2 #include <NTL/mat_ZZ.h>
3 #include <NTL/vec_ZZ.h>
4 #include <barvinok/barvinok.h>
5 #include <barvinok/evalue.h>
6 #include <barvinok/util.h>
8 #include "conversion.h"
9 #include "lattice_point.h"
10 #include "param_util.h"
15 #define ALLOC(type) (type*)malloc(sizeof(type))
17 /* returns an evalue that corresponds to
21 static evalue
*term(int param
, ZZ
& c
, Value
*den
= NULL
)
23 evalue
*EP
= new evalue();
25 value_set_si(EP
->d
,0);
26 EP
->x
.p
= new_enode(polynomial
, 2, param
+ 1);
27 evalue_set_si(&EP
->x
.p
->arr
[0], 0, 1);
28 value_init(EP
->x
.p
->arr
[1].x
.n
);
30 value_set_si(EP
->x
.p
->arr
[1].d
, 1);
32 value_assign(EP
->x
.p
->arr
[1].d
, *den
);
33 zz2value(c
, EP
->x
.p
->arr
[1].x
.n
);
37 /* returns an evalue that corresponds to
41 evalue
*multi_monom(vec_ZZ
& p
)
43 evalue
*X
= new evalue();
46 unsigned nparam
= p
.length()-1;
47 zz2value(p
[nparam
], X
->x
.n
);
48 value_set_si(X
->d
, 1);
49 for (int i
= 0; i
< nparam
; ++i
) {
52 evalue
*T
= term(i
, p
[i
]);
61 * Check whether mapping polyhedron P on the affine combination
62 * num yields a range that has a fixed quotient on integer
64 * If zero is true, then we are only interested in the quotient
65 * for the cases where the remainder is zero.
66 * Returns NULL if false and a newly allocated value if true.
68 static Value
*fixed_quotient(Polyhedron
*P
, vec_ZZ
& num
, Value d
, bool zero
)
71 int len
= num
.length();
72 Matrix
*T
= Matrix_Alloc(2, len
);
73 zz2values(num
, T
->p
[0]);
74 value_set_si(T
->p
[1][len
-1], 1);
75 Polyhedron
*I
= Polyhedron_Image(P
, T
, P
->NbConstraints
);
79 for (i
= 0; i
< I
->NbRays
; ++i
)
80 if (value_zero_p(I
->Ray
[i
][2])) {
88 int bounded
= line_minmax(I
, &min
, &max
);
92 mpz_cdiv_q(min
, min
, d
);
94 mpz_fdiv_q(min
, min
, d
);
95 mpz_fdiv_q(max
, max
, d
);
97 if (value_eq(min
, max
)) {
100 value_assign(*ret
, min
);
108 * Normalize linear expression coef modulo m
109 * Removes common factor and reduces coefficients
110 * Returns index of first non-zero coefficient or len
112 int normal_mod(Value
*coef
, int len
, Value
*m
)
117 Vector_Gcd(coef
, len
, &gcd
);
118 value_gcd(gcd
, gcd
, *m
);
119 Vector_AntiScale(coef
, coef
, gcd
, len
);
121 value_division(*m
, *m
, gcd
);
128 for (j
= 0; j
< len
; ++j
)
129 mpz_fdiv_r(coef
[j
], coef
[j
], *m
);
130 for (j
= 0; j
< len
; ++j
)
131 if (value_notzero_p(coef
[j
]))
137 static bool mod_needed(Polyhedron
*PD
, vec_ZZ
& num
, Value d
, evalue
*E
)
139 Value
*q
= fixed_quotient(PD
, num
, d
, false);
144 value_oppose(*q
, *q
);
147 value_set_si(EV
.d
, 1);
149 value_multiply(EV
.x
.n
, *q
, d
);
151 free_evalue_refs(&EV
);
157 /* Computes the fractional part of the affine expression specified
158 * by coef (of length nvar+1) and the denominator denom.
159 * If PD is not NULL, then it specifies additional constraints
160 * on the variables that may be used to simplify the resulting
161 * fractional part expression.
163 * Modifies coef argument !
165 evalue
*fractional_part(Value
*coef
, Value denom
, int nvar
, Polyhedron
*PD
)
169 evalue
*EP
= evalue_zero();
172 value_assign(m
, denom
);
173 int j
= normal_mod(coef
, nvar
+1, &m
);
181 values2zz(coef
, num
, nvar
+1);
188 evalue_set_si(&tmp
, 0, 1);
192 while (j
< nvar
&& (num
[j
] == g
/2 || num
[j
] == 0))
194 if ((j
< nvar
&& num
[j
] > g
/2) || (j
== nvar
&& num
[j
] >= (g
+1)/2)) {
195 for (int k
= j
; k
< nvar
; ++k
)
198 num
[nvar
] = g
- 1 - num
[nvar
];
199 value_assign(tmp
.d
, m
);
201 zz2value(t
, tmp
.x
.n
);
207 ZZ t
= num
[nvar
] * sign
;
208 zz2value(t
, tmp
.x
.n
);
209 value_assign(tmp
.d
, m
);
212 evalue
*E
= multi_monom(num
);
216 if (PD
&& !mod_needed(PD
, num
, m
, E
)) {
218 value_set_si(EV
.x
.n
, sign
);
219 value_assign(EV
.d
, m
);
224 value_set_si(EV
.x
.n
, 1);
225 value_assign(EV
.d
, m
);
228 value_set_si(EV
.d
, 0);
229 EV
.x
.p
= new_enode(fractional
, 3, -1);
230 evalue_copy(&EV
.x
.p
->arr
[0], E
);
231 evalue_set_si(&EV
.x
.p
->arr
[1], 0, 1);
232 value_init(EV
.x
.p
->arr
[2].x
.n
);
233 value_set_si(EV
.x
.p
->arr
[2].x
.n
, sign
);
234 value_set_si(EV
.x
.p
->arr
[2].d
, 1);
239 free_evalue_refs(&EV
);
244 free_evalue_refs(&tmp
);
252 static evalue
*ceil(Value
*coef
, int len
, Value d
,
253 barvinok_options
*options
)
257 Vector_Oppose(coef
, coef
, len
);
258 c
= fractional_part(coef
, d
, len
-1, NULL
);
259 if (options
->lookup_table
)
260 evalue_mod2table(c
, len
-1);
264 evalue
* bv_ceil3(Value
*coef
, int len
, Value d
, Polyhedron
*P
)
266 Vector
*val
= Vector_Alloc(len
);
271 Vector_Scale(coef
, val
->p
, t
, len
);
272 value_absolute(t
, d
);
275 values2zz(val
->p
, num
, len
);
276 evalue
*EP
= multi_monom(num
);
281 value_set_si(tmp
.x
.n
, 1);
282 value_assign(tmp
.d
, t
);
286 Vector_Oppose(val
->p
, val
->p
, len
);
287 evalue
*f
= fractional_part(val
->p
, t
, len
-1, P
);
293 /* copy EP to malloc'ed evalue */
294 evalue
*E
= ALLOC(evalue
);
298 free_evalue_refs(&tmp
);
304 void lattice_point_fixed(Value
*vertex
, Value
*vertex_res
,
305 Matrix
*Rays
, Matrix
*Rays_res
,
308 unsigned dim
= Rays
->NbRows
;
309 if (value_one_p(vertex
[dim
]))
310 Vector_Copy(vertex_res
, point
, Rays_res
->NbColumns
);
312 Matrix
*R2
= Matrix_Copy(Rays
);
313 Matrix
*inv
= Matrix_Alloc(Rays
->NbRows
, Rays
->NbColumns
);
314 int ok
= Matrix_Inverse(R2
, inv
);
317 Vector
*lambda
= Vector_Alloc(dim
);
318 Vector_Matrix_Product(vertex
, inv
, lambda
->p
);
320 for (int j
= 0; j
< dim
; ++j
)
321 mpz_cdiv_q(lambda
->p
[j
], lambda
->p
[j
], vertex
[dim
]);
322 Vector_Matrix_Product(lambda
->p
, Rays_res
, point
);
327 static Matrix
*Matrix_AddRowColumn(Matrix
*M
)
329 Matrix
*M2
= Matrix_Alloc(M
->NbRows
+1, M
->NbColumns
+1);
330 for (int i
= 0; i
< M
->NbRows
; ++i
)
331 Vector_Copy(M
->p
[i
], M2
->p
[i
], M
->NbColumns
);
332 value_set_si(M2
->p
[M
->NbRows
][M
->NbColumns
], 1);
336 #define FORALL_COSETS(det,D,i,k) \
338 Vector *k = Vector_Alloc(D->NbRows+1); \
339 value_set_si(k->p[D->NbRows], 1); \
340 for (unsigned long i = 0; i < det; ++i) { \
341 unsigned long _fc_val = i; \
342 for (int j = 0; j < D->NbRows; ++j) { \
343 value_set_si(k->p[j], _fc_val % mpz_get_ui(D->p[j][j]));\
344 _fc_val /= mpz_get_ui(D->p[j][j]); \
346 #define END_FORALL_COSETS \
351 /* Compute the lattice points in the vertex cone at "values" with rays "rays".
352 * The lattice points are returned in "vertex".
354 * Rays has the generators as rows and so does W.
355 * We first compute { m-v, u_i^* } with m = k W, where k runs through
358 * [k 1] [ d1*W 0 ] [ U' 0 ] = [k 1] T2
360 * where d1 and d2 are the denominators of v and U^{-1}=U'/d2.
361 * Then lambda = { k } (componentwise)
362 * We compute x - floor(x) = {x} = { a/b } as fdiv_r(a,b)/b
363 * For open rays/facets, we need values in (0,1] rather than [0,1),
364 * so we compute {{x}} = x - ceil(x-1) = a/b - ceil((a-b)/b)
365 * = (a - b cdiv_q(a-b,b) - b + b)/b
366 * = (cdiv_r(a,b)+b)/b
367 * Finally, we compute v + lambda * U
368 * The denominator of lambda can be d1*d2, that of lambda2 = lambda*U
369 * can be at most d1, since it is integer if v = 0.
370 * The denominator of v + lambda2 is 1.
372 * The _res variants of the input variables may have been multiplied with
373 * a (list of) nonorthogonal vector(s) and may therefore have fewer columns
374 * than their original counterparts.
376 void lattice_points_fixed(Value
*vertex
, Value
*vertex_res
,
377 Matrix
*Rays
, Matrix
*Rays_res
, Matrix
*points
,
380 unsigned dim
= Rays
->NbRows
;
382 lattice_point_fixed(vertex
, vertex_res
, Rays
, Rays_res
,
387 Smith(Rays
, &U
, &W
, &D
);
391 unsigned long det2
= 1;
392 for (int i
= 0 ; i
< D
->NbRows
; ++i
)
393 det2
*= mpz_get_ui(D
->p
[i
][i
]);
396 Matrix
*T
= Matrix_Alloc(W
->NbRows
+1, W
->NbColumns
+1);
397 for (int i
= 0; i
< W
->NbRows
; ++i
)
398 Vector_Scale(W
->p
[i
], T
->p
[i
], vertex
[dim
], W
->NbColumns
);
400 Vector_Oppose(vertex
, T
->p
[dim
], dim
);
401 value_assign(T
->p
[dim
][dim
], vertex
[dim
]);
403 Matrix
*R2
= Matrix_AddRowColumn(Rays
);
404 Matrix
*inv
= Matrix_Alloc(R2
->NbRows
, R2
->NbColumns
);
405 int ok
= Matrix_Inverse(R2
, inv
);
409 Matrix
*T2
= Matrix_Alloc(dim
+1, dim
+1);
410 Matrix_Product(T
, inv
, T2
);
413 Vector
*lambda
= Vector_Alloc(dim
+1);
414 Vector
*lambda2
= Vector_Alloc(Rays_res
->NbColumns
);
415 FORALL_COSETS(det
, D
, i
, k
)
416 Vector_Matrix_Product(k
->p
, T2
, lambda
->p
);
417 for (int j
= 0; j
< dim
; ++j
)
418 mpz_fdiv_r(lambda
->p
[j
], lambda
->p
[j
], lambda
->p
[dim
]);
419 Vector_Matrix_Product(lambda
->p
, Rays_res
, lambda2
->p
);
420 for (int j
= 0; j
< lambda2
->Size
; ++j
)
421 assert(mpz_divisible_p(lambda2
->p
[j
], inv
->p
[dim
][dim
]));
422 Vector_AntiScale(lambda2
->p
, lambda2
->p
, inv
->p
[dim
][dim
], lambda2
->Size
);
423 Vector_Add(lambda2
->p
, vertex_res
, lambda2
->p
, lambda2
->Size
);
424 for (int j
= 0; j
< lambda2
->Size
; ++j
)
425 assert(mpz_divisible_p(lambda2
->p
[j
], vertex
[dim
]));
426 Vector_AntiScale(lambda2
->p
, points
->p
[i
], vertex
[dim
], lambda2
->Size
);
429 Vector_Free(lambda2
);
436 /* Returns the power of (t+1) in the term of a rational generating function,
437 * i.e., the scalar product of the actual lattice point and lambda.
438 * The lattice point is the unique lattice point in the fundamental parallelepiped
439 * of the unimodual cone i shifted to the parametric vertex W/lcm.
441 * The rows of W refer to the coordinates of the vertex
442 * The first nparam columns are the coefficients of the parameters
443 * and the final column is the constant term.
444 * lcm is the common denominator of all coefficients.
446 static evalue
**lattice_point_fractional(const mat_ZZ
& rays
, vec_ZZ
& lambda
,
450 unsigned nparam
= V
->NbColumns
-2;
451 evalue
**E
= new evalue
*[det
];
453 Matrix
* Rays
= zz2matrix(rays
);
454 Matrix
*T
= Transpose(Rays
);
455 Matrix
*T2
= Matrix_Copy(T
);
456 Matrix
*inv
= Matrix_Alloc(T2
->NbRows
, T2
->NbColumns
);
457 int ok
= Matrix_Inverse(T2
, inv
);
461 matrix2zz(V
, vertex
, V
->NbRows
, V
->NbColumns
-1);
464 num
= lambda
* vertex
;
466 evalue
*EP
= multi_monom(num
);
468 evalue_div(EP
, V
->p
[0][nparam
+1]);
470 Matrix
*L
= Matrix_Alloc(inv
->NbRows
, V
->NbColumns
);
471 Matrix_Product(inv
, V
, L
);
474 matrix2zz(T
, RT
, T
->NbRows
, T
->NbColumns
);
477 vec_ZZ p
= lambda
* RT
;
483 for (int i
= 0; i
< L
->NbRows
; ++i
) {
485 Vector_Oppose(L
->p
[i
], L
->p
[i
], nparam
+1);
486 f
= fractional_part(L
->p
[i
], V
->p
[i
][nparam
+1], nparam
, NULL
);
494 for (int i
= 0; i
< L
->NbRows
; ++i
)
495 value_assign(L
->p
[i
][nparam
+1], V
->p
[i
][nparam
+1]);
499 mpz_set_ui(denom
, det
);
500 value_multiply(denom
, L
->p
[0][nparam
+1], denom
);
503 Smith(Rays
, &U
, &W
, &D
);
507 unsigned long det2
= 1;
508 for (int i
= 0 ; i
< D
->NbRows
; ++i
)
509 det2
*= mpz_get_ui(D
->p
[i
][i
]);
512 Matrix_Transposition(inv
);
513 Matrix
*T2
= Matrix_Alloc(W
->NbRows
, inv
->NbColumns
);
514 Matrix_Product(W
, inv
, T2
);
517 unsigned dim
= D
->NbRows
;
518 Vector
*lambda
= Vector_Alloc(dim
);
520 Vector
*row
= Vector_Alloc(nparam
+1);
521 FORALL_COSETS(det
, D
, i
, k
)
522 Vector_Matrix_Product(k
->p
, T2
, lambda
->p
);
525 evalue_copy(E
[i
], EP
);
526 for (int j
= 0; j
< L
->NbRows
; ++j
) {
528 Vector_Oppose(L
->p
[j
], row
->p
, nparam
+1);
529 value_addmul(row
->p
[nparam
], L
->p
[j
][nparam
+1], lambda
->p
[j
]);
530 f
= fractional_part(row
->p
, denom
, nparam
, NULL
);
544 free_evalue_refs(EP
);
556 static evalue
**lattice_point(const mat_ZZ
& rays
, vec_ZZ
& lambda
,
559 barvinok_options
*options
)
561 evalue
**lp
= lattice_point_fractional(rays
, lambda
, V
->Vertex
, det
);
562 if (options
->lookup_table
) {
563 for (int i
= 0; i
< det
; ++i
)
564 evalue_mod2table(lp
[i
], V
->Vertex
->NbColumns
-2);
569 /* returns the unique lattice point in the fundamental parallelepiped
570 * of the unimodual cone C shifted to the parametric vertex V.
572 * The return values num and E_vertex are such that
573 * coordinate i of this lattice point is equal to
575 * num[i] + E_vertex[i]
577 void lattice_point(Param_Vertices
*V
, const mat_ZZ
& rays
, vec_ZZ
& num
,
578 evalue
**E_vertex
, barvinok_options
*options
)
580 unsigned nparam
= V
->Vertex
->NbColumns
- 2;
581 unsigned dim
= rays
.NumCols();
583 /* It doesn't really make sense to call lattice_point when dim == 0,
584 * but apparently it happens from indicator_constructor in lexmin.
590 vertex
.SetLength(nparam
+1);
595 assert(V
->Vertex
->NbRows
> 0);
596 Param_Vertex_Common_Denominator(V
);
598 if (value_notone_p(V
->Vertex
->p
[0][nparam
+1])) {
599 Matrix
* Rays
= zz2matrix(rays
);
600 Matrix
*T
= Transpose(Rays
);
601 Matrix
*T2
= Matrix_Copy(T
);
602 Matrix
*inv
= Matrix_Alloc(T2
->NbRows
, T2
->NbColumns
);
603 int ok
= Matrix_Inverse(T2
, inv
);
607 /* temporarily ignore (common) denominator */
608 V
->Vertex
->NbColumns
--;
609 Matrix
*L
= Matrix_Alloc(inv
->NbRows
, V
->Vertex
->NbColumns
);
610 Matrix_Product(inv
, V
->Vertex
, L
);
611 V
->Vertex
->NbColumns
++;
618 evalue
*remainders
[dim
];
619 for (int i
= 0; i
< dim
; ++i
)
620 remainders
[i
] = ceil(L
->p
[i
], nparam
+1, V
->Vertex
->p
[0][nparam
+1],
625 for (int i
= 0; i
< V
->Vertex
->NbRows
; ++i
) {
626 values2zz(V
->Vertex
->p
[i
], vertex
, nparam
+1);
627 E_vertex
[i
] = multi_monom(vertex
);
630 value_set_si(f
.x
.n
, 1);
631 value_assign(f
.d
, V
->Vertex
->p
[0][nparam
+1]);
633 emul(&f
, E_vertex
[i
]);
635 for (int j
= 0; j
< dim
; ++j
) {
636 if (value_zero_p(T
->p
[i
][j
]))
640 evalue_copy(&cp
, remainders
[j
]);
641 if (value_notone_p(T
->p
[i
][j
])) {
642 value_set_si(f
.d
, 1);
643 value_assign(f
.x
.n
, T
->p
[i
][j
]);
646 eadd(&cp
, E_vertex
[i
]);
647 free_evalue_refs(&cp
);
650 for (int i
= 0; i
< dim
; ++i
)
651 evalue_free(remainders
[i
]);
653 free_evalue_refs(&f
);
661 for (int i
= 0; i
< V
->Vertex
->NbRows
; ++i
) {
663 if (First_Non_Zero(V
->Vertex
->p
[i
], nparam
) == -1) {
665 value2zz(V
->Vertex
->p
[i
][nparam
], num
[i
]);
667 values2zz(V
->Vertex
->p
[i
], vertex
, nparam
+1);
668 E_vertex
[i
] = multi_monom(vertex
);
674 static int lattice_point_fixed(Param_Vertices
* V
, const mat_ZZ
& rays
,
675 vec_ZZ
& lambda
, term_info
* term
, unsigned long det
)
677 unsigned nparam
= V
->Vertex
->NbColumns
- 2;
678 unsigned dim
= rays
.NumCols();
680 for (int i
= 0; i
< dim
; ++i
)
681 if (First_Non_Zero(V
->Vertex
->p
[i
], nparam
) != -1)
684 Vector
*fixed
= Vector_Alloc(dim
+1);
685 for (int i
= 0; i
< dim
; ++i
)
686 value_assign(fixed
->p
[i
], V
->Vertex
->p
[i
][nparam
]);
687 value_assign(fixed
->p
[dim
], V
->Vertex
->p
[0][nparam
+1]);
690 Matrix
*points
= Matrix_Alloc(det
, dim
);
691 Matrix
* Rays
= zz2matrix(rays
);
692 lattice_points_fixed(fixed
->p
, fixed
->p
, Rays
, Rays
, points
, det
);
694 matrix2zz(points
, vertex
, points
->NbRows
, points
->NbColumns
);
697 term
->constant
= vertex
* lambda
;
703 /* Returns the power of (t+1) in the term of a rational generating function,
704 * i.e., the scalar product of the actual lattice point and lambda.
705 * The lattice point is the unique lattice point in the fundamental parallelepiped
706 * of the unimodual cone i shifted to the parametric vertex V.
708 * The result is returned in term.
710 void lattice_point(Param_Vertices
* V
, const mat_ZZ
& rays
, vec_ZZ
& lambda
,
711 term_info
* term
, unsigned long det
,
712 barvinok_options
*options
)
714 unsigned nparam
= V
->Vertex
->NbColumns
- 2;
715 unsigned dim
= rays
.NumCols();
717 vertex
.SetDims(V
->Vertex
->NbRows
, nparam
+1);
719 Param_Vertex_Common_Denominator(V
);
721 if (lattice_point_fixed(V
, rays
, lambda
, term
, det
))
724 if (det
!= 1 || value_notone_p(V
->Vertex
->p
[0][nparam
+1])) {
725 term
->E
= lattice_point(rays
, lambda
, V
, det
, options
);
728 for (int i
= 0; i
< V
->Vertex
->NbRows
; ++i
) {
729 assert(value_one_p(V
->Vertex
->p
[i
][nparam
+1])); // for now
730 values2zz(V
->Vertex
->p
[i
], vertex
[i
], nparam
+1);
734 num
= lambda
* vertex
;
737 for (int j
= 0; j
< nparam
; ++j
)
741 term
->E
= new evalue
*[1];
742 term
->E
[0] = multi_monom(num
);
745 term
->constant
.SetLength(1);
746 term
->constant
[0] = num
[nparam
];