libstdc++-v3/doc/html/manual/policy_data_structures_design.html

   1 <?xml version="1.0" encoding="UTF-8" standalone="no"?>
   2 <!DOCTYPE html PUBLIC "-//W3C//DTD XHTML 1.0 Transitional//EN" "http://www.w3.org/TR/xhtml1/DTD/xhtml1-transitional.dtd"><html xmlns="http://www.w3.org/1999/xhtml"><head><meta http-equiv="Content-Type" content="text/html; charset=UTF-8" /><title>Design</title><meta name="generator" content="DocBook XSL-NS Stylesheets V1.78.1" /><meta name="keywords" content="ISO C++, policy, container, data, structure, associated, tree, trie, hash, metaprogramming" /><meta name="keywords" content="ISO C++, library" /><meta name="keywords" content="ISO C++, runtime, library" /><link rel="home" href="../index.html" title="The GNU C++ Library" /><link rel="up" href="policy_data_structures.html" title="Chapter 22. Policy-Based Data Structures" /><link rel="prev" href="policy_data_structures_using.html" title="Using" /><link rel="next" href="policy_based_data_structures_test.html" title="Testing" /></head><body><div class="navheader"><table width="100%" summary="Navigation header"><tr><th colspan="3" align="center">Design</th></tr><tr><td width="20%" align="left"><a accesskey="p" href="policy_data_structures_using.html">Prev</a> </td><th width="60%" align="center">Chapter 22. Policy-Based Data Structures</th><td width="20%" align="right"> <a accesskey="n" href="policy_based_data_structures_test.html">Next</a></td></tr></table><hr /></div><div class="section"><div class="titlepage"><div><div><h2 class="title" style="clear: both"><a id="containers.pbds.design"></a>Design</h2></div></div></div><p></p><div class="section"><div class="titlepage"><div><div><h3 class="title"><a id="pbds.design.concepts"></a>Concepts</h3></div></div></div><div class="section"><div class="titlepage"><div><div><h4 class="title"><a id="pbds.design.concepts.null_type"></a>Null Policy Classes</h4></div></div></div><p>
   3         Associative containers are typically parametrized by various
   4         policies. For example, a hash-based associative container is
   5         parametrized by a hash-functor, transforming each key into an
   6         non-negative numerical type. Each such value is then further mapped
   7         into a position within the table. The mapping of a key into a
   8         position within the table is therefore a two-step process.
   9       </p><p>
  10         In some cases, instantiations are redundant. For example, when the
  11         keys are integers, it is possible to use a redundant hash policy,
  12         which transforms each key into its value.
  13       </p><p>
  14         In some other cases, these policies are irrelevant.  For example, a
  15         hash-based associative container might transform keys into positions
  16         within a table by a different method than the two-step method
  17         described above. In such a case, the hash functor is simply
  18         irrelevant.
  19       </p><p>
  20         When a policy is either redundant or irrelevant, it can be replaced
  21         by <code class="classname">null_type</code>.
  22       </p><p>
  23         For example, a <span class="emphasis"><em>set</em></span> is an associative
  24         container with one of its template parameters (the one for the
  25         mapped type) replaced with <code class="classname">null_type</code>. Other
  26         places simplifications are made possible with this technique
  27         include node updates in tree and trie data structures, and hash
  28         and probe functions for hash data structures.
  29       </p></div><div class="section"><div class="titlepage"><div><div><h4 class="title"><a id="pbds.design.concepts.associative_semantics"></a>Map and Set Semantics</h4></div></div></div><div class="section"><div class="titlepage"><div><div><h5 class="title"><a id="concepts.associative_semantics.set_vs_map"></a>
  30             Distinguishing Between Maps and Sets
  31           </h5></div></div></div><p>
  32           Anyone familiar with the standard knows that there are four kinds
  33           of associative containers: maps, sets, multimaps, and
  34           multisets. The map datatype associates each key to
  35           some data.
  36         </p><p>
  37           Sets are associative containers that simply store keys -
  38           they do not map them to anything. In the standard, each map class
  39           has a corresponding set class. E.g.,
  40           <code class="classname">std::map&lt;int, char&gt;</code> maps each
  41           <code class="classname">int</code> to a <code class="classname">char</code>, but
  42           <code class="classname">std::set&lt;int, char&gt;</code> simply stores
  43           <code class="classname">int</code>s. In this library, however, there are no
  44           distinct classes for maps and sets. Instead, an associative
  45           container's <code class="classname">Mapped</code> template parameter is a policy: if
  46           it is instantiated by <code class="classname">null_type</code>, then it
  47           is a "set"; otherwise, it is a "map". E.g.,
  48         </p><pre class="programlisting">
  49           cc_hash_table&lt;int, char&gt;
  50         </pre><p>
  51           is a "map" mapping each <span class="type">int</span> value to a <span class="type">
  52           char</span>, but
  53         </p><pre class="programlisting">
  54           cc_hash_table&lt;int, null_type&gt;
  55         </pre><p>
  56           is a type that uniquely stores <span class="type">int</span> values.
  57         </p><p>Once the <code class="classname">Mapped</code> template parameter is instantiated
  58         by <code class="classname">null_type</code>, then
  59         the "set" acts very similarly to the standard's sets - it does not
  60         map each key to a distinct <code class="classname">null_type</code> object. Also,
  61         , the container's <span class="type">value_type</span> is essentially
  62         its <span class="type">key_type</span> - just as with the standard's sets
  63         .</p><p>
  64           The standard's multimaps and multisets allow, respectively,
  65           non-uniquely mapping keys and non-uniquely storing keys. As
  66           discussed, the
  67           reasons why this might be necessary are 1) that a key might be
  68           decomposed into a primary key and a secondary key, 2) that a
  69           key might appear more than once, or 3) any arbitrary
  70           combination of 1)s and 2)s. Correspondingly,
  71           one should use 1) "maps" mapping primary keys to secondary
  72           keys, 2) "maps" mapping keys to size types, or 3) any arbitrary
  73           combination of 1)s and 2)s. Thus, for example, an
  74           <code class="classname">std::multiset&lt;int&gt;</code> might be used to store
  75           multiple instances of integers, but using this library's
  76           containers, one might use
  77         </p><pre class="programlisting">
  78           tree&lt;int, size_t&gt;
  79         </pre><p>
  80           i.e., a <code class="classname">map</code> of <span class="type">int</span>s to
  81           <span class="type">size_t</span>s.
  82         </p><p>
  83           These "multimaps" and "multisets" might be confusing to
  84           anyone familiar with the standard's <code class="classname">std::multimap</code> and
  85           <code class="classname">std::multiset</code>, because there is no clear
  86           correspondence between the two. For example, in some cases
  87           where one uses <code class="classname">std::multiset</code> in the standard, one might use
  88           in this library a "multimap" of "multisets" - i.e., a
  89           container that maps primary keys each to an associative
  90           container that maps each secondary key to the number of times
  91           it occurs.
  92         </p><p>
  93           When one uses a "multimap," one should choose with care the
  94           type of container used for secondary keys.
  95         </p></div><div class="section"><div class="titlepage"><div><div><h5 class="title"><a id="concepts.associative_semantics.multi"></a>Alternatives to <code class="classname">std::multiset</code> and <code class="classname">std::multimap</code></h5></div></div></div><p>
  96           Brace onself: this library does not contain containers like
  97           <code class="classname">std::multimap</code> or
  98           <code class="classname">std::multiset</code>. Instead, these data
  99           structures can be synthesized via manipulation of the
 100           <code class="classname">Mapped</code> template parameter.
 101         </p><p>
 102           One maps the unique part of a key - the primary key, into an
 103           associative-container of the (originally) non-unique parts of
 104           the key - the secondary key. A primary associative-container
 105           is an associative container of primary keys; a secondary
 106           associative-container is an associative container of
 107           secondary keys.
 108         </p><p>
 109           Stepping back a bit, and starting in from the beginning.
 110         </p><p>
 111           Maps (or sets) allow mapping (or storing) unique-key values.
 112           The standard library also supplies associative containers which
 113           map (or store) multiple values with equivalent keys:
 114           <code class="classname">std::multimap</code>, <code class="classname">std::multiset</code>,
 115           <code class="classname">std::tr1::unordered_multimap</code>, and
 116           <code class="classname">unordered_multiset</code>. We first discuss how these might
 117           be used, then why we think it is best to avoid them.
 118         </p><p>
 119           Suppose one builds a simple bank-account application that
 120           records for each client (identified by an <code class="classname">std::string</code>)
 121           and account-id (marked by an <span class="type">unsigned long</span>) -
 122           the balance in the account (described by a
 123           <span class="type">float</span>). Suppose further that ordering this
 124           information is not useful, so a hash-based container is
 125           preferable to a tree based container. Then one can use
 126         </p><pre class="programlisting">
 127           std::tr1::unordered_map&lt;std::pair&lt;std::string, unsigned long&gt;, float, ...&gt;
 128         </pre><p>
 129           which hashes every combination of client and account-id. This
 130           might work well, except for the fact that it is now impossible
 131           to efficiently list all of the accounts of a specific client
 132           (this would practically require iterating over all
 133           entries). Instead, one can use
 134         </p><pre class="programlisting">
 135           std::tr1::unordered_multimap&lt;std::pair&lt;std::string, unsigned long&gt;, float, ...&gt;
 136         </pre><p>
 137           which hashes every client, and decides equivalence based on
 138           client only. This will ensure that all accounts belonging to a
 139           specific user are stored consecutively.
 140         </p><p>
 141           Also, suppose one wants an integers' priority queue
 142           (a container that supports <code class="function">push</code>,
 143           <code class="function">pop</code>, and <code class="function">top</code> operations, the last of which
 144           returns the largest <span class="type">int</span>) that also supports
 145           operations such as <code class="function">find</code> and <code class="function">lower_bound</code>. A
 146           reasonable solution is to build an adapter over
 147           <code class="classname">std::set&lt;int&gt;</code>. In this adapter,
 148           <code class="function">push</code> will just call the tree-based
 149           associative container's <code class="function">insert</code> method; <code class="function">pop</code>
 150           will call its <code class="function">end</code> method, and use it to return the
 151           preceding element (which must be the largest). Then this might
 152           work well, except that the container object cannot hold
 153           multiple instances of the same integer (<code class="function">push(4)</code>,
 154           will be a no-op if <code class="constant">4</code> is already in the
 155           container object). If multiple keys are necessary, then one
 156           might build the adapter over an
 157           <code class="classname">std::multiset&lt;int&gt;</code>.
 158         </p><p>
 159           The standard library's non-unique-mapping containers are useful
 160           when (1) a key can be decomposed in to a primary key and a
 161           secondary key, (2) a key is needed multiple times, or (3) any
 162           combination of (1) and (2).
 163         </p><p>
 164           The graphic below shows how the standard library's container
 165           design works internally; in this figure nodes shaded equally
 166           represent equivalent-key values. Equivalent keys are stored
 167           consecutively using the properties of the underlying data
 168           structure: binary search trees (label A) store equivalent-key
 169           values consecutively (in the sense of an in-order walk)
 170           naturally; collision-chaining hash tables (label B) store
 171           equivalent-key values in the same bucket, the bucket can be
 172           arranged so that equivalent-key values are consecutive.
 173         </p><div class="figure"><a id="id-1.3.5.9.4.3.3.3.14"></a><p class="title"><strong>Figure 22.8. Non-unique Mapping Standard Containers</strong></p><div class="figure-contents"><div class="mediaobject" align="center"><img src="../images/pbds_embedded_lists_1.png" align="middle" alt="Non-unique Mapping Standard Containers" /></div></div></div><br class="figure-break" /><p>
 174           Put differently, the standards' non-unique mapping
 175           associative-containers are associative containers that map
 176           primary keys to linked lists that are embedded into the
 177           container. The graphic below shows again the two
 178           containers from the first graphic above, this time with
 179           the embedded linked lists of the grayed nodes marked
 180           explicitly.
 181         </p><div class="figure"><a id="fig.pbds_embedded_lists_2"></a><p class="title"><strong>Figure 22.9.
 182             Effect of embedded lists in
 183             <code class="classname">std::multimap</code>
 184           </strong></p><div class="figure-contents"><div class="mediaobject" align="center"><img src="../images/pbds_embedded_lists_2.png" align="middle" alt="Effect of embedded lists in std::multimap" /></div></div></div><br class="figure-break" /><p>
 185           These embedded linked lists have several disadvantages.
 186         </p><div class="orderedlist"><ol class="orderedlist" type="1"><li class="listitem"><p>
 187               The underlying data structure embeds the linked lists
 188               according to its own consideration, which means that the
 189               search path for a value might include several different
 190               equivalent-key values. For example, the search path for the
 191               the black node in either of the first graphic, labels A or B,
 192               includes more than a single gray node.
 193             </p></li><li class="listitem"><p>
 194               The links of the linked lists are the underlying data
 195               structures' nodes, which typically are quite structured.  In
 196               the case of tree-based containers (the grapic above, label
 197               B), each "link" is actually a node with three pointers (one
 198               to a parent and two to children), and a
 199               relatively-complicated iteration algorithm. The linked
 200               lists, therefore, can take up quite a lot of memory, and
 201               iterating over all values equal to a given key (through the
 202               return value of the standard
 203               library's <code class="function">equal_range</code>) can be
 204               expensive.
 205             </p></li><li class="listitem"><p>
 206               The primary key is stored multiply; this uses more memory.
 207             </p></li><li class="listitem"><p>
 208               Finally, the interface of this design excludes several
 209               useful underlying data structures. Of all the unordered
 210               self-organizing data structures, practically only
 211               collision-chaining hash tables can (efficiently) guarantee
 212               that equivalent-key values are stored consecutively.
 213             </p></li></ol></div><p>
 214           The above reasons hold even when the ratio of secondary keys to
 215           primary keys (or average number of identical keys) is small, but
 216           when it is large, there are more severe problems:
 217         </p><div class="orderedlist"><ol class="orderedlist" type="1"><li class="listitem"><p>
 218               The underlying data structures order the links inside each
 219               embedded linked-lists according to their internal
 220               considerations, which effectively means that each of the
 221               links is unordered. Irrespective of the underlying data
 222               structure, searching for a specific value can degrade to
 223               linear complexity.
 224             </p></li><li class="listitem"><p>
 225               Similarly to the above point, it is impossible to apply
 226               to the secondary keys considerations that apply to primary
 227               keys. For example, it is not possible to maintain secondary
 228               keys by sorted order.
 229             </p></li><li class="listitem"><p>
 230               While the interface "understands" that all equivalent-key
 231               values constitute a distinct list (through
 232               <code class="function">equal_range</code>), the underlying data
 233               structure typically does not. This means that operations such
 234               as erasing from a tree-based container all values whose keys
 235               are equivalent to a a given key can be super-linear in the
 236               size of the tree; this is also true also for several other
 237               operations that target a specific list.
 238             </p></li></ol></div><p>
 239           In this library, all associative containers map
 240           (or store) unique-key values. One can (1) map primary keys to
 241           secondary associative-containers (containers of
 242           secondary keys) or non-associative containers (2) map identical
 243           keys to a size-type representing the number of times they
 244           occur, or (3) any combination of (1) and (2). Instead of
 245           allowing multiple equivalent-key values, this library
 246           supplies associative containers based on underlying
 247           data structures that are suitable as secondary
 248           associative-containers.
 249         </p><p>
 250           In the figure below, labels A and B show the equivalent
 251           underlying data structures in this library, as mapped to the
 252           first graphic above. Labels A and B, respectively. Each shaded
 253           box represents some size-type or secondary
 254           associative-container.
 255         </p><div class="figure"><a id="id-1.3.5.9.4.3.3.3.23"></a><p class="title"><strong>Figure 22.10. Non-unique Mapping Containers</strong></p><div class="figure-contents"><div class="mediaobject" align="center"><img src="../images/pbds_embedded_lists_3.png" align="middle" alt="Non-unique Mapping Containers" /></div></div></div><br class="figure-break" /><p>
 256           In the first example above, then, one would use an associative
 257           container mapping each user to an associative container which
 258           maps each application id to a start time (see
 259           <code class="filename">example/basic_multimap.cc</code>); in the second
 260           example, one would use an associative container mapping
 261           each <code class="classname">int</code> to some size-type indicating the
 262           number of times it logically occurs
 263           (see <code class="filename">example/basic_multiset.cc</code>.
 264         </p><p>
 265           See the discussion in list-based container types for containers
 266           especially suited as secondary associative-containers.
 267         </p></div></div><div class="section"><div class="titlepage"><div><div><h4 class="title"><a id="pbds.design.concepts.iterator_semantics"></a>Iterator Semantics</h4></div></div></div><div class="section"><div class="titlepage"><div><div><h5 class="title"><a id="concepts.iterator_semantics.point_and_range"></a>Point and Range Iterators</h5></div></div></div><p>
 268           Iterator concepts are bifurcated in this design, and are
 269           comprised of point-type and range-type iteration.
 270         </p><p>
 271           A point-type iterator is an iterator that refers to a specific
 272           element as returned through an
 273           associative-container's <code class="function">find</code> method.
 274         </p><p>
 275           A range-type iterator is an iterator that is used to go over a
 276           sequence of elements, as returned by a container's
 277           <code class="function">find</code> method.
 278         </p><p>
 279           A point-type method is a method that
 280           returns a point-type iterator; a range-type method is a method
 281           that returns a range-type iterator.
 282         </p><p>For most containers, these types are synonymous; for
 283         self-organizing containers, such as hash-based containers or
 284         priority queues, these are inherently different (in any
 285         implementation, including that of C++ standard library
 286         components), but in this design, it is made explicit. They are
 287         distinct types.
 288         </p></div><div class="section"><div class="titlepage"><div><div><h5 class="title"><a id="concepts.iterator_semantics.both"></a>Distinguishing Point and Range Iterators</h5></div></div></div><p>When using this library, is necessary to differentiate
 289         between two types of methods and iterators: point-type methods and
 290         iterators, and range-type methods and iterators. Each associative
 291         container's interface includes the methods:</p><pre class="programlisting">
 292           point_const_iterator
 293           find(const_key_reference r_key) const;
 294
 295           point_iterator
 296           find(const_key_reference r_key);
 297
 298           std::pair&lt;point_iterator,bool&gt;
 299           insert(const_reference r_val);
 300         </pre><p>The relationship between these iterator types varies between
 301         container types. The figure below
 302         shows the most general invariant between point-type and
 303         range-type iterators: In <span class="emphasis"><em>A</em></span> <code class="literal">iterator</code>, can
 304         always be converted to <code class="literal">point_iterator</code>. In <span class="emphasis"><em>B</em></span>
 305         shows invariants for order-preserving containers: point-type
 306         iterators are synonymous with range-type iterators.
 307         Orthogonally,  <span class="emphasis"><em>C</em></span>shows invariants for "set"
 308         containers: iterators are synonymous with const iterators.</p><div class="figure"><a id="id-1.3.5.9.4.3.4.3.5"></a><p class="title"><strong>Figure 22.11. Point Iterator Hierarchy</strong></p><div class="figure-contents"><div class="mediaobject" align="center"><img src="../images/pbds_point_iterator_hierarchy.png" align="middle" alt="Point Iterator Hierarchy" /></div></div></div><br class="figure-break" /><p>Note that point-type iterators in self-organizing containers
 309         (hash-based associative containers) lack movement
 310         operators, such as <code class="literal">operator++</code> - in fact, this
 311         is the reason why this library differentiates from the standard C++ librarys
 312         design on this point.</p><p>Typically, one can determine an iterator's movement
 313         capabilities using
 314         <code class="literal">std::iterator_traits&lt;It&gt;iterator_category</code>,
 315         which is a <code class="literal">struct</code> indicating the iterator's
 316         movement capabilities. Unfortunately, none of the standard predefined
 317         categories reflect a pointer's <span class="emphasis"><em>not</em></span> having any
 318         movement capabilities whatsoever. Consequently,
 319         <code class="literal">pb_ds</code> adds a type
 320         <code class="literal">trivial_iterator_tag</code> (whose name is taken from
 321         a concept in C++ standardese, which is the category of iterators
 322         with no movement capabilities.) All other standard C++ library
 323         tags, such as <code class="literal">forward_iterator_tag</code> retain their
 324         common use.</p></div><div class="section"><div class="titlepage"><div><div><h5 class="title"><a id="pbds.design.concepts.invalidation"></a>Invalidation Guarantees</h5></div></div></div><p>
 325           If one manipulates a container object, then iterators previously
 326           obtained from it can be invalidated. In some cases a
 327           previously-obtained iterator cannot be de-referenced; in other cases,
 328           the iterator's next or previous element might have changed
 329           unpredictably. This corresponds exactly to the question whether a
 330           point-type or range-type iterator (see previous concept) is valid or
 331           not. In this design, one can query a container (in compile time) about
 332           its invalidation guarantees.
 333         </p><p>
 334           Given three different types of associative containers, a modifying
 335           operation (in that example, <code class="function">erase</code>) invalidated
 336           iterators in three different ways: the iterator of one container
 337           remained completely valid - it could be de-referenced and
 338           incremented; the iterator of a different container could not even be
 339           de-referenced; the iterator of the third container could be
 340           de-referenced, but its "next" iterator changed unpredictably.
 341         </p><p>
 342           Distinguishing between find and range types allows fine-grained
 343           invalidation guarantees, because these questions correspond exactly
 344           to the question of whether point-type iterators and range-type
 345           iterators are valid. The graphic below shows tags corresponding to
 346           different types of invalidation guarantees.
 347         </p><div class="figure"><a id="id-1.3.5.9.4.3.4.4.5"></a><p class="title"><strong>Figure 22.12. Invalidation Guarantee Tags Hierarchy</strong></p><div class="figure-contents"><div class="mediaobject" align="center"><img src="../images/pbds_invalidation_tag_hierarchy.png" align="middle" alt="Invalidation Guarantee Tags Hierarchy" /></div></div></div><br class="figure-break" /><div class="itemizedlist"><ul class="itemizedlist" style="list-style-type: disc; "><li class="listitem"><p>
 348               <code class="classname">basic_invalidation_guarantee</code>
 349               corresponds to a basic guarantee that a point-type iterator,
 350               a found pointer, or a found reference, remains valid as long
 351               as the container object is not modified.
 352             </p></li><li class="listitem"><p>
 353               <code class="classname">point_invalidation_guarantee</code>
 354               corresponds to a guarantee that a point-type iterator, a
 355               found pointer, or a found reference, remains valid even if
 356               the container object is modified.
 357             </p></li><li class="listitem"><p>
 358               <code class="classname">range_invalidation_guarantee</code>
 359               corresponds to a guarantee that a range-type iterator remains
 360               valid even if the container object is modified.
 361             </p></li></ul></div><p>To find the invalidation guarantee of a
 362         container, one can use</p><pre class="programlisting">
 363           typename container_traits&lt;Cntnr&gt;::invalidation_guarantee
 364         </pre><p>Note that this hierarchy corresponds to the logic it
 365         represents: if a container has range-invalidation guarantees,
 366         then it must also have find invalidation guarantees;
 367         correspondingly, its invalidation guarantee (in this case
 368         <code class="classname">range_invalidation_guarantee</code>)
 369         can be cast to its base class (in this case <code class="classname">point_invalidation_guarantee</code>).
 370         This means that this this hierarchy can be used easily using
 371         standard metaprogramming techniques, by specializing on the
 372         type of <code class="literal">invalidation_guarantee</code>.</p><p>
 373           These types of problems were addressed, in a more general
 374           setting, in <a class="xref" href="policy_data_structures.html#biblio.meyers96more" title="More Effective C++: 35 New Ways to Improve Your Programs and Designs">[biblio.meyers96more]</a> - Item 2. In
 375           our opinion, an invalidation-guarantee hierarchy would solve
 376           these problems in all container types - not just associative
 377           containers.
 378         </p></div></div><div class="section"><div class="titlepage"><div><div><h4 class="title"><a id="pbds.design.concepts.genericity"></a>Genericity</h4></div></div></div><p>
 379         The design attempts to address the following problem of
 380         data-structure genericity. When writing a function manipulating
 381         a generic container object, what is the behavior of the object?
 382         Suppose one writes
 383       </p><pre class="programlisting">
 384         template&lt;typename Cntnr&gt;
 385         void
 386         some_op_sequence(Cntnr &amp;r_container)
 387         {
 388         ...
 389         }
 390       </pre><p>
 391         then one needs to address the following questions in the body
 392         of <code class="function">some_op_sequence</code>:
 393       </p><div class="itemizedlist"><ul class="itemizedlist" style="list-style-type: disc; "><li class="listitem"><p>
 394             Which types and methods does <code class="literal">Cntnr</code> support?
 395             Containers based on hash tables can be queries for the
 396             hash-functor type and object; this is meaningless for tree-based
 397             containers. Containers based on trees can be split, joined, or
 398             can erase iterators and return the following iterator; this
 399             cannot be done by hash-based containers.
 400           </p></li><li class="listitem"><p>
 401             What are the exception and invalidation guarantees
 402             of <code class="literal">Cntnr</code>? A container based on a probing
 403             hash-table invalidates all iterators when it is modified; this
 404             is not the case for containers based on node-based
 405             trees. Containers based on a node-based tree can be split or
 406             joined without exceptions; this is not the case for containers
 407             based on vector-based trees.
 408           </p></li><li class="listitem"><p>
 409             How does the container maintain its elements? Tree-based and
 410             Trie-based containers store elements by key order; others,
 411             typically, do not. A container based on a splay trees or lists
 412             with update policies "cache" "frequently accessed" elements;
 413             containers based on most other underlying data structures do
 414             not.
 415           </p></li><li class="listitem"><p>
 416             How does one query a container about characteristics and
 417             capabilities? What is the relationship between two different
 418             data structures, if anything?
 419           </p></li></ul></div><p>The remainder of this section explains these issues in
 420       detail.</p><div class="section"><div class="titlepage"><div><div><h5 class="title"><a id="concepts.genericity.tag"></a>Tag</h5></div></div></div><p>
 421           Tags are very useful for manipulating generic types. For example, if
 422           <code class="literal">It</code> is an iterator class, then <code class="literal">typename
 423           It::iterator_category</code> or <code class="literal">typename
 424           std::iterator_traits&lt;It&gt;::iterator_category</code> will
 425           yield its category, and <code class="literal">typename
 426           std::iterator_traits&lt;It&gt;::value_type</code> will yield its
 427           value type.
 428         </p><p>
 429           This library contains a container tag hierarchy corresponding to the
 430           diagram below.
 431         </p><div class="figure"><a id="id-1.3.5.9.4.3.5.7.4"></a><p class="title"><strong>Figure 22.13. Container Tag Hierarchy</strong></p><div class="figure-contents"><div class="mediaobject" align="center"><img src="../images/pbds_container_tag_hierarchy.png" align="middle" alt="Container Tag Hierarchy" /></div></div></div><br class="figure-break" /><p>
 432           Given any container <span class="type">Cntnr</span>, the tag of
 433           the underlying data structure can be found via <code class="literal">typename
 434           Cntnr::container_category</code>.
 435         </p></div><div class="section"><div class="titlepage"><div><div><h5 class="title"><a id="concepts.genericity.traits"></a>Traits</h5></div></div></div><p></p><p>Additionally, a traits mechanism can be used to query a
 436         container type for its attributes. Given any container
 437         <code class="literal">Cntnr</code>, then <code class="literal">&lt;Cntnr&gt;</code>
 438         is a traits class identifying the properties of the
 439         container.</p><p>To find if a container can throw when a key is erased (which
 440         is true for vector-based trees, for example), one can
 441         use
 442         </p><pre class="programlisting">container_traits&lt;Cntnr&gt;::erase_can_throw</pre><p>
 443           Some of the definitions in <code class="classname">container_traits</code>
 444           are dependent on other
 445           definitions. If <code class="classname">container_traits&lt;Cntnr&gt;::order_preserving</code>
 446           is <code class="constant">true</code> (which is the case for containers
 447           based on trees and tries), then the container can be split or
 448           joined; in this
 449           case, <code class="classname">container_traits&lt;Cntnr&gt;::split_join_can_throw</code>
 450           indicates whether splits or joins can throw exceptions (which is
 451           true for vector-based trees);
 452           otherwise <code class="classname">container_traits&lt;Cntnr&gt;::split_join_can_throw</code>
 453           will yield a compilation error. (This is somewhat similar to a
 454           compile-time version of the COM model).
 455         </p></div></div></div><div class="section"><div class="titlepage"><div><div><h3 class="title"><a id="pbds.design.container"></a>By Container</h3></div></div></div><div class="section"><div class="titlepage"><div><div><h4 class="title"><a id="pbds.design.container.hash"></a>hash</h4></div></div></div><div class="section"><div class="titlepage"><div><div><h5 class="title"><a id="container.hash.interface"></a>Interface</h5></div></div></div><p>
 456           The collision-chaining hash-based container has the
 457         following declaration.</p><pre class="programlisting">
 458           template&lt;
 459           typename Key,
 460           typename Mapped,
 461           typename Hash_Fn = std::hash&lt;Key&gt;,
 462           typename Eq_Fn = std::equal_to&lt;Key&gt;,
 463           typename Comb_Hash_Fn =  direct_mask_range_hashing&lt;&gt;
 464           typename Resize_Policy = default explained below.
 465           bool Store_Hash = false,
 466           typename Allocator = std::allocator&lt;char&gt; &gt;
 467           class cc_hash_table;
 468         </pre><p>The parameters have the following meaning:</p><div class="orderedlist"><ol class="orderedlist" type="1"><li class="listitem"><p><code class="classname">Key</code> is the key type.</p></li><li class="listitem"><p><code class="classname">Mapped</code> is the mapped-policy.</p></li><li class="listitem"><p><code class="classname">Hash_Fn</code> is a key hashing functor.</p></li><li class="listitem"><p><code class="classname">Eq_Fn</code> is a key equivalence functor.</p></li><li class="listitem"><p><code class="classname">Comb_Hash_Fn</code> is a range-hashing_functor;
 469           it describes how to translate hash values into positions
 470           within the table. </p></li><li class="listitem"><p><code class="classname">Resize_Policy</code> describes how a container object
 471           should change its internal size. </p></li><li class="listitem"><p><code class="classname">Store_Hash</code> indicates whether the hash value
 472           should be stored with each entry. </p></li><li class="listitem"><p><code class="classname">Allocator</code> is an allocator
 473           type.</p></li></ol></div><p>The probing hash-based container has the following
 474         declaration.</p><pre class="programlisting">
 475           template&lt;
 476           typename Key,
 477           typename Mapped,
 478           typename Hash_Fn = std::hash&lt;Key&gt;,
 479           typename Eq_Fn = std::equal_to&lt;Key&gt;,
 480           typename Comb_Probe_Fn = direct_mask_range_hashing&lt;&gt;
 481           typename Probe_Fn = default explained below.
 482           typename Resize_Policy = default explained below.
 483           bool Store_Hash = false,
 484           typename Allocator =  std::allocator&lt;char&gt; &gt;
 485           class gp_hash_table;
 486         </pre><p>The parameters are identical to those of the
 487         collision-chaining container, except for the following.</p><div class="orderedlist"><ol class="orderedlist" type="1"><li class="listitem"><p><code class="classname">Comb_Probe_Fn</code> describes how to transform a probe
 488           sequence into a sequence of positions within the table.</p></li><li class="listitem"><p><code class="classname">Probe_Fn</code> describes a probe sequence policy.</p></li></ol></div><p>Some of the default template values depend on the values of
 489         other parameters, and are explained below.</p></div><div class="section"><div class="titlepage"><div><div><h5 class="title"><a id="container.hash.details"></a>Details</h5></div></div></div><div class="section"><div class="titlepage"><div><div><h6 class="title"><a id="container.hash.details.hash_policies"></a>Hash Policies</h6></div></div></div><div class="section"><div class="titlepage"><div><div><h6 class="title"><a id="details.hash_policies.general"></a>General</h6></div></div></div><p>Following is an explanation of some functions which hashing
 490             involves. The graphic below illustrates the discussion.</p><div class="figure"><a id="id-1.3.5.9.4.4.2.3.2.2.3"></a><p class="title"><strong>Figure 22.14. Hash functions, ranged-hash functions, and
 491               range-hashing functions</strong></p><div class="figure-contents"><div class="mediaobject" align="center"><img src="../images/pbds_hash_ranged_hash_range_hashing_fns.png" align="middle" alt="Hash functions, ranged-hash functions, and range-hashing functions" /></div></div></div><br class="figure-break" /><p>Let U be a domain (e.g., the integers, or the
 492             strings of 3 characters). A hash-table algorithm needs to map
 493             elements of U "uniformly" into the range [0,..., m -
 494             1] (where m is a non-negative integral value, and
 495             is, in general, time varying). I.e., the algorithm needs
 496             a ranged-hash function</p><p>
 497               f : U × Z<sub>+</sub> → Z<sub>+</sub>
 498             </p><p>such that for any u in U ,</p><p>0 ≤ f(u, m) ≤ m - 1</p><p>and which has "good uniformity" properties (say
 499             <a class="xref" href="policy_data_structures.html#biblio.knuth98sorting" title="The Art of Computer Programming - Sorting and Searching">[biblio.knuth98sorting]</a>.)
 500             One
 501             common solution is to use the composition of the hash
 502             function</p><p>h : U → Z<sub>+</sub> ,</p><p>which maps elements of U into the non-negative
 503             integrals, and</p><p>g : Z<sub>+</sub> × Z<sub>+</sub> →
 504             Z<sub>+</sub>,</p><p>which maps a non-negative hash value, and a non-negative
 505             range upper-bound into a non-negative integral in the range
 506             between 0 (inclusive) and the range upper bound (exclusive),
 507             i.e., for any r in Z<sub>+</sub>,</p><p>0 ≤ g(r, m) ≤ m - 1</p><p>The resulting ranged-hash function, is</p><div class="equation"><a id="id-1.3.5.9.4.4.2.3.2.2.15"></a><p class="title"><strong>Equation 22.1. Ranged Hash Function</strong></p><div class="equation-contents"><span class="mathphrase">
 508                 f(u , m) = g(h(u), m)
 509               </span></div></div><br class="equation-break" /><p>From the above, it is obvious that given g and
 510             h, f can always be composed (however the converse
 511             is not true). The standard's hash-based containers allow specifying
 512             a hash function, and use a hard-wired range-hashing function;
 513             the ranged-hash function is implicitly composed.</p><p>The above describes the case where a key is to be mapped
 514             into a single position within a hash table, e.g.,
 515             in a collision-chaining table. In other cases, a key is to be
 516             mapped into a sequence of positions within a table,
 517             e.g., in a probing table. Similar terms apply in this
 518             case: the table requires a ranged probe function,
 519             mapping a key into a sequence of positions withing the table.
 520             This is typically achieved by composing a hash function
 521             mapping the key into a non-negative integral type, a
 522             probe function transforming the hash value into a
 523             sequence of hash values, and a range-hashing function
 524             transforming the sequence of hash values into a sequence of
 525             positions.</p></div><div class="section"><div class="titlepage"><div><div><h6 class="title"><a id="details.hash_policies.range"></a>Range Hashing</h6></div></div></div><p>Some common choices for range-hashing functions are the
 526             division, multiplication, and middle-square methods (<a class="xref" href="policy_data_structures.html#biblio.knuth98sorting" title="The Art of Computer Programming - Sorting and Searching">[biblio.knuth98sorting]</a>), defined
 527             as</p><div class="equation"><a id="id-1.3.5.9.4.4.2.3.2.3.3"></a><p class="title"><strong>Equation 22.2. Range-Hashing, Division Method</strong></p><div class="equation-contents"><span class="mathphrase">
 528                 g(r, m) = r mod m
 529               </span></div></div><br class="equation-break" /><p>g(r, m) = ⌈ u/v ( a r mod v ) ⌉</p><p>and</p><p>g(r, m) = ⌈ u/v ( r<sup>2</sup> mod v ) ⌉</p><p>respectively, for some positive integrals u and
 530             v (typically powers of 2), and some a. Each of
 531             these range-hashing functions works best for some different
 532             setting.</p><p>The division method (see above) is a
 533             very common choice. However, even this single method can be
 534             implemented in two very different ways. It is possible to
 535             implement using the low
 536             level % (modulo) operation (for any m), or the
 537             low level &amp; (bit-mask) operation (for the case where
 538             m is a power of 2), i.e.,</p><div class="equation"><a id="id-1.3.5.9.4.4.2.3.2.3.9"></a><p class="title"><strong>Equation 22.3. Division via Prime Modulo</strong></p><div class="equation-contents"><span class="mathphrase">
 539                 g(r, m) = r % m
 540               </span></div></div><br class="equation-break" /><p>and</p><div class="equation"><a id="id-1.3.5.9.4.4.2.3.2.3.11"></a><p class="title"><strong>Equation 22.4. Division via Bit Mask</strong></p><div class="equation-contents"><span class="mathphrase">
 541                 g(r, m) = r &amp; m - 1, (with m =
 542                 2<sup>k</sup> for some k)
 543               </span></div></div><br class="equation-break" /><p>respectively.</p><p>The % (modulo) implementation has the advantage that for
 544             m a prime far from a power of 2, g(r, m) is
 545             affected by all the bits of r (minimizing the chance of
 546             collision). It has the disadvantage of using the costly modulo
 547             operation. This method is hard-wired into SGI's implementation
 548             .</p><p>The &amp; (bit-mask) implementation has the advantage of
 549             relying on the fast bit-wise and operation. It has the
 550             disadvantage that for g(r, m) is affected only by the
 551             low order bits of r. This method is hard-wired into
 552             Dinkumware's implementation.</p></div><div class="section"><div class="titlepage"><div><div><h6 class="title"><a id="details.hash_policies.ranged"></a>Ranged Hash</h6></div></div></div><p>In cases it is beneficial to allow the
 553             client to directly specify a ranged-hash hash function. It is
 554             true, that the writer of the ranged-hash function cannot rely
 555             on the values of m having specific numerical properties
 556             suitable for hashing (in the sense used in <a class="xref" href="policy_data_structures.html#biblio.knuth98sorting" title="The Art of Computer Programming - Sorting and Searching">[biblio.knuth98sorting]</a>), since
 557             the values of m are determined by a resize policy with
 558             possibly orthogonal considerations.</p><p>There are two cases where a ranged-hash function can be
 559             superior. The firs is when using perfect hashing: the
 560             second is when the values of m can be used to estimate
 561             the "general" number of distinct values required. This is
 562             described in the following.</p><p>Let</p><p>
 563               s = [ s<sub>0</sub>,..., s<sub>t - 1</sub>]
 564             </p><p>be a string of t characters, each of which is from
 565             domain S. Consider the following ranged-hash
 566             function:</p><div class="equation"><a id="id-1.3.5.9.4.4.2.3.2.4.7"></a><p class="title"><strong>Equation 22.5.
 567                 A Standard String Hash Function
 568               </strong></p><div class="equation-contents"><span class="mathphrase">
 569                 f<sub>1</sub>(s, m) = ∑ <sub>i =
 570                 0</sub><sup>t - 1</sup> s<sub>i</sub> a<sup>i</sup> mod m
 571               </span></div></div><br class="equation-break" /><p>where a is some non-negative integral value. This is
 572             the standard string-hashing function used in SGI's
 573             implementation (with a = 5). Its advantage is that
 574             it takes into account all of the characters of the string.</p><p>Now assume that s is the string representation of a
 575             of a long DNA sequence (and so S = {'A', 'C', 'G',
 576             'T'}). In this case, scanning the entire string might be
 577             prohibitively expensive. A possible alternative might be to use
 578             only the first k characters of the string, where</p><p>|S|<sup>k</sup> ≥ m ,</p><p>i.e., using the hash function</p><div class="equation"><a id="id-1.3.5.9.4.4.2.3.2.4.12"></a><p class="title"><strong>Equation 22.6.
 579                 Only k String DNA Hash
 580               </strong></p><div class="equation-contents"><span class="mathphrase">
 581                 f<sub>2</sub>(s, m) = ∑ <sub>i
 582                 = 0</sub><sup>k - 1</sup> s<sub>i</sub> a<sup>i</sup> mod m
 583               </span></div></div><br class="equation-break" /><p>requiring scanning over only</p><p>k = log<sub>4</sub>( m )</p><p>characters.</p><p>Other more elaborate hash-functions might scan k
 584             characters starting at a random position (determined at each
 585             resize), or scanning k random positions (determined at
 586             each resize), i.e., using</p><p>f<sub>3</sub>(s, m) = ∑ <sub>i =
 587             r</sub>0<sup>r<sub>0</sub> + k - 1</sup> s<sub>i</sub>
 588             a<sup>i</sup> mod m ,</p><p>or</p><p>f<sub>4</sub>(s, m) = ∑ <sub>i = 0</sub><sup>k -
 589             1</sup> s<sub>r</sub>i a<sup>r<sub>i</sub></sup> mod
 590             m ,</p><p>respectively, for r<sub>0</sub>,..., r<sub>k-1</sub>
 591             each in the (inclusive) range [0,...,t-1].</p><p>It should be noted that the above functions cannot be
 592             decomposed as per a ranged hash composed of hash and range hashing.</p></div><div class="section"><div class="titlepage"><div><div><h6 class="title"><a id="details.hash_policies.implementation"></a>Implementation</h6></div></div></div><p>This sub-subsection describes the implementation of
 593             the above in this library. It first explains range-hashing
 594             functions in collision-chaining tables, then ranged-hash
 595             functions in collision-chaining tables, then probing-based
 596             tables, and finally lists the relevant classes in this
 597             library.</p><div class="section"><div class="titlepage"><div><div><h6 class="title"><a id="hash_policies.implementation.collision-chaining"></a>
 598                 Range-Hashing and Ranged-Hashes in Collision-Chaining Tables
 599               </h6></div></div></div><p><code class="classname">cc_hash_table</code> is
 600               parametrized by <code class="classname">Hash_Fn</code> and <code class="classname">Comb_Hash_Fn</code>, a
 601               hash functor and a combining hash functor, respectively.</p><p>In general, <code class="classname">Comb_Hash_Fn</code> is considered a
 602               range-hashing functor. <code class="classname">cc_hash_table</code>
 603               synthesizes a ranged-hash function from <code class="classname">Hash_Fn</code> and
 604               <code class="classname">Comb_Hash_Fn</code>. The figure below shows an <code class="classname">insert</code> sequence
 605               diagram for this case. The user inserts an element (point A),
 606               the container transforms the key into a non-negative integral
 607               using the hash functor (points B and C), and transforms the
 608               result into a position using the combining functor (points D
 609               and E).</p><div class="figure"><a id="id-1.3.5.9.4.4.2.3.2.5.3.4"></a><p class="title"><strong>Figure 22.15. Insert hash sequence diagram</strong></p><div class="figure-contents"><div class="mediaobject" align="center"><img src="../images/pbds_hash_range_hashing_seq_diagram.png" align="middle" alt="Insert hash sequence diagram" /></div></div></div><br class="figure-break" /><p>If <code class="classname">cc_hash_table</code>'s
 610               hash-functor, <code class="classname">Hash_Fn</code> is instantiated by <code class="classname">null_type</code> , then <code class="classname">Comb_Hash_Fn</code> is taken to be
 611               a ranged-hash function. The graphic below shows an <code class="function">insert</code> sequence
 612               diagram. The user inserts an element (point A), the container
 613               transforms the key into a position using the combining functor
 614               (points B and C).</p><div class="figure"><a id="id-1.3.5.9.4.4.2.3.2.5.3.6"></a><p class="title"><strong>Figure 22.16. Insert hash sequence diagram with a null policy</strong></p><div class="figure-contents"><div class="mediaobject" align="center"><img src="../images/pbds_hash_range_hashing_seq_diagram2.png" align="middle" alt="Insert hash sequence diagram with a null policy" /></div></div></div><br class="figure-break" /></div><div class="section"><div class="titlepage"><div><div><h6 class="title"><a id="hash_policies.implementation.probe"></a>
 615                 Probing tables
 616               </h6></div></div></div><p><code class="classname">gp_hash_table</code> is parametrized by
 617               <code class="classname">Hash_Fn</code>, <code class="classname">Probe_Fn</code>,
 618               and <code class="classname">Comb_Probe_Fn</code>. As before, if
 619               <code class="classname">Hash_Fn</code> and <code class="classname">Probe_Fn</code>
 620               are both <code class="classname">null_type</code>, then
 621               <code class="classname">Comb_Probe_Fn</code> is a ranged-probe
 622               functor. Otherwise, <code class="classname">Hash_Fn</code> is a hash
 623               functor, <code class="classname">Probe_Fn</code> is a functor for offsets
 624               from a hash value, and <code class="classname">Comb_Probe_Fn</code>
 625               transforms a probe sequence into a sequence of positions within
 626               the table.</p></div><div class="section"><div class="titlepage"><div><div><h6 class="title"><a id="hash_policies.implementation.predefined"></a>
 627                 Pre-Defined Policies
 628               </h6></div></div></div><p>This library contains some pre-defined classes
 629               implementing range-hashing and probing functions:</p><div class="orderedlist"><ol class="orderedlist" type="1"><li class="listitem"><p><code class="classname">direct_mask_range_hashing</code>
 630                 and <code class="classname">direct_mod_range_hashing</code>
 631                 are range-hashing functions based on a bit-mask and a modulo
 632                 operation, respectively.</p></li><li class="listitem"><p><code class="classname">linear_probe_fn</code>, and
 633                 <code class="classname">quadratic_probe_fn</code> are
 634                 a linear probe and a quadratic probe function,
 635                 respectively.</p></li></ol></div><p>
 636                 The graphic below shows the relationships.
 637               </p><div class="figure"><a id="id-1.3.5.9.4.4.2.3.2.5.5.5"></a><p class="title"><strong>Figure 22.17. Hash policy class diagram</strong></p><div class="figure-contents"><div class="mediaobject" align="center"><img src="../images/pbds_hash_policy_cd.png" align="middle" alt="Hash policy class diagram" /></div></div></div><br class="figure-break" /></div></div></div><div class="section"><div class="titlepage"><div><div><h6 class="title"><a id="container.hash.details.resize_policies"></a>Resize Policies</h6></div></div></div><div class="section"><div class="titlepage"><div><div><h6 class="title"><a id="resize_policies.general"></a>General</h6></div></div></div><p>Hash-tables, as opposed to trees, do not naturally grow or
 638             shrink. It is necessary to specify policies to determine how
 639             and when a hash table should change its size. Usually, resize
 640             policies can be decomposed into orthogonal policies:</p><div class="orderedlist"><ol class="orderedlist" type="1"><li class="listitem"><p>A size policy indicating how a hash table
 641               should grow (e.g., it should multiply by powers of
 642               2).</p></li><li class="listitem"><p>A trigger policy indicating when a hash
 643               table should grow (e.g., a load factor is
 644               exceeded).</p></li></ol></div></div><div class="section"><div class="titlepage"><div><div><h6 class="title"><a id="resize_policies.size"></a>Size Policies</h6></div></div></div><p>Size policies determine how a hash table changes size. These
 645             policies are simple, and there are relatively few sensible
 646             options. An exponential-size policy (with the initial size and
 647             growth factors both powers of 2) works well with a mask-based
 648             range-hashing function, and is the
 649             hard-wired policy used by Dinkumware. A
 650             prime-list based policy works well with a modulo-prime range
 651             hashing function and is the hard-wired policy used by SGI's
 652             implementation.</p></div><div class="section"><div class="titlepage"><div><div><h6 class="title"><a id="resize_policies.trigger"></a>Trigger Policies</h6></div></div></div><p>Trigger policies determine when a hash table changes size.
 653             Following is a description of two policies: load-check
 654             policies, and collision-check policies.</p><p>Load-check policies are straightforward. The user specifies
 655             two factors, Α<sub>min</sub> and
 656             Α<sub>max</sub>, and the hash table maintains the
 657             invariant that</p><p>Α<sub>min</sub> ≤ (number of
 658             stored elements) / (hash-table size) ≤
 659             Α<sub>max</sub><em><span class="remark">load factor min max</span></em></p><p>Collision-check policies work in the opposite direction of
 660             load-check policies. They focus on keeping the number of
 661             collisions moderate and hoping that the size of the table will
 662             not grow very large, instead of keeping a moderate load-factor
 663             and hoping that the number of collisions will be small. A
 664             maximal collision-check policy resizes when the longest
 665             probe-sequence grows too large.</p><p>Consider the graphic below. Let the size of the hash table
 666             be denoted by m, the length of a probe sequence be denoted by k,
 667             and some load factor be denoted by Α. We would like to
 668             calculate the minimal length of k, such that if there were Α
 669             m elements in the hash table, a probe sequence of length k would
 670             be found with probability at most 1/m.</p><div class="figure"><a id="id-1.3.5.9.4.4.2.3.3.4.7"></a><p class="title"><strong>Figure 22.18. Balls and bins</strong></p><div class="figure-contents"><div class="mediaobject" align="center"><img src="../images/pbds_balls_and_bins.png" align="middle" alt="Balls and bins" /></div></div></div><br class="figure-break" /><p>Denote the probability that a probe sequence of length
 671             k appears in bin i by p<sub>i</sub>, the
 672             length of the probe sequence of bin i by
 673             l<sub>i</sub>, and assume uniform distribution. Then</p><div class="equation"><a id="id-1.3.5.9.4.4.2.3.3.4.9"></a><p class="title"><strong>Equation 22.7.
 674                 Probability of Probe Sequence of Length k
 675               </strong></p><div class="equation-contents"><span class="mathphrase">
 676                 p<sub>1</sub> =
 677               </span></div></div><br class="equation-break" /><p>P(l<sub>1</sub> ≥ k) =</p><p>
 678               P(l<sub>1</sub> ≥ α ( 1 + k / α - 1) ≤ (a)
 679             </p><p>
 680               e ^ ( - ( α ( k / α - 1 )<sup>2</sup> ) /2)
 681             </p><p>where (a) follows from the Chernoff bound (<a class="xref" href="policy_data_structures.html#biblio.motwani95random" title="Randomized Algorithms">[biblio.motwani95random]</a>). To
 682             calculate the probability that some bin contains a probe
 683             sequence greater than k, we note that the
 684             l<sub>i</sub> are negatively-dependent
 685             (<a class="xref" href="policy_data_structures.html#biblio.dubhashi98neg" title="Balls and bins: A study in negative dependence">[biblio.dubhashi98neg]</a>)
 686             . Let
 687             I(.) denote the indicator function. Then</p><div class="equation"><a id="id-1.3.5.9.4.4.2.3.3.4.14"></a><p class="title"><strong>Equation 22.8.
 688                 Probability Probe Sequence in Some Bin
 689               </strong></p><div class="equation-contents"><span class="mathphrase">
 690                 P( exists<sub>i</sub> l<sub>i</sub> ≥ k ) =
 691               </span></div></div><br class="equation-break" /><p>P ( ∑ <sub>i = 1</sub><sup>m</sup>
 692             I(l<sub>i</sub> ≥ k) ≥ 1 ) =</p><p>P ( ∑ <sub>i = 1</sub><sup>m</sup> I (
 693             l<sub>i</sub> ≥ k ) ≥ m p<sub>1</sub> ( 1 + 1 / (m
 694             p<sub>1</sub>) - 1 ) ) ≤ (a)</p><p>e ^ ( ( - m p<sub>1</sub> ( 1 / (m p<sub>1</sub>)
 695             - 1 ) <sup>2</sup> ) / 2 ) ,</p><p>where (a) follows from the fact that the Chernoff bound can
 696             be applied to negatively-dependent variables (<a class="xref" href="policy_data_structures.html#biblio.dubhashi98neg" title="Balls and bins: A study in negative dependence">[biblio.dubhashi98neg]</a>). Inserting the first probability
 697             equation into the second one, and equating with 1/m, we
 698             obtain</p><p>k ~ √ ( 2 α ln 2 m ln(m) )
 699             ) .</p></div><div class="section"><div class="titlepage"><div><div><h6 class="title"><a id="resize_policies.impl"></a>Implementation</h6></div></div></div><p>This sub-subsection describes the implementation of the
 700             above in this library. It first describes resize policies and
 701             their decomposition into trigger and size policies, then
 702             describes pre-defined classes, and finally discusses controlled
 703             access the policies' internals.</p><div class="section"><div class="titlepage"><div><div><h6 class="title"><a id="resize_policies.impl.decomposition"></a>Decomposition</h6></div></div></div><p>Each hash-based container is parametrized by a
 704               <code class="classname">Resize_Policy</code> parameter; the container derives
 705               <code class="classname">public</code>ly from <code class="classname">Resize_Policy</code>. For
 706               example:</p><pre class="programlisting">
 707                 cc_hash_table&lt;typename Key,
 708                 typename Mapped,
 709                 ...
 710                 typename Resize_Policy
 711                 ...&gt; : public Resize_Policy
 712               </pre><p>As a container object is modified, it continuously notifies
 713               its <code class="classname">Resize_Policy</code> base of internal changes
 714               (e.g., collisions encountered and elements being
 715               inserted). It queries its <code class="classname">Resize_Policy</code> base whether
 716               it needs to be resized, and if so, to what size.</p><p>The graphic below shows a (possible) sequence diagram
 717               of an insert operation. The user inserts an element; the hash
 718               table notifies its resize policy that a search has started
 719               (point A); in this case, a single collision is encountered -
 720               the table notifies its resize policy of this (point B); the
 721               container finally notifies its resize policy that the search
 722               has ended (point C); it then queries its resize policy whether
 723               a resize is needed, and if so, what is the new size (points D
 724               to G); following the resize, it notifies the policy that a
 725               resize has completed (point H); finally, the element is
 726               inserted, and the policy notified (point I).</p><div class="figure"><a id="id-1.3.5.9.4.4.2.3.3.5.3.6"></a><p class="title"><strong>Figure 22.19. Insert resize sequence diagram</strong></p><div class="figure-contents"><div class="mediaobject" align="center"><img src="../images/pbds_insert_resize_sequence_diagram1.png" align="middle" alt="Insert resize sequence diagram" /></div></div></div><br class="figure-break" /><p>In practice, a resize policy can be usually orthogonally
 727               decomposed to a size policy and a trigger policy. Consequently,
 728               the library contains a single class for instantiating a resize
 729               policy: <code class="classname">hash_standard_resize_policy</code>
 730               is parametrized by <code class="classname">Size_Policy</code> and
 731               <code class="classname">Trigger_Policy</code>, derives <code class="classname">public</code>ly from
 732               both, and acts as a standard delegate (<a class="xref" href="policy_data_structures.html#biblio.gof" title="Design Patterns - Elements of Reusable Object-Oriented Software">[biblio.gof]</a>)
 733               to these policies.</p><p>The two graphics immediately below show sequence diagrams
 734               illustrating the interaction between the standard resize policy
 735               and its trigger and size policies, respectively.</p><div class="figure"><a id="id-1.3.5.9.4.4.2.3.3.5.3.9"></a><p class="title"><strong>Figure 22.20. Standard resize policy trigger sequence
 736                 diagram</strong></p><div class="figure-contents"><div class="mediaobject" align="center"><img src="../images/pbds_insert_resize_sequence_diagram2.png" align="middle" alt="Standard resize policy trigger sequence diagram" /></div></div></div><br class="figure-break" /><div class="figure"><a id="id-1.3.5.9.4.4.2.3.3.5.3.10"></a><p class="title"><strong>Figure 22.21. Standard resize policy size sequence
 737                 diagram</strong></p><div class="figure-contents"><div class="mediaobject" align="center"><img src="../images/pbds_insert_resize_sequence_diagram3.png" align="middle" alt="Standard resize policy size sequence diagram" /></div></div></div><br class="figure-break" /></div><div class="section"><div class="titlepage"><div><div><h6 class="title"><a id="resize_policies.impl.predefined"></a>Predefined Policies</h6></div></div></div><p>The library includes the following
 738               instantiations of size and trigger policies:</p><div class="orderedlist"><ol class="orderedlist" type="1"><li class="listitem"><p><code class="classname">hash_load_check_resize_trigger</code>
 739                 implements a load check trigger policy.</p></li><li class="listitem"><p><code class="classname">cc_hash_max_collision_check_resize_trigger</code>
 740                 implements a collision check trigger policy.</p></li><li class="listitem"><p><code class="classname">hash_exponential_size_policy</code>
 741                 implements an exponential-size policy (which should be used
 742                 with mask range hashing).</p></li><li class="listitem"><p><code class="classname">hash_prime_size_policy</code>
 743                 implementing a size policy based on a sequence of primes
 744                 (which should
 745                 be used with mod range hashing</p></li></ol></div><p>The graphic below gives an overall picture of the resize-related
 746               classes. <code class="classname">basic_hash_table</code>
 747               is parametrized by <code class="classname">Resize_Policy</code>, which it subclasses
 748               publicly. This class is currently instantiated only by <code class="classname">hash_standard_resize_policy</code>.
 749               <code class="classname">hash_standard_resize_policy</code>
 750               itself is parametrized by <code class="classname">Trigger_Policy</code> and
 751               <code class="classname">Size_Policy</code>. Currently, <code class="classname">Trigger_Policy</code> is
 752               instantiated by <code class="classname">hash_load_check_resize_trigger</code>,
 753               or <code class="classname">cc_hash_max_collision_check_resize_trigger</code>;
 754               <code class="classname">Size_Policy</code> is instantiated by <code class="classname">hash_exponential_size_policy</code>,
 755               or <code class="classname">hash_prime_size_policy</code>.</p></div><div class="section"><div class="titlepage"><div><div><h6 class="title"><a id="resize_policies.impl.internals"></a>Controling Access to Internals</h6></div></div></div><p>There are cases where (controlled) access to resize
 756               policies' internals is beneficial. E.g., it is sometimes
 757               useful to query a hash-table for the table's actual size (as
 758               opposed to its <code class="function">size()</code> - the number of values it
 759               currently holds); it is sometimes useful to set a table's
 760               initial size, externally resize it, or change load factors.</p><p>Clearly, supporting such methods both decreases the
 761               encapsulation of hash-based containers, and increases the
 762               diversity between different associative-containers' interfaces.
 763               Conversely, omitting such methods can decrease containers'
 764               flexibility.</p><p>In order to avoid, to the extent possible, the above
 765               conflict, the hash-based containers themselves do not address
 766               any of these questions; this is deferred to the resize policies,
 767               which are easier to change or replace. Thus, for example,
 768               neither <code class="classname">cc_hash_table</code> nor
 769               <code class="classname">gp_hash_table</code>
 770               contain methods for querying the actual size of the table; this
 771               is deferred to <code class="classname">hash_standard_resize_policy</code>.</p><p>Furthermore, the policies themselves are parametrized by
 772               template arguments that determine the methods they support
 773               (
 774               <a class="xref" href="policy_data_structures.html#biblio.alexandrescu01modern" title="Modern C++ Design: Generic Programming and Design Patterns Applied">[biblio.alexandrescu01modern]</a>
 775               shows techniques for doing so). <code class="classname">hash_standard_resize_policy</code>
 776               is parametrized by <code class="classname">External_Size_Access</code> that
 777               determines whether it supports methods for querying the actual
 778               size of the table or resizing it. <code class="classname">hash_load_check_resize_trigger</code>
 779               is parametrized by <code class="classname">External_Load_Access</code> that
 780               determines whether it supports methods for querying or
 781               modifying the loads. <code class="classname">cc_hash_max_collision_check_resize_trigger</code>
 782               is parametrized by <code class="classname">External_Load_Access</code> that
 783               determines whether it supports methods for querying the
 784               load.</p><p>Some operations, for example, resizing a container at
 785               run time, or changing the load factors of a load-check trigger
 786               policy, require the container itself to resize. As mentioned
 787               above, the hash-based containers themselves do not contain
 788               these types of methods, only their resize policies.
 789               Consequently, there must be some mechanism for a resize policy
 790               to manipulate the hash-based container. As the hash-based
 791               container is a subclass of the resize policy, this is done
 792               through virtual methods. Each hash-based container has a
 793               <code class="classname">private</code> <code class="classname">virtual</code> method:</p><pre class="programlisting">
 794                 virtual void
 795                 do_resize
 796                 (size_type new_size);
 797               </pre><p>which resizes the container. Implementations of
 798               <code class="classname">Resize_Policy</code> can export public methods for resizing
 799               the container externally; these methods internally call
 800               <code class="classname">do_resize</code> to resize the table.</p></div></div></div><div class="section"><div class="titlepage"><div><div><h6 class="title"><a id="container.hash.details.policy_interaction"></a>Policy Interactions</h6></div></div></div><p>
 801           </p><p>Hash-tables are unfortunately especially susceptible to
 802           choice of policies. One of the more complicated aspects of this
 803           is that poor combinations of good policies can form a poor
 804           container. Following are some considerations.</p><div class="section"><div class="titlepage"><div><div><h6 class="title"><a id="policy_interaction.probesizetrigger"></a>probe/size/trigger</h6></div></div></div><p>Some combinations do not work well for probing containers.
 805             For example, combining a quadratic probe policy with an
 806             exponential size policy can yield a poor container: when an
 807             element is inserted, a trigger policy might decide that there
 808             is no need to resize, as the table still contains unused
 809             entries; the probe sequence, however, might never reach any of
 810             the unused entries.</p><p>Unfortunately, this library cannot detect such problems at
 811             compilation (they are halting reducible). It therefore defines
 812             an exception class <code class="classname">insert_error</code> to throw an
 813             exception in this case.</p></div><div class="section"><div class="titlepage"><div><div><h6 class="title"><a id="policy_interaction.hashtrigger"></a>hash/trigger</h6></div></div></div><p>Some trigger policies are especially susceptible to poor
 814             hash functions. Suppose, as an extreme case, that the hash
 815             function transforms each key to the same hash value. After some
 816             inserts, a collision detecting policy will always indicate that
 817             the container needs to grow.</p><p>The library, therefore, by design, limits each operation to
 818             one resize. For each <code class="classname">insert</code>, for example, it queries
 819             only once whether a resize is needed.</p></div><div class="section"><div class="titlepage"><div><div><h6 class="title"><a id="policy_interaction.eqstorehash"></a>equivalence functors/storing hash values/hash</h6></div></div></div><p><code class="classname">cc_hash_table</code> and
 820             <code class="classname">gp_hash_table</code> are
 821             parametrized by an equivalence functor and by a
 822             <code class="classname">Store_Hash</code> parameter. If the latter parameter is
 823             <code class="classname">true</code>, then the container stores with each entry
 824             a hash value, and uses this value in case of collisions to
 825             determine whether to apply a hash value. This can lower the
 826             cost of collision for some types, but increase the cost of
 827             collisions for other types.</p><p>If a ranged-hash function or ranged probe function is
 828             directly supplied, however, then it makes no sense to store the
 829             hash value with each entry. This library's container will
 830             fail at compilation, by design, if this is attempted.</p></div><div class="section"><div class="titlepage"><div><div><h6 class="title"><a id="policy_interaction.sizeloadtrigger"></a>size/load-check trigger</h6></div></div></div><p>Assume a size policy issues an increasing sequence of sizes
 831             a, a q, a q<sup>1</sup>, a q<sup>2</sup>, ... For
 832             example, an exponential size policy might issue the sequence of
 833             sizes 8, 16, 32, 64, ...</p><p>If a load-check trigger policy is used, with loads
 834             α<sub>min</sub> and α<sub>max</sub>,
 835             respectively, then it is a good idea to have:</p><div class="orderedlist"><ol class="orderedlist" type="1"><li class="listitem"><p>α<sub>max</sub> ~ 1 / q</p></li><li class="listitem"><p>α<sub>min</sub> &lt; 1 / (2 q)</p></li></ol></div><p>This will ensure that the amortized hash cost of each
 836             modifying operation is at most approximately 3.</p><p>α<sub>min</sub> ~ α<sub>max</sub> is, in
 837             any case, a bad choice, and α<sub>min</sub> &gt;
 838             α <sub>max</sub> is horrendous.</p></div></div></div></div><div class="section"><div class="titlepage"><div><div><h4 class="title"><a id="pbds.design.container.tree"></a>tree</h4></div></div></div><div class="section"><div class="titlepage"><div><div><h5 class="title"><a id="container.tree.interface"></a>Interface</h5></div></div></div><p>The tree-based container has the following declaration:</p><pre class="programlisting">
 839           template&lt;
 840           typename Key,
 841           typename Mapped,
 842           typename Cmp_Fn = std::less&lt;Key&gt;,
 843           typename Tag = rb_tree_tag,
 844           template&lt;
 845           typename Const_Node_Iterator,
 846           typename Node_Iterator,
 847           typename Cmp_Fn_,
 848           typename Allocator_&gt;
 849           class Node_Update = null_node_update,
 850           typename Allocator = std::allocator&lt;char&gt; &gt;
 851           class tree;
 852         </pre><p>The parameters have the following meaning:</p><div class="orderedlist"><ol class="orderedlist" type="1"><li class="listitem"><p><code class="classname">Key</code> is the key type.</p></li><li class="listitem"><p><code class="classname">Mapped</code> is the mapped-policy.</p></li><li class="listitem"><p><code class="classname">Cmp_Fn</code> is a key comparison functor</p></li><li class="listitem"><p><code class="classname">Tag</code> specifies which underlying data structure
 853           to use.</p></li><li class="listitem"><p><code class="classname">Node_Update</code> is a policy for updating node
 854           invariants.</p></li><li class="listitem"><p><code class="classname">Allocator</code> is an allocator
 855           type.</p></li></ol></div><p>The <code class="classname">Tag</code> parameter specifies which underlying
 856         data structure to use. Instantiating it by <code class="classname">rb_tree_tag</code>, <code class="classname">splay_tree_tag</code>, or
 857         <code class="classname">ov_tree_tag</code>,
 858         specifies an underlying red-black tree, splay tree, or
 859         ordered-vector tree, respectively; any other tag is illegal.
 860         Note that containers based on the former two contain more types
 861         and methods than the latter (e.g.,
 862         <code class="classname">reverse_iterator</code> and <code class="classname">rbegin</code>), and different
 863         exception and invalidation guarantees.</p></div><div class="section"><div class="titlepage"><div><div><h5 class="title"><a id="container.tree.details"></a>Details</h5></div></div></div><div class="section"><div class="titlepage"><div><div><h6 class="title"><a id="container.tree.node"></a>Node Invariants</h6></div></div></div><p>Consider the two trees in the graphic below, labels A and B. The first
 864           is a tree of floats; the second is a tree of pairs, each
 865           signifying a geometric line interval. Each element in a tree is referred to as a node of the tree. Of course, each of
 866           these trees can support the usual queries: the first can easily
 867           search for <code class="classname">0.4</code>; the second can easily search for
 868           <code class="classname">std::make_pair(10, 41)</code>.</p><p>Each of these trees can efficiently support other queries.
 869           The first can efficiently determine that the 2rd key in the
 870           tree is <code class="constant">0.3</code>; the second can efficiently determine
 871           whether any of its intervals overlaps
 872           </p><pre class="programlisting">std::make_pair(29,42)</pre><p> (useful in geometric
 873           applications or distributed file systems with leases, for
 874           example).  It should be noted that an <code class="classname">std::set</code> can
 875           only solve these types of problems with linear complexity.</p><p>In order to do so, each tree stores some metadata in
 876           each node, and maintains node invariants (see <a class="xref" href="policy_data_structures.html#biblio.clrs2001" title="Introduction to Algorithms, 2nd edition">[biblio.clrs2001]</a>.) The first stores in
 877           each node the size of the sub-tree rooted at the node; the
 878           second stores at each node the maximal endpoint of the
 879           intervals at the sub-tree rooted at the node.</p><div class="figure"><a id="id-1.3.5.9.4.4.3.3.2.5"></a><p class="title"><strong>Figure 22.22. Tree node invariants</strong></p><div class="figure-contents"><div class="mediaobject" align="center"><img src="../images/pbds_tree_node_invariants.png" align="middle" alt="Tree node invariants" /></div></div></div><br class="figure-break" /><p>Supporting such trees is difficult for a number of
 880           reasons:</p><div class="orderedlist"><ol class="orderedlist" type="1"><li class="listitem"><p>There must be a way to specify what a node's metadata
 881             should be (if any).</p></li><li class="listitem"><p>Various operations can invalidate node
 882             invariants.  The graphic below shows how a right rotation,
 883             performed on A, results in B, with nodes x and y having
 884             corrupted invariants (the grayed nodes in C). The graphic shows
 885             how an insert, performed on D, results in E, with nodes x and y
 886             having corrupted invariants (the grayed nodes in F). It is not
 887             feasible to know outside the tree the effect of an operation on
 888             the nodes of the tree.</p></li><li class="listitem"><p>The search paths of standard associative containers are
 889             defined by comparisons between keys, and not through
 890             metadata.</p></li><li class="listitem"><p>It is not feasible to know in advance which methods trees
 891             can support. Besides the usual <code class="classname">find</code> method, the
 892             first tree can support a <code class="classname">find_by_order</code> method, while
 893             the second can support an <code class="classname">overlaps</code> method.</p></li></ol></div><div class="figure"><a id="id-1.3.5.9.4.4.3.3.2.8"></a><p class="title"><strong>Figure 22.23. Tree node invalidation</strong></p><div class="figure-contents"><div class="mediaobject" align="center"><img src="../images/pbds_tree_node_invalidations.png" align="middle" alt="Tree node invalidation" /></div></div></div><br class="figure-break" /><p>These problems are solved by a combination of two means:
 894           node iterators, and template-template node updater
 895           parameters.</p><div class="section"><div class="titlepage"><div><div><h6 class="title"><a id="container.tree.node.iterators"></a>Node Iterators</h6></div></div></div><p>Each tree-based container defines two additional iterator
 896             types, <code class="classname">const_node_iterator</code>
 897             and <code class="classname">node_iterator</code>.
 898             These iterators allow descending from a node to one of its
 899             children. Node iterator allow search paths different than those
 900             determined by the comparison functor. The <code class="classname">tree</code>
 901             supports the methods:</p><pre class="programlisting">
 902               const_node_iterator
 903               node_begin() const;
 904
 905               node_iterator
 906               node_begin();
 907
 908               const_node_iterator
 909               node_end() const;
 910
 911               node_iterator
 912               node_end();
 913             </pre><p>The first pairs return node iterators corresponding to the
 914             root node of the tree; the latter pair returns node iterators
 915             corresponding to a just-after-leaf node.</p></div><div class="section"><div class="titlepage"><div><div><h6 class="title"><a id="container.tree.node.updator"></a>Node Updator</h6></div></div></div><p>The tree-based containers are parametrized by a
 916             <code class="classname">Node_Update</code> template-template parameter. A
 917             tree-based container instantiates
 918             <code class="classname">Node_Update</code> to some
 919             <code class="classname">node_update</code> class, and publicly subclasses
 920             <code class="classname">node_update</code>. The graphic below shows this
 921             scheme, as well as some predefined policies (which are explained
 922             below).</p><div class="figure"><a id="id-1.3.5.9.4.4.3.3.2.11.3"></a><p class="title"><strong>Figure 22.24. A tree and its update policy</strong></p><div class="figure-contents"><div class="mediaobject" align="center"><img src="../images/pbds_tree_node_updator_policy_cd.png" align="middle" alt="A tree and its update policy" /></div></div></div><br class="figure-break" /><p><code class="classname">node_update</code> (an instantiation of
 923             <code class="classname">Node_Update</code>) must define <code class="classname">metadata_type</code> as
 924             the type of metadata it requires. For order statistics,
 925             e.g., <code class="classname">metadata_type</code> might be <code class="classname">size_t</code>.
 926             The tree defines within each node a <code class="classname">metadata_type</code>
 927             object.</p><p><code class="classname">node_update</code> must also define the following method
 928             for restoring node invariants:</p><pre class="programlisting">
 929               void
 930               operator()(node_iterator nd_it, const_node_iterator end_nd_it)
 931             </pre><p>In this method, <code class="varname">nd_it</code> is a
 932             <code class="classname">node_iterator</code> corresponding to a node whose
 933             A) all descendants have valid invariants, and B) its own
 934             invariants might be violated; <code class="classname">end_nd_it</code> is
 935             a <code class="classname">const_node_iterator</code> corresponding to a
 936             just-after-leaf node. This method should correct the node
 937             invariants of the node pointed to by
 938             <code class="classname">nd_it</code>. For example, say node x in the
 939             graphic below label A has an invalid invariant, but its' children,
 940             y and z have valid invariants. After the invocation, all three
 941             nodes should have valid invariants, as in label B.</p><div class="figure"><a id="id-1.3.5.9.4.4.3.3.2.11.8"></a><p class="title"><strong>Figure 22.25. Restoring node invariants</strong></p><div class="figure-contents"><div class="mediaobject" align="center"><img src="../images/pbds_restoring_node_invariants.png" align="middle" alt="Restoring node invariants" /></div></div></div><br class="figure-break" /><p>When a tree operation might invalidate some node invariant,
 942             it invokes this method in its <code class="classname">node_update</code> base to
 943             restore the invariant. For example, the graphic below shows
 944             an <code class="function">insert</code> operation (point A); the tree performs some
 945             operations, and calls the update functor three times (points B,
 946             C, and D). (It is well known that any <code class="function">insert</code>,
 947             <code class="function">erase</code>, <code class="function">split</code> or <code class="function">join</code>, can restore
 948             all node invariants by a small number of node invariant updates (<a class="xref" href="policy_data_structures.html#biblio.clrs2001" title="Introduction to Algorithms, 2nd edition">[biblio.clrs2001]</a>)
 949             .</p><div class="figure"><a id="id-1.3.5.9.4.4.3.3.2.11.10"></a><p class="title"><strong>Figure 22.26. Insert update sequence</strong></p><div class="figure-contents"><div class="mediaobject" align="center"><img src="../images/pbds_update_seq_diagram.png" align="middle" alt="Insert update sequence" /></div></div></div><br class="figure-break" /><p>To complete the description of the scheme, three questions
 950             need to be answered:</p><div class="orderedlist"><ol class="orderedlist" type="1"><li class="listitem"><p>How can a tree which supports order statistics define a
 951               method such as <code class="classname">find_by_order</code>?</p></li><li class="listitem"><p>How can the node updater base access methods of the
 952               tree?</p></li><li class="listitem"><p>How can the following cyclic dependency be resolved?
 953               <code class="classname">node_update</code> is a base class of the tree, yet it
 954               uses node iterators defined in the tree (its child).</p></li></ol></div><p>The first two questions are answered by the fact that
 955             <code class="classname">node_update</code> (an instantiation of
 956             <code class="classname">Node_Update</code>) is a <span class="emphasis"><em>public</em></span> base class
 957             of the tree. Consequently:</p><div class="orderedlist"><ol class="orderedlist" type="1"><li class="listitem"><p>Any public methods of
 958               <code class="classname">node_update</code> are automatically methods of
 959               the tree (<a class="xref" href="policy_data_structures.html#biblio.alexandrescu01modern" title="Modern C++ Design: Generic Programming and Design Patterns Applied">[biblio.alexandrescu01modern]</a>).
 960               Thus an order-statistics node updater,
 961               <code class="classname">tree_order_statistics_node_update</code> defines
 962               the <code class="function">find_by_order</code> method; any tree
 963               instantiated by this policy consequently supports this method as
 964               well.</p></li><li class="listitem"><p>In C++, if a base class declares a method as
 965               <code class="literal">virtual</code>, it is
 966               <code class="literal">virtual</code> in its subclasses. If
 967               <code class="classname">node_update</code> needs to access one of the
 968               tree's methods, say the member function
 969               <code class="function">end</code>, it simply declares that method as
 970               <code class="literal">virtual</code> abstract.</p></li></ol></div><p>The cyclic dependency is solved through template-template
 971             parameters. <code class="classname">Node_Update</code> is parametrized by
 972             the tree's node iterators, its comparison functor, and its
 973             allocator type. Thus, instantiations of
 974             <code class="classname">Node_Update</code> have all information
 975             required.</p><p>This library assumes that constructing a metadata object and
 976             modifying it are exception free. Suppose that during some method,
 977             say <code class="classname">insert</code>, a metadata-related operation
 978             (e.g., changing the value of a metadata) throws an exception. Ack!
 979             Rolling back the method is unusually complex.</p><p>Previously, a distinction was made between redundant
 980             policies and null policies. Node invariants show a
 981             case where null policies are required.</p><p>Assume a regular tree is required, one which need not
 982             support order statistics or interval overlap queries.
 983             Seemingly, in this case a redundant policy - a policy which
 984             doesn't affect nodes' contents would suffice. This, would lead
 985             to the following drawbacks:</p><div class="orderedlist"><ol class="orderedlist" type="1"><li class="listitem"><p>Each node would carry a useless metadata object, wasting
 986               space.</p></li><li class="listitem"><p>The tree cannot know if its
 987               <code class="classname">Node_Update</code> policy actually modifies a
 988               node's metadata (this is halting reducible). In the graphic
 989               below, assume the shaded node is inserted. The tree would have
 990               to traverse the useless path shown to the root, applying
 991               redundant updates all the way.</p></li></ol></div><div class="figure"><a id="id-1.3.5.9.4.4.3.3.2.11.20"></a><p class="title"><strong>Figure 22.27. Useless update path</strong></p><div class="figure-contents"><div class="mediaobject" align="center"><img src="../images/pbds_rationale_null_node_updator.png" align="middle" alt="Useless update path" /></div></div></div><br class="figure-break" /><p>A null policy class, <code class="classname">null_node_update</code>
 992             solves both these problems. The tree detects that node
 993             invariants are irrelevant, and defines all accordingly.</p></div></div><div class="section"><div class="titlepage"><div><div><h6 class="title"><a id="container.tree.details.split"></a>Split and Join</h6></div></div></div><p>Tree-based containers support split and join methods.
 994           It is possible to split a tree so that it passes
 995           all nodes with keys larger than a given key to a different
 996           tree. These methods have the following advantages over the
 997           alternative of externally inserting to the destination
 998           tree and erasing from the source tree:</p><div class="orderedlist"><ol class="orderedlist" type="1"><li class="listitem"><p>These methods are efficient - red-black trees are split
 999             and joined in poly-logarithmic complexity; ordered-vector
1000             trees are split and joined at linear complexity. The
1001             alternatives have super-linear complexity.</p></li><li class="listitem"><p>Aside from orders of growth, these operations perform
1002             few allocations and de-allocations. For red-black trees, allocations are not performed,
1003             and the methods are exception-free. </p></li></ol></div></div></div></div><div class="section"><div class="titlepage"><div><div><h4 class="title"><a id="pbds.design.container.trie"></a>Trie</h4></div></div></div><div class="section"><div class="titlepage"><div><div><h5 class="title"><a id="container.trie.interface"></a>Interface</h5></div></div></div><p>The trie-based container has the following declaration:</p><pre class="programlisting">
1004           template&lt;typename Key,
1005           typename Mapped,
1006           typename Cmp_Fn = std::less&lt;Key&gt;,
1007           typename Tag = pat_trie_tag,
1008           template&lt;typename Const_Node_Iterator,
1009           typename Node_Iterator,
1010           typename E_Access_Traits_,
1011           typename Allocator_&gt;
1012           class Node_Update = null_node_update,
1013           typename Allocator = std::allocator&lt;char&gt; &gt;
1014           class trie;
1015         </pre><p>The parameters have the following meaning:</p><div class="orderedlist"><ol class="orderedlist" type="1"><li class="listitem"><p><code class="classname">Key</code> is the key type.</p></li><li class="listitem"><p><code class="classname">Mapped</code> is the mapped-policy.</p></li><li class="listitem"><p><code class="classname">E_Access_Traits</code> is described in below.</p></li><li class="listitem"><p><code class="classname">Tag</code> specifies which underlying data structure
1016           to use, and is described shortly.</p></li><li class="listitem"><p><code class="classname">Node_Update</code> is a policy for updating node
1017           invariants. This is described below.</p></li><li class="listitem"><p><code class="classname">Allocator</code> is an allocator
1018           type.</p></li></ol></div><p>The <code class="classname">Tag</code> parameter specifies which underlying
1019         data structure to use. Instantiating it by <code class="classname">pat_trie_tag</code>, specifies an
1020         underlying PATRICIA trie (explained shortly); any other tag is
1021         currently illegal.</p><p>Following is a description of a (PATRICIA) trie
1022         (this implementation follows <a class="xref" href="policy_data_structures.html#biblio.okasaki98mereable" title="Fast mergeable integer maps">[biblio.okasaki98mereable]</a> and
1023         <a class="xref" href="policy_data_structures.html#biblio.filliatre2000ptset" title="Ptset: Sets of integers implemented as Patricia trees">[biblio.filliatre2000ptset]</a>).
1024         </p><p>A (PATRICIA) trie is similar to a tree, but with the
1025         following differences:</p><div class="orderedlist"><ol class="orderedlist" type="1"><li class="listitem"><p>It explicitly views keys as a sequence of elements.
1026           E.g., a trie can view a string as a sequence of
1027           characters; a trie can view a number as a sequence of
1028           bits.</p></li><li class="listitem"><p>It is not (necessarily) binary. Each node has fan-out n
1029           + 1, where n is the number of distinct
1030           elements.</p></li><li class="listitem"><p>It stores values only at leaf nodes.</p></li><li class="listitem"><p>Internal nodes have the properties that A) each has at
1031           least two children, and B) each shares the same prefix with
1032           any of its descendant.</p></li></ol></div><p>A (PATRICIA) trie has some useful properties:</p><div class="orderedlist"><ol class="orderedlist" type="1"><li class="listitem"><p>It can be configured to use large node fan-out, giving it
1033           very efficient find performance (albeit at insertion
1034           complexity and size).</p></li><li class="listitem"><p>It works well for common-prefix keys.</p></li><li class="listitem"><p>It can support efficiently queries such as which
1035           keys match a certain prefix. This is sometimes useful in file
1036           systems and routers, and for "type-ahead" aka predictive text matching
1037           on mobile devices.</p></li></ol></div></div><div class="section"><div class="titlepage"><div><div><h5 class="title"><a id="container.trie.details"></a>Details</h5></div></div></div><div class="section"><div class="titlepage"><div><div><h6 class="title"><a id="container.trie.details.etraits"></a>Element Access Traits</h6></div></div></div><p>A trie inherently views its keys as sequences of elements.
1038           For example, a trie can view a string as a sequence of
1039           characters. A trie needs to map each of n elements to a
1040           number in {0, n - 1}. For example, a trie can map a
1041           character <code class="varname">c</code> to
1042           </p><pre class="programlisting">static_cast&lt;size_t&gt;(c)</pre><p>.</p><p>Seemingly, then, a trie can assume that its keys support
1043           (const) iterators, and that the <code class="classname">value_type</code> of this
1044           iterator can be cast to a <code class="classname">size_t</code>. There are several
1045           reasons, though, to decouple the mechanism by which the trie
1046           accesses its keys' elements from the trie:</p><div class="orderedlist"><ol class="orderedlist" type="1"><li class="listitem"><p>In some cases, the numerical value of an element is
1047             inappropriate. Consider a trie storing DNA strings. It is
1048             logical to use a trie with a fan-out of 5 = 1 + |{'A', 'C',
1049             'G', 'T'}|. This requires mapping 'T' to 3, though.</p></li><li class="listitem"><p>In some cases the keys' iterators are different than what
1050             is needed. For example, a trie can be used to search for
1051             common suffixes, by using strings'
1052             <code class="classname">reverse_iterator</code>. As another example, a trie mapping
1053             UNICODE strings would have a huge fan-out if each node would
1054             branch on a UNICODE character; instead, one can define an
1055             iterator iterating over 8-bit (or less) groups.</p></li></ol></div><p>trie is,
1056           consequently, parametrized by <code class="classname">E_Access_Traits</code> -
1057           traits which instruct how to access sequences' elements.
1058           <code class="classname">string_trie_e_access_traits</code>
1059           is a traits class for strings. Each such traits define some
1060           types, like:</p><pre class="programlisting">
1061             typename E_Access_Traits::const_iterator
1062           </pre><p>is a const iterator iterating over a key's elements. The
1063           traits class must also define methods for obtaining an iterator
1064           to the first and last element of a key.</p><p>The graphic below shows a
1065           (PATRICIA) trie resulting from inserting the words: "I wish
1066           that I could ever see a poem lovely as a trie" (which,
1067           unfortunately, does not rhyme).</p><p>The leaf nodes contain values; each internal node contains
1068           two <code class="classname">typename E_Access_Traits::const_iterator</code>
1069           objects, indicating the maximal common prefix of all keys in
1070           the sub-tree. For example, the shaded internal node roots a
1071           sub-tree with leafs "a" and "as". The maximal common prefix is
1072           "a". The internal node contains, consequently, to const
1073           iterators, one pointing to <code class="varname">'a'</code>, and the other to
1074           <code class="varname">'s'</code>.</p><div class="figure"><a id="id-1.3.5.9.4.4.4.3.2.10"></a><p class="title"><strong>Figure 22.28. A PATRICIA trie</strong></p><div class="figure-contents"><div class="mediaobject" align="center"><img src="../images/pbds_pat_trie.png" align="middle" alt="A PATRICIA trie" /></div></div></div><br class="figure-break" /></div><div class="section"><div class="titlepage"><div><div><h6 class="title"><a id="container.trie.details.node"></a>Node Invariants</h6></div></div></div><p>Trie-based containers support node invariants, as do
1075           tree-based containers. There are two minor
1076           differences, though, which, unfortunately, thwart sharing them
1077           sharing the same node-updating policies:</p><div class="orderedlist"><ol class="orderedlist" type="1"><li class="listitem"><p>A trie's <code class="classname">Node_Update</code> template-template
1078               parameter is parametrized by <code class="classname">E_Access_Traits</code>, while
1079               a tree's <code class="classname">Node_Update</code> template-template parameter is
1080             parametrized by <code class="classname">Cmp_Fn</code>.</p></li><li class="listitem"><p>Tree-based containers store values in all nodes, while
1081             trie-based containers (at least in this implementation) store
1082             values in leafs.</p></li></ol></div><p>The graphic below shows the scheme, as well as some predefined
1083           policies (which are explained below).</p><div class="figure"><a id="id-1.3.5.9.4.4.4.3.3.5"></a><p class="title"><strong>Figure 22.29. A trie and its update policy</strong></p><div class="figure-contents"><div class="mediaobject" align="center"><img src="../images/pbds_trie_node_updator_policy_cd.png" align="middle" alt="A trie and its update policy" /></div></div></div><br class="figure-break" /><p>This library offers the following pre-defined trie node
1084           updating policies:</p><div class="orderedlist"><ol class="orderedlist" type="1"><li class="listitem"><p>
1085                 <code class="classname">trie_order_statistics_node_update</code>
1086                 supports order statistics.
1087               </p></li><li class="listitem"><p><code class="classname">trie_prefix_search_node_update</code>
1088             supports searching for ranges that match a given prefix.</p></li><li class="listitem"><p><code class="classname">null_node_update</code>
1089             is the null node updater.</p></li></ol></div></div><div class="section"><div class="titlepage"><div><div><h6 class="title"><a id="container.trie.details.split"></a>Split and Join</h6></div></div></div><p>Trie-based containers support split and join methods; the
1090           rationale is equal to that of tree-based containers supporting
1091           these methods.</p></div></div></div><div class="section"><div class="titlepage"><div><div><h4 class="title"><a id="pbds.design.container.list"></a>List</h4></div></div></div><div class="section"><div class="titlepage"><div><div><h5 class="title"><a id="container.list.interface"></a>Interface</h5></div></div></div><p>The list-based container has the following declaration:</p><pre class="programlisting">
1092           template&lt;typename Key,
1093           typename Mapped,
1094           typename Eq_Fn = std::equal_to&lt;Key&gt;,
1095           typename Update_Policy = move_to_front_lu_policy&lt;&gt;,
1096           typename Allocator = std::allocator&lt;char&gt; &gt;
1097           class list_update;
1098         </pre><p>The parameters have the following meaning:</p><div class="orderedlist"><ol class="orderedlist" type="1"><li class="listitem"><p>
1099               <code class="classname">Key</code> is the key type.
1100             </p></li><li class="listitem"><p>
1101               <code class="classname">Mapped</code> is the mapped-policy.
1102             </p></li><li class="listitem"><p>
1103               <code class="classname">Eq_Fn</code> is a key equivalence functor.
1104             </p></li><li class="listitem"><p>
1105               <code class="classname">Update_Policy</code> is a policy updating positions in
1106               the list based on access patterns. It is described in the
1107               following subsection.
1108             </p></li><li class="listitem"><p>
1109               <code class="classname">Allocator</code> is an allocator type.
1110             </p></li></ol></div><p>A list-based associative container is a container that
1111         stores elements in a linked-list. It does not order the elements
1112         by any particular order related to the keys.  List-based
1113         containers are primarily useful for creating "multimaps". In fact,
1114         list-based containers are designed in this library expressly for
1115         this purpose.</p><p>List-based containers might also be useful for some rare
1116         cases, where a key is encapsulated to the extent that only
1117         key-equivalence can be tested. Hash-based containers need to know
1118         how to transform a key into a size type, and tree-based containers
1119         need to know if some key is larger than another.  List-based
1120         associative containers, conversely, only need to know if two keys
1121         are equivalent.</p><p>Since a list-based associative container does not order
1122         elements by keys, is it possible to order the list in some
1123         useful manner? Remarkably, many on-line competitive
1124         algorithms exist for reordering lists to reflect access
1125         prediction. (See <a class="xref" href="policy_data_structures.html#biblio.motwani95random" title="Randomized Algorithms">[biblio.motwani95random]</a> and <a class="xref" href="policy_data_structures.html#biblio.andrew04mtf" title="MTF, Bit, and COMB: A Guide to Deterministic and Randomized Algorithms for the List Update Problem">[biblio.andrew04mtf]</a>).
1126         </p></div><div class="section"><div class="titlepage"><div><div><h5 class="title"><a id="container.list.details"></a>Details</h5></div></div></div><p>
1127         </p><div class="section"><div class="titlepage"><div><div><h6 class="title"><a id="container.list.details.ds"></a>Underlying Data Structure</h6></div></div></div><p>The graphic below shows a
1128           simple list of integer keys. If we search for the integer 6, we
1129           are paying an overhead: the link with key 6 is only the fifth
1130           link; if it were the first link, it could be accessed
1131           faster.</p><div class="figure"><a id="id-1.3.5.9.4.4.5.3.3.3"></a><p class="title"><strong>Figure 22.30. A simple list</strong></p><div class="figure-contents"><div class="mediaobject" align="center"><img src="../images/pbds_simple_list.png" align="middle" alt="A simple list" /></div></div></div><br class="figure-break" /><p>List-update algorithms reorder lists as elements are
1132           accessed. They try to determine, by the access history, which
1133           keys to move to the front of the list. Some of these algorithms
1134           require adding some metadata alongside each entry.</p><p>For example, in the graphic below label A shows the counter
1135           algorithm. Each node contains both a key and a count metadata
1136           (shown in bold). When an element is accessed (e.g. 6) its count is
1137           incremented, as shown in label B. If the count reaches some
1138           predetermined value, say 10, as shown in label C, the count is set
1139           to 0 and the node is moved to the front of the list, as in label
1140           D.
1141           </p><div class="figure"><a id="id-1.3.5.9.4.4.5.3.3.6"></a><p class="title"><strong>Figure 22.31. The counter algorithm</strong></p><div class="figure-contents"><div class="mediaobject" align="center"><img src="../images/pbds_list_update.png" align="middle" alt="The counter algorithm" /></div></div></div><br class="figure-break" /></div><div class="section"><div class="titlepage"><div><div><h6 class="title"><a id="container.list.details.policies"></a>Policies</h6></div></div></div><p>this library allows instantiating lists with policies
1142           implementing any algorithm moving nodes to the front of the
1143           list (policies implementing algorithms interchanging nodes are
1144           unsupported).</p><p>Associative containers based on lists are parametrized by a
1145           <code class="classname">Update_Policy</code> parameter. This parameter defines the
1146           type of metadata each node contains, how to create the
1147           metadata, and how to decide, using this metadata, whether to
1148           move a node to the front of the list. A list-based associative
1149           container object derives (publicly) from its update policy.
1150           </p><p>An instantiation of <code class="classname">Update_Policy</code> must define
1151           internally <code class="classname">update_metadata</code> as the metadata it
1152           requires. Internally, each node of the list contains, besides
1153           the usual key and data, an instance of <code class="classname">typename
1154           Update_Policy::update_metadata</code>.</p><p>An instantiation of <code class="classname">Update_Policy</code> must define
1155           internally two operators:</p><pre class="programlisting">
1156             update_metadata
1157             operator()();
1158
1159             bool
1160             operator()(update_metadata &amp;);
1161           </pre><p>The first is called by the container object, when creating a
1162           new node, to create the node's metadata. The second is called
1163           by the container object, when a node is accessed (
1164           when a find operation's key is equivalent to the key of the
1165           node), to determine whether to move the node to the front of
1166           the list.
1167           </p><p>The library contains two predefined implementations of
1168           list-update policies. The first
1169           is <code class="classname">lu_counter_policy</code>, which implements the
1170           counter algorithm described above. The second is
1171           <code class="classname">lu_move_to_front_policy</code>,
1172           which unconditionally move an accessed element to the front of
1173           the list. The latter type is very useful in this library,
1174           since there is no need to associate metadata with each element.
1175           (See <a class="xref" href="policy_data_structures.html#biblio.andrew04mtf" title="MTF, Bit, and COMB: A Guide to Deterministic and Randomized Algorithms for the List Update Problem">[biblio.andrew04mtf]</a>
1176           </p></div><div class="section"><div class="titlepage"><div><div><h6 class="title"><a id="container.list.details.mapped"></a>Use in Multimaps</h6></div></div></div><p>In this library, there are no equivalents for the standard's
1177           multimaps and multisets; instead one uses an associative
1178           container mapping primary keys to secondary keys.</p><p>List-based containers are especially useful as associative
1179           containers for secondary keys. In fact, they are implemented
1180           here expressly for this purpose.</p><p>To begin with, these containers use very little per-entry
1181           structure memory overhead, since they can be implemented as
1182           singly-linked lists. (Arrays use even lower per-entry memory
1183           overhead, but they are less flexible in moving around entries,
1184           and have weaker invalidation guarantees).</p><p>More importantly, though, list-based containers use very
1185           little per-container memory overhead. The memory overhead of an
1186           empty list-based container is practically that of a pointer.
1187           This is important for when they are used as secondary
1188           associative-containers in situations where the average ratio of
1189           secondary keys to primary keys is low (or even 1).</p><p>In order to reduce the per-container memory overhead as much
1190           as possible, they are implemented as closely as possible to
1191           singly-linked lists.</p><div class="orderedlist"><ol class="orderedlist" type="1"><li class="listitem"><p>
1192                 List-based containers do not store internally the number
1193                 of values that they hold. This means that their <code class="function">size</code>
1194                 method has linear complexity (just like <code class="classname">std::list</code>).
1195                 Note that finding the number of equivalent-key values in a
1196                 standard multimap also has linear complexity (because it must be
1197                 done,  via <code class="function">std::distance</code> of the
1198                 multimap's <code class="function">equal_range</code> method), but usually with
1199                 higher constants.
1200               </p></li><li class="listitem"><p>
1201                 Most associative-container objects each hold a policy
1202                 object (a hash-based container object holds a
1203                 hash functor). List-based containers, conversely, only have
1204                 class-wide policy objects.
1205               </p></li></ol></div></div></div></div><div class="section"><div class="titlepage"><div><div><h4 class="title"><a id="pbds.design.container.priority_queue"></a>Priority Queue</h4></div></div></div><div class="section"><div class="titlepage"><div><div><h5 class="title"><a id="container.priority_queue.interface"></a>Interface</h5></div></div></div><p>The priority queue container has the following
1206         declaration:
1207         </p><pre class="programlisting">
1208           template&lt;typename  Value_Type,
1209           typename  Cmp_Fn = std::less&lt;Value_Type&gt;,
1210           typename  Tag = pairing_heap_tag,
1211           typename  Allocator = std::allocator&lt;char &gt; &gt;
1212           class priority_queue;
1213         </pre><p>The parameters have the following meaning:</p><div class="orderedlist"><ol class="orderedlist" type="1"><li class="listitem"><p><code class="classname">Value_Type</code> is the value type.</p></li><li class="listitem"><p><code class="classname">Cmp_Fn</code> is a value comparison functor</p></li><li class="listitem"><p><code class="classname">Tag</code> specifies which underlying data structure
1214           to use.</p></li><li class="listitem"><p><code class="classname">Allocator</code> is an allocator
1215           type.</p></li></ol></div><p>The <code class="classname">Tag</code> parameter specifies which underlying
1216         data structure to use. Instantiating it by<code class="classname">pairing_heap_tag</code>,<code class="classname">binary_heap_tag</code>,
1217         <code class="classname">binomial_heap_tag</code>,
1218         <code class="classname">rc_binomial_heap_tag</code>,
1219         or <code class="classname">thin_heap_tag</code>,
1220         specifies, respectively,
1221         an underlying pairing heap (<a class="xref" href="policy_data_structures.html#biblio.fredman86pairing" title="The pairing heap: a new form of self-adjusting heap">[biblio.fredman86pairing]</a>),
1222         binary heap (<a class="xref" href="policy_data_structures.html#biblio.clrs2001" title="Introduction to Algorithms, 2nd edition">[biblio.clrs2001]</a>),
1223         binomial heap (<a class="xref" href="policy_data_structures.html#biblio.clrs2001" title="Introduction to Algorithms, 2nd edition">[biblio.clrs2001]</a>),
1224         a binomial heap with a redundant binary counter (<a class="xref" href="policy_data_structures.html#biblio.maverik_lowerbounds" title="Deamortization - Part 2: Binomial Heaps">[biblio.maverik_lowerbounds]</a>),
1225         or a thin heap (<a class="xref" href="policy_data_structures.html#biblio.kt99fat_heaps" title="New Heap Data Structures">[biblio.kt99fat_heaps]</a>).
1226         </p><p>
1227           As mentioned in the tutorial,
1228           <code class="classname">__gnu_pbds::priority_queue</code> shares most of the
1229           same interface with <code class="classname">std::priority_queue</code>.
1230           E.g. if <code class="varname">q</code> is a priority queue of type
1231           <code class="classname">Q</code>, then <code class="function">q.top()</code> will
1232           return the "largest" value in the container (according to
1233           <code class="classname">typename
1234           Q::cmp_fn</code>). <code class="classname">__gnu_pbds::priority_queue</code>
1235           has a larger (and very slightly different) interface than
1236           <code class="classname">std::priority_queue</code>, however, since typically
1237           <code class="classname">push</code> and <code class="classname">pop</code> are deemed
1238         insufficient for manipulating priority-queues. </p><p>Different settings require different priority-queue
1239         implementations which are described in later; see traits
1240         discusses ways to differentiate between the different traits of
1241         different implementations.</p></div><div class="section"><div class="titlepage"><div><div><h5 class="title"><a id="container.priority_queue.details"></a>Details</h5></div></div></div><div class="section"><div class="titlepage"><div><div><h6 class="title"><a id="container.priority_queue.details.iterators"></a>Iterators</h6></div></div></div><p>There are many different underlying-data structures for
1242           implementing priority queues. Unfortunately, most such
1243           structures are oriented towards making <code class="function">push</code> and
1244           <code class="function">top</code> efficient, and consequently don't allow efficient
1245           access of other elements: for instance, they cannot support an efficient
1246           <code class="function">find</code> method. In the use case where it
1247           is important to both access and "do something with" an
1248           arbitrary value, one would be out of luck. For example, many graph algorithms require
1249           modifying a value (typically increasing it in the sense of the
1250           priority queue's comparison functor).</p><p>In order to access and manipulate an arbitrary value in a
1251           priority queue, one needs to reference the internals of the
1252           priority queue from some form of an associative container -
1253           this is unavoidable. Of course, in order to maintain the
1254           encapsulation of the priority queue, this needs to be done in a
1255           way that minimizes exposure to implementation internals.</p><p>In this library the priority queue's <code class="function">insert</code>
1256           method returns an iterator, which if valid can be used for subsequent <code class="function">modify</code> and
1257           <code class="function">erase</code> operations. This both preserves the priority
1258           queue's encapsulation, and allows accessing arbitrary values (since the
1259           returned iterators from the <code class="function">push</code> operation can be
1260           stored in some form of associative container).</p><p>Priority queues' iterators present a problem regarding their
1261           invalidation guarantees. One assumes that calling
1262           <code class="function">operator++</code> on an iterator will associate it
1263           with the "next" value. Priority-queues are
1264           self-organizing: each operation changes what the "next" value
1265           means. Consequently, it does not make sense that <code class="function">push</code>
1266           will return an iterator that can be incremented - this can have
1267           no possible use. Also, as in the case of hash-based containers,
1268           it is awkward to define if a subsequent <code class="function">push</code> operation
1269           invalidates a prior returned iterator: it invalidates it in the
1270           sense that its "next" value is not related to what it
1271           previously considered to be its "next" value. However, it might not
1272           invalidate it, in the sense that it can be
1273           de-referenced and used for <code class="function">modify</code> and <code class="function">erase</code>
1274           operations.</p><p>Similarly to the case of the other unordered associative
1275           containers, this library uses a distinction between
1276           point-type and range type iterators. A priority queue's <code class="classname">iterator</code> can always be
1277           converted to a <code class="classname">point_iterator</code>, and a
1278           <code class="classname">const_iterator</code> can always be converted to a
1279           <code class="classname">point_const_iterator</code>.</p><p>The following snippet demonstrates manipulating an arbitrary
1280           value:</p><pre class="programlisting">
1281             // A priority queue of integers.
1282             priority_queue&lt;int &gt; p;
1283
1284             // Insert some values into the priority queue.
1285             priority_queue&lt;int &gt;::point_iterator it = p.push(0);
1286
1287             p.push(1);
1288             p.push(2);
1289
1290             // Now modify a value.
1291             p.modify(it, 3);
1292
1293             assert(p.top() == 3);
1294           </pre><p>It should be noted that an alternative design could embed an
1295           associative container in a priority queue. Could, but most
1296           probably should not. To begin with, it should be noted that one
1297           could always encapsulate a priority queue and an associative
1298           container mapping values to priority queue iterators with no
1299           performance loss. One cannot, however, "un-encapsulate" a priority
1300           queue embedding an associative container, which might lead to
1301           performance loss. Assume, that one needs to associate each value
1302           with some data unrelated to priority queues. Then using
1303           this library's design, one could use an
1304           associative container mapping each value to a pair consisting of
1305           this data and a priority queue's iterator. Using the embedded
1306           method would need to use two associative containers. Similar
1307           problems might arise in cases where a value can reside
1308           simultaneously in many priority queues.</p></div><div class="section"><div class="titlepage"><div><div><h6 class="title"><a id="container.priority_queue.details.d"></a>Underlying Data Structure</h6></div></div></div><p>There are three main implementations of priority queues: the
1309           first employs a binary heap, typically one which uses a
1310           sequence; the second uses a tree (or forest of trees), which is
1311           typically less structured than an associative container's tree;
1312           the third simply uses an associative container. These are
1313           shown in the graphic below, in labels A1 and A2, label B, and label C.</p><div class="figure"><a id="id-1.3.5.9.4.4.6.3.3.3"></a><p class="title"><strong>Figure 22.32. Underlying Priority-Queue Data-Structures.</strong></p><div class="figure-contents"><div class="mediaobject" align="center"><img src="../images/pbds_priority_queue_different_underlying_dss.png" align="middle" alt="Underlying Priority-Queue Data-Structures." /></div></div></div><br class="figure-break" /><p>Roughly speaking, any value that is both pushed and popped
1314           from a priority queue must incur a logarithmic expense (in the
1315           amortized sense). Any priority queue implementation that would
1316           avoid this, would violate known bounds on comparison-based
1317           sorting (see <a class="xref" href="policy_data_structures.html#biblio.clrs2001" title="Introduction to Algorithms, 2nd edition">[biblio.clrs2001]</a> and <a class="xref" href="policy_data_structures.html#biblio.brodal96priority" title="Worst-case efficient priority queues">[biblio.brodal96priority]</a>).
1318           </p><p>Most implementations do
1319           not differ in the asymptotic amortized complexity of
1320           <code class="function">push</code> and <code class="function">pop</code> operations, but they differ in
1321           the constants involved, in the complexity of other operations
1322           (e.g., <code class="function">modify</code>), and in the worst-case
1323           complexity of single operations. In general, the more
1324           "structured" an implementation (i.e., the more internal
1325           invariants it possesses) - the higher its amortized complexity
1326           of <code class="function">push</code> and <code class="function">pop</code> operations.</p><p>This library implements different algorithms using a
1327           single class: <code class="classname">priority_queue</code>.
1328           Instantiating the <code class="classname">Tag</code> template parameter, "selects"
1329           the implementation:</p><div class="orderedlist"><ol class="orderedlist" type="1"><li class="listitem"><p>
1330               Instantiating <code class="classname">Tag = binary_heap_tag</code> creates
1331               a binary heap of the form in represented in the graphic with labels A1 or A2. The former is internally
1332               selected by priority_queue
1333               if <code class="classname">Value_Type</code> is instantiated by a primitive type
1334               (e.g., an <span class="type">int</span>); the latter is
1335               internally selected for all other types (e.g.,
1336               <code class="classname">std::string</code>). This implementations is relatively
1337               unstructured, and so has good <code class="classname">push</code> and <code class="classname">pop</code>
1338               performance; it is the "best-in-kind" for primitive
1339               types, e.g., <span class="type">int</span>s. Conversely, it has
1340               high worst-case performance, and can support only linear-time
1341             <code class="function">modify</code> and <code class="function">erase</code> operations.</p></li><li class="listitem"><p>Instantiating <code class="classname">Tag =
1342             pairing_heap_tag</code> creates a pairing heap of the form
1343             in represented by label B in the graphic above. This
1344             implementations too is relatively unstructured, and so has good
1345             <code class="function">push</code> and <code class="function">pop</code>
1346             performance; it is the "best-in-kind" for non-primitive types,
1347             e.g., <code class="classname">std:string</code>s. It also has very good
1348             worst-case <code class="function">push</code> and
1349             <code class="function">join</code> performance (O(1)), but has high
1350             worst-case <code class="function">pop</code>
1351             complexity.</p></li><li class="listitem"><p>Instantiating <code class="classname">Tag =
1352             binomial_heap_tag</code> creates a binomial heap of the
1353             form repsented by label B in the graphic above. This
1354             implementations is more structured than a pairing heap, and so
1355             has worse <code class="function">push</code> and <code class="function">pop</code>
1356             performance. Conversely, it has sub-linear worst-case bounds for
1357             <code class="function">pop</code>, e.g., and so it might be preferred in
1358             cases where responsiveness is important.</p></li><li class="listitem"><p>Instantiating <code class="classname">Tag =
1359             rc_binomial_heap_tag</code> creates a binomial heap of the
1360             form represented in label B above, accompanied by a redundant
1361             counter which governs the trees. This implementations is
1362             therefore more structured than a binomial heap, and so has worse
1363             <code class="function">push</code> and <code class="function">pop</code>
1364             performance. Conversely, it guarantees O(1)
1365             <code class="function">push</code> complexity, and so it might be
1366             preferred in cases where the responsiveness of a binomial heap
1367             is insufficient.</p></li><li class="listitem"><p>Instantiating <code class="classname">Tag =
1368             thin_heap_tag</code> creates a thin heap of the form
1369             represented by the label B in the graphic above. This
1370             implementations too is more structured than a pairing heap, and
1371             so has worse <code class="function">push</code> and
1372             <code class="function">pop</code> performance. Conversely, it has better
1373             worst-case and identical amortized complexities than a Fibonacci
1374             heap, and so might be more appropriate for some graph
1375             algorithms.</p></li></ol></div><p>Of course, one can use any order-preserving associative
1376           container as a priority queue, as in the graphic above label C, possibly by creating an adapter class
1377           over the associative container (much as
1378           <code class="classname">std::priority_queue</code> can adapt <code class="classname">std::vector</code>).
1379           This has the advantage that no cross-referencing is necessary
1380           at all; the priority queue itself is an associative container.
1381           Most associative containers are too structured to compete with
1382           priority queues in terms of <code class="function">push</code> and <code class="function">pop</code>
1383           performance.</p></div><div class="section"><div class="titlepage"><div><div><h6 class="title"><a id="container.priority_queue.details.traits"></a>Traits</h6></div></div></div><p>It would be nice if all priority queues could
1384           share exactly the same behavior regardless of implementation. Sadly, this is not possible. Just one for instance is in join operations: joining
1385           two binary heaps might throw an exception (not corrupt
1386           any of the heaps on which it operates), but joining two pairing
1387           heaps is exception free.</p><p>Tags and traits are very useful for manipulating generic
1388           types. <code class="classname">__gnu_pbds::priority_queue</code>
1389           publicly defines <code class="classname">container_category</code> as one of the tags. Given any
1390           container <code class="classname">Cntnr</code>, the tag of the underlying
1391           data structure can be found via <code class="classname">typename
1392           Cntnr::container_category</code>; this is one of the possible tags shown in the graphic below.
1393           </p><div class="figure"><a id="id-1.3.5.9.4.4.6.3.4.4"></a><p class="title"><strong>Figure 22.33. Priority-Queue Data-Structure Tags.</strong></p><div class="figure-contents"><div class="mediaobject" align="center"><img src="../images/pbds_priority_queue_tag_hierarchy.png" align="middle" alt="Priority-Queue Data-Structure Tags." /></div></div></div><br class="figure-break" /><p>Additionally, a traits mechanism can be used to query a
1394           container type for its attributes. Given any container
1395           <code class="classname">Cntnr</code>, then </p><pre class="programlisting">__gnu_pbds::container_traits&lt;Cntnr&gt;</pre><p>
1396           is a traits class identifying the properties of the
1397           container.</p><p>To find if a container might throw if two of its objects are
1398           joined, one can use
1399           </p><pre class="programlisting">
1400             container_traits&lt;Cntnr&gt;::split_join_can_throw
1401           </pre><p>
1402           </p><p>
1403             Different priority-queue implementations have different invalidation guarantees. This is
1404             especially important, since there is no way to access an arbitrary
1405             value of priority queues except for iterators. Similarly to
1406             associative containers, one can use
1407             </p><pre class="programlisting">
1408               container_traits&lt;Cntnr&gt;::invalidation_guarantee
1409             </pre><p>
1410           to get the invalidation guarantee type of a priority queue.</p><p>It is easy to understand from the graphic above, what <code class="classname">container_traits&lt;Cntnr&gt;::invalidation_guarantee</code>
1411           will be for different implementations. All implementations of
1412           type represented by label B have <code class="classname">point_invalidation_guarantee</code>:
1413           the container can freely internally reorganize the nodes -
1414           range-type iterators are invalidated, but point-type iterators
1415           are always valid. Implementations of type represented by labels A1 and A2 have <code class="classname">basic_invalidation_guarantee</code>:
1416           the container can freely internally reallocate the array - both
1417           point-type and range-type iterators might be invalidated.</p><p>
1418             This has major implications, and constitutes a good reason to avoid
1419             using binary heaps. A binary heap can perform <code class="function">modify</code>
1420             or <code class="function">erase</code> efficiently given a valid point-type
1421             iterator. However, in order to supply it with a valid point-type
1422             iterator, one needs to iterate (linearly) over all
1423             values, then supply the relevant iterator (recall that a
1424             range-type iterator can always be converted to a point-type
1425             iterator). This means that if the number of <code class="function">modify</code> or
1426             <code class="function">erase</code> operations is non-negligible (say
1427             super-logarithmic in the total sequence of operations) - binary
1428             heaps will perform badly.
1429           </p></div></div></div></div></div><div class="navfooter"><hr /><table width="100%" summary="Navigation footer"><tr><td width="40%" align="left"><a accesskey="p" href="policy_data_structures_using.html">Prev</a> </td><td width="20%" align="center"><a accesskey="u" href="policy_data_structures.html">Up</a></td><td width="40%" align="right"> <a accesskey="n" href="policy_based_data_structures_test.html">Next</a></td></tr><tr><td width="40%" align="left" valign="top">Using </td><td width="20%" align="center"><a accesskey="h" href="../index.html">Home</a></td><td width="40%" align="right" valign="top"> Testing</td></tr></table></div></body></html>