doc/applications.tex

   1 \section{\texorpdfstring{Applications included
   2 in the \protect\ai[\tt]{barvinok} distribution}
   3 {Applications included in the barvinok distribution}}
   4 \label{a:usage}
   5
   6 \index{barvinok@{\tt  barvinok}!availability}
   7 {\sloppy
   8 This section describes some application programs
   9 provided by the \barvinok/ library,
  10 available from {\tt http://freshmeat.net/projects/barvinok/.}
  11 For compilation instructions we refer to the \verb+README+ file
  12 included in the distribution.
  13 }
  14
  15 Common option to all programs:\\
  16 \begin{tabular}{lll}
  17 \ai[\tt]{--version} & \ai[\tt]{-V} & print version
  18 \\
  19 \ai[\tt]{--help} & \ai[\tt]{-?} & list available options
  20 \end{tabular}
  21
  22 \subsection{\texorpdfstring{\protect\ai[\tt]{barvinok\_count}}{barvinok\_count}}
  23
  24 The program \ai[\tt]{barvinok\_count} enumerates a
  25 non-parametric polytope.  It takes one polytope
  26 in \PolyLib/ notation as input and prints the number
  27 of integer points in the polytope.
  28 The \PolyLib/ notation corresponds to the internal
  29 representation of \ai[\tt]{Polyhedron}s as explained
  30 in Section~\ref{a:existing}.
  31 \index{input format!constraints}
  32 The first line of the input contains the number of rows
  33 and the number of columns in the \ai[\tt]{Constraint} matrix.
  34 The rest of the input is composed of the elements of the matrix.
  35 Recall that the number of columns is two more than the number
  36 of variables, where the extra first columns is one or zero
  37 depending on whether the constraint is an inequality ($\ge 0$)
  38 or an equality ($= 0$).  The next columns contain
  39 the coefficients of the variables and the final column contains
  40 the constant in the constraint.
  41 E.g., the set
  42 $S = \lb\, s \mid s \geq 0 \wedge  2 s \leq 13 \, \rb$
  43 from \citeN[Example~38 on page~134]{Verdoolaege2005PhD}
  44 corresponds to the following input and
  45 output.
  46 \begin{verbatim}
  47 > cat S
  48 2 3
  49
  50 1 1 0
  51 1 -2 13
  52 > ./barvinok_count  < S
  53 POLYHEDRON Dimension:1
  54            Constraints:2  Equations:0  Rays:2  Lines:0
  55 Constraints 2 3
  56 Inequality: [   1   0 ]
  57 Inequality: [  -2  13 ]
  58 Rays 2 3
  59 Vertex: [   0 ]/1
  60 Vertex: [  13 ]/2
  61    7
  62 \end{verbatim}
  63 \index{PolyLib@{\tt  PolyLib}!version 5.22.0 or newer}
  64 Note that if you use \PolyLib/ version 5.22.0 or newer then the output
  65 may look slightly different as the computation of the \ai[\tt]{Rays}
  66 may have been postponed to a later stage.
  67 The program \ai[\tt]{latte2polylib.pl} can be used to
  68 convert a polytope from \ai[\tt]{LattE} \shortcite{latte1.1}
  69 notation to \PolyLib/ notation.
  70
  71 \index{input format!vertices}
  72 As an alternative to the constraints based input, the input polytope
  73 may also be specified by its \ai[\tt]{Ray} matrix.
  74 The first line of the input contains the single word \ai[\tt]{vertices}.
  75 The second line contains the number of rows
  76 and the number of columns in the \ai[\tt]{Ray} matrix.
  77 The rest of the input is composed of the elements of the matrix.
  78 Recall that the number of columns is two more than the number
  79 of variables, where the extra first columns is one or zero
  80 depending on whether the ray is a line or not.
  81 The next columns contain
  82 the numerators of the coordinates and the final column contains
  83 the denominator of the vertex or $0$ for a ray.
  84 E.g., the above set can also be described as
  85 \begin{verbatim}
  86 vertices
  87
  88 2 3
  89
  90 1 0 1
  91 1 13 2
  92 \end{verbatim}
  93
  94 \subsection{\texorpdfstring{\protect\ai[\tt]{barvinok\_enumerate}}
  95 {barvinok\_enumerate}}
  96
  97 The program \ai[\tt]{barvinok\_enumerate} enumerates a
  98 parametric polytope as a \psp/ or \rgf/.  It takes two polytopes in \PolyLib/
  99 notation as input, optionally followed by a list of parameter
 100 names.
 101 The two polytopes refer to arguments \verb+P+ and \verb+C+
 102 of the corresponding function. (See Section~\ref{a:counting:functions}.)
 103 The following example was taken by \shortciteN{Loechner1999}
 104 from \shortciteN[Chapter II.2]{Loechner97phd}.
 105 \begin{verbatim}
 106 > cat loechner
 107 # Dimension of the matrix:
 108 7 7
 109 # Constraints:
 110 # i j k P Q cte
 111 1 1 0 0 0 0 0 # 0 <= i
 112 1 -1 0 0 1 0 0 # i <= P
 113 1 0 1 0 0 0 0 # 0 <= j
 114 1 1 -1 0 0 0 0 # j <= i
 115 1 0 0 1 0 0 0 # 0 <= k
 116 1 1 -1 -1 0 0 0 # k <= i-j
 117 0 1 1 1 0 -1 0 # Q = i + j + k
 118
 119 # 2 parameters, no constraints.
 120 0 4
 121 > ./barvinok_enumerate < loechner
 122 POLYHEDRON Dimension:5
 123            Constraints:6  Equations:1  Rays:5  Lines:0
 124 Constraints 6 7
 125 Equality:   [   1   1   1   0  -1   0 ]
 126 Inequality: [   0   1   1   1  -1   0 ]
 127 Inequality: [   0   1   0   0   0   0 ]
 128 Inequality: [   0   0   1   0   0   0 ]
 129 Inequality: [   0  -2  -2   0   1   0 ]
 130 Inequality: [   0   0   0   0   0   1 ]
 131 Rays 5 7
 132 Ray:    [   1   0   1   1   2 ]
 133 Ray:    [   1   1   0   1   2 ]
 134 Vertex: [   0   0   0   0   0 ]/1
 135 Ray:    [   0   0   0   1   0 ]
 136 Ray:    [   1   0   0   1   1 ]
 137 POLYHEDRON Dimension:2
 138            Constraints:1  Equations:0  Rays:3  Lines:2
 139 Constraints 1 4
 140 Inequality: [   0   0   1 ]
 141 Rays 3 4
 142 Line:   [   1   0 ]
 143 Line:   [   0   1 ]
 144 Vertex: [   0   0 ]/1
 145          - P + Q  >= 0
 146          2P - Q  >= 0
 147           1 >= 0
 148
 149 ( -1/2 * P^2 + ( 1 * Q + 1/2 )
 150  * P + ( -3/8 * Q^2 + ( -1/2 * {( 1/2 * Q + 0 )
 151 } + 1/4 )
 152  * Q + ( -5/4 * {( 1/2 * Q + 0 )
 153 } + 1 )
 154  )
 155  )
 156          Q  >= 0
 157          P - Q  -1 >= 0
 158           1 >= 0
 159
 160 ( 1/8 * Q^2 + ( -1/2 * {( 1/2 * Q + 0 )
 161 } + 3/4 )
 162  * Q + ( -5/4 * {( 1/2 * Q + 0 )
 163 } + 1 )
 164  )
 165 \end{verbatim}
 166 The output corresponds to
 167 $$
 168 \begin{cases}
 169 \makebox[0pt][l]{$-\frac 1 2 P^2 + P Q + \frac 1 2 P - \frac 3 8 Q^2
 170 + \left( \frac 1 4 - \frac 1 2 \left\{ \frac 1 2 Q \right\} \right)
 171                 Q + 1
 172 - \frac 5 4 \left\{ \frac 1 2 Q \right\}$} \\
 173 &
 174 \hbox{if $P \le Q \le 2 P$}
 175 \\
 176 \frac 1 8 Q^2 +
 177 \left( \frac 3 4 - \frac 1 2 \left\{ \frac 1 2 Q \right\} \right)
 178 - \frac 5 4 \left\{ \frac 1 2 Q \right\}
 179 \qquad
 180 \qquad
 181 \qquad
 182 \qquad
 183 \qquad
 184 &
 185 \hbox{if $0 \le Q \le P-1$}
 186 .
 187 \end{cases}
 188 $$
 189 The following is an example of Petr Lison\u{e}k\index{Lison\u{e}k, P.}.
 190 \begin{verbatim}
 191 > cat petr
 192 4 6
 193 1 -1 -1 -1 1 0
 194 1 1 -1 0 0 0
 195 1 0 1 -1 0 0
 196 1 0 0 1 0 -1
 197
 198 0 3
 199 n
 200 > ./barvinok_enumerate --series < petr
 201 POLYHEDRON Dimension:4
 202            Constraints:5  Equations:0  Rays:5  Lines:0
 203 Constraints 5 6
 204 Inequality: [  -1  -1  -1   1   0 ]
 205 Inequality: [   1  -1   0   0   0 ]
 206 Inequality: [   0   1  -1   0   0 ]
 207 Inequality: [   0   0   1   0  -1 ]
 208 Inequality: [   0   0   0   0   1 ]
 209 Rays 5 6
 210 Ray:    [   1   1   1   3 ]
 211 Ray:    [   1   1   0   2 ]
 212 Ray:    [   1   0   0   1 ]
 213 Ray:    [   0   0   0   1 ]
 214 Vertex: [   1   1   1   3 ]/1
 215 POLYHEDRON Dimension:1
 216            Constraints:1  Equations:0  Rays:2  Lines:1
 217 Constraints 1 3
 218 Inequality: [   0   1 ]
 219 Rays 2 3
 220 Line:   [   1 ]
 221 Vertex: [   0 ]/1
 222 (n^3)/((1-n) * (1-n) * (1-n^2) * (1-n^3))
 223 \end{verbatim}
 224
 225 Options:\\
 226 \begin{tabular}{llp{0.7\textwidth}}
 227 \ai[\tt]{--floor} & \ai[\tt]{-f} &
 228 convert \ai[\tt]{fractional}s to \ai[\tt]{flooring}s
 229 \\
 230 \ai[\tt]{--convert} & \ai[\tt]{-c} &
 231 convert \ai[\tt]{fractional}s to \ai[\tt]{periodic}s
 232 \\
 233 \ai[\tt]{--series} & \ai[\tt]{-s} &
 234 compute \rgf/ instead of \psp/
 235 \\
 236 \ai[\tt]{--explicit} & \ai[\tt]{-e} &
 237 convert computed \rgf/ to a \psp/
 238 \end{tabular}
 239
 240 \subsection{\texorpdfstring{\protect\ai[\tt]{barvinok\_enumerate\_e}}
 241 {barvinok\_enumerate\_e}}
 242
 243 The program \ai[\tt]{barvinok\_enumerate\_e} enumerates a
 244 parametric projected set.  It takes a single polytope in \PolyLib/
 245 notation as input, followed by two lines indicating the number
 246 or existential variables and the number of parameters and
 247 optionally followed by a list of parameter names.
 248 The syntax for the line indicating the number of existential
 249 variables is the letter \verb+E+ followed by a space and the actual number.
 250 For indicating the number of parameters, the letter \verb+P+ is used.
 251 The following example corresponds to
 252 \citeN[Example~36 on page~129]{Verdoolaege2005PhD}.
 253 \begin{verbatim}
 254 > cat projected
 255 5 6
 256 #   k   i   j   p   cst
 257 1   0   1   0   0   -1
 258 1   0   -1  0   0   8
 259 1   0   0   1   0   -1
 260 1   0   0   -1  1   0
 261 0   -1  6   9   0   -7
 262
 263 E 2
 264 P 1
 265 > ./barvinok_enumerate_e <projected
 266 POLYHEDRON Dimension:4
 267            Constraints:5  Equations:1  Rays:4  Lines:0
 268 Constraints 5 6
 269 Equality:   [   1  -6  -9   0   7 ]
 270 Inequality: [   0   1   0   0  -1 ]
 271 Inequality: [   0  -1   0   0   8 ]
 272 Inequality: [   0   0   1   0  -1 ]
 273 Inequality: [   0   0  -1   1   0 ]
 274 Rays 4 6
 275 Vertex: [  50   8   1   1 ]/1
 276 Ray:    [   0   0   0   1 ]
 277 Ray:    [   9   0   1   1 ]
 278 Vertex: [   8   1   1   1 ]/1
 279 exist: 2, nparam: 1
 280          P  -3 >= 0
 281           1 >= 0
 282
 283 ( 3 * P + 10 )
 284          P  -1 >= 0
 285          - P + 2 >= 0
 286
 287 ( 8 * P + 0 )
 288 \end{verbatim}
 289
 290 Options:\\
 291 \begin{tabular}{llp{0.7\textwidth}}
 292 \ai[\tt]{--floor} & \ai[\tt]{-f} &
 293 convert \ai[\tt]{fractional}s to \ai[\tt]{flooring}s
 294 \\
 295 \ai[\tt]{--convert} & \ai[\tt]{-c} &
 296 convert \ai[\tt]{fractional}s to \ai[\tt]{periodic}s
 297 \\
 298 \ai[\tt]{--omega} & \ai[\tt]{-o} &
 299 use \Omegalib/ as a preprocessor
 300 \\
 301 \ai[\tt]{--pip} & \ai[\tt]{-p} &
 302 \raggedright
 303 call \ai[\tt]{barvinok\_enumerate\_pip} instead of \ai[\tt]{barvinok\_enumerate\_e}
 304 \end{tabular}
 305
 306 \subsection{\texorpdfstring{\protect\ai[\tt]{barvinok\_union}}
 307 {barvinok\_union}}
 308
 309 The program \ai[\tt]{barvinok\_union} enumerates a \ai{union} of
 310 parametric polytopes.  It takes as input the number of parametric
 311 polytopes in the union, the polytopes in combined data and
 312 parameter space in \PolyLib/ notation, the context in parameter space
 313 in \PolyLib/ notation and optionally a list of parameter names.
 314
 315 Options:\\
 316 \begin{tabular}{llp{0.7\textwidth}}
 317 \ai[\tt]{--series} & \ai[\tt]{-s} &
 318 compute \rgf/ instead of \psp/
 319 \end{tabular}
 320
 321 \subsection{\texorpdfstring{\protect\ai[\tt]{barvinok\_ehrhart}}
 322 {barvinok\_ehrhart}}
 323
 324 \sindex{Ehrhart}{quasi-polynomial}
 325 The program \ai[\tt]{barvinok\_ehrhart} computes the
 326 \ai{Ehrhart quasi-polynomial} of a polytope $P$, i.e., a quasi-polynomial
 327 in $n$ that evaluates to the number of integer points in the dilation
 328 of $P$ by a factor $n$.
 329 The input is the same as that of \ai[\tt]{barvinok\_count}, except that
 330 it may be followed by the variable name.
 331 The functionality is the same as running \ai[\tt]{barvinok\_enumerate}
 332 on the cone over $P$ placed at $n=1$.
 333
 334 Options:\\
 335 \begin{tabular}{llp{0.7\textwidth}}
 336 \ai[\tt]{--floor} & \ai[\tt]{-f} &
 337 convert \ai[\tt]{fractional}s to \ai[\tt]{flooring}s
 338 \\
 339 \ai[\tt]{--convert} & \ai[\tt]{-c} &
 340 convert \ai[\tt]{fractional}s to \ai[\tt]{periodic}s
 341 \\
 342 \ai[\tt]{--series} & \ai[\tt]{-s} &
 343 compute \ai{Ehrhart series} instead of \ai{Ehrhart quasi-polynomial}
 344 \end{tabular}
 345
 346 \subsection{\texorpdfstring{\protect\ai[\tt]{polyhedron\_sample}}
 347 {polyhedron\_sample}}
 348
 349 The program \ai[\tt]{polyhedron\_sample} takes a polytope
 350 in \PolyLib/ notation and prints an integer point in the polytope
 351 if there is one.  The point is computed using
 352 \ai[\tt]{Polyhedron\_Sample}.
 353
 354 \subsection{\texorpdfstring{\protect\ai[\tt]{polytope\_scan}}
 355 {polytope\_scan}}
 356
 357 The program \ai[\tt]{polytope\_scan} takes a polytope in
 358 \PolyLib/ notation and prints a list of all integer points in the polytope.
 359 Unless the \ai[\tt]{--direct} options is given, the order is based
 360 on the \ai{reduced basis} computed with
 361 \ai[\tt]{Polyhedron\_Reduced\_Basis}.
 362
 363 Options:\\
 364 \begin{tabular}{llp{0.7\textwidth}}
 365 \ai[\tt]{--direct} & \ai[\tt]{-d} &
 366 list the points in the lexicographical order
 367 \end{tabular}
 368
 369 \subsection{\texorpdfstring{\protect\ai[\tt]{lexmin}}{lexmin}}
 370
 371 The program \ai[\tt]{lexmin} implements an algorithm for performing
 372 \indac{PIP} based on \rgf/s and provides an alternative for the
 373 technique of \shortciteN{Feautrier88parametric}, which is based
 374 on \ai{cutting plane}s \shortcite{Gomory1963}.
 375 The input is the same as that of the \ai[\tt]{example} program
 376 from \piplib/ \cite{Feautrier:PIP}, except that the value
 377 for the ``\ai{big parameter}'' needs to be $-1$, since there is
 378 no need for big parameters, and it does not read any options
 379 from the input file.
 380
 381 \subsection{\texorpdfstring{\protect\ai[\tt]{barvinok\_summate}}
 382 {barvinok\_summate}}
 383
 384 Given a piecewise quasi-polynomial,
 385 the program \ai[\tt]{barvinok\_summate} computes the \ai{sum} of
 386 the piecewise quasi-polynomial evaluated in all (integer) values of
 387 a subset of the variables.  The result is an expression in the
 388 remaining variables.
 389
 390 The input format corresponds to the {\em output\/} format
 391 of \ai[\tt]{barvinok\_enumerate} and \ai[\tt]{barvinok\_enumerate\_e}.
 392 That is, the program expects a list of guarded quasi-polynomials.
 393 Each guarded quasi-polynomial consists of a domain and a quasi-polynomial,
 394 separated by an empty line.
 395 The domain is specified as a list of constraints, each on a separate
 396 line, consisting of an affine expression in the variables followed
 397 by \verb+>= 0+.
 398 Use the \ai[\tt]{--verbose} option to check that your input
 399 was parsed correctly.
 400 The list of guarded quasi-polynomials may be preceded by
 401 a line specifying the variables over which to sum as
 402 \verb+#variables+ followed by a comma separated list of variable names.
 403
 404 For example
 405 \begin{verbatim}
 406 > cat square_p3
 407 #variables x,y
 408 x -2 >= 0
 409 -3x + n + 9 >= 0
 410 y -4 >= 0
 411 -y +5 >= 0
 412
 413 x * y
 414 > ./barvinok_summate < square_p3
 415          n + 3 >= 0
 416          - n  -1 >= 0
 417
 418 18         n  >= 0
 419           1 >= 0
 420
 421 ( 1/2 * n^2 + ( -3 * {( 1/3 * n + 0 )
 422 } + 21/2 )
 423  * n + ( 9/2 * {( 1/3 * n + 0 )
 424 }^2 + -63/2 * {( 1/3 * n + 0 )
 425 } + 45 )
 426  )
 427 \end{verbatim}
 428
 429 Options:\\
 430 \begin{tabular}{llp{0.7\textwidth}}
 431 \ai[\tt]{--variables} & &
 432 comma separated list of variables over which to sum
 433 \\
 434 \ai[\tt]{--verbose} & \ai[\tt]{-v} &
 435 print parsed piecewise quasi-polynomial
 436 \\
 437 \ai[\tt]{--summation} & &
 438 specifies which summation method to use;
 439 \ai[\tt]{box} refers to the method of
 440 \shortciteN[Section~4.5.4]{Verdoolaege2005PhD}, while
 441 \ai[\tt]{euler} refers to the method of
 442 \autoref{s:euler}
 443 \end{tabular}
 444
 445 \subsection{\texorpdfstring{\protect\ai[\tt]{barvinok\_bound}}
 446 {barvinok\_bound}}
 447
 448 Given a piecewise quasi-polynomial,
 449 the program \ai[\tt]{barvinok\_bound} computes an \ai{upper bound}
 450 (or \ai{lower bound}) for
 451 the values attained by the piecewise quasi-polynomial
 452 over all (integer) values of a subset of the variables.
 453 The result is an expression in the remaining variables.
 454
 455 The input format corresponds to the {\em output\/} format
 456 of \ai[\tt]{barvinok\_enumerate} and \ai[\tt]{barvinok\_enumerate\_e}.
 457 That is, the program expects a list of guarded quasi-polynomials.
 458 Each guarded quasi-polynomial consists of a domain and a quasi-polynomial,
 459 separated by an empty line.
 460 The domain is specified as a list of constraints, each on a separate
 461 line, consisting of an affine expression in the variables followed
 462 by \verb+>= 0+.
 463 Use the \ai[\tt]{--verbose} option to check that your input
 464 was parsed correctly.
 465 The list of guarded quasi-polynomials may be preceded by
 466 a line specifying the variables over which to compute the upper bound as
 467 \verb+#variables+ followed by a comma separated list of variable names.
 468
 469 \begin{verbatim}
 470 > cat devos
 471 #variables V
 472          U + 2V + 3 >= 0
 473          - U -2V  >= 0
 474          - U  10 >= 0
 475          U  >= 0
 476
 477 ( {( 1/3 * U + ( 2/3 * V + 0 ) ) } )
 478 > ./barvinok_bound < devos
 479 (1*U >= 0 && -1*U + 10 >= 0) ? ((2.0/3.0)) : 0
 480 \end{verbatim}
 481
 482 Options:\\
 483 \begin{tabular}{llp{0.7\textwidth}}
 484 \ai[\tt]{--variables} & &
 485 comma separated list of variables over which to compute a bound
 486 \\
 487 \ai[\tt]{--verbose} & \ai[\tt]{-v} &
 488 print parsed piecewise quasi-polynomial
 489 \\
 490 \ai[\tt]{--lower} & &
 491 compute lower bound instead of upper bound
 492 \end{tabular}
 493
 494 \subsection{\texorpdfstring{\protect\ai[\tt]{polytope\_minimize}}
 495 {polytope\_minimize}}
 496
 497 The program \ai[\tt]{polytope\_minimize} takes a polytope
 498 in \PolyLib/ notation and a linear objective function
 499 as input and prints an integer point in the polytope attaining the minimial
 500 value of the objective function.
 501 The objective function is specified as the length of
 502 the vector (the number of variables) followed by
 503 the coefficients of the variables.
 504 The point is computed as explained in \autoref{s:optimization}.
 505
 506 For example
 507 \begin{verbatim}
 508 > cat min_test
 509 8 8
 510    1   34    0    0    0    1    0    0
 511    1    0  -82   -1    0    0    0    0
 512    1    0  -82    0    0    0   -1    0
 513    1    0   31    0    0    1    0    0
 514    1    0    0    0    2   -3    0    0
 515    1    0    0    0    0   -1    0    0
 516    1    0    0    0    0    0    0    1
 517    1   -34 4676  34   -34  21  34  34
 518
 519 6
 520   34 -4676  -34   34  -21  -34
 521 > ./polytope_minimize < min_test
 522 7
 523    2    2 -164  -93  -62 -164    1
 524 \end{verbatim}
 525
 526 \subsection{\texorpdfstring{\protect\ai[\tt]{polyhedron\_integer\_hull}}
 527 {polyhedron\_integer\_hull}}
 528
 529 The program \ai[\tt]{polyhedron\_integer\_hull} takes a polyhedron
 530 in \PolyLib/ notation and
 531 prints its \ai{integer hull}.
 532 The integer hull is computed as explained in \autoref{s:integer:hull}.
 533
 534 \subsection{\texorpdfstring{\protect\ai[\tt]{polytope\_lattice\_width}}
 535 {polytope\_lattice\_width}}
 536
 537 The program \ai[\tt]{polytope\_lattice\_width} computes
 538 the \ai{lattice width} of a parametric polytope.
 539 The input is the same as that of \ai[\tt]{barvinok\_enumerate}.
 540 The lattice width is computed as explained
 541 in \autoref{s:width}.
 542
 543 Options:\\
 544 \begin{tabular}{llp{0.7\textwidth}}
 545 \ai[\tt]{--direction} & \ai[\tt]{-d} &
 546 print the lattice width directions
 547 \end{tabular}