2 * Copyright 2008-2009 Katholieke Universiteit Leuven
4 * Use of this software is governed by the GNU LGPLv2.1 license
6 * Written by Sven Verdoolaege, K.U.Leuven, Departement
7 * Computerwetenschappen, Celestijnenlaan 200A, B-3001 Leuven, Belgium
12 #include "isl_map_private.h"
16 #include "isl_equalities.h"
19 static struct isl_basic_set *uset_convex_hull_wrap_bounded(struct isl_set *set);
21 static void swap_ineq(struct isl_basic_map *bmap, unsigned i, unsigned j)
27 bmap->ineq[i] = bmap->ineq[j];
32 /* Return 1 if constraint c is redundant with respect to the constraints
33 * in bmap. If c is a lower [upper] bound in some variable and bmap
34 * does not have a lower [upper] bound in that variable, then c cannot
35 * be redundant and we do not need solve any lp.
37 int isl_basic_map_constraint_is_redundant(struct isl_basic_map **bmap,
38 isl_int *c, isl_int *opt_n, isl_int *opt_d)
40 enum isl_lp_result res;
47 total = isl_basic_map_total_dim(*bmap);
48 for (i = 0; i < total; ++i) {
50 if (isl_int_is_zero(c[1+i]))
52 sign = isl_int_sgn(c[1+i]);
53 for (j = 0; j < (*bmap)->n_ineq; ++j)
54 if (sign == isl_int_sgn((*bmap)->ineq[j][1+i]))
56 if (j == (*bmap)->n_ineq)
62 res = isl_basic_map_solve_lp(*bmap, 0, c, (*bmap)->ctx->one,
64 if (res == isl_lp_unbounded)
66 if (res == isl_lp_error)
68 if (res == isl_lp_empty) {
69 *bmap = isl_basic_map_set_to_empty(*bmap);
72 return !isl_int_is_neg(*opt_n);
75 int isl_basic_set_constraint_is_redundant(struct isl_basic_set **bset,
76 isl_int *c, isl_int *opt_n, isl_int *opt_d)
78 return isl_basic_map_constraint_is_redundant(
79 (struct isl_basic_map **)bset, c, opt_n, opt_d);
83 * constraints. If the minimal value along the normal of a constraint
84 * is the same if the constraint is removed, then the constraint is redundant.
86 * Alternatively, we could have intersected the basic map with the
87 * corresponding equality and the checked if the dimension was that
90 __isl_give isl_basic_map *isl_basic_map_remove_redundancies(
91 __isl_take isl_basic_map *bmap)
98 bmap = isl_basic_map_gauss(bmap, NULL);
99 if (ISL_F_ISSET(bmap, ISL_BASIC_MAP_EMPTY))
101 if (ISL_F_ISSET(bmap, ISL_BASIC_MAP_NO_REDUNDANT))
103 if (bmap->n_ineq <= 1)
106 tab = isl_tab_from_basic_map(bmap);
107 if (isl_tab_detect_implicit_equalities(tab) < 0)
109 if (isl_tab_detect_redundant(tab) < 0)
111 bmap = isl_basic_map_update_from_tab(bmap, tab);
113 ISL_F_SET(bmap, ISL_BASIC_MAP_NO_IMPLICIT);
114 ISL_F_SET(bmap, ISL_BASIC_MAP_NO_REDUNDANT);
118 isl_basic_map_free(bmap);
122 __isl_give isl_basic_set *isl_basic_set_remove_redundancies(
123 __isl_take isl_basic_set *bset)
125 return (struct isl_basic_set *)
126 isl_basic_map_remove_redundancies((struct isl_basic_map *)bset);
129 /* Check if the set set is bound in the direction of the affine
130 * constraint c and if so, set the constant term such that the
131 * resulting constraint is a bounding constraint for the set.
133 static int uset_is_bound(struct isl_set *set, isl_int *c, unsigned len)
141 isl_int_init(opt_denom);
143 for (j = 0; j < set->n; ++j) {
144 enum isl_lp_result res;
146 if (ISL_F_ISSET(set->p[j], ISL_BASIC_SET_EMPTY))
149 res = isl_basic_set_solve_lp(set->p[j],
150 0, c, set->ctx->one, &opt, &opt_denom, NULL);
151 if (res == isl_lp_unbounded)
153 if (res == isl_lp_error)
155 if (res == isl_lp_empty) {
156 set->p[j] = isl_basic_set_set_to_empty(set->p[j]);
161 if (first || isl_int_is_neg(opt)) {
162 if (!isl_int_is_one(opt_denom))
163 isl_seq_scale(c, c, opt_denom, len);
164 isl_int_sub(c[0], c[0], opt);
169 isl_int_clear(opt_denom);
173 isl_int_clear(opt_denom);
177 struct isl_basic_set *isl_basic_set_set_rational(struct isl_basic_set *bset)
182 if (ISL_F_ISSET(bset, ISL_BASIC_MAP_RATIONAL))
185 bset = isl_basic_set_cow(bset);
189 ISL_F_SET(bset, ISL_BASIC_MAP_RATIONAL);
191 return isl_basic_set_finalize(bset);
194 static struct isl_set *isl_set_set_rational(struct isl_set *set)
198 set = isl_set_cow(set);
201 for (i = 0; i < set->n; ++i) {
202 set->p[i] = isl_basic_set_set_rational(set->p[i]);
212 static struct isl_basic_set *isl_basic_set_add_equality(
213 struct isl_basic_set *bset, isl_int *c)
221 if (ISL_F_ISSET(bset, ISL_BASIC_SET_EMPTY))
224 isl_assert(bset->ctx, isl_basic_set_n_param(bset) == 0, goto error);
225 isl_assert(bset->ctx, bset->n_div == 0, goto error);
226 dim = isl_basic_set_n_dim(bset);
227 bset = isl_basic_set_cow(bset);
228 bset = isl_basic_set_extend(bset, 0, dim, 0, 1, 0);
229 i = isl_basic_set_alloc_equality(bset);
232 isl_seq_cpy(bset->eq[i], c, 1 + dim);
235 isl_basic_set_free(bset);
239 static struct isl_set *isl_set_add_basic_set_equality(struct isl_set *set, isl_int *c)
243 set = isl_set_cow(set);
246 for (i = 0; i < set->n; ++i) {
247 set->p[i] = isl_basic_set_add_equality(set->p[i], c);
257 /* Given a union of basic sets, construct the constraints for wrapping
258 * a facet around one of its ridges.
259 * In particular, if each of n the d-dimensional basic sets i in "set"
260 * contains the origin, satisfies the constraints x_1 >= 0 and x_2 >= 0
261 * and is defined by the constraints
265 * then the resulting set is of dimension n*(1+d) and has as constraints
274 static struct isl_basic_set *wrap_constraints(struct isl_set *set)
276 struct isl_basic_set *lp;
280 unsigned dim, lp_dim;
285 dim = 1 + isl_set_n_dim(set);
288 for (i = 0; i < set->n; ++i) {
289 n_eq += set->p[i]->n_eq;
290 n_ineq += set->p[i]->n_ineq;
292 lp = isl_basic_set_alloc(set->ctx, 0, dim * set->n, 0, n_eq, n_ineq);
295 lp_dim = isl_basic_set_n_dim(lp);
296 k = isl_basic_set_alloc_equality(lp);
297 isl_int_set_si(lp->eq[k][0], -1);
298 for (i = 0; i < set->n; ++i) {
299 isl_int_set_si(lp->eq[k][1+dim*i], 0);
300 isl_int_set_si(lp->eq[k][1+dim*i+1], 1);
301 isl_seq_clr(lp->eq[k]+1+dim*i+2, dim-2);
303 for (i = 0; i < set->n; ++i) {
304 k = isl_basic_set_alloc_inequality(lp);
305 isl_seq_clr(lp->ineq[k], 1+lp_dim);
306 isl_int_set_si(lp->ineq[k][1+dim*i], 1);
308 for (j = 0; j < set->p[i]->n_eq; ++j) {
309 k = isl_basic_set_alloc_equality(lp);
310 isl_seq_clr(lp->eq[k], 1+dim*i);
311 isl_seq_cpy(lp->eq[k]+1+dim*i, set->p[i]->eq[j], dim);
312 isl_seq_clr(lp->eq[k]+1+dim*(i+1), dim*(set->n-i-1));
315 for (j = 0; j < set->p[i]->n_ineq; ++j) {
316 k = isl_basic_set_alloc_inequality(lp);
317 isl_seq_clr(lp->ineq[k], 1+dim*i);
318 isl_seq_cpy(lp->ineq[k]+1+dim*i, set->p[i]->ineq[j], dim);
319 isl_seq_clr(lp->ineq[k]+1+dim*(i+1), dim*(set->n-i-1));
325 /* Given a facet "facet" of the convex hull of "set" and a facet "ridge"
326 * of that facet, compute the other facet of the convex hull that contains
329 * We first transform the set such that the facet constraint becomes
333 * I.e., the facet lies in
337 * and on that facet, the constraint that defines the ridge is
341 * (This transformation is not strictly needed, all that is needed is
342 * that the ridge contains the origin.)
344 * Since the ridge contains the origin, the cone of the convex hull
345 * will be of the form
350 * with this second constraint defining the new facet.
351 * The constant a is obtained by settting x_1 in the cone of the
352 * convex hull to 1 and minimizing x_2.
353 * Now, each element in the cone of the convex hull is the sum
354 * of elements in the cones of the basic sets.
355 * If a_i is the dilation factor of basic set i, then the problem
356 * we need to solve is
369 * the constraints of each (transformed) basic set.
370 * If a = n/d, then the constraint defining the new facet (in the transformed
373 * -n x_1 + d x_2 >= 0
375 * In the original space, we need to take the same combination of the
376 * corresponding constraints "facet" and "ridge".
378 * If a = -infty = "-1/0", then we just return the original facet constraint.
379 * This means that the facet is unbounded, but has a bounded intersection
380 * with the union of sets.
382 isl_int *isl_set_wrap_facet(__isl_keep isl_set *set,
383 isl_int *facet, isl_int *ridge)
387 struct isl_mat *T = NULL;
388 struct isl_basic_set *lp = NULL;
390 enum isl_lp_result res;
397 set = isl_set_copy(set);
398 set = isl_set_set_rational(set);
400 dim = 1 + isl_set_n_dim(set);
401 T = isl_mat_alloc(ctx, 3, dim);
404 isl_int_set_si(T->row[0][0], 1);
405 isl_seq_clr(T->row[0]+1, dim - 1);
406 isl_seq_cpy(T->row[1], facet, dim);
407 isl_seq_cpy(T->row[2], ridge, dim);
408 T = isl_mat_right_inverse(T);
409 set = isl_set_preimage(set, T);
413 lp = wrap_constraints(set);
414 obj = isl_vec_alloc(ctx, 1 + dim*set->n);
417 isl_int_set_si(obj->block.data[0], 0);
418 for (i = 0; i < set->n; ++i) {
419 isl_seq_clr(obj->block.data + 1 + dim*i, 2);
420 isl_int_set_si(obj->block.data[1 + dim*i+2], 1);
421 isl_seq_clr(obj->block.data + 1 + dim*i+3, dim-3);
425 res = isl_basic_set_solve_lp(lp, 0,
426 obj->block.data, ctx->one, &num, &den, NULL);
427 if (res == isl_lp_ok) {
428 isl_int_neg(num, num);
429 isl_seq_combine(facet, num, facet, den, ridge, dim);
434 isl_basic_set_free(lp);
436 if (res == isl_lp_error)
438 isl_assert(ctx, res == isl_lp_ok || res == isl_lp_unbounded,
442 isl_basic_set_free(lp);
448 /* Compute the constraint of a facet of "set".
450 * We first compute the intersection with a bounding constraint
451 * that is orthogonal to one of the coordinate axes.
452 * If the affine hull of this intersection has only one equality,
453 * we have found a facet.
454 * Otherwise, we wrap the current bounding constraint around
455 * one of the equalities of the face (one that is not equal to
456 * the current bounding constraint).
457 * This process continues until we have found a facet.
458 * The dimension of the intersection increases by at least
459 * one on each iteration, so termination is guaranteed.
461 static __isl_give isl_mat *initial_facet_constraint(__isl_keep isl_set *set)
463 struct isl_set *slice = NULL;
464 struct isl_basic_set *face = NULL;
466 unsigned dim = isl_set_n_dim(set);
470 isl_assert(set->ctx, set->n > 0, goto error);
471 bounds = isl_mat_alloc(set->ctx, 1, 1 + dim);
475 isl_seq_clr(bounds->row[0], dim);
476 isl_int_set_si(bounds->row[0][1 + dim - 1], 1);
477 is_bound = uset_is_bound(set, bounds->row[0], 1 + dim);
480 isl_assert(set->ctx, is_bound, goto error);
481 isl_seq_normalize(set->ctx, bounds->row[0], 1 + dim);
485 slice = isl_set_copy(set);
486 slice = isl_set_add_basic_set_equality(slice, bounds->row[0]);
487 face = isl_set_affine_hull(slice);
490 if (face->n_eq == 1) {
491 isl_basic_set_free(face);
494 for (i = 0; i < face->n_eq; ++i)
495 if (!isl_seq_eq(bounds->row[0], face->eq[i], 1 + dim) &&
496 !isl_seq_is_neg(bounds->row[0],
497 face->eq[i], 1 + dim))
499 isl_assert(set->ctx, i < face->n_eq, goto error);
500 if (!isl_set_wrap_facet(set, bounds->row[0], face->eq[i]))
502 isl_seq_normalize(set->ctx, bounds->row[0], bounds->n_col);
503 isl_basic_set_free(face);
508 isl_basic_set_free(face);
509 isl_mat_free(bounds);
513 /* Given the bounding constraint "c" of a facet of the convex hull of "set",
514 * compute a hyperplane description of the facet, i.e., compute the facets
517 * We compute an affine transformation that transforms the constraint
526 * by computing the right inverse U of a matrix that starts with the rows
539 * Since z_1 is zero, we can drop this variable as well as the corresponding
540 * column of U to obtain
548 * with Q' equal to Q, but without the corresponding row.
549 * After computing the facets of the facet in the z' space,
550 * we convert them back to the x space through Q.
552 static struct isl_basic_set *compute_facet(struct isl_set *set, isl_int *c)
554 struct isl_mat *m, *U, *Q;
555 struct isl_basic_set *facet = NULL;
560 set = isl_set_copy(set);
561 dim = isl_set_n_dim(set);
562 m = isl_mat_alloc(set->ctx, 2, 1 + dim);
565 isl_int_set_si(m->row[0][0], 1);
566 isl_seq_clr(m->row[0]+1, dim);
567 isl_seq_cpy(m->row[1], c, 1+dim);
568 U = isl_mat_right_inverse(m);
569 Q = isl_mat_right_inverse(isl_mat_copy(U));
570 U = isl_mat_drop_cols(U, 1, 1);
571 Q = isl_mat_drop_rows(Q, 1, 1);
572 set = isl_set_preimage(set, U);
573 facet = uset_convex_hull_wrap_bounded(set);
574 facet = isl_basic_set_preimage(facet, Q);
576 isl_assert(ctx, facet->n_eq == 0, goto error);
579 isl_basic_set_free(facet);
584 /* Given an initial facet constraint, compute the remaining facets.
585 * We do this by running through all facets found so far and computing
586 * the adjacent facets through wrapping, adding those facets that we
587 * hadn't already found before.
589 * For each facet we have found so far, we first compute its facets
590 * in the resulting convex hull. That is, we compute the ridges
591 * of the resulting convex hull contained in the facet.
592 * We also compute the corresponding facet in the current approximation
593 * of the convex hull. There is no need to wrap around the ridges
594 * in this facet since that would result in a facet that is already
595 * present in the current approximation.
597 * This function can still be significantly optimized by checking which of
598 * the facets of the basic sets are also facets of the convex hull and
599 * using all the facets so far to help in constructing the facets of the
602 * using the technique in section "3.1 Ridge Generation" of
603 * "Extended Convex Hull" by Fukuda et al.
605 static struct isl_basic_set *extend(struct isl_basic_set *hull,
610 struct isl_basic_set *facet = NULL;
611 struct isl_basic_set *hull_facet = NULL;
617 isl_assert(set->ctx, set->n > 0, goto error);
619 dim = isl_set_n_dim(set);
621 for (i = 0; i < hull->n_ineq; ++i) {
622 facet = compute_facet(set, hull->ineq[i]);
623 facet = isl_basic_set_add_equality(facet, hull->ineq[i]);
624 facet = isl_basic_set_gauss(facet, NULL);
625 facet = isl_basic_set_normalize_constraints(facet);
626 hull_facet = isl_basic_set_copy(hull);
627 hull_facet = isl_basic_set_add_equality(hull_facet, hull->ineq[i]);
628 hull_facet = isl_basic_set_gauss(hull_facet, NULL);
629 hull_facet = isl_basic_set_normalize_constraints(hull_facet);
630 if (!facet || !hull_facet)
632 hull = isl_basic_set_cow(hull);
633 hull = isl_basic_set_extend_dim(hull,
634 isl_dim_copy(hull->dim), 0, 0, facet->n_ineq);
637 for (j = 0; j < facet->n_ineq; ++j) {
638 for (f = 0; f < hull_facet->n_ineq; ++f)
639 if (isl_seq_eq(facet->ineq[j],
640 hull_facet->ineq[f], 1 + dim))
642 if (f < hull_facet->n_ineq)
644 k = isl_basic_set_alloc_inequality(hull);
647 isl_seq_cpy(hull->ineq[k], hull->ineq[i], 1+dim);
648 if (!isl_set_wrap_facet(set, hull->ineq[k], facet->ineq[j]))
651 isl_basic_set_free(hull_facet);
652 isl_basic_set_free(facet);
654 hull = isl_basic_set_simplify(hull);
655 hull = isl_basic_set_finalize(hull);
658 isl_basic_set_free(hull_facet);
659 isl_basic_set_free(facet);
660 isl_basic_set_free(hull);
664 /* Special case for computing the convex hull of a one dimensional set.
665 * We simply collect the lower and upper bounds of each basic set
666 * and the biggest of those.
668 static struct isl_basic_set *convex_hull_1d(struct isl_set *set)
670 struct isl_mat *c = NULL;
671 isl_int *lower = NULL;
672 isl_int *upper = NULL;
675 struct isl_basic_set *hull;
677 for (i = 0; i < set->n; ++i) {
678 set->p[i] = isl_basic_set_simplify(set->p[i]);
682 set = isl_set_remove_empty_parts(set);
685 isl_assert(set->ctx, set->n > 0, goto error);
686 c = isl_mat_alloc(set->ctx, 2, 2);
690 if (set->p[0]->n_eq > 0) {
691 isl_assert(set->ctx, set->p[0]->n_eq == 1, goto error);
694 if (isl_int_is_pos(set->p[0]->eq[0][1])) {
695 isl_seq_cpy(lower, set->p[0]->eq[0], 2);
696 isl_seq_neg(upper, set->p[0]->eq[0], 2);
698 isl_seq_neg(lower, set->p[0]->eq[0], 2);
699 isl_seq_cpy(upper, set->p[0]->eq[0], 2);
702 for (j = 0; j < set->p[0]->n_ineq; ++j) {
703 if (isl_int_is_pos(set->p[0]->ineq[j][1])) {
705 isl_seq_cpy(lower, set->p[0]->ineq[j], 2);
708 isl_seq_cpy(upper, set->p[0]->ineq[j], 2);
715 for (i = 0; i < set->n; ++i) {
716 struct isl_basic_set *bset = set->p[i];
720 for (j = 0; j < bset->n_eq; ++j) {
724 isl_int_mul(a, lower[0], bset->eq[j][1]);
725 isl_int_mul(b, lower[1], bset->eq[j][0]);
726 if (isl_int_lt(a, b) && isl_int_is_pos(bset->eq[j][1]))
727 isl_seq_cpy(lower, bset->eq[j], 2);
728 if (isl_int_gt(a, b) && isl_int_is_neg(bset->eq[j][1]))
729 isl_seq_neg(lower, bset->eq[j], 2);
732 isl_int_mul(a, upper[0], bset->eq[j][1]);
733 isl_int_mul(b, upper[1], bset->eq[j][0]);
734 if (isl_int_lt(a, b) && isl_int_is_pos(bset->eq[j][1]))
735 isl_seq_neg(upper, bset->eq[j], 2);
736 if (isl_int_gt(a, b) && isl_int_is_neg(bset->eq[j][1]))
737 isl_seq_cpy(upper, bset->eq[j], 2);
740 for (j = 0; j < bset->n_ineq; ++j) {
741 if (isl_int_is_pos(bset->ineq[j][1]))
743 if (isl_int_is_neg(bset->ineq[j][1]))
745 if (lower && isl_int_is_pos(bset->ineq[j][1])) {
746 isl_int_mul(a, lower[0], bset->ineq[j][1]);
747 isl_int_mul(b, lower[1], bset->ineq[j][0]);
748 if (isl_int_lt(a, b))
749 isl_seq_cpy(lower, bset->ineq[j], 2);
751 if (upper && isl_int_is_neg(bset->ineq[j][1])) {
752 isl_int_mul(a, upper[0], bset->ineq[j][1]);
753 isl_int_mul(b, upper[1], bset->ineq[j][0]);
754 if (isl_int_gt(a, b))
755 isl_seq_cpy(upper, bset->ineq[j], 2);
766 hull = isl_basic_set_alloc(set->ctx, 0, 1, 0, 0, 2);
767 hull = isl_basic_set_set_rational(hull);
771 k = isl_basic_set_alloc_inequality(hull);
772 isl_seq_cpy(hull->ineq[k], lower, 2);
775 k = isl_basic_set_alloc_inequality(hull);
776 isl_seq_cpy(hull->ineq[k], upper, 2);
778 hull = isl_basic_set_finalize(hull);
788 /* Project out final n dimensions using Fourier-Motzkin */
789 static struct isl_set *set_project_out(struct isl_ctx *ctx,
790 struct isl_set *set, unsigned n)
792 return isl_set_remove_dims(set, isl_set_n_dim(set) - n, n);
795 static struct isl_basic_set *convex_hull_0d(struct isl_set *set)
797 struct isl_basic_set *convex_hull;
802 if (isl_set_is_empty(set))
803 convex_hull = isl_basic_set_empty(isl_dim_copy(set->dim));
805 convex_hull = isl_basic_set_universe(isl_dim_copy(set->dim));
810 /* Compute the convex hull of a pair of basic sets without any parameters or
811 * integer divisions using Fourier-Motzkin elimination.
812 * The convex hull is the set of all points that can be written as
813 * the sum of points from both basic sets (in homogeneous coordinates).
814 * We set up the constraints in a space with dimensions for each of
815 * the three sets and then project out the dimensions corresponding
816 * to the two original basic sets, retaining only those corresponding
817 * to the convex hull.
819 static struct isl_basic_set *convex_hull_pair_elim(struct isl_basic_set *bset1,
820 struct isl_basic_set *bset2)
823 struct isl_basic_set *bset[2];
824 struct isl_basic_set *hull = NULL;
827 if (!bset1 || !bset2)
830 dim = isl_basic_set_n_dim(bset1);
831 hull = isl_basic_set_alloc(bset1->ctx, 0, 2 + 3 * dim, 0,
832 1 + dim + bset1->n_eq + bset2->n_eq,
833 2 + bset1->n_ineq + bset2->n_ineq);
836 for (i = 0; i < 2; ++i) {
837 for (j = 0; j < bset[i]->n_eq; ++j) {
838 k = isl_basic_set_alloc_equality(hull);
841 isl_seq_clr(hull->eq[k], (i+1) * (1+dim));
842 isl_seq_clr(hull->eq[k]+(i+2)*(1+dim), (1-i)*(1+dim));
843 isl_seq_cpy(hull->eq[k]+(i+1)*(1+dim), bset[i]->eq[j],
846 for (j = 0; j < bset[i]->n_ineq; ++j) {
847 k = isl_basic_set_alloc_inequality(hull);
850 isl_seq_clr(hull->ineq[k], (i+1) * (1+dim));
851 isl_seq_clr(hull->ineq[k]+(i+2)*(1+dim), (1-i)*(1+dim));
852 isl_seq_cpy(hull->ineq[k]+(i+1)*(1+dim),
853 bset[i]->ineq[j], 1+dim);
855 k = isl_basic_set_alloc_inequality(hull);
858 isl_seq_clr(hull->ineq[k], 1+2+3*dim);
859 isl_int_set_si(hull->ineq[k][(i+1)*(1+dim)], 1);
861 for (j = 0; j < 1+dim; ++j) {
862 k = isl_basic_set_alloc_equality(hull);
865 isl_seq_clr(hull->eq[k], 1+2+3*dim);
866 isl_int_set_si(hull->eq[k][j], -1);
867 isl_int_set_si(hull->eq[k][1+dim+j], 1);
868 isl_int_set_si(hull->eq[k][2*(1+dim)+j], 1);
870 hull = isl_basic_set_set_rational(hull);
871 hull = isl_basic_set_remove_dims(hull, dim, 2*(1+dim));
872 hull = isl_basic_set_remove_redundancies(hull);
873 isl_basic_set_free(bset1);
874 isl_basic_set_free(bset2);
877 isl_basic_set_free(bset1);
878 isl_basic_set_free(bset2);
879 isl_basic_set_free(hull);
883 /* Is the set bounded for each value of the parameters?
885 int isl_basic_set_is_bounded(__isl_keep isl_basic_set *bset)
892 if (isl_basic_set_fast_is_empty(bset))
895 tab = isl_tab_from_recession_cone(bset, 1);
896 bounded = isl_tab_cone_is_bounded(tab);
901 /* Is the set bounded for each value of the parameters?
903 int isl_set_is_bounded(__isl_keep isl_set *set)
910 for (i = 0; i < set->n; ++i) {
911 int bounded = isl_basic_set_is_bounded(set->p[i]);
912 if (!bounded || bounded < 0)
918 /* Compute the lineality space of the convex hull of bset1 and bset2.
920 * We first compute the intersection of the recession cone of bset1
921 * with the negative of the recession cone of bset2 and then compute
922 * the linear hull of the resulting cone.
924 static struct isl_basic_set *induced_lineality_space(
925 struct isl_basic_set *bset1, struct isl_basic_set *bset2)
928 struct isl_basic_set *lin = NULL;
931 if (!bset1 || !bset2)
934 dim = isl_basic_set_total_dim(bset1);
935 lin = isl_basic_set_alloc_dim(isl_basic_set_get_dim(bset1), 0,
936 bset1->n_eq + bset2->n_eq,
937 bset1->n_ineq + bset2->n_ineq);
938 lin = isl_basic_set_set_rational(lin);
941 for (i = 0; i < bset1->n_eq; ++i) {
942 k = isl_basic_set_alloc_equality(lin);
945 isl_int_set_si(lin->eq[k][0], 0);
946 isl_seq_cpy(lin->eq[k] + 1, bset1->eq[i] + 1, dim);
948 for (i = 0; i < bset1->n_ineq; ++i) {
949 k = isl_basic_set_alloc_inequality(lin);
952 isl_int_set_si(lin->ineq[k][0], 0);
953 isl_seq_cpy(lin->ineq[k] + 1, bset1->ineq[i] + 1, dim);
955 for (i = 0; i < bset2->n_eq; ++i) {
956 k = isl_basic_set_alloc_equality(lin);
959 isl_int_set_si(lin->eq[k][0], 0);
960 isl_seq_neg(lin->eq[k] + 1, bset2->eq[i] + 1, dim);
962 for (i = 0; i < bset2->n_ineq; ++i) {
963 k = isl_basic_set_alloc_inequality(lin);
966 isl_int_set_si(lin->ineq[k][0], 0);
967 isl_seq_neg(lin->ineq[k] + 1, bset2->ineq[i] + 1, dim);
970 isl_basic_set_free(bset1);
971 isl_basic_set_free(bset2);
972 return isl_basic_set_affine_hull(lin);
974 isl_basic_set_free(lin);
975 isl_basic_set_free(bset1);
976 isl_basic_set_free(bset2);
980 static struct isl_basic_set *uset_convex_hull(struct isl_set *set);
982 /* Given a set and a linear space "lin" of dimension n > 0,
983 * project the linear space from the set, compute the convex hull
984 * and then map the set back to the original space.
990 * describe the linear space. We first compute the Hermite normal
991 * form H = M U of M = H Q, to obtain
995 * The last n rows of H will be zero, so the last n variables of x' = Q x
996 * are the one we want to project out. We do this by transforming each
997 * basic set A x >= b to A U x' >= b and then removing the last n dimensions.
998 * After computing the convex hull in x'_1, i.e., A' x'_1 >= b',
999 * we transform the hull back to the original space as A' Q_1 x >= b',
1000 * with Q_1 all but the last n rows of Q.
1002 static struct isl_basic_set *modulo_lineality(struct isl_set *set,
1003 struct isl_basic_set *lin)
1005 unsigned total = isl_basic_set_total_dim(lin);
1007 struct isl_basic_set *hull;
1008 struct isl_mat *M, *U, *Q;
1012 lin_dim = total - lin->n_eq;
1013 M = isl_mat_sub_alloc(set->ctx, lin->eq, 0, lin->n_eq, 1, total);
1014 M = isl_mat_left_hermite(M, 0, &U, &Q);
1018 isl_basic_set_free(lin);
1020 Q = isl_mat_drop_rows(Q, Q->n_row - lin_dim, lin_dim);
1022 U = isl_mat_lin_to_aff(U);
1023 Q = isl_mat_lin_to_aff(Q);
1025 set = isl_set_preimage(set, U);
1026 set = isl_set_remove_dims(set, total - lin_dim, lin_dim);
1027 hull = uset_convex_hull(set);
1028 hull = isl_basic_set_preimage(hull, Q);
1032 isl_basic_set_free(lin);
1037 /* Given two polyhedra with as constraints h_{ij} x >= 0 in homegeneous space,
1038 * set up an LP for solving
1040 * \sum_j \alpha_{1j} h_{1j} = \sum_j \alpha_{2j} h_{2j}
1042 * \alpha{i0} corresponds to the (implicit) positivity constraint 1 >= 0
1043 * The next \alpha{ij} correspond to the equalities and come in pairs.
1044 * The final \alpha{ij} correspond to the inequalities.
1046 static struct isl_basic_set *valid_direction_lp(
1047 struct isl_basic_set *bset1, struct isl_basic_set *bset2)
1049 struct isl_dim *dim;
1050 struct isl_basic_set *lp;
1055 if (!bset1 || !bset2)
1057 d = 1 + isl_basic_set_total_dim(bset1);
1059 2 * bset1->n_eq + bset1->n_ineq + 2 * bset2->n_eq + bset2->n_ineq;
1060 dim = isl_dim_set_alloc(bset1->ctx, 0, n);
1061 lp = isl_basic_set_alloc_dim(dim, 0, d, n);
1064 for (i = 0; i < n; ++i) {
1065 k = isl_basic_set_alloc_inequality(lp);
1068 isl_seq_clr(lp->ineq[k] + 1, n);
1069 isl_int_set_si(lp->ineq[k][0], -1);
1070 isl_int_set_si(lp->ineq[k][1 + i], 1);
1072 for (i = 0; i < d; ++i) {
1073 k = isl_basic_set_alloc_equality(lp);
1077 isl_int_set_si(lp->eq[k][n++], 0);
1078 /* positivity constraint 1 >= 0 */
1079 isl_int_set_si(lp->eq[k][n++], i == 0);
1080 for (j = 0; j < bset1->n_eq; ++j) {
1081 isl_int_set(lp->eq[k][n++], bset1->eq[j][i]);
1082 isl_int_neg(lp->eq[k][n++], bset1->eq[j][i]);
1084 for (j = 0; j < bset1->n_ineq; ++j)
1085 isl_int_set(lp->eq[k][n++], bset1->ineq[j][i]);
1086 /* positivity constraint 1 >= 0 */
1087 isl_int_set_si(lp->eq[k][n++], -(i == 0));
1088 for (j = 0; j < bset2->n_eq; ++j) {
1089 isl_int_neg(lp->eq[k][n++], bset2->eq[j][i]);
1090 isl_int_set(lp->eq[k][n++], bset2->eq[j][i]);
1092 for (j = 0; j < bset2->n_ineq; ++j)
1093 isl_int_neg(lp->eq[k][n++], bset2->ineq[j][i]);
1095 lp = isl_basic_set_gauss(lp, NULL);
1096 isl_basic_set_free(bset1);
1097 isl_basic_set_free(bset2);
1100 isl_basic_set_free(bset1);
1101 isl_basic_set_free(bset2);
1105 /* Compute a vector s in the homogeneous space such that <s, r> > 0
1106 * for all rays in the homogeneous space of the two cones that correspond
1107 * to the input polyhedra bset1 and bset2.
1109 * We compute s as a vector that satisfies
1111 * s = \sum_j \alpha_{ij} h_{ij} for i = 1,2 (*)
1113 * with h_{ij} the normals of the facets of polyhedron i
1114 * (including the "positivity constraint" 1 >= 0) and \alpha_{ij}
1115 * strictly positive numbers. For simplicity we impose \alpha_{ij} >= 1.
1116 * We first set up an LP with as variables the \alpha{ij}.
1117 * In this formulation, for each polyhedron i,
1118 * the first constraint is the positivity constraint, followed by pairs
1119 * of variables for the equalities, followed by variables for the inequalities.
1120 * We then simply pick a feasible solution and compute s using (*).
1122 * Note that we simply pick any valid direction and make no attempt
1123 * to pick a "good" or even the "best" valid direction.
1125 static struct isl_vec *valid_direction(
1126 struct isl_basic_set *bset1, struct isl_basic_set *bset2)
1128 struct isl_basic_set *lp;
1129 struct isl_tab *tab;
1130 struct isl_vec *sample = NULL;
1131 struct isl_vec *dir;
1136 if (!bset1 || !bset2)
1138 lp = valid_direction_lp(isl_basic_set_copy(bset1),
1139 isl_basic_set_copy(bset2));
1140 tab = isl_tab_from_basic_set(lp);
1141 sample = isl_tab_get_sample_value(tab);
1143 isl_basic_set_free(lp);
1146 d = isl_basic_set_total_dim(bset1);
1147 dir = isl_vec_alloc(bset1->ctx, 1 + d);
1150 isl_seq_clr(dir->block.data + 1, dir->size - 1);
1152 /* positivity constraint 1 >= 0 */
1153 isl_int_set(dir->block.data[0], sample->block.data[n++]);
1154 for (i = 0; i < bset1->n_eq; ++i) {
1155 isl_int_sub(sample->block.data[n],
1156 sample->block.data[n], sample->block.data[n+1]);
1157 isl_seq_combine(dir->block.data,
1158 bset1->ctx->one, dir->block.data,
1159 sample->block.data[n], bset1->eq[i], 1 + d);
1163 for (i = 0; i < bset1->n_ineq; ++i)
1164 isl_seq_combine(dir->block.data,
1165 bset1->ctx->one, dir->block.data,
1166 sample->block.data[n++], bset1->ineq[i], 1 + d);
1167 isl_vec_free(sample);
1168 isl_seq_normalize(bset1->ctx, dir->el, dir->size);
1169 isl_basic_set_free(bset1);
1170 isl_basic_set_free(bset2);
1173 isl_vec_free(sample);
1174 isl_basic_set_free(bset1);
1175 isl_basic_set_free(bset2);
1179 /* Given a polyhedron b_i + A_i x >= 0 and a map T = S^{-1},
1180 * compute b_i' + A_i' x' >= 0, with
1182 * [ b_i A_i ] [ y' ] [ y' ]
1183 * [ 1 0 ] S^{-1} [ x' ] >= 0 or [ b_i' A_i' ] [ x' ] >= 0
1185 * In particular, add the "positivity constraint" and then perform
1188 static struct isl_basic_set *homogeneous_map(struct isl_basic_set *bset,
1195 bset = isl_basic_set_extend_constraints(bset, 0, 1);
1196 k = isl_basic_set_alloc_inequality(bset);
1199 isl_seq_clr(bset->ineq[k] + 1, isl_basic_set_total_dim(bset));
1200 isl_int_set_si(bset->ineq[k][0], 1);
1201 bset = isl_basic_set_preimage(bset, T);
1205 isl_basic_set_free(bset);
1209 /* Compute the convex hull of a pair of basic sets without any parameters or
1210 * integer divisions, where the convex hull is known to be pointed,
1211 * but the basic sets may be unbounded.
1213 * We turn this problem into the computation of a convex hull of a pair
1214 * _bounded_ polyhedra by "changing the direction of the homogeneous
1215 * dimension". This idea is due to Matthias Koeppe.
1217 * Consider the cones in homogeneous space that correspond to the
1218 * input polyhedra. The rays of these cones are also rays of the
1219 * polyhedra if the coordinate that corresponds to the homogeneous
1220 * dimension is zero. That is, if the inner product of the rays
1221 * with the homogeneous direction is zero.
1222 * The cones in the homogeneous space can also be considered to
1223 * correspond to other pairs of polyhedra by chosing a different
1224 * homogeneous direction. To ensure that both of these polyhedra
1225 * are bounded, we need to make sure that all rays of the cones
1226 * correspond to vertices and not to rays.
1227 * Let s be a direction such that <s, r> > 0 for all rays r of both cones.
1228 * Then using s as a homogeneous direction, we obtain a pair of polytopes.
1229 * The vector s is computed in valid_direction.
1231 * Note that we need to consider _all_ rays of the cones and not just
1232 * the rays that correspond to rays in the polyhedra. If we were to
1233 * only consider those rays and turn them into vertices, then we
1234 * may inadvertently turn some vertices into rays.
1236 * The standard homogeneous direction is the unit vector in the 0th coordinate.
1237 * We therefore transform the two polyhedra such that the selected
1238 * direction is mapped onto this standard direction and then proceed
1239 * with the normal computation.
1240 * Let S be a non-singular square matrix with s as its first row,
1241 * then we want to map the polyhedra to the space
1243 * [ y' ] [ y ] [ y ] [ y' ]
1244 * [ x' ] = S [ x ] i.e., [ x ] = S^{-1} [ x' ]
1246 * We take S to be the unimodular completion of s to limit the growth
1247 * of the coefficients in the following computations.
1249 * Let b_i + A_i x >= 0 be the constraints of polyhedron i.
1250 * We first move to the homogeneous dimension
1252 * b_i y + A_i x >= 0 [ b_i A_i ] [ y ] [ 0 ]
1253 * y >= 0 or [ 1 0 ] [ x ] >= [ 0 ]
1255 * Then we change directoin
1257 * [ b_i A_i ] [ y' ] [ y' ]
1258 * [ 1 0 ] S^{-1} [ x' ] >= 0 or [ b_i' A_i' ] [ x' ] >= 0
1260 * Then we compute the convex hull of the polytopes b_i' + A_i' x' >= 0
1261 * resulting in b' + A' x' >= 0, which we then convert back
1264 * [ b' A' ] S [ x ] >= 0 or [ b A ] [ x ] >= 0
1266 * The polyhedron b + A x >= 0 is then the convex hull of the input polyhedra.
1268 static struct isl_basic_set *convex_hull_pair_pointed(
1269 struct isl_basic_set *bset1, struct isl_basic_set *bset2)
1271 struct isl_ctx *ctx = NULL;
1272 struct isl_vec *dir = NULL;
1273 struct isl_mat *T = NULL;
1274 struct isl_mat *T2 = NULL;
1275 struct isl_basic_set *hull;
1276 struct isl_set *set;
1278 if (!bset1 || !bset2)
1281 dir = valid_direction(isl_basic_set_copy(bset1),
1282 isl_basic_set_copy(bset2));
1285 T = isl_mat_alloc(bset1->ctx, dir->size, dir->size);
1288 isl_seq_cpy(T->row[0], dir->block.data, dir->size);
1289 T = isl_mat_unimodular_complete(T, 1);
1290 T2 = isl_mat_right_inverse(isl_mat_copy(T));
1292 bset1 = homogeneous_map(bset1, isl_mat_copy(T2));
1293 bset2 = homogeneous_map(bset2, T2);
1294 set = isl_set_alloc_dim(isl_basic_set_get_dim(bset1), 2, 0);
1295 set = isl_set_add_basic_set(set, bset1);
1296 set = isl_set_add_basic_set(set, bset2);
1297 hull = uset_convex_hull(set);
1298 hull = isl_basic_set_preimage(hull, T);
1305 isl_basic_set_free(bset1);
1306 isl_basic_set_free(bset2);
1310 static struct isl_basic_set *uset_convex_hull_wrap(struct isl_set *set);
1311 static struct isl_basic_set *modulo_affine_hull(
1312 struct isl_set *set, struct isl_basic_set *affine_hull);
1314 /* Compute the convex hull of a pair of basic sets without any parameters or
1315 * integer divisions.
1317 * This function is called from uset_convex_hull_unbounded, which
1318 * means that the complete convex hull is unbounded. Some pairs
1319 * of basic sets may still be bounded, though.
1320 * They may even lie inside a lower dimensional space, in which
1321 * case they need to be handled inside their affine hull since
1322 * the main algorithm assumes that the result is full-dimensional.
1324 * If the convex hull of the two basic sets would have a non-trivial
1325 * lineality space, we first project out this lineality space.
1327 static struct isl_basic_set *convex_hull_pair(struct isl_basic_set *bset1,
1328 struct isl_basic_set *bset2)
1330 isl_basic_set *lin, *aff;
1331 int bounded1, bounded2;
1333 aff = isl_set_affine_hull(isl_basic_set_union(isl_basic_set_copy(bset1),
1334 isl_basic_set_copy(bset2)));
1338 return modulo_affine_hull(isl_basic_set_union(bset1, bset2), aff);
1339 isl_basic_set_free(aff);
1341 bounded1 = isl_basic_set_is_bounded(bset1);
1342 bounded2 = isl_basic_set_is_bounded(bset2);
1344 if (bounded1 < 0 || bounded2 < 0)
1347 if (bounded1 && bounded2)
1348 uset_convex_hull_wrap(isl_basic_set_union(bset1, bset2));
1350 if (bounded1 || bounded2)
1351 return convex_hull_pair_pointed(bset1, bset2);
1353 lin = induced_lineality_space(isl_basic_set_copy(bset1),
1354 isl_basic_set_copy(bset2));
1357 if (isl_basic_set_is_universe(lin)) {
1358 isl_basic_set_free(bset1);
1359 isl_basic_set_free(bset2);
1362 if (lin->n_eq < isl_basic_set_total_dim(lin)) {
1363 struct isl_set *set;
1364 set = isl_set_alloc_dim(isl_basic_set_get_dim(bset1), 2, 0);
1365 set = isl_set_add_basic_set(set, bset1);
1366 set = isl_set_add_basic_set(set, bset2);
1367 return modulo_lineality(set, lin);
1369 isl_basic_set_free(lin);
1371 return convex_hull_pair_pointed(bset1, bset2);
1373 isl_basic_set_free(bset1);
1374 isl_basic_set_free(bset2);
1378 /* Compute the lineality space of a basic set.
1379 * We currently do not allow the basic set to have any divs.
1380 * We basically just drop the constants and turn every inequality
1383 struct isl_basic_set *isl_basic_set_lineality_space(struct isl_basic_set *bset)
1386 struct isl_basic_set *lin = NULL;
1391 isl_assert(bset->ctx, bset->n_div == 0, goto error);
1392 dim = isl_basic_set_total_dim(bset);
1394 lin = isl_basic_set_alloc_dim(isl_basic_set_get_dim(bset), 0, dim, 0);
1397 for (i = 0; i < bset->n_eq; ++i) {
1398 k = isl_basic_set_alloc_equality(lin);
1401 isl_int_set_si(lin->eq[k][0], 0);
1402 isl_seq_cpy(lin->eq[k] + 1, bset->eq[i] + 1, dim);
1404 lin = isl_basic_set_gauss(lin, NULL);
1407 for (i = 0; i < bset->n_ineq && lin->n_eq < dim; ++i) {
1408 k = isl_basic_set_alloc_equality(lin);
1411 isl_int_set_si(lin->eq[k][0], 0);
1412 isl_seq_cpy(lin->eq[k] + 1, bset->ineq[i] + 1, dim);
1413 lin = isl_basic_set_gauss(lin, NULL);
1417 isl_basic_set_free(bset);
1420 isl_basic_set_free(lin);
1421 isl_basic_set_free(bset);
1425 /* Compute the (linear) hull of the lineality spaces of the basic sets in the
1426 * "underlying" set "set".
1428 static struct isl_basic_set *uset_combined_lineality_space(struct isl_set *set)
1431 struct isl_set *lin = NULL;
1436 struct isl_dim *dim = isl_set_get_dim(set);
1438 return isl_basic_set_empty(dim);
1441 lin = isl_set_alloc_dim(isl_set_get_dim(set), set->n, 0);
1442 for (i = 0; i < set->n; ++i)
1443 lin = isl_set_add_basic_set(lin,
1444 isl_basic_set_lineality_space(isl_basic_set_copy(set->p[i])));
1446 return isl_set_affine_hull(lin);
1449 /* Compute the convex hull of a set without any parameters or
1450 * integer divisions.
1451 * In each step, we combined two basic sets until only one
1452 * basic set is left.
1453 * The input basic sets are assumed not to have a non-trivial
1454 * lineality space. If any of the intermediate results has
1455 * a non-trivial lineality space, it is projected out.
1457 static struct isl_basic_set *uset_convex_hull_unbounded(struct isl_set *set)
1459 struct isl_basic_set *convex_hull = NULL;
1461 convex_hull = isl_set_copy_basic_set(set);
1462 set = isl_set_drop_basic_set(set, convex_hull);
1465 while (set->n > 0) {
1466 struct isl_basic_set *t;
1467 t = isl_set_copy_basic_set(set);
1470 set = isl_set_drop_basic_set(set, t);
1473 convex_hull = convex_hull_pair(convex_hull, t);
1476 t = isl_basic_set_lineality_space(isl_basic_set_copy(convex_hull));
1479 if (isl_basic_set_is_universe(t)) {
1480 isl_basic_set_free(convex_hull);
1484 if (t->n_eq < isl_basic_set_total_dim(t)) {
1485 set = isl_set_add_basic_set(set, convex_hull);
1486 return modulo_lineality(set, t);
1488 isl_basic_set_free(t);
1494 isl_basic_set_free(convex_hull);
1498 /* Compute an initial hull for wrapping containing a single initial
1500 * This function assumes that the given set is bounded.
1502 static struct isl_basic_set *initial_hull(struct isl_basic_set *hull,
1503 struct isl_set *set)
1505 struct isl_mat *bounds = NULL;
1511 bounds = initial_facet_constraint(set);
1514 k = isl_basic_set_alloc_inequality(hull);
1517 dim = isl_set_n_dim(set);
1518 isl_assert(set->ctx, 1 + dim == bounds->n_col, goto error);
1519 isl_seq_cpy(hull->ineq[k], bounds->row[0], bounds->n_col);
1520 isl_mat_free(bounds);
1524 isl_basic_set_free(hull);
1525 isl_mat_free(bounds);
1529 struct max_constraint {
1535 static int max_constraint_equal(const void *entry, const void *val)
1537 struct max_constraint *a = (struct max_constraint *)entry;
1538 isl_int *b = (isl_int *)val;
1540 return isl_seq_eq(a->c->row[0] + 1, b, a->c->n_col - 1);
1543 static void update_constraint(struct isl_ctx *ctx, struct isl_hash_table *table,
1544 isl_int *con, unsigned len, int n, int ineq)
1546 struct isl_hash_table_entry *entry;
1547 struct max_constraint *c;
1550 c_hash = isl_seq_get_hash(con + 1, len);
1551 entry = isl_hash_table_find(ctx, table, c_hash, max_constraint_equal,
1557 isl_hash_table_remove(ctx, table, entry);
1561 if (isl_int_gt(c->c->row[0][0], con[0]))
1563 if (isl_int_eq(c->c->row[0][0], con[0])) {
1568 c->c = isl_mat_cow(c->c);
1569 isl_int_set(c->c->row[0][0], con[0]);
1573 /* Check whether the constraint hash table "table" constains the constraint
1576 static int has_constraint(struct isl_ctx *ctx, struct isl_hash_table *table,
1577 isl_int *con, unsigned len, int n)
1579 struct isl_hash_table_entry *entry;
1580 struct max_constraint *c;
1583 c_hash = isl_seq_get_hash(con + 1, len);
1584 entry = isl_hash_table_find(ctx, table, c_hash, max_constraint_equal,
1591 return isl_int_eq(c->c->row[0][0], con[0]);
1594 /* Check for inequality constraints of a basic set without equalities
1595 * such that the same or more stringent copies of the constraint appear
1596 * in all of the basic sets. Such constraints are necessarily facet
1597 * constraints of the convex hull.
1599 * If the resulting basic set is by chance identical to one of
1600 * the basic sets in "set", then we know that this basic set contains
1601 * all other basic sets and is therefore the convex hull of set.
1602 * In this case we set *is_hull to 1.
1604 static struct isl_basic_set *common_constraints(struct isl_basic_set *hull,
1605 struct isl_set *set, int *is_hull)
1608 int min_constraints;
1610 struct max_constraint *constraints = NULL;
1611 struct isl_hash_table *table = NULL;
1616 for (i = 0; i < set->n; ++i)
1617 if (set->p[i]->n_eq == 0)
1621 min_constraints = set->p[i]->n_ineq;
1623 for (i = best + 1; i < set->n; ++i) {
1624 if (set->p[i]->n_eq != 0)
1626 if (set->p[i]->n_ineq >= min_constraints)
1628 min_constraints = set->p[i]->n_ineq;
1631 constraints = isl_calloc_array(hull->ctx, struct max_constraint,
1635 table = isl_alloc_type(hull->ctx, struct isl_hash_table);
1636 if (isl_hash_table_init(hull->ctx, table, min_constraints))
1639 total = isl_dim_total(set->dim);
1640 for (i = 0; i < set->p[best]->n_ineq; ++i) {
1641 constraints[i].c = isl_mat_sub_alloc(hull->ctx,
1642 set->p[best]->ineq + i, 0, 1, 0, 1 + total);
1643 if (!constraints[i].c)
1645 constraints[i].ineq = 1;
1647 for (i = 0; i < min_constraints; ++i) {
1648 struct isl_hash_table_entry *entry;
1650 c_hash = isl_seq_get_hash(constraints[i].c->row[0] + 1, total);
1651 entry = isl_hash_table_find(hull->ctx, table, c_hash,
1652 max_constraint_equal, constraints[i].c->row[0] + 1, 1);
1655 isl_assert(hull->ctx, !entry->data, goto error);
1656 entry->data = &constraints[i];
1660 for (s = 0; s < set->n; ++s) {
1664 for (i = 0; i < set->p[s]->n_eq; ++i) {
1665 isl_int *eq = set->p[s]->eq[i];
1666 for (j = 0; j < 2; ++j) {
1667 isl_seq_neg(eq, eq, 1 + total);
1668 update_constraint(hull->ctx, table,
1672 for (i = 0; i < set->p[s]->n_ineq; ++i) {
1673 isl_int *ineq = set->p[s]->ineq[i];
1674 update_constraint(hull->ctx, table, ineq, total, n,
1675 set->p[s]->n_eq == 0);
1680 for (i = 0; i < min_constraints; ++i) {
1681 if (constraints[i].count < n)
1683 if (!constraints[i].ineq)
1685 j = isl_basic_set_alloc_inequality(hull);
1688 isl_seq_cpy(hull->ineq[j], constraints[i].c->row[0], 1 + total);
1691 for (s = 0; s < set->n; ++s) {
1692 if (set->p[s]->n_eq)
1694 if (set->p[s]->n_ineq != hull->n_ineq)
1696 for (i = 0; i < set->p[s]->n_ineq; ++i) {
1697 isl_int *ineq = set->p[s]->ineq[i];
1698 if (!has_constraint(hull->ctx, table, ineq, total, n))
1701 if (i == set->p[s]->n_ineq)
1705 isl_hash_table_clear(table);
1706 for (i = 0; i < min_constraints; ++i)
1707 isl_mat_free(constraints[i].c);
1712 isl_hash_table_clear(table);
1715 for (i = 0; i < min_constraints; ++i)
1716 isl_mat_free(constraints[i].c);
1721 /* Create a template for the convex hull of "set" and fill it up
1722 * obvious facet constraints, if any. If the result happens to
1723 * be the convex hull of "set" then *is_hull is set to 1.
1725 static struct isl_basic_set *proto_hull(struct isl_set *set, int *is_hull)
1727 struct isl_basic_set *hull;
1732 for (i = 0; i < set->n; ++i) {
1733 n_ineq += set->p[i]->n_eq;
1734 n_ineq += set->p[i]->n_ineq;
1736 hull = isl_basic_set_alloc_dim(isl_dim_copy(set->dim), 0, 0, n_ineq);
1737 hull = isl_basic_set_set_rational(hull);
1740 return common_constraints(hull, set, is_hull);
1743 static struct isl_basic_set *uset_convex_hull_wrap(struct isl_set *set)
1745 struct isl_basic_set *hull;
1748 hull = proto_hull(set, &is_hull);
1749 if (hull && !is_hull) {
1750 if (hull->n_ineq == 0)
1751 hull = initial_hull(hull, set);
1752 hull = extend(hull, set);
1759 /* Compute the convex hull of a set without any parameters or
1760 * integer divisions. Depending on whether the set is bounded,
1761 * we pass control to the wrapping based convex hull or
1762 * the Fourier-Motzkin elimination based convex hull.
1763 * We also handle a few special cases before checking the boundedness.
1765 static struct isl_basic_set *uset_convex_hull(struct isl_set *set)
1767 struct isl_basic_set *convex_hull = NULL;
1768 struct isl_basic_set *lin;
1770 if (isl_set_n_dim(set) == 0)
1771 return convex_hull_0d(set);
1773 set = isl_set_coalesce(set);
1774 set = isl_set_set_rational(set);
1781 convex_hull = isl_basic_set_copy(set->p[0]);
1785 if (isl_set_n_dim(set) == 1)
1786 return convex_hull_1d(set);
1788 if (isl_set_is_bounded(set))
1789 return uset_convex_hull_wrap(set);
1791 lin = uset_combined_lineality_space(isl_set_copy(set));
1794 if (isl_basic_set_is_universe(lin)) {
1798 if (lin->n_eq < isl_basic_set_total_dim(lin))
1799 return modulo_lineality(set, lin);
1800 isl_basic_set_free(lin);
1802 return uset_convex_hull_unbounded(set);
1805 isl_basic_set_free(convex_hull);
1809 /* This is the core procedure, where "set" is a "pure" set, i.e.,
1810 * without parameters or divs and where the convex hull of set is
1811 * known to be full-dimensional.
1813 static struct isl_basic_set *uset_convex_hull_wrap_bounded(struct isl_set *set)
1815 struct isl_basic_set *convex_hull = NULL;
1820 if (isl_set_n_dim(set) == 0) {
1821 convex_hull = isl_basic_set_universe(isl_dim_copy(set->dim));
1823 convex_hull = isl_basic_set_set_rational(convex_hull);
1827 set = isl_set_set_rational(set);
1828 set = isl_set_coalesce(set);
1832 convex_hull = isl_basic_set_copy(set->p[0]);
1836 if (isl_set_n_dim(set) == 1)
1837 return convex_hull_1d(set);
1839 return uset_convex_hull_wrap(set);
1845 /* Compute the convex hull of set "set" with affine hull "affine_hull",
1846 * We first remove the equalities (transforming the set), compute the
1847 * convex hull of the transformed set and then add the equalities back
1848 * (after performing the inverse transformation.
1850 static struct isl_basic_set *modulo_affine_hull(
1851 struct isl_set *set, struct isl_basic_set *affine_hull)
1855 struct isl_basic_set *dummy;
1856 struct isl_basic_set *convex_hull;
1858 dummy = isl_basic_set_remove_equalities(
1859 isl_basic_set_copy(affine_hull), &T, &T2);
1862 isl_basic_set_free(dummy);
1863 set = isl_set_preimage(set, T);
1864 convex_hull = uset_convex_hull(set);
1865 convex_hull = isl_basic_set_preimage(convex_hull, T2);
1866 convex_hull = isl_basic_set_intersect(convex_hull, affine_hull);
1869 isl_basic_set_free(affine_hull);
1874 /* Compute the convex hull of a map.
1876 * The implementation was inspired by "Extended Convex Hull" by Fukuda et al.,
1877 * specifically, the wrapping of facets to obtain new facets.
1879 struct isl_basic_map *isl_map_convex_hull(struct isl_map *map)
1881 struct isl_basic_set *bset;
1882 struct isl_basic_map *model = NULL;
1883 struct isl_basic_set *affine_hull = NULL;
1884 struct isl_basic_map *convex_hull = NULL;
1885 struct isl_set *set = NULL;
1886 struct isl_ctx *ctx;
1893 convex_hull = isl_basic_map_empty_like_map(map);
1898 map = isl_map_detect_equalities(map);
1899 map = isl_map_align_divs(map);
1902 model = isl_basic_map_copy(map->p[0]);
1903 set = isl_map_underlying_set(map);
1907 affine_hull = isl_set_affine_hull(isl_set_copy(set));
1910 if (affine_hull->n_eq != 0)
1911 bset = modulo_affine_hull(set, affine_hull);
1913 isl_basic_set_free(affine_hull);
1914 bset = uset_convex_hull(set);
1917 convex_hull = isl_basic_map_overlying_set(bset, model);
1921 ISL_F_SET(convex_hull, ISL_BASIC_MAP_NO_IMPLICIT);
1922 ISL_F_SET(convex_hull, ISL_BASIC_MAP_ALL_EQUALITIES);
1923 ISL_F_CLR(convex_hull, ISL_BASIC_MAP_RATIONAL);
1927 isl_basic_map_free(model);
1931 struct isl_basic_set *isl_set_convex_hull(struct isl_set *set)
1933 return (struct isl_basic_set *)
1934 isl_map_convex_hull((struct isl_map *)set);
1937 struct sh_data_entry {
1938 struct isl_hash_table *table;
1939 struct isl_tab *tab;
1942 /* Holds the data needed during the simple hull computation.
1944 * n the number of basic sets in the original set
1945 * hull_table a hash table of already computed constraints
1946 * in the simple hull
1947 * p for each basic set,
1948 * table a hash table of the constraints
1949 * tab the tableau corresponding to the basic set
1952 struct isl_ctx *ctx;
1954 struct isl_hash_table *hull_table;
1955 struct sh_data_entry p[1];
1958 static void sh_data_free(struct sh_data *data)
1964 isl_hash_table_free(data->ctx, data->hull_table);
1965 for (i = 0; i < data->n; ++i) {
1966 isl_hash_table_free(data->ctx, data->p[i].table);
1967 isl_tab_free(data->p[i].tab);
1972 struct ineq_cmp_data {
1977 static int has_ineq(const void *entry, const void *val)
1979 isl_int *row = (isl_int *)entry;
1980 struct ineq_cmp_data *v = (struct ineq_cmp_data *)val;
1982 return isl_seq_eq(row + 1, v->p + 1, v->len) ||
1983 isl_seq_is_neg(row + 1, v->p + 1, v->len);
1986 static int hash_ineq(struct isl_ctx *ctx, struct isl_hash_table *table,
1987 isl_int *ineq, unsigned len)
1990 struct ineq_cmp_data v;
1991 struct isl_hash_table_entry *entry;
1995 c_hash = isl_seq_get_hash(ineq + 1, len);
1996 entry = isl_hash_table_find(ctx, table, c_hash, has_ineq, &v, 1);
2003 /* Fill hash table "table" with the constraints of "bset".
2004 * Equalities are added as two inequalities.
2005 * The value in the hash table is a pointer to the (in)equality of "bset".
2007 static int hash_basic_set(struct isl_hash_table *table,
2008 struct isl_basic_set *bset)
2011 unsigned dim = isl_basic_set_total_dim(bset);
2013 for (i = 0; i < bset->n_eq; ++i) {
2014 for (j = 0; j < 2; ++j) {
2015 isl_seq_neg(bset->eq[i], bset->eq[i], 1 + dim);
2016 if (hash_ineq(bset->ctx, table, bset->eq[i], dim) < 0)
2020 for (i = 0; i < bset->n_ineq; ++i) {
2021 if (hash_ineq(bset->ctx, table, bset->ineq[i], dim) < 0)
2027 static struct sh_data *sh_data_alloc(struct isl_set *set, unsigned n_ineq)
2029 struct sh_data *data;
2032 data = isl_calloc(set->ctx, struct sh_data,
2033 sizeof(struct sh_data) +
2034 (set->n - 1) * sizeof(struct sh_data_entry));
2037 data->ctx = set->ctx;
2039 data->hull_table = isl_hash_table_alloc(set->ctx, n_ineq);
2040 if (!data->hull_table)
2042 for (i = 0; i < set->n; ++i) {
2043 data->p[i].table = isl_hash_table_alloc(set->ctx,
2044 2 * set->p[i]->n_eq + set->p[i]->n_ineq);
2045 if (!data->p[i].table)
2047 if (hash_basic_set(data->p[i].table, set->p[i]) < 0)
2056 /* Check if inequality "ineq" is a bound for basic set "j" or if
2057 * it can be relaxed (by increasing the constant term) to become
2058 * a bound for that basic set. In the latter case, the constant
2060 * Return 1 if "ineq" is a bound
2061 * 0 if "ineq" may attain arbitrarily small values on basic set "j"
2062 * -1 if some error occurred
2064 static int is_bound(struct sh_data *data, struct isl_set *set, int j,
2067 enum isl_lp_result res;
2070 if (!data->p[j].tab) {
2071 data->p[j].tab = isl_tab_from_basic_set(set->p[j]);
2072 if (!data->p[j].tab)
2078 res = isl_tab_min(data->p[j].tab, ineq, data->ctx->one,
2080 if (res == isl_lp_ok && isl_int_is_neg(opt))
2081 isl_int_sub(ineq[0], ineq[0], opt);
2085 return (res == isl_lp_ok || res == isl_lp_empty) ? 1 :
2086 res == isl_lp_unbounded ? 0 : -1;
2089 /* Check if inequality "ineq" from basic set "i" can be relaxed to
2090 * become a bound on the whole set. If so, add the (relaxed) inequality
2093 * We first check if "hull" already contains a translate of the inequality.
2094 * If so, we are done.
2095 * Then, we check if any of the previous basic sets contains a translate
2096 * of the inequality. If so, then we have already considered this
2097 * inequality and we are done.
2098 * Otherwise, for each basic set other than "i", we check if the inequality
2099 * is a bound on the basic set.
2100 * For previous basic sets, we know that they do not contain a translate
2101 * of the inequality, so we directly call is_bound.
2102 * For following basic sets, we first check if a translate of the
2103 * inequality appears in its description and if so directly update
2104 * the inequality accordingly.
2106 static struct isl_basic_set *add_bound(struct isl_basic_set *hull,
2107 struct sh_data *data, struct isl_set *set, int i, isl_int *ineq)
2110 struct ineq_cmp_data v;
2111 struct isl_hash_table_entry *entry;
2117 v.len = isl_basic_set_total_dim(hull);
2119 c_hash = isl_seq_get_hash(ineq + 1, v.len);
2121 entry = isl_hash_table_find(hull->ctx, data->hull_table, c_hash,
2126 for (j = 0; j < i; ++j) {
2127 entry = isl_hash_table_find(hull->ctx, data->p[j].table,
2128 c_hash, has_ineq, &v, 0);
2135 k = isl_basic_set_alloc_inequality(hull);
2136 isl_seq_cpy(hull->ineq[k], ineq, 1 + v.len);
2140 for (j = 0; j < i; ++j) {
2142 bound = is_bound(data, set, j, hull->ineq[k]);
2149 isl_basic_set_free_inequality(hull, 1);
2153 for (j = i + 1; j < set->n; ++j) {
2156 entry = isl_hash_table_find(hull->ctx, data->p[j].table,
2157 c_hash, has_ineq, &v, 0);
2159 ineq_j = entry->data;
2160 neg = isl_seq_is_neg(ineq_j + 1,
2161 hull->ineq[k] + 1, v.len);
2163 isl_int_neg(ineq_j[0], ineq_j[0]);
2164 if (isl_int_gt(ineq_j[0], hull->ineq[k][0]))
2165 isl_int_set(hull->ineq[k][0], ineq_j[0]);
2167 isl_int_neg(ineq_j[0], ineq_j[0]);
2170 bound = is_bound(data, set, j, hull->ineq[k]);
2177 isl_basic_set_free_inequality(hull, 1);
2181 entry = isl_hash_table_find(hull->ctx, data->hull_table, c_hash,
2185 entry->data = hull->ineq[k];
2189 isl_basic_set_free(hull);
2193 /* Check if any inequality from basic set "i" can be relaxed to
2194 * become a bound on the whole set. If so, add the (relaxed) inequality
2197 static struct isl_basic_set *add_bounds(struct isl_basic_set *bset,
2198 struct sh_data *data, struct isl_set *set, int i)
2201 unsigned dim = isl_basic_set_total_dim(bset);
2203 for (j = 0; j < set->p[i]->n_eq; ++j) {
2204 for (k = 0; k < 2; ++k) {
2205 isl_seq_neg(set->p[i]->eq[j], set->p[i]->eq[j], 1+dim);
2206 bset = add_bound(bset, data, set, i, set->p[i]->eq[j]);
2209 for (j = 0; j < set->p[i]->n_ineq; ++j)
2210 bset = add_bound(bset, data, set, i, set->p[i]->ineq[j]);
2214 /* Compute a superset of the convex hull of set that is described
2215 * by only translates of the constraints in the constituents of set.
2217 static struct isl_basic_set *uset_simple_hull(struct isl_set *set)
2219 struct sh_data *data = NULL;
2220 struct isl_basic_set *hull = NULL;
2228 for (i = 0; i < set->n; ++i) {
2231 n_ineq += 2 * set->p[i]->n_eq + set->p[i]->n_ineq;
2234 hull = isl_basic_set_alloc_dim(isl_dim_copy(set->dim), 0, 0, n_ineq);
2238 data = sh_data_alloc(set, n_ineq);
2242 for (i = 0; i < set->n; ++i)
2243 hull = add_bounds(hull, data, set, i);
2251 isl_basic_set_free(hull);
2256 /* Compute a superset of the convex hull of map that is described
2257 * by only translates of the constraints in the constituents of map.
2259 struct isl_basic_map *isl_map_simple_hull(struct isl_map *map)
2261 struct isl_set *set = NULL;
2262 struct isl_basic_map *model = NULL;
2263 struct isl_basic_map *hull;
2264 struct isl_basic_map *affine_hull;
2265 struct isl_basic_set *bset = NULL;
2270 hull = isl_basic_map_empty_like_map(map);
2275 hull = isl_basic_map_copy(map->p[0]);
2280 map = isl_map_detect_equalities(map);
2281 affine_hull = isl_map_affine_hull(isl_map_copy(map));
2282 map = isl_map_align_divs(map);
2283 model = isl_basic_map_copy(map->p[0]);
2285 set = isl_map_underlying_set(map);
2287 bset = uset_simple_hull(set);
2289 hull = isl_basic_map_overlying_set(bset, model);
2291 hull = isl_basic_map_intersect(hull, affine_hull);
2292 hull = isl_basic_map_remove_redundancies(hull);
2293 ISL_F_SET(hull, ISL_BASIC_MAP_NO_IMPLICIT);
2294 ISL_F_SET(hull, ISL_BASIC_MAP_ALL_EQUALITIES);
2299 struct isl_basic_set *isl_set_simple_hull(struct isl_set *set)
2301 return (struct isl_basic_set *)
2302 isl_map_simple_hull((struct isl_map *)set);
2305 /* Given a set "set", return parametric bounds on the dimension "dim".
2307 static struct isl_basic_set *set_bounds(struct isl_set *set, int dim)
2309 unsigned set_dim = isl_set_dim(set, isl_dim_set);
2310 set = isl_set_copy(set);
2311 set = isl_set_eliminate_dims(set, dim + 1, set_dim - (dim + 1));
2312 set = isl_set_eliminate_dims(set, 0, dim);
2313 return isl_set_convex_hull(set);
2316 /* Computes a "simple hull" and then check if each dimension in the
2317 * resulting hull is bounded by a symbolic constant. If not, the
2318 * hull is intersected with the corresponding bounds on the whole set.
2320 struct isl_basic_set *isl_set_bounded_simple_hull(struct isl_set *set)
2323 struct isl_basic_set *hull;
2324 unsigned nparam, left;
2325 int removed_divs = 0;
2327 hull = isl_set_simple_hull(isl_set_copy(set));
2331 nparam = isl_basic_set_dim(hull, isl_dim_param);
2332 for (i = 0; i < isl_basic_set_dim(hull, isl_dim_set); ++i) {
2333 int lower = 0, upper = 0;
2334 struct isl_basic_set *bounds;
2336 left = isl_basic_set_total_dim(hull) - nparam - i - 1;
2337 for (j = 0; j < hull->n_eq; ++j) {
2338 if (isl_int_is_zero(hull->eq[j][1 + nparam + i]))
2340 if (isl_seq_first_non_zero(hull->eq[j]+1+nparam+i+1,
2347 for (j = 0; j < hull->n_ineq; ++j) {
2348 if (isl_int_is_zero(hull->ineq[j][1 + nparam + i]))
2350 if (isl_seq_first_non_zero(hull->ineq[j]+1+nparam+i+1,
2352 isl_seq_first_non_zero(hull->ineq[j]+1+nparam,
2355 if (isl_int_is_pos(hull->ineq[j][1 + nparam + i]))
2366 if (!removed_divs) {
2367 set = isl_set_remove_divs(set);
2372 bounds = set_bounds(set, i);
2373 hull = isl_basic_set_intersect(hull, bounds);