002
8003           <para>This test inserts a number of values with keys from an
8004           arbitrary text ([ wickland96thirty ]) into
8005           a container, then pops them until only one is left in the
8006           container. It measures the memory use as a function of the
8007           number of values pushed to the container.</para>
8008           <para>
8009             It uses the test file:
8010             <filename>performance/ext/pb_ds/priority_queue_text_pop_mem_usage.cc</filename>
8011           </para>
8012
8013           <para>The test checks the effect of different underlying data
8014           structures.
8015           </para>
8016
8017         </section>
8018
8019         <section xml:id="priority_queue.text_pop.results">
8020           <info><title>
8021             Results
8022           </title></info>
8023
8024           <para>The graphic immediately below shows the results for the
8025           native priority_queue type instantiated with different underlying
8026           container types versus several different versions of library's
8027           priority_queues.
8028           </para>
8029
8030           <!-- results graphic -->
8031           <informalfigure>
8032             <mediaobject>
8033               <imageobject>
8034                 <imagedata align="center" format="PDF" scale="75"
8035                            fileref="../images/pbds_priority_queue_text_pop_mem.pdf"/>
8036               </imageobject>
8037               <imageobject>
8038                 <imagedata align="center" format="PNG" scale="100"
8039                            fileref="../images/pbds_priority_queue_text_pop_mem.png"/>
8040               </imageobject>
8041             </mediaobject>
8042           </informalfigure>
8043
8044           <para>
8045             The abbreviated names in the legend of the graphic above are
8046             instantiated with the types in the following table.
8047           </para>
8048
8049
8050           <informaltable frame="all">
8051
8052             <tgroup cols="3" align="left" colsep="1" rowsep="1">
8053               <colspec colname="c1"/>
8054               <colspec colname="c2"/>
8055               <colspec colname="c3"/>
8056               <thead>
8057                 <row>
8058                   <entry><emphasis>Name/Instantiating Type</emphasis></entry>
8059                   <entry><emphasis>Parameter</emphasis></entry>
8060                   <entry><emphasis>Details</emphasis></entry>
8061                 </row>
8062               </thead>
8063
8064               <tbody>
8065
8066                 <!-- native 01 -->
8067                 <row>
8068                   <?dbhtml bgcolor="#B0B0B0" ?>
8069                   <entry namest="c1" nameend="c3">
8070                     n_pq_vector
8071                   </entry>
8072                 </row>
8073
8074                 <row>
8075                   <entry>
8076                     <classname>std::priority_queue</classname>
8077                   </entry>
8078                   <entry>
8079                     <classname>Sequence</classname>
8080                   </entry>
8081                   <entry>
8082                     <classname>std::vector</classname>
8083                   </entry>
8084                 </row>
8085
8086                 <!-- native 02 -->
8087                 <row>
8088                   <?dbhtml bgcolor="#B0B0B0" ?>
8089                   <entry namest="c1" nameend="c3">
8090                     n_pq_deque
8091                   </entry>
8092                 </row>
8093
8094                 <row>
8095                   <entry>
8096                     <classname>std::priority_queue</classname>
8097                   </entry>
8098                   <entry>
8099                     <classname>Sequence</classname>
8100                   </entry>
8101                   <entry>
8102                     <classname>std::deque</classname>
8103                   </entry>
8104
8105                 </row>
8106
8107                 <!-- priority_queue 01 -->
8108                 <row>
8109                   <?dbhtml bgcolor="#B0B0B0" ?>
8110                   <entry namest="c1" nameend="c3">
8111                     binary_heap
8112                   </entry>
8113                 </row>
8114
8115                 <row>
8116                   <entry>
8117                     <classname>priority_queue</classname>
8118                   </entry>
8119                   <entry>
8120                     <classname>Tag</classname>
8121                   </entry>
8122                   <entry>
8123                     <classname>binary_heap_tag</classname>
8124                   </entry>
8125                 </row>
8126
8127                 <!-- priority_queue 02 -->
8128                 <row>
8129                   <?dbhtml bgcolor="#B0B0B0" ?>
8130                   <entry namest="c1" nameend="c3">
8131                     binomial_heap
8132                   </entry>
8133                 </row>
8134
8135                 <row>
8136                   <entry>
8137                     <classname>priority_queue</classname>
8138                   </entry>
8139                   <entry>
8140                     <classname>Tag</classname>
8141                   </entry>
8142                   <entry>
8143                     <classname>binomial_heap_tag</classname>
8144                   </entry>
8145                 </row>
8146
8147                 <!-- priority_queue 03 -->
8148                 <row>
8149                   <?dbhtml bgcolor="#B0B0B0" ?>
8150                   <entry namest="c1" nameend="c3">
8151                     rc_binomial_heap
8152                   </entry>
8153                 </row>
8154
8155                 <row>
8156                   <entry>
8157                     <classname>priority_queue</classname>
8158                   </entry>
8159                   <entry>
8160                     <classname>Tag</classname>
8161                   </entry>
8162                   <entry>
8163                     <classname>rc_binomial_heap_tag</classname>
8164                   </entry>
8165                 </row>
8166
8167                 <!-- priority_queue 04 -->
8168                 <row>
8169                   <?dbhtml bgcolor="#B0B0B0" ?>
8170                   <entry namest="c1" nameend="c3">
8171                     thin_heap
8172                   </entry>
8173                 </row>
8174
8175                 <row>
8176                   <entry>
8177                     <classname>priority_queue</classname>
8178                   </entry>
8179                   <entry>
8180                     <classname>Tag</classname>
8181                   </entry>
8182                   <entry>
8183                     <classname>thin_heap_tag</classname>
8184                   </entry>
8185                 </row>
8186
8187                 <!-- priority_queue 05 -->
8188                 <row>
8189                   <?dbhtml bgcolor="#B0B0B0" ?>
8190                   <entry namest="c1" nameend="c3">
8191                     pairing_heap
8192                   </entry>
8193                 </row>
8194
8195                 <row>
8196                   <entry>
8197                     <classname>priority_queue</classname>
8198                   </entry>
8199                   <entry>
8200                     <classname>Tag</classname>
8201                   </entry>
8202                   <entry>
8203                     <classname>pairing_heap_tag</classname>
8204                   </entry>
8205                 </row>
8206
8207               </tbody>
8208             </tgroup>
8209           </informaltable>
8210
8211
8212
8213         </section>
8214
8215         <section xml:id="priority_queue.text_pop.observations">
8216           <info><title>
8217             Observations
8218           </title></info>
8219
8220
8221           <para>The priority queue implementations (excluding the standard's) use
8222           memory proportionally to the number of values they hold:
8223           node-based implementations (e.g., a pairing heap) do so
8224           naturally; this library's binary heap de-allocates memory when
8225           a certain lower threshold is exceeded.</para>
8226
8227           <para>Note from Priority Queue Text <function>push</function>
8228           and <function>pop</function> Timing Test and Priority Queue
8229           Random Integer <function>push</function>
8230           and <function>pop</function> Timing Test that this does not
8231           impede performance compared to the standard's priority
8232           queues.</para>
8233           <para>See Hash-Based Erase
8234           Memory Use Test for a similar phenomenon regarding priority
8235           queues.</para>
8236         </section>
8237       </section>
8238
8239       <!-- 06 <a href="priority_queue_text_join_timing_test"> -->
8240       <section xml:id="performance.priority_queue.text_join">
8241         <info><title>
8242           Text <function>join</function>
8243         </title></info>
8244         <para></para>
8245
8246         <section xml:id="priority_queue.text_join.info">
8247           <info><title>
8248             Description
8249           </title></info>
8250
8251           <para>This test inserts a number of values with keys from an
8252           arbitrary text ([ wickland96thirty ]) into
8253           two containers, then merges the containers. It uses
8254           <function>join</function> for this library's priority queues; for
8255           the standard's priority queues, it successively pops values from
8256           one container and pushes them into the other. The test measures
8257           the average time as a function of the number of values.</para>
8258           <para>
8259             It uses the test file:
8260             <filename>performance/ext/pb_ds/priority_queue_text_join_timing.cc</filename>
8261           </para>
8262
8263           <para>The test checks the effect of different underlying data
8264           structures.
8265           </para>
8266
8267         </section>
8268
8269         <section xml:id="priority_queue.text_join.results">
8270           <info><title>
8271             Results
8272           </title></info>
8273
8274           <para>The graphic immediately below shows the results for the
8275           native priority_queue type instantiated with different underlying
8276           container types versus several different versions of library's
8277           priority_queues.
8278           </para>
8279
8280           <!-- results graphic -->
8281           <informalfigure>
8282             <mediaobject>
8283               <imageobject>
8284                 <imagedata align="center" format="PDF" scale="75"
8285                            fileref="../images/pbds_priority_queue_text_join.pdf"/>
8286               </imageobject>
8287               <imageobject>
8288                 <imagedata align="center" format="PNG" scale="100"
8289                            fileref="../images/pbds_priority_queue_text_join.png"/>
8290               </imageobject>
8291             </mediaobject>
8292           </informalfigure>
8293
8294           <para>
8295             The abbreviated names in the legend of the graphic above are
8296             instantiated with the types in the following table.
8297           </para>
8298
8299
8300           <informaltable frame="all">
8301
8302             <tgroup cols="3" align="left" colsep="1" rowsep="1">
8303               <colspec colname="c1"/>
8304               <colspec colname="c2"/>
8305               <colspec colname="c3"/>
8306               <thead>
8307                 <row>
8308                   <entry><emphasis>Name/Instantiating Type</emphasis></entry>
8309                   <entry><emphasis>Parameter</emphasis></entry>
8310                   <entry><emphasis>Details</emphasis></entry>
8311                 </row>
8312               </thead>
8313
8314               <tbody>
8315
8316                 <!-- native 01 -->
8317                 <row>
8318                   <?dbhtml bgcolor="#B0B0B0" ?>
8319                   <entry namest="c1" nameend="c3">
8320                     n_pq_vector
8321                   </entry>
8322                 </row>
8323
8324                 <row>
8325                   <entry>
8326                     <classname>std::priority_queue</classname>
8327                   </entry>
8328                   <entry>
8329                     <classname>Sequence</classname>
8330                   </entry>
8331                   <entry>
8332                     <classname>std::vector</classname>
8333                   </entry>
8334                 </row>
8335
8336                 <!-- native 02 -->
8337                 <row>
8338                   <?dbhtml bgcolor="#B0B0B0" ?>
8339                   <entry namest="c1" nameend="c3">
8340                     n_pq_deque
8341                   </entry>
8342                 </row>
8343
8344                 <row>
8345                   <entry>
8346                     <classname>std::priority_queue</classname>
8347                   </entry>
8348                   <entry>
8349                     <classname>Sequence</classname>
8350                   </entry>
8351                   <entry>
8352                     <classname>std::deque</classname>
8353                   </entry>
8354
8355                 </row>
8356
8357                 <!-- priority_queue 01 -->
8358                 <row>
8359                   <?dbhtml bgcolor="#B0B0B0" ?>
8360                   <entry namest="c1" nameend="c3">
8361                     binary_heap
8362                   </entry>
8363                 </row>
8364
8365                 <row>
8366                   <entry>
8367                     <classname>priority_queue</classname>
8368                   </entry>
8369                   <entry>
8370                     <classname>Tag</classname>
8371                   </entry>
8372                   <entry>
8373                     <classname>binary_heap_tag</classname>
8374                   </entry>
8375                 </row>
8376
8377                 <!-- priority_queue 02 -->
8378                 <row>
8379                   <?dbhtml bgcolor="#B0B0B0" ?>
8380                   <entry namest="c1" nameend="c3">
8381                     binomial_heap
8382                   </entry>
8383                 </row>
8384
8385                 <row>
8386                   <entry>
8387                     <classname>priority_queue</classname>
8388                   </entry>
8389                   <entry>
8390                     <classname>Tag</classname>
8391                   </entry>
8392                   <entry>
8393                     <classname>binomial_heap_tag</classname>
8394                   </entry>
8395                 </row>
8396
8397                 <!-- priority_queue 03 -->
8398                 <row>
8399                   <?dbhtml bgcolor="#B0B0B0" ?>
8400                   <entry namest="c1" nameend="c3">
8401                     rc_binomial_heap
8402                   </entry>
8403                 </row>
8404
8405                 <row>
8406                   <entry>
8407                     <classname>priority_queue</classname>
8408                   </entry>
8409                   <entry>
8410                     <classname>Tag</classname>
8411                   </entry>
8412                   <entry>
8413                     <classname>rc_binomial_heap_tag</classname>
8414                   </entry>
8415                 </row>
8416
8417                 <!-- priority_queue 04 -->
8418                 <row>
8419                   <?dbhtml bgcolor="#B0B0B0" ?>
8420                   <entry namest="c1" nameend="c3">
8421                     thin_heap
8422                   </entry>
8423                 </row>
8424
8425                 <row>
8426                   <entry>
8427                     <classname>priority_queue</classname>
8428                   </entry>
8429                   <entry>
8430                     <classname>Tag</classname>
8431                   </entry>
8432                   <entry>
8433                     <classname>thin_heap_tag</classname>
8434                   </entry>
8435                 </row>
8436
8437                 <!-- priority_queue 05 -->
8438                 <row>
8439                   <?dbhtml bgcolor="#B0B0B0" ?>
8440                   <entry namest="c1" nameend="c3">
8441                     pairing_heap
8442                   </entry>
8443                 </row>
8444
8445                 <row>
8446                   <entry>
8447                     <classname>priority_queue</classname>
8448                   </entry>
8449                   <entry>
8450                     <classname>Tag</classname>
8451                   </entry>
8452                   <entry>
8453                     <classname>pairing_heap_tag</classname>
8454                   </entry>
8455                 </row>
8456
8457               </tbody>
8458             </tgroup>
8459           </informaltable>
8460
8461
8462
8463         </section>
8464
8465         <section xml:id="priority_queue.text_join.observations">
8466           <info><title>
8467             Observations
8468           </title></info>
8469
8470           <para>In this test the node-based heaps perform <function>join</function> in
8471           either logarithmic or constant time. The binary heap requires
8472           linear time, since the well-known heapify algorithm [clrs2001] is linear.</para>
8473           <para>It would be possible to apply the heapify algorithm to the
8474           standard containers, if they would support iteration (which they
8475           don't). Barring iterators, it is still somehow possible to perform
8476           linear-time merge on a <classname>std::vector</classname>-based
8477           standard priority queue, using <function>top()</function>
8478           and <function>size()</function> (since they are enough to expose
8479           the underlying array), but this is impossible for
8480           a <classname>std::deque</classname>-based standard priority queue.
8481           Without heapify, the cost is super-linear.</para>
8482         </section>
8483       </section>
8484
8485
8486       <!-- 07 <a href="priority_queue_text_push_timing_test"> -->
8487       <section xml:id="performance.priority_queue.text_modify_up">
8488         <info><title>
8489           Text <function>modify</function> Up
8490         </title></info>
8491         <para></para>
8492
8493         <section xml:id="priority_queue.text_modify_up.info">
8494           <info><title>
8495             Description
8496           </title></info>
8497
8498           <para>This test inserts a number of values with keys from an
8499           arbitrary text ([ wickland96thirty ]) into
8500           into a container then modifies each one "up" (i.e., it
8501           makes it larger). It uses <function>modify</function> for this library's
8502           priority queues; for the standard's priority queues, it pops values
8503           from a container until it reaches the value that should be
8504           modified, then pushes values back in. It measures the average
8505           time for <function>modify</function> as a function of the number of
8506           values.</para>
8507
8508           <para>
8509             It uses the test file:
8510             <filename>performance/ext/pb_ds/priority_queue_text_modify_up_timing.cc</filename>
8511           </para>
8512
8513           <para>The test checks the effect of different underlying data
8514           structures for graph algorithms settings.  Note that making an
8515           arbitrary value larger (in the sense of the priority queue's
8516           comparison functor) corresponds to decrease-key in standard graph
8517           algorithms [clrs2001].
8518           </para>
8519
8520         </section>
8521
8522         <section xml:id="priority_queue.text_modify_up.results">
8523           <info><title>
8524             Results
8525           </title></info>
8526
8527           <para>The two graphics below show the results for the native
8528           priority_queues and this library's priority_queues.
8529           </para>
8530
8531           <para>The graphic immediately below shows the results for the
8532           native priority_queue type instantiated with different underlying
8533           container types versus several different versions of library's
8534           priority_queues.
8535           </para>
8536
8537           <!-- results graphic -->
8538           <informalfigure>
8539             <mediaobject>
8540               <imageobject>
8541                 <imagedata align="center" format="PDF" scale="75"
8542                            fileref="../images/pbds_priority_queue_text_modify_up.pdf"/>
8543               </imageobject>
8544               <imageobject>
8545                 <imagedata align="center" format="PNG" scale="100"
8546                            fileref="../images/pbds_priority_queue_text_modify_up.png"/>
8547               </imageobject>
8548             </mediaobject>
8549           </informalfigure>
8550
8551           <para>
8552             The abbreviated names in the legend of the graphic above are
8553             instantiated with the types in the following table.
8554           </para>
8555
8556
8557           <informaltable frame="all">
8558
8559             <tgroup cols="3" align="left" colsep="1" rowsep="1">
8560               <colspec colname="c1"/>
8561               <colspec colname="c2"/>
8562               <colspec colname="c3"/>
8563               <thead>
8564                 <row>
8565                   <entry><emphasis>Name/Instantiating Type</emphasis></entry>
8566                   <entry><emphasis>Parameter</emphasis></entry>
8567                   <entry><emphasis>Details</emphasis></entry>
8568                 </row>
8569               </thead>
8570
8571               <tbody>
8572
8573                 <!-- native 01 -->
8574                 <row>
8575                   <?dbhtml bgcolor="#B0B0B0" ?>
8576                   <entry namest="c1" nameend="c3">
8577                     n_pq_vector
8578                   </entry>
8579                 </row>
8580
8581                 <row>
8582                   <entry>
8583                     <classname>std::priority_queue</classname>
8584                   </entry>
8585                   <entry>
8586                     <classname>Sequence</classname>
8587                   </entry>
8588                   <entry>
8589                     <classname>std::vector</classname>
8590                   </entry>
8591                 </row>
8592
8593                 <!-- native 02 -->
8594                 <row>
8595                   <?dbhtml bgcolor="#B0B0B0" ?>
8596                   <entry namest="c1" nameend="c3">
8597                     n_pq_deque
8598                   </entry>
8599                 </row>
8600
8601                 <row>
8602                   <entry>
8603                     <classname>std::priority_queue</classname>
8604                   </entry>
8605                   <entry>
8606                     <classname>Sequence</classname>
8607                   </entry>
8608                   <entry>
8609                     <classname>std::deque</classname>
8610                   </entry>
8611                 </row>
8612
8613                 <!-- priority_queue 01 -->
8614                 <row>
8615                   <?dbhtml bgcolor="#B0B0B0" ?>
8616                   <entry namest="c1" nameend="c3">
8617                     binary_heap
8618                   </entry>
8619                 </row>
8620
8621                 <row>
8622                   <entry>
8623                     <classname>priority_queue</classname>
8624                   </entry>
8625                   <entry>
8626                     <classname>Tag</classname>
8627                   </entry>
8628                   <entry>
8629                     <classname>binary_heap_tag</classname>
8630                   </entry>
8631                 </row>
8632
8633                 <!-- priority_queue 02 -->
8634                 <row>
8635                   <?dbhtml bgcolor="#B0B0B0" ?>
8636                   <entry namest="c1" nameend="c3">
8637                     binomial_heap
8638                   </entry>
8639                 </row>
8640
8641                 <row>
8642                   <entry>
8643                     <classname>priority_queue</classname>
8644                   </entry>
8645                   <entry>
8646                     <classname>Tag</classname>
8647                   </entry>
8648                   <entry>
8649                     <classname>binomial_heap_tag</classname>
8650                   </entry>
8651                 </row>
8652
8653                 <!-- priority_queue 03 -->
8654                 <row>
8655                   <?dbhtml bgcolor="#B0B0B0" ?>
8656                   <entry namest="c1" nameend="c3">
8657                     rc_binomial_heap
8658                   </entry>
8659                 </row>
8660
8661                 <row>
8662                   <entry>
8663                     <classname>priority_queue</classname>
8664                   </entry>
8665                   <entry>
8666                     <classname>Tag</classname>
8667                   </entry>
8668                   <entry>
8669                     <classname>rc_binomial_heap_tag</classname>
8670                   </entry>
8671                 </row>
8672
8673                 <!-- priority_queue 04 -->
8674                 <row>
8675                   <?dbhtml bgcolor="#B0B0B0" ?>
8676                   <entry namest="c1" nameend="c3">
8677                     thin_heap
8678                   </entry>
8679                 </row>
8680
8681                 <row>
8682                   <entry>
8683                     <classname>priority_queue</classname>
8684                   </entry>
8685                   <entry>
8686                     <classname>Tag</classname>
8687                   </entry>
8688                   <entry>
8689                     <classname>thin_heap_tag</classname>
8690                   </entry>
8691                 </row>
8692
8693
8694                 <!-- priority_queue 05 -->
8695                 <row>
8696                   <?dbhtml bgcolor="#B0B0B0" ?>
8697                   <entry namest="c1" nameend="c3">
8698                     pairing_heap
8699                   </entry>
8700                 </row>
8701
8702                 <row>
8703                   <entry>
8704                     <classname>priority_queue</classname>
8705                   </entry>
8706                   <entry>
8707                     <classname>Tag</classname>
8708                   </entry>
8709                   <entry>
8710                     <classname>pairing_heap_tag</classname>
8711                   </entry>
8712                 </row>
8713
8714               </tbody>
8715             </tgroup>
8716           </informaltable>
8717
8718
8719
8720           <para>The graphic below shows the results for the
8721           native priority queues and this library's pairing and thin heap
8722           priority_queue data structures.
8723           </para>
8724
8725           <!-- results graphic -->
8726           <informalfigure>
8727             <mediaobject>
8728               <imageobject>
8729                 <imagedata align="center" format="PDF" scale="75"
8730                            fileref="../images/pbds_pairing_priority_queue_text_modify_up_thin.pdf"/>
8731               </imageobject>
8732               <imageobject>
8733                 <imagedata align="center" format="PNG" scale="100"
8734                            fileref="../images/pbds_pairing_priority_queue_text_modify_up_thin.png"/>
8735               </imageobject>
8736             </mediaobject>
8737           </informalfigure>
8738
8739           <para>
8740             The abbreviated names in the legend of the graphic above are
8741             instantiated with the types in the following table.
8742           </para>
8743
8744
8745           <informaltable frame="all">
8746
8747             <tgroup cols="3" align="left" colsep="1" rowsep="1">
8748               <colspec colname="c1"/>
8749               <colspec colname="c2"/>
8750               <colspec colname="c3"/>
8751               <thead>
8752                 <row>
8753                   <entry><emphasis>Name/Instantiating Type</emphasis></entry>
8754                   <entry><emphasis>Parameter</emphasis></entry>
8755                   <entry><emphasis>Details</emphasis></entry>
8756                 </row>
8757               </thead>
8758
8759               <tbody>
8760
8761                 <!-- priority_queue 01 -->
8762                 <row>
8763                   <?dbhtml bgcolor="#B0B0B0" ?>
8764                   <entry namest="c1" nameend="c3">
8765                     thin_heap
8766                   </entry>
8767                 </row>
8768
8769                 <row>
8770                   <entry>
8771                     <classname>priority_queue</classname>
8772                   </entry>
8773                   <entry>
8774                     <classname>Tag</classname>
8775                   </entry>
8776                   <entry>
8777                     <classname>thin_heap_tag</classname>
8778                   </entry>
8779                 </row>
8780
8781
8782                 <!-- priority_queue 02 -->
8783                 <row>
8784                   <?dbhtml bgcolor="#B0B0B0" ?>
8785                   <entry namest="c1" nameend="c3">
8786                     pairing_heap
8787                   </entry>
8788                 </row>
8789
8790                 <row>
8791                   <entry>
8792                     <classname>priority_queue</classname>
8793                   </entry>
8794                   <entry>
8795                     <classname>Tag</classname>
8796                   </entry>
8797                   <entry>
8798                     <classname>pairing_heap_tag</classname>
8799                   </entry>
8800                 </row>
8801
8802               </tbody>
8803             </tgroup>
8804           </informaltable>
8805
8806
8807         </section>
8808
8809         <section xml:id="priority_queue.text_modify_up.observations">
8810           <info><title>
8811             Observations
8812           </title></info>
8813
8814           <para>As noted above, increasing an arbitrary value (in the sense of
8815           the priority queue's comparison functor) is very common in
8816           graph-related algorithms. In this case, a thin heap
8817           (<classname>priority_queue</classname> with
8818           <classname>Tag</classname> = <classname>thin_heap_tag</classname>)
8819           outperforms a pairing heap (<classname>priority_queue</classname> with
8820           <classname>Tag</classname> = <classname>pairing_heap_tag</classname>).
8821           Conversely, Priority Queue Text
8822           <function>push</function> Timing Test, Priority Queue
8823           Text <function>push</function> and <function>pop</function> Timing Test, Priority
8824           Queue Random Integer <function>push</function> Timing Test, and
8825           Priority
8826           Queue Random Integer <function>push</function> and <function>pop</function> Timing
8827           Test show that the situation is reversed for other
8828           operations. It is not clear when to prefer one of these two
8829           different types.</para>
8830
8831           <para>In this test this library's binary heaps
8832           effectively perform modify in linear time. As explained in
8833           Priority Queue Design::Traits, given a valid point-type iterator,
8834           a binary heap can perform
8835           <function>modify</function> logarithmically. The problem is that binary
8836           heaps invalidate their find iterators with each modifying
8837           operation, and so the only way to obtain a valid point-type
8838           iterator is to iterate using a range-type iterator until
8839           finding the appropriate value, then use the range-type iterator
8840           for the <function>modify</function> operation.</para>
8841           <para>The explanation for the standard's priority queues' performance
8842           is similar to that in Priority Queue Text
8843           <function>join</function> Timing Test.</para>
8844
8845
8846         </section>
8847       </section>
8848
8849       <!-- 08 <a href="priority_queue_text_modify_down_timing_test"> -->
8850
8851       <section xml:id="performance.priority_queue.text_modify_down">
8852         <info><title>
8853           Text <function>modify</function> Down
8854         </title></info>
8855         <para></para>
8856
8857         <section xml:id="priority_queue.text_modify_down.info">
8858           <info><title>
8859             Description
8860           </title></info>
8861
8862           <para>This test inserts a number of values with keys from an
8863           arbitrary text ([ wickland96thirty ]) into
8864           into a container then modifies each one "down" (i.e., it
8865           makes it smaller). It uses <function>modify</function> for this library's
8866           priority queues; for the standard's priority queues, it pops values
8867           from a container until it reaches the value that should be
8868           modified, then pushes values back in. It measures the average
8869           time for <function>modify</function> as a function of the number of
8870           values.</para>
8871
8872           <para>
8873             It uses the test file:
8874             <filename>performance/ext/pb_ds/priority_queue_text_modify_down_timing.cc</filename>
8875           </para>
8876
8877           <para>The main purpose of this test is to contrast Priority Queue
8878           Text <classname>modify</classname> Up Timing Test.</para>
8879
8880         </section>
8881
8882         <section xml:id="priority_queue.text_modify_down.results">
8883           <info><title>
8884             Results
8885           </title></info>
8886
8887           <para>The two graphics below show the results for the native
8888           priority_queues and this library's priority_queues.
8889           </para>
8890
8891           <para>The graphic immediately below shows the results for the
8892           native priority_queue type instantiated with different underlying
8893           container types versus several different versions of library's
8894           priority_queues.
8895           </para>
8896
8897           <!-- results graphic -->
8898           <informalfigure>
8899             <mediaobject>
8900               <imageobject>
8901                 <imagedata align="center" format="PDF" scale="75"
8902                            fileref="../images/pbds_priority_queue_text_modify_down.pdf"/>
8903               </imageobject>
8904               <imageobject>
8905                 <imagedata align="center" format="PNG" scale="100"
8906                            fileref="../images/pbds_priority_queue_text_modify_down.png"/>
8907               </imageobject>
8908             </mediaobject>
8909           </informalfigure>
8910
8911           <para>
8912             The abbreviated names in the legend of the graphic above are
8913             instantiated with the types in the following table.
8914           </para>
8915
8916
8917           <informaltable frame="all">
8918
8919             <tgroup cols="3" align="left" colsep="1" rowsep="1">
8920               <colspec colname="c1"/>
8921               <colspec colname="c2"/>
8922               <colspec colname="c3"/>
8923               <thead>
8924                 <row>
8925                   <entry><emphasis>Name/Instantiating Type</emphasis></entry>
8926                   <entry><emphasis>Parameter</emphasis></entry>
8927                   <entry><emphasis>Details</emphasis></entry>
8928                 </row>
8929               </thead>
8930
8931               <tbody>
8932
8933                 <!-- native 01 -->
8934                 <row>
8935                   <?dbhtml bgcolor="#B0B0B0" ?>
8936                   <entry namest="c1" nameend="c3">
8937                     n_pq_vector
8938                   </entry>
8939                 </row>
8940
8941                 <row>
8942                   <entry>
8943                     <classname>std::priority_queue</classname>
8944                   </entry>
8945                   <entry>
8946                     <classname>Sequence</classname>
8947                   </entry>
8948                   <entry>
8949                     <classname>std::vector</classname>
8950                   </entry>
8951                 </row>
8952
8953                 <!-- native 02 -->
8954                 <row>
8955                   <?dbhtml bgcolor="#B0B0B0" ?>
8956                   <entry namest="c1" nameend="c3">
8957                     n_pq_deque
8958                   </entry>
8959                 </row>
8960
8961                 <row>
8962                   <entry>
8963                     <classname>std::priority_queue</classname>
8964                   </entry>
8965                   <entry>
8966                     <classname>Sequence</classname>
8967                   </entry>
8968                   <entry>
8969                     <classname>std::deque</classname>
8970                   </entry>
8971                 </row>
8972
8973                 <!-- priority_queue 01 -->
8974                 <row>
8975                   <?dbhtml bgcolor="#B0B0B0" ?>
8976                   <entry namest="c1" nameend="c3">
8977                     binary_heap
8978                   </entry>
8979                 </row>
8980
8981                 <row>
8982                   <entry>
8983                     <classname>priority_queue</classname>
8984                   </entry>
8985                   <entry>
8986                     <classname>Tag</classname>
8987                   </entry>
8988                   <entry>
8989                     <classname>binary_heap_tag</classname>
8990                   </entry>
8991                 </row>
8992
8993                 <!-- priority_queue 02 -->
8994                 <row>
8995                   <?dbhtml bgcolor="#B0B0B0" ?>
8996                   <entry namest="c1" nameend="c3">
8997                     binomial_heap
8998                   </entry>
8999                 </row>
9000
9001                 <row>
9002                   <entry>
9003                     <classname>priority_queue</classname>
9004                   </entry>
9005                   <entry>
9006                     <classname>Tag</classname>
9007                   </entry>
9008                   <entry>
9009                     <classname>binomial_heap_tag</classname>
9010                   </entry>
9011                 </row>
9012
9013                 <!-- priority_queue 03 -->
9014                 <row>
9015                   <?dbhtml bgcolor="#B0B0B0" ?>
9016                   <entry namest="c1" nameend="c3">
9017                     rc_binomial_heap
9018                   </entry>
9019                 </row>
9020
9021                 <row>
9022                   <entry>
9023                     <classname>priority_queue</classname>
9024                   </entry>
9025                   <entry>
9026                     <classname>Tag</classname>
9027                   </entry>
9028                   <entry>
9029                     <classname>rc_binomial_heap_tag</classname>
9030                   </entry>
9031                 </row>
9032
9033                 <!-- priority_queue 04 -->
9034                 <row>
9035                   <?dbhtml bgcolor="#B0B0B0" ?>
9036                   <entry namest="c1" nameend="c3">
9037                     thin_heap
9038                   </entry>
9039                 </row>
9040
9041                 <row>
9042                   <entry>
9043                     <classname>priority_queue</classname>
9044                   </entry>
9045                   <entry>
9046                     <classname>Tag</classname>
9047                   </entry>
9048                   <entry>
9049                     <classname>thin_heap_tag</classname>
9050                   </entry>
9051                 </row>
9052
9053
9054                 <!-- priority_queue 05 -->
9055                 <row>
9056                   <?dbhtml bgcolor="#B0B0B0" ?>
9057                   <entry namest="c1" nameend="c3">
9058                     pairing_heap
9059                   </entry>
9060                 </row>
9061
9062                 <row>
9063                   <entry>
9064                     <classname>priority_queue</classname>
9065                   </entry>
9066                   <entry>
9067                     <classname>Tag</classname>
9068                   </entry>
9069                   <entry>
9070                     <classname>pairing_heap_tag</classname>
9071                   </entry>
9072                 </row>
9073
9074               </tbody>
9075             </tgroup>
9076           </informaltable>
9077
9078
9079
9080           <para>The graphic below shows the results for the
9081           native priority queues and this library's pairing and thin heap
9082           priority_queue data structures.
9083           </para>
9084
9085           <!-- results graphic -->
9086           <informalfigure>
9087             <mediaobject>
9088               <imageobject>
9089                 <imagedata align="center" format="PDF" scale="75"
9090                            fileref="../images/pbds_pairing_priority_queue_text_modify_down_thin.pdf"/>
9091               </imageobject>
9092               <imageobject>
9093                 <imagedata align="center" format="PNG" scale="100"
9094                            fileref="../images/pbds_pairing_priority_queue_text_modify_down_thin.png"/>
9095               </imageobject>
9096             </mediaobject>
9097           </informalfigure>
9098
9099           <para>
9100             The abbreviated names in the legend of the graphic above are
9101             instantiated with the types in the following table.
9102           </para>
9103
9104
9105           <informaltable frame="all">
9106
9107             <tgroup cols="3" align="left" colsep="1" rowsep="1">
9108               <colspec colname="c1"/>
9109               <colspec colname="c2"/>
9110               <colspec colname="c3"/>
9111               <thead>
9112                 <row>
9113                   <entry><emphasis>Name/Instantiating Type</emphasis></entry>
9114                   <entry><emphasis>Parameter</emphasis></entry>
9115                   <entry><emphasis>Details</emphasis></entry>
9116                 </row>
9117               </thead>
9118
9119               <tbody>
9120
9121                 <!-- priority_queue 01 -->
9122                 <row>
9123                   <?dbhtml bgcolor="#B0B0B0" ?>
9124                   <entry namest="c1" nameend="c3">
9125                     thin_heap
9126                   </entry>
9127                 </row>
9128
9129                 <row>
9130                   <entry>
9131                     <classname>priority_queue</classname>
9132                   </entry>
9133                   <entry>
9134                     <classname>Tag</classname>
9135                   </entry>
9136                   <entry>
9137                     <classname>thin_heap_tag</classname>
9138                   </entry>
9139                 </row>
9140
9141
9142                 <!-- priority_queue 02 -->
9143                 <row>
9144                   <?dbhtml bgcolor="#B0B0B0" ?>
9145                   <entry namest="c1" nameend="c3">
9146                     pairing_heap
9147                   </entry>
9148                 </row>
9149
9150                 <row>
9151                   <entry>
9152                     <classname>priority_queue</classname>
9153                   </entry>
9154                   <entry>
9155                     <classname>Tag</classname>
9156                   </entry>
9157                   <entry>
9158                     <classname>pairing_heap_tag</classname>
9159                   </entry>
9160                 </row>
9161
9162               </tbody>
9163             </tgroup>
9164           </informaltable>
9165
9166
9167         </section>
9168
9169         <section xml:id="priority_queue.text_modify_down.observations">
9170           <info><title>
9171             Observations
9172           </title></info>
9173
9174           <para>Most points in these results are similar to Priority Queue
9175           Text <function>modify</function> Up Timing Test.</para>
9176
9177           <para>It is interesting to note, however, that as opposed to that
9178           test, a thin heap (<classname>priority_queue</classname> with
9179           <classname>Tag</classname> = <classname>thin_heap_tag</classname>) is
9180           outperformed by a pairing heap (<classname>priority_queue</classname> with
9181           <classname>Tag</classname> = <classname>pairing_heap_tag</classname>).
9182           In this case, both heaps essentially perform an <function>erase</function>
9183           operation followed by a <function>push</function> operation. As the other
9184           tests show, a pairing heap is usually far more efficient than a
9185           thin heap, so this is not surprising.</para>
9186           <para>Most algorithms that involve priority queues increase values
9187           (in the sense of the priority queue's comparison functor), and
9188           so Priority Queue
9189           Text <classname>modify</classname> Up Timing Test - is more interesting
9190           than this test.</para>
9191         </section>
9192       </section>
9193
9194
9195     </section> <!-- priority_queue -->
9196
9197     <section xml:id="pbds.test.performance.observations">
9198       <info><title>Observations</title></info>
9199
9200       <section xml:id="observations.associative">
9201         <info><title>Associative</title></info>
9202
9203         <section xml:id="observations.associative.underlying">
9204           <info><title>
9205             Underlying Data-Structure Families
9206           </title></info>
9207
9208           <para>In general, hash-based containers have better timing performance
9209           than containers based on different underlying-data structures. The
9210           main reason to choose a tree-based or trie-based container is if a
9211           byproduct of the tree-like structure is required: either
9212           order-preservation, or the ability to utilize node invariants. If
9213           memory-use is the major factor, an ordered-vector tree gives
9214           optimal results (albeit with high modificiation costs), and a
9215           list-based container gives reasonable results.</para>
9216
9217         </section>
9218
9219         <section xml:id="observations.associative.hash">
9220           <info><title>
9221             Hash-Based Containers
9222           </title></info>
9223
9224           <para>Hash-based containers are typically either collision
9225           chaining or probing. Collision-chaining
9226           containers are more flexible internally, and so offer better
9227           timing performance. Probing containers, if used for simple
9228           value-types, manage memory more efficiently (they perform far
9229           fewer allocation-related calls). In general, therefore, a
9230           collision-chaining table should be used. A probing container,
9231           conversely, might be used efficiently for operations such as
9232           eliminating duplicates in a sequence, or counting the number of
9233           occurrences within a sequence. Probing containers might be more
9234           useful also in multithreaded applications where each thread
9235           manipulates a hash-based container: in the standard, allocators have
9236           class-wise semantics (see [meyers96more] - Item 10); a
9237           probing container might incur less contention in this case.</para>
9238         </section>
9239
9240         <section xml:id="observations.associative.hash_policies">
9241           <info><title>
9242             Hash Policies
9243           </title></info>
9244
9245           <para>In hash-based containers, the range-hashing scheme seems to
9246           affect performance more than other considerations. In most
9247           settings, a mask-based scheme works well (or can be made to
9248           work well). If the key-distribution can be estimated a-priori,
9249           a simple hash function can produce nearly uniform hash-value
9250           distribution. In many other cases (e.g., text hashing,
9251           floating-point hashing), the hash function is powerful enough
9252           to generate hash values with good uniformity properties
9253           [knuth98sorting];
9254           a modulo-based scheme, taking into account all bits of the hash
9255           value, appears to overlap the hash function in its effort.</para>
9256           <para>The range-hashing scheme determines many of the other
9257           policies. A mask-based scheme works
9258           well with an exponential-size policy; for
9259           probing-based containers, it goes well with a linear-probe
9260           function.</para>
9261           <para>An orthogonal consideration is the trigger policy. This
9262           presents difficult tradeoffs. E.g., different load
9263           factors in a load-check trigger policy yield a
9264           space/amortized-cost tradeoff.</para>
9265         </section>
9266
9267         <section xml:id="observations.associative.branch">
9268           <info><title>
9269             Branch-Based Containers
9270           </title></info>
9271           <para>In general, there are several families of tree-based
9272           underlying data structures: balanced node-based trees
9273           (e.g., red-black or AVL trees), high-probability
9274           balanced node-based trees (e.g., random treaps or
9275           skip-lists), competitive node-based trees (e.g., splay
9276           trees), vector-based "trees", and tries. (Additionally, there
9277           are disk-residing or network-residing trees, such as B-Trees
9278           and their numerous variants. An interface for this would have
9279           to deal with the execution model and ACID guarantees; this is
9280           out of the scope of this library.) Following are some
9281           observations on their application to different settings.</para>
9282
9283           <para>Of the balanced node-based trees, this library includes a
9284           red-black tree, as does standard (in
9285           practice). This type of tree is the "workhorse" of tree-based
9286           containers: it offers both reasonable modification and
9287           reasonable lookup time. Unfortunately, this data structure
9288           stores a huge amount of metadata. Each node must contain,
9289           besides a value, three pointers and a boolean. This type might
9290           be avoided if space is at a premium [austern00noset].</para>
9291           <para>High-probability balanced node-based trees suffer the
9292           drawbacks of deterministic balanced trees. Although they are
9293           fascinating data structures, preliminary tests with them showed
9294           their performance was worse than red-black trees. The library
9295           does not contain any such trees, therefore.</para>
9296           <para>Competitive node-based trees have two drawbacks. They are
9297           usually somewhat unbalanced, and they perform a large number of
9298           comparisons. Balanced trees perform one comparison per each
9299           node they encounter on a search path; a splay tree performs two
9300           comparisons. If the keys are complex objects, e.g.,
9301           <classname>std::string</classname>, this can increase the running time.
9302           Conversely, such trees do well when there is much locality of
9303           reference. It is difficult to determine in which case to prefer
9304           such trees over balanced trees. This library includes a splay
9305           tree.</para>
9306           <para>Ordered-vector trees use very little space
9307           [austern00noset].
9308           They do not have any other advantages (at least in this
9309           implementation).</para>
9310           <para>Large-fan-out PATRICIA tries have excellent lookup
9311           performance, but they do so through maintaining, for each node,
9312           a miniature "hash-table". Their space efficiency is low, and
9313           their modification performance is bad. These tries might be
9314           used for semi-static settings, where order preservation is
9315           important. Alternatively, red-black trees cross-referenced with
9316           hash tables can be used. [okasaki98mereable]
9317           discusses small-fan-out PATRICIA tries for integers, but the
9318           cited results seem to indicate that the amortized cost of
9319           maintaining such trees is higher than that of balanced trees.
9320           Moderate-fan-out trees might be useful for sequences where each
9321           element has a limited number of choices, e.g., DNA
9322           strings.</para>
9323         </section>
9324
9325         <section xml:id="observations.associative.mapping_semantics">
9326           <info><title>
9327             Mapping-Semantics
9328           </title></info>
9329           <para>Different mapping semantics were discussed in the introduction and design sections.Here
9330           the focus will be on the case where a keys can be composed into
9331           primary keys and secondary keys. (In the case where some keys
9332           are completely identical, it is trivial that one should use an
9333           associative container mapping values to size types.) In this
9334           case there are (at least) five possibilities:</para>
9335           <orderedlist>
9336             <listitem><para>Use an associative container that allows equivalent-key
9337             values (such as <classname>std::multimap</classname>)</para></listitem>
9338             <listitem><para>Use a unique-key value associative container that maps
9339             each primary key to some complex associative container of
9340             secondary keys, say a tree-based or hash-based container.
9341             </para></listitem>
9342             <listitem><para>Use a unique-key value associative container that maps
9343             each primary key to some simple associative container of
9344             secondary keys, say a list-based container.</para></listitem>
9345             <listitem><para>Use a unique-key value associative container that maps
9346             each primary key to some non-associative container
9347             (e.g., <classname>std::vector</classname>)</para></listitem>
9348             <listitem><para>Use a unique-key value associative container that takes
9349             into account both primary and secondary keys.</para></listitem>
9350           </orderedlist>
9351           <para>Stated simply: there is a simple answer for this. (Excluding
9352           option 1, which should be avoided in all cases).</para>
9353           <para>If the expected ratio of secondary keys to primary keys is
9354           small, then 3 and 4 seem reasonable. Both types of secondary
9355           containers are relatively lightweight (in terms of memory use
9356           and construction time), and so creating an entire container
9357           object for each primary key is not too expensive. Option 4
9358           might be preferable to option 3 if changing the secondary key
9359           of some primary key is frequent - one cannot modify an
9360           associative container's key, and the only possibility,
9361           therefore, is erasing the secondary key and inserting another
9362           one instead; a non-associative container, conversely, can
9363           support in-place modification. The actual cost of erasing a
9364           secondary key and inserting another one depends also on the
9365           allocator used for secondary associative-containers (The tests
9366           above used the standard allocator, but in practice one might
9367           choose to use, e.g., [boost_pool]). Option 2 is
9368           definitely an overkill in this case. Option 1 loses out either
9369           immediately (when there is one secondary key per primary key)
9370           or almost immediately after that. Option 5 has the same
9371           drawbacks as option 2, but it has the additional drawback that
9372           finding all values whose primary key is equivalent to some key,
9373           might be linear in the total number of values stored (for
9374           example, if using a hash-based container).</para>
9375           <para>If the expected ratio of secondary keys to primary keys is
9376           large, then the answer is more complicated. It depends on the
9377           distribution of secondary keys to primary keys, the
9378           distribution of accesses according to primary keys, and the
9379           types of operations most frequent.</para>
9380           <para>To be more precise, assume there are m primary keys,
9381           primary key i is mapped to n<subscript>i</subscript>
9382           secondary keys, and each primary key is mapped, on average, to
9383           n secondary keys (i.e.,
9384           E(n<subscript>i</subscript>) = n).</para>
9385           <para>Suppose one wants to find a specific pair of primary and
9386           secondary keys. Using 1 with a tree based container
9387           (<classname>std::multimap</classname>), the expected cost is
9388           E(Θ(log(m) + n<subscript>i</subscript>)) = Θ(log(m) +
9389           n); using 1 with a hash-based container
9390           (<classname>std::tr1::unordered_multimap</classname>), the expected cost is
9391           Θ(n). Using 2 with a primary hash-based container
9392           and secondary hash-based containers, the expected cost is
9393           O(1); using 2 with a primary tree-based container and
9394           secondary tree-based containers, the expected cost is (using
9395           the Jensen inequality [motwani95random])
9396           E(O(log(m) + log(n<subscript>i</subscript>)) = O(log(m)) +
9397           E(O(log(n<subscript>i</subscript>)) = O(log(m)) + O(log(n)),
9398           assuming that primary keys are accessed equiprobably. 3 and 4
9399           are similar to 1, but with lower constants. Using 5 with a
9400           hash-based container, the expected cost is O(1); using 5
9401           with a tree based container, the cost is
9402           E(Θ(log(mn))) = Θ(log(m) +
9403           log(n)).</para>
9404           <para>Suppose one needs the values whose primary key matches some
9405           given key. Using 1 with a hash-based container, the expected
9406           cost is Θ(n), but the values will not be ordered
9407           by secondary keys (which may or may not be required); using 1
9408           with a tree-based container, the expected cost is
9409           Θ(log(m) + n), but with high constants; again the
9410           values will not be ordered by secondary keys. 2, 3, and 4 are
9411           similar to 1, but typically with lower constants (and,
9412           additionally, if one uses a tree-based container for secondary
9413           keys, they will be ordered). Using 5 with a hash-based
9414           container, the cost is Θ(mn).</para>
9415           <para>Suppose one wants to assign to a primary key all secondary
9416           keys assigned to a different primary key. Using 1 with a
9417           hash-based container, the expected cost is Θ(n),
9418           but with very high constants; using 1 with a tree-based
9419           container, the cost is Θ(nlog(mn)). Using 2, 3,
9420           and 4, the expected cost is Θ(n), but typically
9421           with far lower costs than 1. 5 is similar to 1.</para>
9422
9423         </section>
9424
9425       </section>
9426
9427
9428       <section xml:id="observations.priority_queue">
9429         <info><title>Priority_Queue</title></info>
9430
9431         <section xml:id="observations.priority_queue.complexity">
9432           <info><title>Complexity</title></info>
9433
9434           <para>The following table shows the complexities of the different
9435           underlying data structures in terms of orders of growth. It is
9436           interesting to note that this table implies something about the
9437           constants of the operations as well (see Amortized <function>push</function>
9438           and <function>pop</function> operations).</para>
9439
9440           <informaltable frame="all">
9441
9442             <tgroup cols="6" align="left" colsep="1" rowsep="1">
9443               <colspec colname="c1"/>
9444               <colspec colname="c2"/>
9445               <colspec colname="c3"/>
9446               <colspec colname="c4"/>
9447               <colspec colname="c5"/>
9448               <colspec colname="c6"/>
9449               <thead>
9450                 <row>
9451                   <entry></entry>
9452                   <entry><emphasis><function>push</function></emphasis></entry>
9453                   <entry><emphasis><function>pop</function></emphasis></entry>
9454                   <entry><emphasis><function>modify</function></emphasis></entry>
9455                   <entry><emphasis><function>erase</function></emphasis></entry>
9456                   <entry><emphasis><function>join</function></emphasis></entry>
9457                 </row>
9458               </thead>
9459
9460               <tbody>
9461
9462                 <row>
9463                   <entry>
9464                     <classname>std::priority_queue</classname>
9465                   </entry>
9466                   <entry>
9467                     Θ(n) worst
9468                     Θ(log(n)) amortized
9469                   </entry>
9470                   <entry>
9471                     Θ(log(n)) Worst
9472                   </entry>
9473                   <entry>
9474                     Θ(n log(n)) Worst
9475                     <subscript>[std note 1]</subscript>
9476                   </entry>
9477                   <entry>
9478                     Θ(n log(n))
9479                     <subscript>[std note 2]</subscript>
9480                   </entry>
9481                   <entry>
9482                     Θ(n log(n))
9483                     <subscript>[std note 1]</subscript>
9484                   </entry>
9485                 </row>
9486                 <row>
9487                   <entry>
9488                     <classname>priority_queue</classname>
9489                     &lt;<classname>Tag</classname> =
9490                     <classname>pairing_heap_tag</classname>&gt;
9491                   </entry>
9492                   <entry>
9493                     O(1)
9494                   </entry>
9495                   <entry>
9496                     Θ(n) worst
9497                     Θ(log(n)) amortized
9498                   </entry>
9499                   <entry>
9500                     Θ(n) worst
9501                     Θ(log(n)) amortized
9502                   </entry>
9503                   <entry>
9504                     Θ(n) worst
9505                     Θ(log(n)) amortized
9506                   </entry>
9507                   <entry>
9508                     O(1)
9509                   </entry>
9510                 </row>
9511                 <row>
9512                   <entry>
9513                     <classname>priority_queue</classname>
9514                     &lt;<classname>Tag</classname> =
9515                     <classname>binary_heap_tag</classname>&gt;
9516                   </entry>
9517                   <entry>
9518                     Θ(n) worst
9519                     Θ(log(n)) amortized
9520                   </entry>
9521                   <entry>
9522                     Θ(n) worst
9523                     Θ(log(n)) amortized
9524                   </entry>
9525                   <entry>
9526                     Θ(n)
9527                   </entry>
9528                   <entry>
9529                     Θ(n)
9530                   </entry>
9531                   <entry>
9532                     Θ(n)
9533                   </entry>
9534                 </row>
9535                 <row>
9536                   <entry>
9537                     <classname>priority_queue</classname>
9538                     &lt;<classname>Tag</classname> =
9539                     <classname>binomial_heap_tag</classname>&gt;
9540                   </entry>
9541                   <entry>
9542                     Θ(log(n)) worst
9543                     O(1) amortized
9544                   </entry>
9545                   <entry>
9546                     Θ(log(n))
9547                   </entry>
9548                   <entry>
9549                     Θ(log(n))
9550                   </entry>
9551                   <entry>
9552                     Θ(log(n))
9553                   </entry>
9554                   <entry>
9555                     Θ(log(n))
9556                   </entry>
9557                 </row>
9558                 <row>
9559                   <entry>
9560                     <classname>priority_queue</classname>
9561                     &lt;<classname>Tag</classname> =
9562                     <classname>rc_binomial_heap_tag</classname>&gt;
9563                   </entry>
9564                   <entry>
9565                     O(1)
9566                   </entry>
9567                   <entry>
9568                     Θ(log(n))
9569                   </entry>
9570                   <entry>
9571                     Θ(log(n))
9572                   </entry>
9573                   <entry>
9574                     Θ(log(n))
9575                   </entry>
9576                   <entry>
9577                     Θ(log(n))
9578                   </entry>
9579                 </row>
9580                 <row>
9581                   <entry>
9582                     <classname>priority_queue</classname>&lt;<classname>Tag</classname> =
9583                     <classname>thin_heap_tag</classname>&gt;
9584                   </entry>
9585                   <entry>
9586                     O(1)
9587                   </entry>
9588                   <entry>
9589                     Θ(n) worst
9590                     Θ(log(n)) amortized
9591                   </entry>
9592                   <entry>
9593                     Θ(log(n)) worst
9594                     O(1) amortized,
9595                     or Θ(log(n)) amortized
9596                     <subscript>[thin_heap_note]</subscript>
9597                   </entry>
9598                   <entry>
9599                     Θ(n) worst
9600                     Θ(log(n)) amortized
9601                   </entry>
9602                   <entry>
9603                     Θ(n)
9604                   </entry>
9605                 </row>
9606               </tbody>
9607             </tgroup>
9608
9609           </informaltable>
9610
9611           <para>[std note 1] This
9612           is not a property of the algorithm, but rather due to the fact
9613           that the standard's priority queue implementation does not support
9614           iterators (and consequently the ability to access a specific
9615           value inside it). If the priority queue is adapting an
9616           <classname>std::vector</classname>, then it is still possible to reduce this
9617           to Θ(n) by adapting over the standard's adapter and
9618           using the fact that <function>top</function> returns a reference to the
9619           first value; if, however, it is adapting an
9620           <classname>std::deque</classname>, then this is impossible.</para>
9621
9622           <para>[std note 2] As
9623           with [std note 1], this is not a
9624           property of the algorithm, but rather the standard's implementation.
9625           Again, if the priority queue is adapting an
9626           <classname>std::vector</classname> then it is possible to reduce this to
9627           Θ(n), but with a very high constant (one must call
9628           <function>std::make_heap</function> which is an expensive linear
9629           operation); if the priority queue is adapting an
9630           <classname>std::deque</classname>, then this is impossible.</para>
9631
9632           <para>[thin_heap_note] A thin heap has
9633           Θ(log(n)) worst case <function>modify</function> time
9634           always, but the amortized time depends on the nature of the
9635           operation: I) if the operation increases the key (in the sense
9636           of the priority queue's comparison functor), then the amortized
9637           time is O(1), but if II) it decreases it, then the
9638           amortized time is the same as the worst case time. Note that
9639           for most algorithms, I) is important and II) is not.</para>
9640
9641         </section>
9642
9643         <section xml:id="observations.priority_queue.amortized_ops">
9644           <info><title>
9645             Amortized <function>push</function>
9646             and <function>pop</function> operations
9647           </title></info>
9648
9649
9650           <para>In many cases, a priority queue is needed primarily for
9651           sequences of <function>push</function> and <function>pop</function> operations. All of
9652           the underlying data structures have the same amortized
9653           logarithmic complexity, but they differ in terms of
9654           constants.</para>
9655           <para>The table above shows that the different data structures are
9656           "constrained" in some respects. In general, if a data structure
9657           has lower worst-case complexity than another, then it will
9658           perform slower in the amortized sense. Thus, for example a
9659           redundant-counter binomial heap (<classname>priority_queue</classname> with
9660           <classname>Tag</classname> = <classname>rc_binomial_heap_tag</classname>)
9661           has lower worst-case <function>push</function> performance than a binomial
9662           heap (<classname>priority_queue</classname>
9663           with <classname>Tag</classname> = <classname>binomial_heap_tag</classname>),
9664           and so its amortized <function>push</function> performance is slower in
9665           terms of constants.</para>
9666           <para>As the table shows, the "least constrained" underlying
9667           data structures are binary heaps and pairing heaps.
9668           Consequently, it is not surprising that they perform best in
9669           terms of amortized constants.</para>
9670           <orderedlist>
9671             <listitem><para>Pairing heaps seem to perform best for non-primitive
9672             types (e.g., <classname>std::string</classname>s), as shown by
9673             Priority
9674             Queue Text <function>push</function> Timing Test and Priority
9675             Queue Text <function>push</function> and <function>pop</function> Timing
9676             Test</para></listitem>
9677             <listitem><para>binary heaps seem to perform best for primitive types
9678             (e.g., <type>int</type>s), as shown by Priority
9679             Queue Random Integer <function>push</function> Timing Test and
9680             Priority
9681             Queue Random Integer <function>push</function> and <function>pop</function> Timing
9682             Test.</para></listitem>
9683           </orderedlist>
9684
9685         </section>
9686
9687         <section xml:id="observations.priority_queue.graphs">
9688           <info><title>
9689             Graph Algorithms
9690           </title></info>
9691
9692           <para>In some graph algorithms, a decrease-key operation is
9693           required [clrs2001];
9694           this operation is identical to <function>modify</function> if a value is
9695           increased (in the sense of the priority queue's comparison
9696           functor). The table above and Priority Queue
9697           Text <function>modify</function> Up Timing Test show that a thin heap
9698           (<classname>priority_queue</classname> with
9699           <classname>Tag</classname> = <classname>thin_heap_tag</classname>)
9700           outperforms a pairing heap (<classname>priority_queue</classname> with
9701           <classname>Tag</classname> = <classname>Tag</classname> = <classname>pairing_heap_tag</classname>),
9702           but the rest of the tests show otherwise.</para>
9703
9704           <para>This makes it difficult to decide which implementation to use in
9705           this case. Dijkstra's shortest-path algorithm, for example, requires
9706           Θ(n) <function>push</function> and <function>pop</function> operations
9707           (in the number of vertices), but O(n<superscript>2</superscript>)
9708           <function>modify</function> operations, which can be in practice Θ(n)
9709           as well. It is difficult to find an a-priori characterization of
9710           graphs in which the actual number of <function>modify</function>
9711           operations will dwarf the number of <function>push</function> and
9712           <function>pop</function> operations.</para>
9713
9714         </section>
9715
9716       </section> <!-- priority_queue -->
9717
9718     </section>
9719
9720
9721   </section> <!-- performance -->
9722
9723 </section>