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 \end{tabular}
  19
  20 \subsection{\texorpdfstring{\protect\ai[\tt]{barvinok\_count}}{barvinok\_count}}
  21
  22 The program \ai[\tt]{barvinok\_count} enumerates a
  23 non-parametric polytope.  It takes one polytope
  24 in \PolyLib/ notation as input and prints the number
  25 of integer points in the polytope.
  26 The \PolyLib/ notation corresponds to the internal
  27 representation of \ai[\tt]{Polyhedron}s as explained
  28 in Section~\ref{a:existing}.
  29 \index{input format!constraints}
  30 The first line of the input contains the number of rows
  31 and the number of columns in the \ai[\tt]{Constraint} matrix.
  32 The rest of the input is composed of the elements of the matrix.
  33 Recall that the number of columns is two more than the number
  34 of variables, where the extra first columns is one or zero
  35 depending on whether the constraint is an inequality ($\ge 0$)
  36 or an equality ($= 0$).  The next columns contain
  37 the coefficients of the variables and the final column contains
  38 the constant in the constraint.
  39 E.g., the set
  40 $S = \lb\, s \mid s \geq 0 \wedge  2 s \leq 13 \, \rb$
  41 from \citeN[Example~38 on page~134]{Verdoolaege2005PhD}
  42 corresponds to the following input and
  43 output.
  44 \begin{verbatim}
  45 > cat S
  46 2 3
  47
  48 1 1 0
  49 1 -2 13
  50 > ./barvinok_count  < S
  51 POLYHEDRON Dimension:1
  52            Constraints:2  Equations:0  Rays:2  Lines:0
  53 Constraints 2 3
  54 Inequality: [   1   0 ]
  55 Inequality: [  -2  13 ]
  56 Rays 2 3
  57 Vertex: [   0 ]/1
  58 Vertex: [  13 ]/2
  59    7
  60 \end{verbatim}
  61 \index{PolyLib@{\tt  PolyLib}!version 5.22.0 or newer}
  62 Note that if you use \PolyLib/ version 5.22.0 or newer then the output
  63 may look slightly different as the computation of the \ai[\tt]{Rays}
  64 may have been postponed to a later stage.
  65 The program \ai[\tt]{latte2polylib.pl} can be used to
  66 convert a polytope from \ai[\tt]{LattE} \shortcite{latte1.1}
  67 notation to \PolyLib/ notation.
  68
  69 \index{input format!vertices}
  70 As an alternative to the constraints based input, the input polytope
  71 may also be specified by its \ai[\tt]{Ray} matrix.
  72 The first line of the input contains the single word \ai[\tt]{vertices}.
  73 The second line contains the number of rows
  74 and the number of columns in the \ai[\tt]{Ray} matrix.
  75 The rest of the input is composed of the elements of the matrix.
  76 Recall that the number of columns is two more than the number
  77 of variables, where the extra first columns is one or zero
  78 depending on whether the ray is a line or not.
  79 The next columns contain
  80 the numerators of the coordinates and the final column contains
  81 the denominator of the vertex or $0$ for a ray.
  82 E.g., the above set can also be described as
  83 \begin{verbatim}
  84 vertices
  85
  86 2 3
  87
  88 1 0 1
  89 1 13 2
  90 \end{verbatim}
  91
  92 \subsection{\texorpdfstring{\protect\ai[\tt]{barvinok\_enumerate}}
  93 {barvinok\_enumerate}}
  94
  95 The program \ai[\tt]{barvinok\_enumerate} enumerates a
  96 parametric polytope.  It takes two polytopes in \PolyLib/
  97 notation as input, optionally followed by a list of parameter
  98 names.
  99 The two polytopes refer to arguments \verb+P+ and \verb+C+
 100 of the corresponding function. (See Section~\ref{a:counting:functions}.)
 101 The following example was taken by \shortciteN{Loechner1999}
 102 from \shortciteN[Chapter II.2]{Loechner97phd}.
 103 \begin{verbatim}
 104 > cat loechner
 105 # Dimension of the matrix:
 106 7 7
 107 # Constraints:
 108 # i j k P Q cte
 109 1 1 0 0 0 0 0 # 0 <= i
 110 1 -1 0 0 1 0 0 # i <= P
 111 1 0 1 0 0 0 0 # 0 <= j
 112 1 1 -1 0 0 0 0 # j <= i
 113 1 0 0 1 0 0 0 # 0 <= k
 114 1 1 -1 -1 0 0 0 # k <= i-j
 115 0 1 1 1 0 -1 0 # Q = i + j + k
 116
 117 # 2 parameters, no constraints.
 118 0 4
 119 > ./barvinok_enumerate < loechner
 120 POLYHEDRON Dimension:5
 121            Constraints:6  Equations:1  Rays:5  Lines:0
 122 Constraints 6 7
 123 Equality:   [   1   1   1   0  -1   0 ]
 124 Inequality: [   0   1   1   1  -1   0 ]
 125 Inequality: [   0   1   0   0   0   0 ]
 126 Inequality: [   0   0   1   0   0   0 ]
 127 Inequality: [   0  -2  -2   0   1   0 ]
 128 Inequality: [   0   0   0   0   0   1 ]
 129 Rays 5 7
 130 Ray:    [   1   0   1   1   2 ]
 131 Ray:    [   1   1   0   1   2 ]
 132 Vertex: [   0   0   0   0   0 ]/1
 133 Ray:    [   0   0   0   1   0 ]
 134 Ray:    [   1   0   0   1   1 ]
 135 POLYHEDRON Dimension:2
 136            Constraints:1  Equations:0  Rays:3  Lines:2
 137 Constraints 1 4
 138 Inequality: [   0   0   1 ]
 139 Rays 3 4
 140 Line:   [   1   0 ]
 141 Line:   [   0   1 ]
 142 Vertex: [   0   0 ]/1
 143          - P + Q  >= 0
 144          2P - Q  >= 0
 145           1 >= 0
 146
 147 ( -1/2 * P^2 + ( 1 * Q + 1/2 )
 148  * P + ( -3/8 * Q^2 + ( -1/2 * {( 1/2 * Q + 0 )
 149 } + 1/4 )
 150  * Q + ( -5/4 * {( 1/2 * Q + 0 )
 151 } + 1 )
 152  )
 153  )
 154          Q  >= 0
 155          P - Q  -1 >= 0
 156           1 >= 0
 157
 158 ( 1/8 * Q^2 + ( -1/2 * {( 1/2 * Q + 0 )
 159 } + 3/4 )
 160  * Q + ( -5/4 * {( 1/2 * Q + 0 )
 161 } + 1 )
 162  )
 163 \end{verbatim}
 164 The output corresponds to
 165 $$
 166 \begin{cases}
 167 \makebox[0pt][l]{$-\frac 1 2 P^2 + P Q + \frac 1 2 P - \frac 3 8 Q^2
 168 + \left( \frac 1 4 - \frac 1 2 \left\{ \frac 1 2 Q \right\} \right)
 169                 Q + 1
 170 - \frac 5 4 \left\{ \frac 1 2 Q \right\}$} \\
 171 &
 172 \hbox{if $P \le Q \le 2 P$}
 173 \\
 174 \frac 1 8 Q^2 +
 175 \left( \frac 3 4 - \frac 1 2 \left\{ \frac 1 2 Q \right\} \right)
 176 - \frac 5 4 \left\{ \frac 1 2 Q \right\}
 177 \qquad
 178 \qquad
 179 \qquad
 180 \qquad
 181 \qquad
 182 &
 183 \hbox{if $0 \le Q \le P-1$}
 184 .
 185 \end{cases}
 186 $$
 187
 188 Options:\\
 189 \begin{tabular}{lll}
 190 \ai[\tt]{--floor} & \ai[\tt]{-f} &
 191 convert \ai[\tt]{fractional}s to \ai[\tt]{flooring}s
 192 \\
 193 \ai[\tt]{--convert} & \ai[\tt]{-c} &
 194 convert \ai[\tt]{fractional}s to \ai[\tt]{periodic}s
 195 \end{tabular}
 196
 197 \subsection{\texorpdfstring{\protect\ai[\tt]{barvinok\_enumerate\_e}}
 198 {barvinok\_enumerate\_e}}
 199
 200 The program \ai[\tt]{barvinok\_enumerate\_e} enumerates a
 201 parametric projected set.  It takes a single polytope in \PolyLib/
 202 notation as input, followed by two lines indicating the number
 203 or existential variables and the number of parameters and
 204 optionally followed by a list of parameter names.
 205 The syntax for the line indicating the number of existential
 206 variables is the letter \verb+E+ followed by a space and the actual number.
 207 For indicating the number of parameters, the letter \verb+P+ is used.
 208 The following example corresponds to
 209 \citeN[Example~36 on page~129]{Verdoolaege2005PhD}.
 210 \begin{verbatim}
 211 > cat projected
 212 5 6
 213 #   k   i   j   p   cst
 214 1   0   1   0   0   -1
 215 1   0   -1  0   0   8
 216 1   0   0   1   0   -1
 217 1   0   0   -1  1   0
 218 0   -1  6   9   0   -7
 219
 220 E 2
 221 P 1
 222 > ./barvinok_enumerate_e <projected
 223 POLYHEDRON Dimension:4
 224            Constraints:5  Equations:1  Rays:4  Lines:0
 225 Constraints 5 6
 226 Equality:   [   1  -6  -9   0   7 ]
 227 Inequality: [   0   1   0   0  -1 ]
 228 Inequality: [   0  -1   0   0   8 ]
 229 Inequality: [   0   0   1   0  -1 ]
 230 Inequality: [   0   0  -1   1   0 ]
 231 Rays 4 6
 232 Vertex: [  50   8   1   1 ]/1
 233 Ray:    [   0   0   0   1 ]
 234 Ray:    [   9   0   1   1 ]
 235 Vertex: [   8   1   1   1 ]/1
 236 exist: 2, nparam: 1
 237          P  -3 >= 0
 238           1 >= 0
 239
 240 ( 3 * P + 10 )
 241          P  -1 >= 0
 242          - P + 2 >= 0
 243
 244 ( 8 * P + 0 )
 245 \end{verbatim}
 246
 247 Options:\\
 248 \begin{tabular}{llp{0.7\textwidth}}
 249 \ai[\tt]{--floor} & \ai[\tt]{-f} &
 250 convert \ai[\tt]{fractional}s to \ai[\tt]{flooring}s
 251 \\
 252 \ai[\tt]{--convert} & \ai[\tt]{-c} &
 253 convert \ai[\tt]{fractional}s to \ai[\tt]{periodic}s
 254 \\
 255 \ai[\tt]{--omega} & \ai[\tt]{-o} &
 256 use \Omegalib/ as a preprocessor
 257 \\
 258 \ai[\tt]{--pip} & \ai[\tt]{-p} &
 259 \raggedright
 260 call \ai[\tt]{barvinok\_enumerate\_pip} instead of \ai[\tt]{barvinok\_enumerate\_e}
 261 \end{tabular}
 262
 263 \subsection{\texorpdfstring{\protect\ai[\tt]{barvinok\_series}}
 264 {barvinok\_series}}
 265
 266 The program \ai[\tt]{barvinok\_series} enumerates a
 267 parametric polytope in the form of a \rgf/.
 268 The input of this program is the same as that of
 269 \ai[\tt]{barvinok\_enumerate}, except that the input polyhedron
 270 is assumed to be full-dimensional.
 271 The following is an example of Petr Lison\u{e}k\index{Lison\u{e}k, P.}.
 272 \begin{verbatim}
 273 > cat petr
 274 4 6
 275 1 -1 -1 -1 1 0
 276 1 1 -1 0 0 0
 277 1 0 1 -1 0 0
 278 1 0 0 1 0 -1
 279
 280 0 3
 281 n
 282 > ./barvinok_series  < petr
 283 POLYHEDRON Dimension:4
 284            Constraints:5  Equations:0  Rays:5  Lines:0
 285 Constraints 5 6
 286 Inequality: [  -1  -1  -1   1   0 ]
 287 Inequality: [   1  -1   0   0   0 ]
 288 Inequality: [   0   1  -1   0   0 ]
 289 Inequality: [   0   0   1   0  -1 ]
 290 Inequality: [   0   0   0   0   1 ]
 291 Rays 5 6
 292 Ray:    [   1   1   1   3 ]
 293 Ray:    [   1   1   0   2 ]
 294 Ray:    [   1   0   0   1 ]
 295 Ray:    [   0   0   0   1 ]
 296 Vertex: [   1   1   1   3 ]/1
 297 POLYHEDRON Dimension:1
 298            Constraints:1  Equations:0  Rays:2  Lines:1
 299 Constraints 1 3
 300 Inequality: [   0   1 ]
 301 Rays 2 3
 302 Line:   [   1 ]
 303 Vertex: [   0 ]/1
 304 (n^3)/((1-n) * (1-n) * (1-n^2) * (1-n^3))
 305 \end{verbatim}
 306
 307 Options:\\
 308 \begin{tabular}{llp{0.7\textwidth}}
 309 \ai[\tt]{--explicit} & \ai[\tt]{-e} &
 310 convert computed \rgf/ to a \psp/
 311 \end{tabular}
 312
 313 \subsection{\texorpdfstring{\protect\ai[\tt]{barvinok\_union}}
 314 {barvinok\_union}}
 315
 316 The program \ai[\tt]{barvinok\_union} enumerates a \ai{union} of
 317 parametric polytopes.  It takes as input the number of parametric
 318 polytopes in the union, the polytopes in combined data and
 319 parameter space in \PolyLib/ notation, the context in parameter space
 320 in \PolyLib/ notation and optionally a list of parameter names.
 321
 322 Options:\\
 323 \begin{tabular}{llp{0.7\textwidth}}
 324 \ai[\tt]{--series} & \ai[\tt]{-s} &
 325 compute \rgf/ instead of \psp/
 326 \end{tabular}
 327
 328 \subsection{\texorpdfstring{\protect\ai[\tt]{barvinok\_ehrhart}}
 329 {barvinok\_ehrhart}}
 330
 331 The program \ai[\tt]{barvinok\_ehrhart} computes the
 332 \ai{Ehrhart quasi-polynomial} of a polytope $P$, i.e., a quasi-polynomial
 333 in $n$ that evaluates to the number of integer points in the dilation
 334 of $P$ by a factor $n$.
 335 The input is the same as that of \ai[\tt]{barvinok\_count}, except that
 336 it may be followed by the variable name.
 337 The functionality is the same as running \ai[\tt]{barvinok\_enumerate}
 338 (or \ai[\tt]{barvinok\_series} if the \ai[\tt]{--series} option was given)
 339 on the cone over $P$ placed at $n=1$.
 340
 341 Options:\\
 342 \begin{tabular}{llp{0.7\textwidth}}
 343 \ai[\tt]{--floor} & \ai[\tt]{-f} &
 344 convert \ai[\tt]{fractional}s to \ai[\tt]{flooring}s
 345 \\
 346 \ai[\tt]{--convert} & \ai[\tt]{-c} &
 347 convert \ai[\tt]{fractional}s to \ai[\tt]{periodic}s
 348 \\
 349 \ai[\tt]{--series} & \ai[\tt]{-s} &
 350 compute \ai{Ehrhart series} instead of \ai{Ehrhart quasi-polynomial}
 351 \end{tabular}
 352
 353 \subsection{\texorpdfstring{\protect\ai[\tt]{polyhedron\_sample}}
 354 {polyhedron\_sample}}
 355
 356 The program \ai[\tt]{polyhedron\_sample} takes a polytope
 357 in \PolyLib/ notation and prints an integer point in the polytope
 358 if there is one.  The point is computed using
 359 \ai[\tt]{Polyhedron\_Sample}.
 360
 361 \subsection{\texorpdfstring{\protect\ai[\tt]{polytope\_scan}}
 362 {polytope\_scan}}
 363
 364 The program \ai[\tt]{polytope\_scan} takes a polytope in
 365 \PolyLib/ notation and prints a list of all integer points in the polytope.
 366 Unless the \ai[\tt]{--direct} options is given, the order is based
 367 on the \ai{reduced basis} computed with
 368 \ai[\tt]{Polyhedron\_Reduced\_Basis}.
 369
 370 Options:\\
 371 \begin{tabular}{llp{0.7\textwidth}}
 372 \ai[\tt]{--direct} & \ai[\tt]{-d} &
 373 list the points in the lexicographical order
 374 \end{tabular}
 375
 376 \subsection{\texorpdfstring{\protect\ai[\tt]{lexmin}}{lexmin}}
 377
 378 The program \ai[\tt]{lexmin} implements an algorithm for performing
 379 \indac{PIP} based on \rgf/s and provides an alternative for the
 380 technique of \shortciteN{Feautrier88parametric}, which is based
 381 on \ai{cutting plane}s \shortcite{Gomory1963}.
 382 The input is the same as that of the \ai[\tt]{example} program
 383 from \piplib/ \cite{Feautrier:PIP}, except that the value
 384 for the ``\ai{big parameter}'' needs to be $-1$, since there is
 385 no need for big parameters, and it does not read any options
 386 from the input file.