1 \section{\protect\PolyLib/ interface of the
\protect\ai[\tt]{barvinok
} library
}
3 Our
\barvinok/ library is built on top of
\PolyLib/
4 \shortcite{Wilde1993,Loechner1999
}.
5 In particular, it reuses the implementations
7 \shortciteN{Loechner97parameterized
}
8 for computing parametric vertices
10 \shortciteN{Clauss1998parametric
}
11 for computing chamber decompositions.
12 Initially, our library was meant to be a replacement
13 for the algorithm of
\shortciteN{Clauss1998parametric
},
14 also implemented in
\PolyLib/, for computing quasi-polynomials.
15 To ease the transition of application programs we
16 tried to reuse the existing data structures as much as possible.
18 \subsection{Existing Data Structures
}
21 Inside
\PolyLib/ integer values are represented by the
22 \ai[\tt]{Value
} data type.
23 Depending on a configure option, the data type may
24 either by a
32-bit integer, a
64-bit integer
25 or an arbitrary precision integer using
\ai[\tt]{GMP
}.
26 The
\barvinok/ library requires that
\PolyLib/ is compiled
27 with support for arbitrary precision integers.
29 The basic structure for representing (unions of) polyhedra is a
32 typedef struct polyhedron
{
33 unsigned Dimension, NbConstraints, NbRays, NbEq, NbBid;
38 struct polyhedron *next;
41 The attribute
\ai[\tt]{Dimension
} is the dimension
42 of the ambient space, i.e., the number of variables.
43 The attributes
\ai[\tt]{Constraint
}
44 and
\ai[\tt]{Ray
} point to two-dimensional arrays
45 of constraints and generators, respectively.
46 The number of rows is stored in
47 \ai[\tt]{NbConstraints
} and
48 \ai[\tt]{NbRays
}, respectively.
49 The number of columns in both arrays is equal
50 to
\verb!
1+Dimension+
1!.
51 The first column of
\ai[\tt]{Constraint
} is either
52 $
0$ or $
1$ depending on whether the constraint
53 is an equality ($
0$) or an inequality ($
1$).
54 The number of equalities is stored in
\ai[\tt]{NbEq
}.
55 If the constraint is $
\sp a x + c
\ge 0$, then
56 the next columns contain the coefficients $a_i$
57 and the final column contains the constant $c$.
58 The first column of
\ai[\tt]{Ray
} is either
59 $
0$ or $
1$ depending on whether the generator
60 is a line ($
0$) or a vertex or ray ($
1$).
61 The number of lines is stored in
\ai[\tt]{NbBid
}.
62 Let $d$ be the
\ac{lcm
} of the denominators of the coordinates
63 of a vertex $
\vec v$, then the next columns contain
64 $d v_i$ and the final column contains $d$.
65 For a ray, the final column contains $
0$.
66 The field
\ai[\tt]{next
} points to the next polyhedron in
67 the union of polyhedra.
68 It is
\verb+
0+ if this is the last (or only) polyhedron in the union.
69 For more information on this structure, we refer to
\shortciteN{Wilde1993
}.
71 Quasi-polynomials are represented using the
72 \ai[\tt]{evalue
} and
\ai[\tt]{enode
} structures.
74 typedef enum
{ polynomial, periodic, evector
} enode_type;
76 typedef struct _evalue
{
77 Value d; /* denominator */
79 Value n; /* numerator (if denominator !=
0) */
80 struct _enode *p; /* pointer (if denominator ==
0) */
84 typedef struct _enode
{
85 enode_type type; /* polynomial or periodic or evector */
86 int size; /* number of attached pointers */
87 int pos; /* parameter position */
88 evalue arr
[1]; /* array of rational/pointer */
91 If the field
\ai[\tt]{d
} of an
\ai[\tt]{evalue
} is zero, then
92 the
\ai[\tt]{evalue
} is a placeholder for a pointer to
93 an
\ai[\tt]{enode
}, stored in
\ai[\tt]{x.p
}.
94 Otherwise, the
\ai[\tt]{evalue
} is a rational number with
95 numerator
\ai[\tt]{x.n
} and denominator
\ai[\tt]{d
}.
96 An
\ai[\tt]{enode
} is either a
\ai[\tt]{polynomial
}
97 or a
\ai[\tt]{periodic
}, depending on the value
99 The length of the array
\ai[\tt]{arr
} is stored in
\ai[\tt]{size
}.
100 For a
\ai[\tt]{polynomial
},
\ai[\tt]{arr
} contains the coefficients.
101 For a
\ai[\tt]{periodic
}, it contains the values for the different
102 residue classes modulo the parameter indicated by
\ai[\tt]{pos
}.
103 For a polynomial,
\ai[\tt]{pos
} refers to the variable
105 The value of
\ai[\tt]{pos
} is
\verb+
1+ for the first parameter.
106 That is, if the value of
\ai[\tt]{pos
} is
\verb+
1+ and the first
107 parameter is $p$, and if the length of the array is $l$,
108 then in case it is a polynomial, the
109 \ai[\tt]{enode
} represents
111 \verb+arr
[0]+ +
\verb+arr
[1]+ p +
\verb+arr
[2]+ p^
2 +
\cdots +
112 \verb+arr
[l-
1]+ p^
{l-
1}
115 If it is a periodic, then it represents
118 \verb+arr
[0]+,
\verb+arr
[1]+,
\verb+arr
[2]+,
\ldots,
123 Note that the elements of a
\ai[\tt]{periodic
} may themselves
124 be other
\ai[\tt]{periodic
}s or even
\ai[\tt]{polynomial
}s.
125 In our library, we only allow the elements of a
\ai[\tt]{periodic
}
126 to be other
\ai[\tt]{periodic
}s or rational numbers.
127 The chambers and their corresponding quasi-polynomial are
128 stored in
\ai[\tt]{Enumeration
} structures.
130 typedef struct _enumeration
{
131 Polyhedron *ValidityDomain; /* constraints on the parameters */
132 evalue EP; /* dimension = combined space */
133 struct _enumeration *next; /* Ehrhart Polynomial,
134 corresponding to parameter
135 values inside the domain
136 ValidityDomain above */
139 For more information on these structures, we refer to
\shortciteN{Loechner1999
}.
142 Figure~
\ref{f:Loechner
} is a skillful reconstruction
143 of Figure~
2 from
\shortciteN{Loechner1999
}.
144 It shows the contents of the
\ai[\tt]{enode
} structures
145 representing the quasi-polynomial
147 [1,
2]_p p^
2 +
3 p +
\frac 5 2
154 \begin{tabular
}{|c|c|c|
}
156 \multicolumn{2}{|c|
}{type
} & polynomial \\
158 \multicolumn{2}{|c|
}{size
} &
3 \\
160 \multicolumn{2}{|c|
}{pos
} &
1 \\
162 \smash{\lower 6.25pt
\hbox{arr
[0]}} & d &
2 \\
166 \smash{\lower 6.25pt
\hbox{arr
[1]}} & d &
1 \\
170 \smash{\lower 6.25pt
\hbox{arr
[2]}} & d &
0 \\
177 +DR*!DR
\hbox{\strut\hskip 1.5\tabcolsep\phantom{\tt polynomial
}\hskip 1.5\tabcolsep}+C="a"
178 \POS(
60,-
15)*!UL
{\hbox{
180 \begin{tabular
}{|c|c|c|
}
182 \multicolumn{2}{|c|
}{type
} & periodic \\
184 \multicolumn{2}{|c|
}{size
} &
2 \\
186 \multicolumn{2}{|c|
}{pos
} &
1 \\
188 \smash{\lower 6.25pt
\hbox{arr
[0]}} & d &
1 \\
192 \smash{\lower 6.25pt
\hbox{arr
[1]}} & d &
1 \\
199 +UL+<
0.5\tabcolsep,
0pt>*!UL
\hbox{\strut}+CL="b"
201 \POS"box1"+UC*++!D
\hbox{\tt enode
}
202 \POS"box2"+UC*++!D
\hbox{\tt enode
}
204 \caption{The quasi-polynomial $
[1,
2]_p p^
2 +
3 p +
\frac 5 2$.
}
212 The
\ai[\tt]{barvinok
\_options} structure contains various
213 options that influence the behavior of the library.
216 struct barvinok_options
{
217 struct barvinok_stats *stats;
219 /* PolyLib options */
223 /* LLL reduction parameter delta=LLL_a/LLL_b */
227 /* barvinok options */
228 #define BV_SPECIALIZATION_BF
2
229 #define BV_SPECIALIZATION_DF
1
230 #define BV_SPECIALIZATION_RANDOM
0
231 #define BV_SPECIALIZATION_TODD
3
232 int incremental_specialization;
234 unsigned long max_index;
237 int count_sample_infinite;
239 int try_Delaunay_triangulation;
241 #define BV_APPROX_SIGN_NONE
0
242 #define BV_APPROX_SIGN_APPROX
1
243 #define BV_APPROX_SIGN_LOWER
2
244 #define BV_APPROX_SIGN_UPPER
3
245 int polynomial_approximation;
246 #define BV_APPROX_NONE
0
247 #define BV_APPROX_DROP
1
248 #define BV_APPROX_SCALE
2
249 #define BV_APPROX_VOLUME
3
250 #define BV_APPROX_BERNOULLI
4
251 int approximation_method;
252 #define BV_APPROX_SCALE_FAST (
1 <<
0)
253 #define BV_APPROX_SCALE_NARROW (
1 <<
1)
254 #define BV_APPROX_SCALE_NARROW2 (
1 <<
2)
255 #define BV_APPROX_SCALE_CHAMBER (
1 <<
3)
257 #define BV_VOL_LIFT
0
258 #define BV_VOL_VERTEX
1
259 #define BV_VOL_BARYCENTER
2
260 int volume_triangulate;
262 /* basis reduction options */
263 #define BV_GBR_NONE
0
264 #define BV_GBR_GLPK
1
268 #define BV_LP_POLYLIB
0
275 #define BV_HULL_GBR
0
276 #define BV_HULL_HILBERT
1
280 struct barvinok_options *barvinok_options_new_with_defaults();
283 The function
\ai[\tt]{barvinok
\_options\_new\_with\_defaults}
284 can be used to create a
\ai[\tt]{barvinok
\_options} structure
288 \item \PolyLib/ options
292 \item \ai[\tt]{MaxRays
}
294 The value of
\ai[\tt]{MaxRays
} is passed to various
\PolyLib/
295 functions and defines the
296 maximum size of a table used in the
\ai{double description
} computation
297 in the
\PolyLib/ function
\ai[\tt]{Chernikova
}.
298 In earlier versions of
\PolyLib/,
299 this parameter had to be conservatively set
300 to a high number to ensure successful operation,
301 resulting in significant memory overhead.
302 Our change to allow this table to grow
303 dynamically is available in recent versions of
\PolyLib/.
304 In these versions, the value no longer indicates the maximal
305 table size, but rather the size of the initial allocation.
306 This value may be set to
\verb+
0+ or left as set
307 by
\ai[\tt]{barvinok
\_options\_new\_with\_defaults}.
311 \item \ai[\tt]{NTL
} options
315 \item \ai[\tt]{LLL
\_a}
316 \item \ai[\tt]{LLL
\_b}
318 The values used for the
\ai{reduction parameter
}
319 in the call to
\ai[\tt]{NTL
}'s implementation of
\indac{LLL
}.
323 \item \ai[\tt]{barvinok
} specific options
327 \item \ai[\tt]{incremental
\_specialization}
329 Selects the
\ai{specialization
} algorithm to be used.
330 If set to
{\tt 0} then a direct specialization is performed
331 using a random vector.
332 Value
{\tt 1} selects a depth first incremental specialization,
333 while value
{\tt 2} selects a breadth first incremental specialization.
334 The default is selected by the
\ai[\tt]{--enable-incremental
}
335 \ai[\tt]{configure
} option.
336 For more information we refer to~
\citeN[Section~
4.4.3]{Verdoolaege2005PhD
}.
342 \subsection{Data Structures for Quasi-polynomials
}
345 Internally, we do not represent our
\ai{quasi-polynomial
}s
346 as step-polynomials, but instead as polynomials of
347 fractional parts of degree-$
1$ polynomials.
348 However, we also allow our quasi-polynomials to be represented
349 as polynomials with periodic numbers for coefficients,
350 similarly to
\shortciteN{Loechner1999
}.
351 By default, the current version of
\barvinok/ uses
352 \ai[\tt]{fractional
}s, but this can be changed through
353 the
\ai[\tt]{--disable-fractional
} configure option.
354 When this option is specified, the periodic numbers
356 an explicit enumeration using the
\ai[\tt]{periodic
} type.
357 A quasi-polynomial based on fractional
358 parts can also be converted to an actual step-polynomial
359 using
\ai[\tt]{evalue
\_frac2floor}, but this is not fully
362 For reasons of compatibility,
%
363 \footnote{Also known as laziness.
}
364 we shoehorned our representations for piecewise quasi-polynomials
365 into the existing data structures.
366 To this effect, we introduced four new types,
367 \ai[\tt]{fractional
},
\ai[\tt]{relation
},
368 \ai[\tt]{partition
} and
\ai[\tt]{flooring
}.
370 typedef enum
{ polynomial, periodic, evector, fractional,
371 relation, partition, flooring
} enode_type;
373 The field
\ai[\tt]{pos
} is not used in most of these
374 additional types and is therefore set to
\verb+-
1+.
376 The types
\ai[\tt]{fractional
} and
\ai[\tt]{flooring
}
377 represent polynomial expressions in a fractional part or a floor respectively.
378 The generator is stored in
\verb+arr
[0]+, while the
379 coefficients are stored in the remaining array elements.
380 That is, an
\ai[\tt]{enode
} of type
\ai[\tt]{fractional
}
383 \verb+arr
[1]+ +
\verb+arr
[2]+ \
{\verb+arr
[0]+\
} +
384 \verb+arr
[3]+ \
{\verb+arr
[0]+\
}^
2 +
\cdots +
385 \verb+arr
[l-
1]+ \
{\verb+arr
[0]+\
}^
{l-
2}
388 An
\ai[\tt]{enode
} of type
\ai[\tt]{flooring
}
391 \verb+arr
[1]+ +
\verb+arr
[2]+
\lfloor\verb+arr
[0]+
\rfloor +
392 \verb+arr
[3]+
\lfloor\verb+arr
[0]+
\rfloor^
2 +
\cdots +
393 \verb+arr
[l-
1]+
\lfloor\verb+arr
[0]+
\rfloor^
{l-
2}
398 The internal representation of the quasi-polynomial
399 $$
\left(
1+
2 \left\
{\frac p
2\right\
}\right) p^
2 +
3 p +
\frac 5 2$$
400 is shown in Figure~
\ref{f:fractional
}.
406 \begin{tabular
}{|c|c|c|
}
408 \multicolumn{2}{|c|
}{type
} & polynomial \\
410 \multicolumn{2}{|c|
}{size
} &
3 \\
412 \multicolumn{2}{|c|
}{pos
} &
1 \\
414 \smash{\lower 6.25pt
\hbox{arr
[0]}} & d &
2 \\
418 \smash{\lower 6.25pt
\hbox{arr
[1]}} & d &
1 \\
422 \smash{\lower 6.25pt
\hbox{arr
[2]}} & d &
0 \\
429 +DR*!DR
\hbox{\strut\hskip 1.5\tabcolsep\phantom{\tt polynomial
}\hskip 1.5\tabcolsep}+C="a"
430 \POS(
60,
0)*!UL
{\hbox{
432 \begin{tabular
}{|c|c|c|
}
434 \multicolumn{2}{|c|
}{type
} & fractional \\
436 \multicolumn{2}{|c|
}{size
} &
3 \\
438 \multicolumn{2}{|c|
}{pos
} & -
1 \\
440 \smash{\lower 6.25pt
\hbox{arr
[0]}} & d &
0 \\
444 \smash{\lower 6.25pt
\hbox{arr
[1]}} & d &
1 \\
448 \smash{\lower 6.25pt
\hbox{arr
[2]}} & d &
1 \\
455 +UL+<
0.5\tabcolsep,
0pt>*!UL
\hbox{\strut}+CL="b"
457 \POS"box2"+UR*!UR
{\hbox{
471 }+CD*!U
{\strut}+C="c"
472 \POS(
60,-
50)*!UL
{\hbox{
474 \begin{tabular
}{|c|c|c|
}
476 \multicolumn{2}{|c|
}{type
} & polynomial \\
478 \multicolumn{2}{|c|
}{size
} &
2 \\
480 \multicolumn{2}{|c|
}{pos
} &
1 \\
482 \smash{\lower 6.25pt
\hbox{arr
[0]}} & d &
1 \\
486 \smash{\lower 6.25pt
\hbox{arr
[1]}} & d &
2 \\
493 +UR-<
0.8\tabcolsep,
0pt>*!UR
\hbox{\strut}+CR="d"
495 \POS"box1"+UC*++!D
\hbox{\tt enode
}
496 \POS"box2"+UC*++!D
\hbox{\tt enode
}
497 \POS"box3"+UC*++!D
\hbox{\tt enode
}
499 \caption{The quasi-polynomial
500 $
\left(
1+
2 \left\
{\frac p
2\right\
}\right) p^
2 +
3 p +
\frac 5 2$.
}
506 The
\ai[\tt]{relation
} type is used to represent
\ai{stride
}s.
507 In particular, if the value of
\ai[\tt]{size
} is
2, then
508 the value of a
\ai[\tt]{relation
} is (in pseudo-code):
510 (value(arr
[0]) ==
0) ? value(arr
[1]) :
0
512 If the size is
3, then the value is:
514 (value(arr
[0]) ==
0) ? value(arr
[1]) : value(arr
[2])
516 The type of
\verb+arr
[0]+ is typically
\ai[\tt]{fractional
}.
518 Finally, the
\ai[\tt]{partition
} type is used to
519 represent piecewise quasi-polynomials.
520 We prefer to encode this information inside
\ai[\tt]{evalue
}s
522 rather than using
\ai[\tt]{Enumeration
}s since we want
523 to perform the same kinds of operations on both quasi-polynomials
524 and piecewise quasi-polynomials.
525 An
\ai[\tt]{enode
} of type
\ai[\tt]{partition
} may not be nested
526 inside another
\ai[\tt]{enode
}.
527 The size of the array is twice the number of ``chambers''.
528 Pointers to chambers are stored in the even slots,
529 whereas pointer to the associated quasi-polynomials
530 are stored in the odd slots.
531 To be able to store pointers to chambers, the
532 definition of
\ai[\tt]{evalue
} was changed as follows.
534 typedef struct _evalue
{
535 Value d; /* denominator */
537 Value n; /* numerator (if denominator >
0) */
538 struct _enode *p; /* pointer (if denominator ==
0) */
539 Polyhedron *D; /* domain (if denominator == -
1) */
543 Note that we allow a ``chamber'' to be a union of polyhedra
544 as discussed in
\citeN[Section~
4.5.1]{Verdoolaege2005PhD
}.
545 Chambers with extra variables, i.e., those of
546 \citeN[Section~
4.6.5]{Verdoolaege2005PhD
},
547 are only partially supported.
548 The field
\ai[\tt]{pos
} is set to the actual dimension,
549 i.e., the number of parameters.
551 \subsection{Operations on Quasi-polynomials
}
554 In this section we discuss some of the more important
555 operations on
\ai[\tt]{evalue
}s provided by the
557 Some of these operations are extensions
558 of the functions from
\PolyLib/ with the same name.
560 Most of these operation are also provided by
\isl/ on
561 \ai[\tt]{isl
\_pw\_qpolynomial}s, which are set to replace
562 \ai[\tt]{evalue
}s. Use
\ai[\tt]{isl
\_pw\_qpolynomial\_from\_evalue} to convert
563 from
\ai[\tt]{evalue
}s to
\ai[\tt]{isl
\_pw\_qpolynomial}s.
565 __isl_give isl_pw_qpolynomial *isl_pw_qpolynomial_from_evalue(
566 __isl_take isl_dim *dim, const evalue *e);
570 void eadd(const evalue *e1,evalue *res);
571 void emul(const evalue *e1, evalue *res);
573 The functions
\ai[\tt]{eadd
} and
\ai[\tt]{emul
} takes
574 two (pointers to)
\ai[\tt]{evalue
}s
\verb+e1+ and
\verb+res+
575 and computes their sum and product respectively.
576 The result is stored in
\verb+res+, overwriting (and deallocating)
577 the original value of
\verb+res+.
578 It is an error if exactly one of
579 the arguments of
\ai[\tt]{eadd
} is of type
\ai[\tt]{partition
}
580 (unless the other argument is
\verb+
0+).
581 The addition and multiplication operations are described
582 in
\citeN[Section~
4.5.1]{Verdoolaege2005PhD
}
583 and~
\citeN[Section~
4.5.2]{Verdoolaege2005PhD
}
586 The function
\ai[\tt]{eadd
} is an extension of the function
587 \ai[\tt]{new
\_eadd} from
\shortciteN{Seghir2002
}.
588 Apart from supporting the additional types from Section~
\ref{a:data
},
589 the new version also additionally imposes an order on the nesting of
590 different
\ai[\tt]{enode
}s.
591 Without such an ordering,
\ai[\tt]{evalue
}s could be constructed
592 representing for example
594 (
0 y^
0 + (
0 x^
0 +
1 x^
1 ) y^
1 ) x^
0 + (
0 y^
0 -
1 y^
1) x^
1
597 which is just a funny way of saying $
0$.
600 void eor(evalue *e1, evalue *res);
602 The function
\ai[\tt]{eor
} implements the
\ai{union
}
603 operation from
\citeN[Section~
4.5.3]{Verdoolaege2005PhD
}. Both arguments
604 are assumed to correspond to indicator functions.
607 evalue *esum(evalue *E, int nvar);
608 evalue *evalue_sum(evalue *E, int nvar, unsigned MaxRays);
610 The function
\ai[\tt]{esum
} has been superseded by
611 \ai[\tt]{evalue
\_sum}.
612 The function
\ai[\tt]{evalue
\_sum} performs the summation
613 operation from
\citeN[Section~
4.5.4]{Verdoolaege2005PhD
}.
614 The piecewise step-polynomial represented by
\verb+E+ is summated
615 over its first
\verb+nvar+ variables.
616 Note that
\verb+E+ must be zero or of type
\ai[\tt]{partition
}.
617 The function returns the result in a newly allocated
619 Note also that
\verb+E+ needs to have been converted
620 from
\ai[\tt]{fractional
}s to
\ai[\tt]{flooring
}s using
621 the function
\ai[\tt]{evalue
\_frac2floor}.
623 void evalue_frac2floor(evalue *e);
625 This function also ensures that the arguments of the
626 \ai[\tt]{flooring
}s are positive in the relevant chambers.
627 It currently assumes that the argument of each
628 \ai[\tt]{fractional
} in the original
\ai[\tt]{evalue
}
629 has a minimum in the corresponding chamber.
632 double compute_evalue(const evalue *e, Value *list_args);
633 Value *compute_poly(Enumeration *en,Value *list_args);
634 evalue *evalue_eval(const evalue *e, Value *values);
636 The functions
\ai[\tt]{compute
\_evalue},
637 \ai[\tt]{compute
\_poly} and
638 \ai[\tt]{evalue
\_eval}
639 evaluate a (piecewise) quasi-polynomial
640 at a certain point. The argument
\verb+list_args+
641 points to an array of
\ai[\tt]{Value
}s that is assumed
643 The
\verb+double+ return value of
\ai[\tt]{compute
\_evalue}
644 is inherited from
\PolyLib/.
647 void print_evalue(FILE *DST, const evalue *e, char **pname);
649 The function
\ai[\tt]{print
\_evalue} dumps a human-readable
650 representation to the stream pointed to by
\verb+DST+.
651 The argument
\verb+pname+ points
652 to an array of character strings representing the parameter names.
653 The array is assumed to be long enough.
656 int eequal(const evalue *e1, const evalue *e2);
658 The function
\ai[\tt]{eequal
} return true (
\verb+
1+) if its
659 two arguments are structurally identical.
660 I.e., it does
{\em not\/
} check whether the two
661 (piecewise) quasi-polynomial represent the same function.
664 void reduce_evalue (evalue *e);
666 The function
\ai[\tt]{reduce
\_evalue} performs some
667 simplifications on
\ai[\tt]{evalue
}s.
668 Here, we only describe the simplifications that are directly
669 related to the internal representation.
670 Some other simplifications are explained in
671 \citeN[Section~
4.7.2]{Verdoolaege2005PhD
}.
672 If the highest order coefficients of a
\ai[\tt]{polynomial
},
673 \ai[\tt]{fractional
} or
\ai[\tt]{flooring
} are zero (possibly
674 after some other simplifications), then the size of the array
675 is reduced. If only the constant term remains, i.e.,
676 the size is reduced to $
1$ for
\ai[\tt]{polynomial
} or to $
2$
677 for the other types, then the whole node is replaced by
679 Additionally, if the argument of a
\ai[\tt]{fractional
}
680 has been reduced to a constant, then the whole node
681 is replaced by its partial evaluation.
682 A
\ai[\tt]{relation
} is similarly reduced if its second
683 branch or both its branches are zero.
684 Chambers with zero associated quasi-polynomials are
685 discarded from a
\ai[\tt]{partition
}.
687 \subsection{Generating Functions
}
689 The representation of
\rgf/s uses
690 some basic types from the
\ai[\tt]{NTL
} library
\shortcite{NTL
}
691 for representing arbitrary precision integers
693 as well as vectors (
\ai[\tt]{vec
\_ZZ}) and matrices (
\ai[\tt]{mat
\_ZZ})
695 We further introduces a type
\ai[\tt]{QQ
} for representing a rational
696 number and use vectors (
\ai[\tt]{vec
\_QQ}) of such numbers.
703 NTL_vector_decl(QQ,vec_QQ);
706 Each term in a
\rgf/ is represented by a
\ai[\tt]{short
\_rat}
711 /* rows: terms in numerator */
716 /* rows: factors in denominator */
721 The fields
\ai[\tt]{n
} and
\ai[\tt]{d
} represent the
722 numerator and the denominator respectively.
723 Note that in our implementation we combine terms
724 with the same denominator.
725 In the numerator, each element of
\ai[\tt]{coeff
} and each row of
\ai[\tt]{power
}
726 represents a single such term.
727 The vector
\ai[\tt]{coeff
} contains the rational coefficients
728 $
\alpha_i$ of each term.
729 The columns of
\ai[\tt]{power
} correspond to the powers
731 In the denominator, each row of
\ai[\tt]{power
}
732 corresponds to the power $
\vec b_
{ij
}$ of a
733 factor in the denominator.
737 shows the internal representation of
739 \frac{\frac 3 2 \, x_0^
2 x_1^
3 +
2 \, x_0^
5 x_1^
{-
7}}
740 { (
1 - x_0 x_1^
{-
3}) (
1 - x_1^
2)
}
746 \begin{minipage
}{0cm
}
750 \begin{tabular
}{|c|c|c|
}
765 }+UC*++!D
\hbox{\tt short
\_rat}
769 \caption{Representation of
771 \left(
\frac 3 2 \, x_0^
2 x_1^
3 +
2 \, x_0^
5 x_1^
{-
7}\right)
772 /
\left( (
1 - x_0 x_1^
{-
3}) (
1 - x_1^
2)
\right)
779 The whole
\rgf/ is represented by a
\ai[\tt]{gen
\_fun}
782 typedef std::set<short_rat *,
783 short_rat_lex_smaller_denominator > short_rat_list;
789 void add(const QQ& c, const vec_ZZ& num, const mat_ZZ& den);
790 void add(short_rat *r);
791 void add(const QQ& c, const gen_fun *gf,
792 barvinok_options *options);
793 void substitute(Matrix *CP);
794 gen_fun *Hadamard_product(const gen_fun *gf,
795 barvinok_options *options);
796 void print(std::ostream& os,
797 unsigned int nparam, char **param_name) const;
798 operator evalue *() const;
799 ZZ coefficient(Value* params, barvinok_options *options) const;
800 void coefficient(Value* params, Value* c) const;
802 gen_fun(Polyhedron *C);
804 gen_fun(const gen_fun *gf);
808 A new
\ai[\tt]{gen
\_fun} can be constructed either as empty
\rgf/ (possibly
809 with a given context
\verb+C+), as a copy of an existing
\rgf/
\verb+gf+, or as
810 constant
\rgf/ with value for the constant term specified by
\verb+c+.
812 The first
\ai[\tt]{gen
\_fun::add
} method adds a new term to the
\rgf/,
813 described by the coefficient
\verb+c+, the numerator
\verb+num+ and the
814 denominator
\verb+den+.
815 It makes all powers in the denominator lexico-positive,
816 orders them in lexicographical order and inserts the new
817 term in
\ai[\tt]{term
} according to the lexicographical
818 order of the combined powers in the denominator.
819 The second
\ai[\tt]{gen
\_fun::add
} method adds
\verb+c+ times
\verb+gf+
822 The method
\ai[\tt]{gen
\_fun::operator evalue *
} performs
823 the conversion from
\rgf/ to
\psp/ explained in
824 \citeN[Section~
4.5.5]{Verdoolaege2005PhD
}.
825 The
\ai[\tt]{Polyhedron
} \ai[\tt]{context
} is the superset
826 of all points where the enumerator is non-zero used during this conversion,
827 i.e., it is the set $Q$ from
\citeN[Equation~
4.31]{Verdoolaege2005PhD
}.
828 If
\ai[\tt]{context
} is
\verb+NULL+ the maximal
829 allowed context is assumed, i.e., the maximal
830 region with lexico-positive rays.
832 The method
\ai[\tt]{gen
\_fun::coefficient
} computes the coefficient
833 of the term with power given by
\verb+params+ and stores the result
835 This method performs essentially the same computations as
836 \ai[\tt]{gen
\_fun::operator evalue *
}, except that it adds extra
837 equality constraints based on the specified values for the power.
839 The method
\ai[\tt]{gen
\_fun::substitute
} performs the
840 \ai{monomial substitution
} specified by the homogeneous matrix
\verb+CP+
841 that maps a set of ``
\ai{compressed parameter
}s''
\shortcite{Meister2004PhD
}
842 to the original set of parameters.
843 That is, if we are given a
\rgf/ $G(
\vec z)$ that encodes the
844 explicit function $g(
\vec i')$, where $
\vec i'$ are the coordinates of
845 the transformed space, and
\verb+CP+ represents the map
846 $
\vec i = A
\vec i' +
\vec a$ back to the original space with coordinates $
\vec i$,
847 then this method transforms the
\rgf/ to $F(
\vec x)$ encoding the
848 same explicit function $f(
\vec i)$, i.e.,
849 $$f(
\vec i) = f(A
\vec i' +
\vec a) = g(
\vec i ').$$
850 This means that the coefficient of the term
851 $
\vec x^
{\vec i
} =
\vec x^
{A
\vec i' +
\vec a
}$ in $F(
\vec x)$ should be equal to the
852 coefficient of the term $
\vec z^
{\vec i'
}$ in $G(
\vec z)$.
856 \sum_i \epsilon_i \frac{\vec z^
{\vec v_i
}}{\prod_j (
1-
\vec z^
{\vec b_
{ij
}})
}
861 \sum_i \epsilon_i \frac{\vec x^
{A
\vec v_i +
\vec a
}}
862 {\prod_j (
1-
\vec x^
{A
\vec b_
{ij
}})
}
866 The method
\ai[\tt]{gen
\_fun::Hadamard
\_product} computes the
867 \ai{Hadamard product
} of the current
\rgf/ with the
\rgf/
\verb+gf+,
868 as explained in
\citeN[Section~
4.5.2]{Verdoolaege2005PhD
}.
870 \subsection{Counting Functions
}
871 \label{a:counting:functions
}
873 Our library provides essentially three different counting functions:
874 one for non-parametric polytopes, one for parametric polytopes
875 and one for parametric sets with existential variables.
876 The old versions of these functions have a ``
\ai[\tt]{MaxRays
}''
877 argument, while the new versions have a more general
878 \ai[\tt]{barvinok
\_options} argument.
879 For more information on
\ai[\tt]{barvinok
\_options}, see Section~
\ref{a:options
}.
882 void barvinok_count(Polyhedron *P, Value* result,
884 void barvinok_count_with_options(Polyhedron *P, Value* result,
885 struct barvinok_options *options);
887 The function
\ai[\tt]{barvinok
\_count} or
888 \ai[\tt]{barvinok
\_count\_with\_options} enumerates the non-parametric
889 polytope
\verb+P+ and returns the result in the
\ai[\tt]{Value
}
890 pointed to by
\verb+result+, which needs to have been allocated
892 If
\verb+P+ is a union, then only the first set in the union will
893 be taken into account.
894 For the meaning of the argument
\verb+NbMaxCons+, see
895 the discussion on
\ai[\tt]{MaxRays
} in Section~
\ref{a:options
}.
897 The function
\ai[\tt]{barvinok
\_enumerate} for enumerating
898 parametric polytopes was meant to be
899 a drop-in replacement of
\PolyLib/'s
\ai[\tt]{Polyhedron
\_Enumerate}
901 Unfortunately, the latter has been changed to
902 accept an extra argument in recent versions of
\PolyLib/ as shown below.
904 Enumeration* barvinok_enumerate(Polyhedron *P, Polyhedron* C,
906 extern Enumeration *Polyhedron_Enumerate(Polyhedron *P,
907 Polyhedron *C, unsigned MAXRAYS, char **pname);
909 The argument
\verb+MaxRays+ has the same meaning as the argument
910 \verb+NbMaxCons+ above.
911 The argument
\verb+P+ refers to the $(d+n)$-dimensional
912 polyhedron defining the parametric polytope.
913 The argument
\verb+C+ is an $n$-dimensional polyhedron containing
914 extra constraints on the parameter space.
915 Its primary use is to indicate how many of the dimensions
916 in
\verb+P+ refer to parameters as any constraint in
\verb+C+
917 could equally well have been added to
\verb+P+ itself.
918 Note that the dimensions referring to the parameters should
920 If either
\verb+P+ or
\verb+C+ is a union,
921 then only the first set in the union will be taken into account.
922 The result is a newly allocated
\ai[\tt]{Enumeration
}.
923 As an alternative we also provide a function
924 (
\ai[\tt]{barvinok
\_enumerate\_ev} or
925 \ai[\tt]{barvinok
\_enumerate\_with\_options}) that returns
928 evalue* barvinok_enumerate_ev(Polyhedron *P, Polyhedron* C,
930 evalue* barvinok_enumerate_with_options(Polyhedron *P,
931 Polyhedron* C, struct barvinok_options *options);
934 For enumerating parametric sets with existentially quantified variables,
935 we provide two functions:
936 \ai[\tt]{barvinok
\_enumerate\_e},
938 \ai[\tt]{barvinok
\_enumerate\_isl}.
940 evalue* barvinok_enumerate_e(Polyhedron *P,
941 unsigned exist, unsigned nparam, unsigned MaxRays);
942 evalue* barvinok_enumerate_e_with_options(Polyhedron *P,
943 unsigned exist, unsigned nparam,
944 struct barvinok_options *options);
945 evalue *barvinok_enumerate_isl(Polyhedron *P,
946 unsigned exist, unsigned nparam,
947 struct barvinok_options *options);
948 evalue *barvinok_enumerate_scarf(Polyhedron *P,
949 unsigned exist, unsigned nparam,
950 struct barvinok_options *options);
952 The first function tries the simplification rules from
953 \citeN[Section~
4.6.2]{Verdoolaege2005PhD
} before resorting to the method
954 based on
\indac{PIP
} from
\citeN[Section~
4.6.3]{Verdoolaege2005PhD
}.
955 The second function immediately applies the technique from
956 \citeN[Section~
4.6.3]{Verdoolaege2005PhD
}.
957 The argument
\verb+exist+ refers to the number of existential variables,
959 the argument
\verb+nparam+ refers to the number of parameters.
960 The order of the dimensions in
\verb+P+ is:
961 counted variables first, then existential variables and finally
963 The function
\ai[\tt]{barvinok
\_enumerate\_scarf} performs the same
964 computation as the function
\ai[\tt]{barvinok
\_enumerate\_scarf\_series}
965 below, but produces an explicit representation instead of a generating function.
968 gen_fun * barvinok_series(Polyhedron *P, Polyhedron* C,
970 gen_fun * barvinok_series_with_options(Polyhedron *P,
971 Polyhedron* C, barvinok_options *options);
972 gen_fun *barvinok_enumerate_e_series(Polyhedron *P,
973 unsigned exist, unsigned nparam,
974 barvinok_options *options);
975 gen_fun *barvinok_enumerate_scarf_series(Polyhedron *P,
976 unsigned exist, unsigned nparam,
977 barvinok_options *options);
980 \ai[\tt]{barvinok
\_series} or
981 \ai[\tt]{barvinok
\_series\_with\_options} enumerates parametric polytopes
982 in the form of a
\rgf/.
983 The polyhedron
\verb+P+ is assumed to have only
984 revlex-positive rays.
986 The function
\ai[\tt]{barvinok
\_enumerate\_e\_series} computes a
987 generating function for the number of point in the parametric set
988 defined by
\verb+P+ with
\verb+exist+ existentially quantified
989 variables using the
\ai{projection theorem
}, as explained
990 in
\autoref{s:projection
}.
991 The function
\ai[\tt]{barvinok
\_enumerate\_scarf\_series} computes a
992 generating function for the number of point in the parametric set
993 defined by
\verb+P+ with
\verb+exist+ existentially quantified
994 variables, which is assumed to be
2.
995 This function implements the technique of
996 \shortciteN{Scarf2006Neighborhood
} using the
\ai{neighborhood complex
}
997 description of
\shortciteN{Scarf1981indivisibilities:II
}.
998 It is currently restricted to problems with
3 or
4 constraints involving
999 the existentially quantified variables.
1001 \subsection{Auxiliary Functions
}
1003 In this section we briefly mention some auxiliary functions
1004 available in the
\barvinok/ library.
1007 void Polyhedron_Polarize(Polyhedron *P);
1009 The function
\ai[\tt]{Polyhedron
\_Polarize}
1010 polarizes its argument and is explained
1011 in
\citeN[Section~
4.4.2]{Verdoolaege2005PhD
}.
1014 int unimodular_complete(Matrix *M, int row);
1016 The function
\ai[\tt]{unimodular
\_complete} extends
1017 the first
\verb+row+ rows of
1018 \verb+M+ with an integral basis of the orthogonal complement
1019 as explained in Section~
\ref{s:completion
}.
1021 if the resulting matrix is unimodular
\index{unimodular matrix
}.
1024 int DomainIncludes(Polyhedron *D1, Polyhedron *D2);
1026 The function
\ai[\tt]{DomainIncludes
} extends
1027 the function
\ai[\tt]{PolyhedronIncludes
}
1028 provided by
\PolyLib/
1029 to unions of polyhedra.
1030 It checks whether every polyhedron in the union
{\tt D2
}
1031 is included in some polyhedron of
{\tt D1
}.
1034 Polyhedron *DomainConstraintSimplify(Polyhedron *P,
1037 The value returned by
1038 \ai[\tt]{DomainConstraintSimplify
} is a pointer to
1039 a newly allocated
\ai[\tt]{Polyhedron
} that contains the
1040 same integer points as its first argument but possibly
1041 has simpler constraints.
1042 Each constraint $ g
\sp a x
\ge c $
1043 is replaced by $
\sp a x
\ge \ceil{ \frac c g
} $,
1044 where $g$ is the
\ac{gcd
} of the coefficients in the original
1046 The
\ai[\tt]{Polyhedron
} pointed to by
\verb+P+ is destroyed.
1049 Polyhedron* Polyhedron_Project(Polyhedron *P, int dim);
1051 The function
\ai[\tt]{Polyhedron
\_Project} projects
1052 \verb+P+ onto its last
\verb+dim+ dimensions.
1055 Matrix *left_inverse(Matrix *M, Matrix **Eq);
1057 The
\ai[\tt]{left
\_inverse} function computes the left inverse
1058 of
\verb+M+ as explained in Section~
\ref{s:inverse
}.
1060 \sindex{reduced
}{basis
}
1061 \sindex{generalized
}{reduced basis
}
1063 Matrix *Polyhedron_Reduced_Basis(Polyhedron *P,
1064 struct barvinok_options *options);
1066 \ai[\tt]{Polyhedron
\_Reduced\_Basis} computes
1067 a
\ai{generalized reduced basis
} of
{\tt P
}, which
1068 is assumed to be a polytope, using the algorithm
1069 of~
\shortciteN{Cook1993implementation
}.
1070 See
\autoref{s:feasibility
} for more information.
1071 The basis vectors are stored in the rows of the matrix returned.
1074 Vector *Polyhedron_Sample(Polyhedron *P,
1075 struct barvinok_options *options);
1077 \ai[\tt]{Polyhedron
\_Sample} returns an
\ai{integer point
} of
{\tt P
}
1078 or
{\tt NULL
} if
{\tt P
} contains no integer points.
1079 The integer point is found using the algorithm
1080 of~
\shortciteN{Cook1993implementation
} and uses
1081 \ai[\tt]{Polyhedron
\_Reduced\_Basis} to compute the reduced bases.
1082 See
\autoref{s:feasibility
} for more information.