1 \section{Internal Representation 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 /* PolyLib options */
221 /* LLL reduction parameter delta=LLL_a/LLL_b */
225 /* barvinok options */
231 int incremental_specialization;
234 struct barvinok_options *barvinok_options_new_with_defaults();
237 The function
\ai[\tt]{barvinok
\_options\_new\_with\_defaults}
238 can be used to create a
\ai[\tt]{barvinok
\_options} structure
242 \item \PolyLib/ options
246 \item \ai[\tt]{MaxRays
}
248 The value of
\ai[\tt]{MaxRays
} is passed to various
\PolyLib/
249 functions and defines the
250 maximum size of a table used in the
\ai{double description
} computation
251 in the
\PolyLib/ function
\ai[\tt]{Chernikova
}.
252 In earlier versions of
\PolyLib/,
253 this parameter had to be conservatively set
254 to a high number to ensure successful operation,
255 resulting in significant memory overhead.
256 Our change to allow this table to grow
257 dynamically is available in recent versions of
\PolyLib/.
258 In these versions, the value no longer indicates the maximal
259 table size, but rather the size of the initial allocation.
260 This value may be set to
\verb+
0+ or left as set
261 by
\ai[\tt]{barvinok
\_options\_new\_with\_defaults}.
265 \item \ai[\tt]{NTL
} options
269 \item \ai[\tt]{LLL
\_a}
270 \item \ai[\tt]{LLL
\_b}
272 The values used for the
\ai{reduction parameter
}
273 in the call to
\ai[\tt]{NTL
}'s implementation of
\indac{LLL
}.
277 \item \ai[\tt]{barvinok
} specific options
281 \item \ai[\tt]{incremental
\_specialization}
283 Selects the
\ai{specialization
} algorithm to be used.
284 If set to
{\tt 0} then a direct specialization is performed
285 using a random vector.
286 Value
{\tt 1} selects a depth first incremental specialization,
287 while value
{\tt 2} selects a breadth first incremental specialization.
288 The default is selected by the
\ai[\tt]{--enable-incremental
}
289 \ai[\tt]{configure
} option.
290 For more information we refer to~
\citeN[Section~
4.4.3]{Verdoolaege2005PhD
}.
296 \subsection{Data Structures for Quasi-polynomials
}
299 Internally, we do not represent our quasi-polynomials
300 as step-polynomials, but, similarly to
\shortciteN{Loechner1999
},
301 as polynomials with periodic numbers for coefficients.
302 However, we also allow our periodic numbers to be represented by
303 fractional parts of degree-$
1$ polynomials rather than
304 an explicit enumeration using the
\ai[\tt]{periodic
} type.
305 By default, the current version of
\barvinok/ uses
306 \ai[\tt]{periodic
}s, but this can be changed through
307 the
\ai[\tt]{--enable-fractional
} configure option.
308 In the latter case, the quasi-polynomial using fractional
309 parts can also be converted to an actual step-polynomial
310 using
\ai[\tt]{evalue
\_frac2floor}, but this is not fully
313 For reasons of compatibility,
%
314 \footnote{Also known as laziness.
}
315 we shoehorned our representations for piecewise quasi-polynomials
316 into the existing data structures.
317 To this effect, we introduced four new types,
318 \ai[\tt]{fractional
},
\ai[\tt]{relation
},
319 \ai[\tt]{partition
} and
\ai[\tt]{flooring
}.
321 typedef enum
{ polynomial, periodic, evector, fractional,
322 relation, partition, flooring
} enode_type;
324 The field
\ai[\tt]{pos
} is not used in most of these
325 additional types and is therefore set to
\verb+-
1+.
327 The types
\ai[\tt]{fractional
} and
\ai[\tt]{flooring
}
328 represent polynomial expressions in a fractional part or a floor respectively.
329 The generator is stored in
\verb+arr
[0]+, while the
330 coefficients are stored in the remaining array elements.
331 That is, an
\ai[\tt]{enode
} of type
\ai[\tt]{fractional
}
334 \verb+arr
[1]+ +
\verb+arr
[2]+ \
{\verb+arr
[0]+\
} +
335 \verb+arr
[3]+ \
{\verb+arr
[0]+\
}^
2 +
\cdots +
336 \verb+arr
[l-
1]+ \
{\verb+arr
[0]+\
}^
{l-
2}
339 An
\ai[\tt]{enode
} of type
\ai[\tt]{flooring
}
342 \verb+arr
[1]+ +
\verb+arr
[2]+
\lfloor\verb+arr
[0]+
\rfloor +
343 \verb+arr
[3]+
\lfloor\verb+arr
[0]+
\rfloor^
2 +
\cdots +
344 \verb+arr
[l-
1]+
\lfloor\verb+arr
[0]+
\rfloor^
{l-
2}
349 The internal representation of the quasi-polynomial
350 $$
\left(
1+
2 \left\
{\frac p
2\right\
}\right) p^
2 +
3 p +
\frac 5 2$$
351 is shown in Figure~
\ref{f:fractional
}.
357 \begin{tabular
}{|c|c|c|
}
359 \multicolumn{2}{|c|
}{type
} & polynomial \\
361 \multicolumn{2}{|c|
}{size
} &
3 \\
363 \multicolumn{2}{|c|
}{pos
} &
1 \\
365 \smash{\lower 6.25pt
\hbox{arr
[0]}} & d &
2 \\
369 \smash{\lower 6.25pt
\hbox{arr
[1]}} & d &
1 \\
373 \smash{\lower 6.25pt
\hbox{arr
[2]}} & d &
0 \\
380 +DR*!DR
\hbox{\strut\hskip 1.5\tabcolsep\phantom{\tt polynomial
}\hskip 1.5\tabcolsep}+C="a"
381 \POS(
60,
0)*!UL
{\hbox{
383 \begin{tabular
}{|c|c|c|
}
385 \multicolumn{2}{|c|
}{type
} & fractional \\
387 \multicolumn{2}{|c|
}{size
} &
3 \\
389 \multicolumn{2}{|c|
}{pos
} & -
1 \\
391 \smash{\lower 6.25pt
\hbox{arr
[0]}} & d &
0 \\
395 \smash{\lower 6.25pt
\hbox{arr
[1]}} & d &
1 \\
399 \smash{\lower 6.25pt
\hbox{arr
[2]}} & d &
1 \\
406 +UL+<
0.5\tabcolsep,
0pt>*!UL
\hbox{\strut}+CL="b"
408 \POS"box2"+UR*!UR
{\hbox{
422 }+CD*!U
{\strut}+C="c"
423 \POS(
60,-
50)*!UL
{\hbox{
425 \begin{tabular
}{|c|c|c|
}
427 \multicolumn{2}{|c|
}{type
} & polynomial \\
429 \multicolumn{2}{|c|
}{size
} &
2 \\
431 \multicolumn{2}{|c|
}{pos
} &
1 \\
433 \smash{\lower 6.25pt
\hbox{arr
[0]}} & d &
1 \\
437 \smash{\lower 6.25pt
\hbox{arr
[1]}} & d &
2 \\
444 +UR-<
0.8\tabcolsep,
0pt>*!UR
\hbox{\strut}+CR="d"
446 \POS"box1"+UC*++!D
\hbox{\tt enode
}
447 \POS"box2"+UC*++!D
\hbox{\tt enode
}
448 \POS"box3"+UC*++!D
\hbox{\tt enode
}
450 \caption{The quasi-polynomial
451 $
\left(
1+
2 \left\
{\frac p
2\right\
}\right) p^
2 +
3 p +
\frac 5 2$.
}
457 The
\ai[\tt]{relation
} type is used to represent
\ai{stride
}s.
458 In particular, if the value of
\ai[\tt]{size
} is
2, then
459 the value of a
\ai[\tt]{relation
} is (in pseudo-code):
461 (value(arr
[0]) ==
0) ? value(arr
[1]) :
0
463 If the size is
3, then the value is:
465 (value(arr
[0]) ==
0) ? value(arr
[1]) : value(arr
[2])
467 The type of
\verb+arr
[0]+ is typically
\ai[\tt]{fractional
}.
469 Finally, the
\ai[\tt]{partition
} type is used to
470 represent piecewise quasi-polynomials.
471 We prefer to encode this information inside
\ai[\tt]{evalue
}s
473 rather than using
\ai[\tt]{Enumeration
}s since we want
474 to perform the same kinds of operations on both quasi-polynomials
475 and piecewise quasi-polynomials.
476 An
\ai[\tt]{enode
} of type
\ai[\tt]{partition
} may not be nested
477 inside another
\ai[\tt]{enode
}.
478 The size of the array is twice the number of ``chambers''.
479 Pointers to chambers are stored in the even slots,
480 whereas pointer to the associated quasi-polynomials
481 are stored in the odd slots.
482 To be able to store pointers to chambers, the
483 definition of
\ai[\tt]{evalue
} was changed as follows.
485 typedef struct _evalue
{
486 Value d; /* denominator */
488 Value n; /* numerator (if denominator >
0) */
489 struct _enode *p; /* pointer (if denominator ==
0) */
490 Polyhedron *D; /* domain (if denominator == -
1) */
494 Note that we allow a ``chamber'' to be a union of polyhedra
495 as discussed in
\citeN[Section~
4.5.1]{Verdoolaege2005PhD
}.
496 Chambers with extra variables, i.e., those of
497 \citeN[Section~
4.6.5]{Verdoolaege2005PhD
},
498 are only partially supported.
499 The field
\ai[\tt]{pos
} is set to the actual dimension,
500 i.e., the number of parameters.
502 \subsection{Operations on Quasi-polynomials
}
505 In this section we discuss some of the more important
506 operations on
\ai[\tt]{evalue
}s provided by the
508 Some of these operations are extensions
509 of the functions from
\PolyLib/ with the same name.
512 void eadd(const evalue *e1,evalue *res);
513 void emul (evalue *e1, evalue *res );
515 The functions
\ai[\tt]{eadd
} and
\ai[\tt]{emul
} takes
516 two (pointers to)
\ai[\tt]{evalue
}s
\verb+e1+ and
\verb+res+
517 and computes their sum and product respectively.
518 The result is stored in
\verb+res+, overwriting (and deallocating)
519 the original value of
\verb+res+.
520 It is an error if exactly one of
521 the arguments of
\ai[\tt]{eadd
} is of type
\ai[\tt]{partition
}
522 (unless the other argument is
\verb+
0+).
523 The addition and multiplication operations are described
524 in
\citeN[Section~
4.5.1]{Verdoolaege2005PhD
}
525 and~
\citeN[Section~
4.5.2]{Verdoolaege2005PhD
}
528 The function
\ai[\tt]{eadd
} is an extension of the function
529 \ai[\tt]{new
\_eadd} from
\shortciteN{Seghir2002
}.
530 Apart from supporting the additional types from Section~
\ref{a:data
},
531 the new version also additionally imposes an order on the nesting of
532 different
\ai[\tt]{enode
}s.
533 Without such an ordering,
\ai[\tt]{evalue
}s could be constructed
534 representing for example
536 (
0 y^
0 + (
0 x^
0 +
1 x^
1 ) y^
1 ) x^
0 + (
0 y^
0 -
1 y^
1) x^
1
539 which is just a funny way of saying $
0$.
542 void eor(evalue *e1, evalue *res);
544 The function
\ai[\tt]{eor
} implements the
\ai{union
}
545 operation from
\citeN[Section~
4.5.3]{Verdoolaege2005PhD
}. Both arguments
546 are assumed to correspond to indicator functions.
549 evalue *esum(evalue *E, int nvar);
551 The function
\ai[\tt]{esum
} performs the summation
552 operation from
\citeN[Section~
4.5.4]{Verdoolaege2005PhD
}.
553 The piecewise step-polynomial represented by
\verb+E+ is summated
554 over its first
\verb+nvar+ variables.
555 Note that
\verb+E+ must be zero or of type
\ai[\tt]{partition
}.
556 The function returns the result in a newly allocated
558 Note also that
\verb+E+ needs to have been converted
559 from
\ai[\tt]{fractional
}s to
\ai[\tt]{flooring
}s using
560 the function
\ai[\tt]{evalue
\_frac2floor}.
562 void evalue_frac2floor(evalue *e);
564 This function also ensures that the arguments of the
565 \ai[\tt]{flooring
}s are positive in the relevant chambers.
566 It currently assumes that the argument of each
567 \ai[\tt]{fractional
} in the original
\ai[\tt]{evalue
}
568 has a minimum in the corresponding chamber.
571 double compute_evalue(evalue *e,Value *list_args);
572 Value *compute_poly(Enumeration *en,Value *list_args);
574 The functions
\ai[\tt]{compute
\_evalue} and
575 \ai[\tt]{compute
\_poly} evaluate a (piecewise) quasi-polynomial
576 at a certain point. The argument
\verb+list_args+
577 points to an array of
\ai[\tt]{Value
}s that is assumed
579 The
\verb+double+ return value of
\ai[\tt]{compute
\_evalue}
580 is inherited from
\PolyLib/.
583 void print_evalue(FILE *DST,evalue *e,char **pname);
585 The function
\ai[\tt]{print
\_evalue} dumps a human-readable
586 representation to the stream pointed to by
\verb+DST+.
587 The argument
\verb+pname+ points
588 to an array of character strings representing the parameter names.
589 The array is assumed to be long enough.
592 int eequal(evalue *e1,evalue *e2);
594 The function
\ai[\tt]{eequal
} return true (
\verb+
1+) if its
595 two arguments are structurally identical.
596 I.e., it does
{\em not\/
} check whether the two
597 (piecewise) quasi-polynomial represent the same function.
600 void reduce_evalue (evalue *e);
602 The function
\ai[\tt]{reduce
\_evalue} performs some
603 simplifications on
\ai[\tt]{evalue
}s.
604 Here, we only describe the simplifications that are directly
605 related to the internal representation.
606 Some other simplifications are explained in
607 \citeN[Section~
4.7.2]{Verdoolaege2005PhD
}.
608 If the highest order coefficients of a
\ai[\tt]{polynomial
},
609 \ai[\tt]{fractional
} or
\ai[\tt]{flooring
} are zero (possibly
610 after some other simplifications), then the size of the array
611 is reduced. If only the constant term remains, i.e.,
612 the size is reduced to $
1$ for
\ai[\tt]{polynomial
} or to $
2$
613 for the other types, then the whole node is replaced by
615 Additionally, if the argument of a
\ai[\tt]{fractional
}
616 has been reduced to a constant, then the whole node
617 is replaced by its partial evaluation.
618 A
\ai[\tt]{relation
} is similarly reduced if its second
619 branch or both its branches are zero.
620 Chambers with zero associated quasi-polynomials are
621 discarded from a
\ai[\tt]{partition
}.
623 \subsection{Generating Functions
}
625 The representation of
\rgf/s uses
626 some basic types from the
\ai[\tt]{NTL
} library
\shortcite{NTL
}
627 for representing arbitrary precision integers
629 as well as vectors (
\ai[\tt]{vec
\_ZZ}) and matrices (
\ai[\tt]{mat
\_ZZ})
631 We further introduces a type
\ai[\tt]{QQ
} for representing a rational
632 number and use vectors (
\ai[\tt]{vec
\_QQ}) of such numbers.
639 NTL_vector_decl(QQ,vec_QQ);
642 Each term in a
\rgf/ is represented by a
\ai[\tt]{short
\_rat}
647 /* rows: terms in numerator */
652 /* rows: factors in denominator */
657 The fields
\ai[\tt]{n
} and
\ai[\tt]{d
} represent the
658 numerator and the denominator respectively.
659 Note that in our implementation we combine terms
660 with the same denominator.
661 In the numerator, each element of
\ai[\tt]{coeff
} and each row of
\ai[\tt]{power
}
662 represents a single such term.
663 The vector
\ai[\tt]{coeff
} contains the rational coefficients
664 $
\alpha_i$ of each term.
665 The columns of
\ai[\tt]{power
} correspond to the powers
667 In the denominator, each row of
\ai[\tt]{power
}
668 corresponds to the power $
\vec b_
{ij
}$ of a
669 factor in the denominator.
673 shows the internal representation of
675 \frac{\frac 3 2 \, x_0^
2 x_1^
3 +
2 \, x_0^
5 x_1^
{-
7}}
676 { (
1 - x_0 x_1^
{-
3}) (
1 - x_1^
2)
}
682 \begin{minipage
}{0cm
}
686 \begin{tabular
}{|c|c|c|
}
701 }+UC*++!D
\hbox{\tt short
\_rat}
705 \caption{Representation of
707 \left(
\frac 3 2 \, x_0^
2 x_1^
3 +
2 \, x_0^
5 x_1^
{-
7}\right)
708 /
\left( (
1 - x_0 x_1^
{-
3}) (
1 - x_1^
2)
\right)
715 The whole
\rgf/ is represented by a
\ai[\tt]{gen
\_fun}
719 std::vector< short_rat * > term;
722 void add(const QQ& c, const vec_ZZ& num, const mat_ZZ& den);
723 void add(const QQ& c, const gen_fun *gf);
724 void substitute(Matrix *CP);
725 gen_fun *Hadamard_product(const gen_fun *gf,
726 barvinok_options *options);
727 void print(std::ostream& os,
728 unsigned int nparam, char **param_name) const;
729 operator evalue *() const;
730 void coefficient(Value* params, Value* c) const;
732 gen_fun(Polyhedron *C = NULL);
734 gen_fun(const gen_fun *gf);
738 A new
\ai[\tt]{gen
\_fun} can be constructed either as empty
\rgf/ (possibly
739 with a given context
\verb+C+), as a copy of an existing
\rgf/
\verb+gf+, or as
740 constant
\rgf/ with value for the constant term specified by
\verb+c+.
742 The first
\ai[\tt]{gen
\_fun::add
} method adds a new term to the
\rgf/,
743 described by the coefficient
\verb+c+, the numerator
\verb+num+ and the
744 denominator
\verb+den+.
745 It makes all powers in the denominator lexico-positive,
746 orders them in lexicographical order and inserts the new
747 term in
\ai[\tt]{term
} according to the lexicographical
748 order of the combined powers in the denominator.
749 The second
\ai[\tt]{gen
\_fun::add
} method adds
\verb+c+ times
\verb+gf+
752 The method
\ai[\tt]{gen
\_fun::operator evalue *
} performs
753 the conversion from
\rgf/ to
\psp/ explained in
754 \citeN[Section~
4.5.5]{Verdoolaege2005PhD
}.
755 The
\ai[\tt]{Polyhedron
} \ai[\tt]{context
} is the superset
756 of all points where the enumerator is non-zero used during this conversion,
757 i.e., it is the set $Q$ from
\citeN[Equation~
4.31]{Verdoolaege2005PhD
}.
758 If
\ai[\tt]{context
} is
\verb+NULL+ the maximal
759 allowed context is assumed, i.e., the maximal
760 region with lexico-positive rays.
762 The method
\ai[\tt]{gen
\_fun::coefficient
} computes the coefficient
763 of the term with power given by
\verb+params+ and stores the result
765 This method performs essentially the same computations as
766 \ai[\tt]{gen
\_fun::operator evalue *
}, except that it adds extra
767 equality constraints based on the specified values for the power.
769 The method
\ai[\tt]{gen
\_fun::substitute
} performs the
770 \ai{monomial substitution
} specified by the homogeneous matrix
\verb+CP+
771 that maps a set of ``
\ai{compressed parameter
}s''
\shortcite{Meister2004PhD
}
772 to the original set of parameters.
773 That is, if we are given a
\rgf/ $G(
\vec z)$ that encodes the
774 explicit function $g(
\vec i')$, where $
\vec i'$ are the coordinates of
775 the transformed space, and
\verb+CP+ represents the map
776 $
\vec i = A
\vec i' +
\vec a$ back to the original space with coordinates $
\vec i$,
777 then this method transforms the
\rgf/ to $F(
\vec x)$ encoding the
778 same explicit function $f(
\vec i)$, i.e.,
779 $$f(
\vec i) = f(A
\vec i' +
\vec a) = g(
\vec i ').$$
780 This means that the coefficient of the term
781 $
\vec x^
{\vec i
} =
\vec x^
{A
\vec i' +
\vec a
}$ in $F(
\vec x)$ should be equal to the
782 coefficient of the term $
\vec z^
{\vec i'
}$ in $G(
\vec z)$.
786 \sum_i \epsilon_i \frac{\vec z^
{\vec v_i
}}{\prod_j (
1-
\vec z^
{\vec b_
{ij
}})
}
791 \sum_i \epsilon_i \frac{\vec x^
{A
\vec v_i +
\vec a
}}
792 {\prod_j (
1-
\vec x^
{A
\vec b_
{ij
}})
}
796 The method
\ai[\tt]{gen
\_fun::Hadamard
\_product} computes the
797 \ai{Hadamard product
} of the current
\rgf/ with the
\rgf/
\verb+gf+,
798 as explained in
\citeN[Section~
4.5.2]{Verdoolaege2005PhD
}.
800 \subsection{Counting Functions
}
801 \label{a:counting:functions
}
803 Our library provides essentially three different counting functions:
804 one for non-parametric polytopes, one for parametric polytopes
805 and one for parametric sets with existential variables.
806 The old versions of these functions have a ``
\ai[\tt]{MaxRays
}''
807 argument, while the new versions have a more general
808 \ai[\tt]{barvinok
\_options} argument.
809 For more information on
\ai[\tt]{barvinok
\_options}, see Section~
\ref{a:options
}.
812 void barvinok_count(Polyhedron *P, Value* result,
814 void barvinok_count_with_options(Polyhedron *P, Value* result,
815 struct barvinok_options *options);
817 The function
\ai[\tt]{barvinok
\_count} or
818 \ai[\tt]{barvinok
\_count\_with\_options} enumerates the non-parametric
819 polytope
\verb+P+ and returns the result in the
\ai[\tt]{Value
}
820 pointed to by
\verb+result+, which needs to have been allocated
822 For the meaning of the argument
\verb+NbMaxCons+, see
823 the discussion on
\ai[\tt]{MaxRays
} in Section~
\ref{a:options
}.
825 The function
\ai[\tt]{barvinok
\_enumerate} for enumerating
826 parametric polytopes was meant to be
827 a drop-in replacement of
\PolyLib/'s
\ai[\tt]{Polyhedron
\_Enumerate}
829 Unfortunately, the latter has been changed to
830 accept an extra argument in recent versions of
\PolyLib/ as shown below.
832 Enumeration* barvinok_enumerate(Polyhedron *P, Polyhedron* C,
834 extern Enumeration *Polyhedron_Enumerate(Polyhedron *P,
835 Polyhedron *C, unsigned MAXRAYS, char **pname);
837 The argument
\verb+MaxRays+ has the same meaning as the argument
838 \verb+NbMaxCons+ above.
839 The argument
\verb+P+ refers to the $(d+n)$-dimensional
840 polyhedron defining the parametric polytope.
841 The argument
\verb+C+ is an $n$-dimensional polyhedron containing
842 extra constraints on the parameter space.
843 Its primary use is to indicate how many of the dimensions
844 in
\verb+P+ refer to parameters as any constraint in
\verb+C+
845 could equally well have been added to
\verb+P+ itself.
846 Note that the dimensions referring to the parameters should
848 The result is a newly allocated
\ai[\tt]{Enumeration
}.
849 As an alternative we also provide a function
850 (
\ai[\tt]{barvinok
\_enumerate\_ev} or
851 \ai[\tt]{barvinok
\_enumerate\_with\_options}) that returns
854 evalue* barvinok_enumerate_ev(Polyhedron *P, Polyhedron* C,
856 evalue* barvinok_enumerate_with_options(Polyhedron *P,
857 Polyhedron* C, struct barvinok_options *options);
860 For enumerating parametric sets with existentially quantified variables,
861 we provide two functions:
862 \ai[\tt]{barvinok
\_enumerate\_e}
864 \ai[\tt]{barvinok
\_enumerate\_pip}.
866 evalue* barvinok_enumerate_e(Polyhedron *P,
867 unsigned exist, unsigned nparam, unsigned MaxRays);
868 evalue* barvinok_enumerate_e_with_options(Polyhedron *P,
869 unsigned exist, unsigned nparam,
870 struct barvinok_options *options);
871 evalue *barvinok_enumerate_pip(Polyhedron *P,
872 unsigned exist, unsigned nparam, unsigned MaxRays);
873 evalue* barvinok_enumerate_pip_with_options(Polyhedron *P,
874 unsigned exist, unsigned nparam,
875 struct barvinok_options *options);
876 evalue *barvinok_enumerate_scarf(Polyhedron *P,
877 unsigned exist, unsigned nparam,
878 struct barvinok_options *options);
880 The first function tries the simplification rules from
881 \citeN[Section~
4.6.2]{Verdoolaege2005PhD
} before resorting to the method
882 based on
\indac{PIP
} from
\citeN[Section~
4.6.3]{Verdoolaege2005PhD
}.
883 The second function immediately applies the technique from
884 \citeN[Section~
4.6.3]{Verdoolaege2005PhD
}.
885 The argument
\verb+exist+ refers to the number of existential variables,
887 the argument
\verb+nparam+ refers to the number of parameters.
888 The order of the dimensions in
\verb+P+ is:
889 counted variables first, then existential variables and finally
891 The function
\ai[\tt]{barvinok
\_enumerate\_scarf} performs the same
892 computation as the function
\ai[\tt]{barvinok
\_enumerate\_scarf\_series}
893 below, but produces an explicit representation instead of a generating function.
896 gen_fun * barvinok_series(Polyhedron *P, Polyhedron* C,
898 gen_fun * barvinok_series_with_options(Polyhedron *P,
899 Polyhedron* C, barvinok_options *options);
900 gen_fun *barvinok_enumerate_scarf_series(Polyhedron *P,
901 unsigned exist, unsigned nparam,
902 barvinok_options *options);
905 \ai[\tt]{barvinok
\_series} or
906 \ai[\tt]{barvinok
\_series\_with\_options} enumerates parametric polytopes
907 in the form of a
\rgf/.
908 The polyhedron
\verb+P+ is assumed to have only
909 revlex-positive rays.
911 The function
\ai[\tt]{barvinok
\_enumerate\_scarf\_series} computes a
912 generating function for the number of point in the parametric set
913 defined by
\verb+P+ with
\verb+exist+ existentially quantified
914 variables, which is assumed to be
2.
915 This function implements the technique of
916 \shortciteN{Scarf2006Neighborhood
} using the
\ai{neighborhood complex
}
917 description of
\shortciteN{Scarf1981indivisibilities:II
}.
918 It is currently restricted to problems with
3 or
4 constraints involving
919 the existentially quantified variables.
921 \subsection{Auxiliary Functions
}
923 In this section we briefly mention some auxiliary functions
924 available in the
\barvinok/ library.
927 void Polyhedron_Polarize(Polyhedron *P);
929 The function
\ai[\tt]{Polyhedron
\_Polarize}
930 polarizes its argument and is explained
931 in
\citeN[Section~
4.4.2]{Verdoolaege2005PhD
}.
934 Matrix * unimodular_complete(Vector *row);
936 The function
\ai[\tt]{unimodular
\_complete} extends
937 \verb+row+ to a
\ai{unimodular matrix
} using the
938 algorithm of
\shortciteN{Bik1996PhD
}.
941 int DomainIncludes(Polyhedron *Pol1, Polyhedron *Pol2);
943 The function
\ai[\tt]{DomainIncludes
} extends
944 the function
\ai[\tt]{PolyhedronIncludes
}
945 provided by
\PolyLib/
946 to unions of polyhedra.
947 It checks whether its first argument is a superset of
951 Polyhedron *DomainConstraintSimplify(Polyhedron *P,
954 The value returned by
955 \ai[\tt]{DomainConstraintSimplify
} is a pointer to
956 a newly allocated
\ai[\tt]{Polyhedron
} that contains the
957 same integer points as its first argument but possibly
958 has simpler constraints.
959 Each constraint $ g
\sp a x
\ge c $
960 is replaced by $
\sp a x
\ge \ceil{ \frac c g
} $,
961 where $g$ is the
\ac{gcd
} of the coefficients in the original
963 The
\ai[\tt]{Polyhedron
} pointed to by
\verb+P+ is destroyed.
966 Polyhedron* Polyhedron_Project(Polyhedron *P, int dim);
968 The function
\ai[\tt]{Polyhedron
\_Project} projects
969 \verb+P+ onto its last
\verb+dim+ dimensions.
971 \sindex{reduced
}{basis
}
972 \sindex{generalized
}{reduced basis
}
974 Matrix *Polyhedron_Reduced_Basis(Polyhedron *P);
976 \ai[\tt]{Polyhedron
\_Reduced\_Basis} computes
977 a
\ai{generalized reduced basis
} of
{\tt P
}, which
978 is assumed to be a polytope, using the algorithm
979 of~
\shortciteN{Cook1993implementation
}.
980 The basis vectors are stored in the rows of the matrix returned.
981 This function currently uses
\ai[\tt]{GLPK
}~
\shortcite{GLPK
}
982 to perform the linear optimizations and so is only available
983 if you have
\ai[\tt]{GLPK
}.
986 Vector *Polyhedron_Sample(Polyhedron *P, unsigned MaxRays);
988 \ai[\tt]{Polyhedron
\_Sample} returns an
\ai{integer point
} of
{\tt P
}
989 or
{\tt NULL
} if
{\tt P
} contains no integer points.
990 The integer point is found using the algorithm
991 of~
\shortciteN{Cook1993implementation
} and uses
992 \ai[\tt]{Polyhedron
\_Reduced\_Basis} to compute the reduced bases
993 and therefore also requires
\ai[\tt]{GLPK
}.
995 \subsection{\protect\ai[\tt]{bernstein
} Data Structures and Functions
}
997 The
\bernstein/ library used
\ai[\tt]{GiNaC
} data structures to
998 represent the data it manipulates.
999 In particular, a polynomial is stored in a
\ai[\tt]{GiNaC::ex
},
1000 a list of variable or parameter names is stored in a
\ai[\tt]{GiNaC::exvector
},
1001 while the parametric vertices or generators are stored in a
\ai[\tt]{GiNaC::matrix
},
1002 where the rows refer to the generators and the columns to the coordinates
1006 namespace bernstein
{
1007 GiNaC::exvector constructParameterVector(
1008 const char * const *param_names, unsigned nbParams);
1009 GiNaC::exvector constructVariableVector(unsigned nbVariables,
1010 const char *prefix);
1013 The functions
\ai[\tt]{constructParameterVector
}
1014 and
\ai[\tt]{constructVariableVector
} construct a list of variable
1015 names either from a list of
{\tt char *
}s or
1016 by suffixing
{\tt prefix
} with a number starting from
0.
1017 Such lists are needed for the functions
1018 \ai[\tt]{domainVertices
},
\ai[\tt]{bernsteinExpansion
}
1019 and
\ai[\tt]{evalue
\_bernstein\_coefficients}.
1022 namespace bernstein
{
1023 GiNaC::matrix domainVertices(Param_Polyhedron *PP, Param_Domain *Q,
1024 const GiNaC::exvector& params);
1027 The function
\ai[\tt]{domainVertices
} constructs a matrix representing the
1028 generators (in this case vertices) of the
\ai[\tt]{Param
\_Polyhedron} {\tt PP
}
1029 for the
\ai[\tt]{Param
\_Domain} {\tt Q
}, to be used
1030 in a call to
\ai[\tt]{bernsteinExpansion
}.
1031 The elements of
{\tt params
} are used in the resulting matrix
1032 to refer to the parameters.
1035 namespace bernstein
{
1036 GiNaC::lst bernsteinExpansion(const GiNaC::matrix& vert,
1037 const GiNaC::ex& poly,
1038 const GiNaC::exvector& vars,
1039 const GiNaC::exvector& params);
1042 The function
\ai[\tt]{bernsteinExpansion
} computes the
1043 \ai{Bernstein coefficient
}s of the polynomial
\verb+poly+
1044 over the
\ai{parametric polytope
} that is the
\ai{convex hull
}
1045 of the rows in
\verb+vert+. The vectors
\verb+vars+
1046 and
\verb+params+ identify the variables (i.e., the coordinates
1047 of the space in which the parametric polytope lives) and
1048 the parameters, respectively.
1051 namespace bernstein
{
1053 typedef std::pair< Polyhedron *, GiNaC::lst > guarded_lst;
1055 struct piecewise_lst
{
1056 const GiNaC::exvector vars;
1057 std::vector<guarded_lst> list;
1059 piecewise_lst::piecewise_lst(const GiNaC::exvector& vars);
1060 piecewise_lst& combine(const piecewise_lst& other);
1062 void simplify_domains(Polyhedron *ctx, unsigned MaxRays);
1063 GiNaC::numeric evaluate(const GiNaC::exvector& values);
1064 void add(const GiNaC::ex& poly);
1069 A
\ai[\tt]{piecewise
\_list} structure represents a list of (disjoint)
1070 polyhedral domains, each with an associated
\ai[\tt]{GiNaC::lst
}
1072 The
\ai[\tt]{vars
} member contains the variable names of the
1073 dimensions of the polyhedral domains.
1075 \ai[\tt]{piecewise
\_lst::combine
} computes the
\ai{common refinement
}
1076 of the polyhedral domains in
\verb+this+ and
\verb+other+ and associates
1077 to each of the resulting subdomains the union of the sets of polynomials
1078 associated to the domains from
\verb+this+ and
\verb+other+ that contain
1080 The result is stored in
\verb+this+.
1082 \ai[\tt]{piecewise
\_lst::maximize
} removes polynomials from domains that evaluate
1083 to a value that is smaller than or equal to the value of some
1084 other polynomial associated to the same domain for each point in the domain.
1086 \ai[\tt]{piecewise
\_lst::evaluate
} ``evaluates'' the
\ai[\tt]{piecewise
\_list}
1087 by looking for the domain (if any) that contains the point given by
1088 \verb+values+ and computing the maximal value attained by any of the
1089 associated polynomials evaluated at that point.
1091 \ai[\tt]{piecewise
\_lst::add
} adds the polynomial
\verb+poly+
1092 to each of the polynomial associated to each of the domains.
1094 \ai[\tt]{piecewise
\_lst::simplify
\_domains} ``simplifies'' the domains
1095 by removing the constraints that are implied by the constraints
1096 in
\verb+ctx+, basically by calling
\PolyLib/'s
1097 \ai[\tt]{DomainSimplify
}. Note that you should only do this
1098 at the end of your computation. In particular, you do not
1099 want to call this method before calling
1100 \ai[\tt]{piecewise
\_lst::maximize
}, since this method will then
1101 have less information on the domains to exploit.
1105 namespace barvinok
{
1106 bernstein::piecewise_lst *evalue_bernstein_coefficients(
1107 bernstein::piecewise_lst *pl_all, evalue *e,
1108 Polyhedron *ctx, const GiNaC::exvector& params);
1111 The
\ai[\tt]{evalue
\_bernstein\_coefficients} function will compute the
1112 \ai{Bernstein coefficient
}s of the piecewise parametric polynomial stored in the
1113 \ai[\tt]{evalue
} \verb+e+.
1114 The
\verb+params+ vector specifies the names to be used for the parameters,
1115 while the context
\ai[\tt]{Polyhedron
} \verb+ctx+ specifies extra constraints
1117 The dimension of
\verb+ctx+ needs to be the same as the length of
\verb+params+.
1118 The
\ai[\tt]{evalue
} \verb+e+ is assumed to be of type
\ai[\tt]{partition
}
1119 and each of the domains in this
\ai[\tt]{partition
} is interpreted
1120 as a parametric polytope in the given parameters. The procedure
1121 will compute the
\ai{Bernstein coefficient
}s of the associated polynomial
1122 over each such parametric polytope.
1123 The resulting
\ai[\tt]{bernstein::piecewise
\_lst} collects the
1124 Bernstein coefficients over all parametric polytopes in
\verb+e+.
1125 If
\verb+pl_all+ is not
\verb+NULL+ then this list will be combined
1126 with the list computed by calling
\ai[\tt]{piecewise
\_lst::combine
}.