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);
82 /* Compute the convex hull of a basic map, by removing the redundant
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 struct isl_basic_map *isl_basic_map_convex_hull(struct isl_basic_map *bmap)
97 bmap = isl_basic_map_gauss(bmap, NULL);
98 if (ISL_F_ISSET(bmap, ISL_BASIC_MAP_EMPTY))
100 if (ISL_F_ISSET(bmap, ISL_BASIC_MAP_NO_REDUNDANT))
102 if (bmap->n_ineq <= 1)
105 tab = isl_tab_from_basic_map(bmap);
106 if (isl_tab_detect_implicit_equalities(tab) < 0)
108 if (isl_tab_detect_redundant(tab) < 0)
110 bmap = isl_basic_map_update_from_tab(bmap, tab);
112 ISL_F_SET(bmap, ISL_BASIC_MAP_NO_IMPLICIT);
113 ISL_F_SET(bmap, ISL_BASIC_MAP_NO_REDUNDANT);
117 isl_basic_map_free(bmap);
121 struct isl_basic_set *isl_basic_set_convex_hull(struct isl_basic_set *bset)
123 return (struct isl_basic_set *)
124 isl_basic_map_convex_hull((struct isl_basic_map *)bset);
127 /* Check if the set set is bound in the direction of the affine
128 * constraint c and if so, set the constant term such that the
129 * resulting constraint is a bounding constraint for the set.
131 static int uset_is_bound(struct isl_set *set, isl_int *c, unsigned len)
139 isl_int_init(opt_denom);
141 for (j = 0; j < set->n; ++j) {
142 enum isl_lp_result res;
144 if (ISL_F_ISSET(set->p[j], ISL_BASIC_SET_EMPTY))
147 res = isl_basic_set_solve_lp(set->p[j],
148 0, c, set->ctx->one, &opt, &opt_denom, NULL);
149 if (res == isl_lp_unbounded)
151 if (res == isl_lp_error)
153 if (res == isl_lp_empty) {
154 set->p[j] = isl_basic_set_set_to_empty(set->p[j]);
159 if (first || isl_int_is_neg(opt)) {
160 if (!isl_int_is_one(opt_denom))
161 isl_seq_scale(c, c, opt_denom, len);
162 isl_int_sub(c[0], c[0], opt);
167 isl_int_clear(opt_denom);
171 isl_int_clear(opt_denom);
175 struct isl_basic_set *isl_basic_set_set_rational(struct isl_basic_set *bset)
180 if (ISL_F_ISSET(bset, ISL_BASIC_MAP_RATIONAL))
183 bset = isl_basic_set_cow(bset);
187 ISL_F_SET(bset, ISL_BASIC_MAP_RATIONAL);
189 return isl_basic_set_finalize(bset);
192 static struct isl_set *isl_set_set_rational(struct isl_set *set)
196 set = isl_set_cow(set);
199 for (i = 0; i < set->n; ++i) {
200 set->p[i] = isl_basic_set_set_rational(set->p[i]);
210 static struct isl_basic_set *isl_basic_set_add_equality(
211 struct isl_basic_set *bset, isl_int *c)
216 if (ISL_F_ISSET(bset, ISL_BASIC_SET_EMPTY))
219 isl_assert(bset->ctx, isl_basic_set_n_param(bset) == 0, goto error);
220 isl_assert(bset->ctx, bset->n_div == 0, goto error);
221 dim = isl_basic_set_n_dim(bset);
222 bset = isl_basic_set_cow(bset);
223 bset = isl_basic_set_extend(bset, 0, dim, 0, 1, 0);
224 i = isl_basic_set_alloc_equality(bset);
227 isl_seq_cpy(bset->eq[i], c, 1 + dim);
230 isl_basic_set_free(bset);
234 static struct isl_set *isl_set_add_basic_set_equality(struct isl_set *set, isl_int *c)
238 set = isl_set_cow(set);
241 for (i = 0; i < set->n; ++i) {
242 set->p[i] = isl_basic_set_add_equality(set->p[i], c);
252 /* Given a union of basic sets, construct the constraints for wrapping
253 * a facet around one of its ridges.
254 * In particular, if each of n the d-dimensional basic sets i in "set"
255 * contains the origin, satisfies the constraints x_1 >= 0 and x_2 >= 0
256 * and is defined by the constraints
260 * then the resulting set is of dimension n*(1+d) and has as constraints
269 static struct isl_basic_set *wrap_constraints(struct isl_set *set)
271 struct isl_basic_set *lp;
275 unsigned dim, lp_dim;
280 dim = 1 + isl_set_n_dim(set);
283 for (i = 0; i < set->n; ++i) {
284 n_eq += set->p[i]->n_eq;
285 n_ineq += set->p[i]->n_ineq;
287 lp = isl_basic_set_alloc(set->ctx, 0, dim * set->n, 0, n_eq, n_ineq);
290 lp_dim = isl_basic_set_n_dim(lp);
291 k = isl_basic_set_alloc_equality(lp);
292 isl_int_set_si(lp->eq[k][0], -1);
293 for (i = 0; i < set->n; ++i) {
294 isl_int_set_si(lp->eq[k][1+dim*i], 0);
295 isl_int_set_si(lp->eq[k][1+dim*i+1], 1);
296 isl_seq_clr(lp->eq[k]+1+dim*i+2, dim-2);
298 for (i = 0; i < set->n; ++i) {
299 k = isl_basic_set_alloc_inequality(lp);
300 isl_seq_clr(lp->ineq[k], 1+lp_dim);
301 isl_int_set_si(lp->ineq[k][1+dim*i], 1);
303 for (j = 0; j < set->p[i]->n_eq; ++j) {
304 k = isl_basic_set_alloc_equality(lp);
305 isl_seq_clr(lp->eq[k], 1+dim*i);
306 isl_seq_cpy(lp->eq[k]+1+dim*i, set->p[i]->eq[j], dim);
307 isl_seq_clr(lp->eq[k]+1+dim*(i+1), dim*(set->n-i-1));
310 for (j = 0; j < set->p[i]->n_ineq; ++j) {
311 k = isl_basic_set_alloc_inequality(lp);
312 isl_seq_clr(lp->ineq[k], 1+dim*i);
313 isl_seq_cpy(lp->ineq[k]+1+dim*i, set->p[i]->ineq[j], dim);
314 isl_seq_clr(lp->ineq[k]+1+dim*(i+1), dim*(set->n-i-1));
320 /* Given a facet "facet" of the convex hull of "set" and a facet "ridge"
321 * of that facet, compute the other facet of the convex hull that contains
324 * We first transform the set such that the facet constraint becomes
328 * I.e., the facet lies in
332 * and on that facet, the constraint that defines the ridge is
336 * (This transformation is not strictly needed, all that is needed is
337 * that the ridge contains the origin.)
339 * Since the ridge contains the origin, the cone of the convex hull
340 * will be of the form
345 * with this second constraint defining the new facet.
346 * The constant a is obtained by settting x_1 in the cone of the
347 * convex hull to 1 and minimizing x_2.
348 * Now, each element in the cone of the convex hull is the sum
349 * of elements in the cones of the basic sets.
350 * If a_i is the dilation factor of basic set i, then the problem
351 * we need to solve is
364 * the constraints of each (transformed) basic set.
365 * If a = n/d, then the constraint defining the new facet (in the transformed
368 * -n x_1 + d x_2 >= 0
370 * In the original space, we need to take the same combination of the
371 * corresponding constraints "facet" and "ridge".
373 * If a = -infty = "-1/0", then we just return the original facet constraint.
374 * This means that the facet is unbounded, but has a bounded intersection
375 * with the union of sets.
377 isl_int *isl_set_wrap_facet(__isl_keep isl_set *set,
378 isl_int *facet, isl_int *ridge)
381 struct isl_mat *T = NULL;
382 struct isl_basic_set *lp = NULL;
384 enum isl_lp_result res;
388 set = isl_set_copy(set);
389 set = isl_set_set_rational(set);
391 dim = 1 + isl_set_n_dim(set);
392 T = isl_mat_alloc(set->ctx, 3, dim);
395 isl_int_set_si(T->row[0][0], 1);
396 isl_seq_clr(T->row[0]+1, dim - 1);
397 isl_seq_cpy(T->row[1], facet, dim);
398 isl_seq_cpy(T->row[2], ridge, dim);
399 T = isl_mat_right_inverse(T);
400 set = isl_set_preimage(set, T);
404 lp = wrap_constraints(set);
405 obj = isl_vec_alloc(set->ctx, 1 + dim*set->n);
408 isl_int_set_si(obj->block.data[0], 0);
409 for (i = 0; i < set->n; ++i) {
410 isl_seq_clr(obj->block.data + 1 + dim*i, 2);
411 isl_int_set_si(obj->block.data[1 + dim*i+2], 1);
412 isl_seq_clr(obj->block.data + 1 + dim*i+3, dim-3);
416 res = isl_basic_set_solve_lp(lp, 0,
417 obj->block.data, set->ctx->one, &num, &den, NULL);
418 if (res == isl_lp_ok) {
419 isl_int_neg(num, num);
420 isl_seq_combine(facet, num, facet, den, ridge, dim);
425 isl_basic_set_free(lp);
427 isl_assert(set->ctx, res == isl_lp_ok || res == isl_lp_unbounded,
431 isl_basic_set_free(lp);
437 /* Compute the constraint of a facet of "set".
439 * We first compute the intersection with a bounding constraint
440 * that is orthogonal to one of the coordinate axes.
441 * If the affine hull of this intersection has only one equality,
442 * we have found a facet.
443 * Otherwise, we wrap the current bounding constraint around
444 * one of the equalities of the face (one that is not equal to
445 * the current bounding constraint).
446 * This process continues until we have found a facet.
447 * The dimension of the intersection increases by at least
448 * one on each iteration, so termination is guaranteed.
450 static __isl_give isl_mat *initial_facet_constraint(__isl_keep isl_set *set)
452 struct isl_set *slice = NULL;
453 struct isl_basic_set *face = NULL;
455 unsigned dim = isl_set_n_dim(set);
459 isl_assert(set->ctx, set->n > 0, goto error);
460 bounds = isl_mat_alloc(set->ctx, 1, 1 + dim);
464 isl_seq_clr(bounds->row[0], dim);
465 isl_int_set_si(bounds->row[0][1 + dim - 1], 1);
466 is_bound = uset_is_bound(set, bounds->row[0], 1 + dim);
467 isl_assert(set->ctx, is_bound == 1, goto error);
468 isl_seq_normalize(set->ctx, bounds->row[0], 1 + dim);
472 slice = isl_set_copy(set);
473 slice = isl_set_add_basic_set_equality(slice, bounds->row[0]);
474 face = isl_set_affine_hull(slice);
477 if (face->n_eq == 1) {
478 isl_basic_set_free(face);
481 for (i = 0; i < face->n_eq; ++i)
482 if (!isl_seq_eq(bounds->row[0], face->eq[i], 1 + dim) &&
483 !isl_seq_is_neg(bounds->row[0],
484 face->eq[i], 1 + dim))
486 isl_assert(set->ctx, i < face->n_eq, goto error);
487 if (!isl_set_wrap_facet(set, bounds->row[0], face->eq[i]))
489 isl_seq_normalize(set->ctx, bounds->row[0], bounds->n_col);
490 isl_basic_set_free(face);
495 isl_basic_set_free(face);
496 isl_mat_free(bounds);
500 /* Given the bounding constraint "c" of a facet of the convex hull of "set",
501 * compute a hyperplane description of the facet, i.e., compute the facets
504 * We compute an affine transformation that transforms the constraint
513 * by computing the right inverse U of a matrix that starts with the rows
526 * Since z_1 is zero, we can drop this variable as well as the corresponding
527 * column of U to obtain
535 * with Q' equal to Q, but without the corresponding row.
536 * After computing the facets of the facet in the z' space,
537 * we convert them back to the x space through Q.
539 static struct isl_basic_set *compute_facet(struct isl_set *set, isl_int *c)
541 struct isl_mat *m, *U, *Q;
542 struct isl_basic_set *facet = NULL;
547 set = isl_set_copy(set);
548 dim = isl_set_n_dim(set);
549 m = isl_mat_alloc(set->ctx, 2, 1 + dim);
552 isl_int_set_si(m->row[0][0], 1);
553 isl_seq_clr(m->row[0]+1, dim);
554 isl_seq_cpy(m->row[1], c, 1+dim);
555 U = isl_mat_right_inverse(m);
556 Q = isl_mat_right_inverse(isl_mat_copy(U));
557 U = isl_mat_drop_cols(U, 1, 1);
558 Q = isl_mat_drop_rows(Q, 1, 1);
559 set = isl_set_preimage(set, U);
560 facet = uset_convex_hull_wrap_bounded(set);
561 facet = isl_basic_set_preimage(facet, Q);
562 isl_assert(ctx, facet->n_eq == 0, goto error);
565 isl_basic_set_free(facet);
570 /* Given an initial facet constraint, compute the remaining facets.
571 * We do this by running through all facets found so far and computing
572 * the adjacent facets through wrapping, adding those facets that we
573 * hadn't already found before.
575 * For each facet we have found so far, we first compute its facets
576 * in the resulting convex hull. That is, we compute the ridges
577 * of the resulting convex hull contained in the facet.
578 * We also compute the corresponding facet in the current approximation
579 * of the convex hull. There is no need to wrap around the ridges
580 * in this facet since that would result in a facet that is already
581 * present in the current approximation.
583 * This function can still be significantly optimized by checking which of
584 * the facets of the basic sets are also facets of the convex hull and
585 * using all the facets so far to help in constructing the facets of the
588 * using the technique in section "3.1 Ridge Generation" of
589 * "Extended Convex Hull" by Fukuda et al.
591 static struct isl_basic_set *extend(struct isl_basic_set *hull,
596 struct isl_basic_set *facet = NULL;
597 struct isl_basic_set *hull_facet = NULL;
603 isl_assert(set->ctx, set->n > 0, goto error);
605 dim = isl_set_n_dim(set);
607 for (i = 0; i < hull->n_ineq; ++i) {
608 facet = compute_facet(set, hull->ineq[i]);
609 facet = isl_basic_set_add_equality(facet, hull->ineq[i]);
610 facet = isl_basic_set_gauss(facet, NULL);
611 facet = isl_basic_set_normalize_constraints(facet);
612 hull_facet = isl_basic_set_copy(hull);
613 hull_facet = isl_basic_set_add_equality(hull_facet, hull->ineq[i]);
614 hull_facet = isl_basic_set_gauss(hull_facet, NULL);
615 hull_facet = isl_basic_set_normalize_constraints(hull_facet);
618 hull = isl_basic_set_cow(hull);
619 hull = isl_basic_set_extend_dim(hull,
620 isl_dim_copy(hull->dim), 0, 0, facet->n_ineq);
621 for (j = 0; j < facet->n_ineq; ++j) {
622 for (f = 0; f < hull_facet->n_ineq; ++f)
623 if (isl_seq_eq(facet->ineq[j],
624 hull_facet->ineq[f], 1 + dim))
626 if (f < hull_facet->n_ineq)
628 k = isl_basic_set_alloc_inequality(hull);
631 isl_seq_cpy(hull->ineq[k], hull->ineq[i], 1+dim);
632 if (!isl_set_wrap_facet(set, hull->ineq[k], facet->ineq[j]))
635 isl_basic_set_free(hull_facet);
636 isl_basic_set_free(facet);
638 hull = isl_basic_set_simplify(hull);
639 hull = isl_basic_set_finalize(hull);
642 isl_basic_set_free(hull_facet);
643 isl_basic_set_free(facet);
644 isl_basic_set_free(hull);
648 /* Special case for computing the convex hull of a one dimensional set.
649 * We simply collect the lower and upper bounds of each basic set
650 * and the biggest of those.
652 static struct isl_basic_set *convex_hull_1d(struct isl_set *set)
654 struct isl_mat *c = NULL;
655 isl_int *lower = NULL;
656 isl_int *upper = NULL;
659 struct isl_basic_set *hull;
661 for (i = 0; i < set->n; ++i) {
662 set->p[i] = isl_basic_set_simplify(set->p[i]);
666 set = isl_set_remove_empty_parts(set);
669 isl_assert(set->ctx, set->n > 0, goto error);
670 c = isl_mat_alloc(set->ctx, 2, 2);
674 if (set->p[0]->n_eq > 0) {
675 isl_assert(set->ctx, set->p[0]->n_eq == 1, goto error);
678 if (isl_int_is_pos(set->p[0]->eq[0][1])) {
679 isl_seq_cpy(lower, set->p[0]->eq[0], 2);
680 isl_seq_neg(upper, set->p[0]->eq[0], 2);
682 isl_seq_neg(lower, set->p[0]->eq[0], 2);
683 isl_seq_cpy(upper, set->p[0]->eq[0], 2);
686 for (j = 0; j < set->p[0]->n_ineq; ++j) {
687 if (isl_int_is_pos(set->p[0]->ineq[j][1])) {
689 isl_seq_cpy(lower, set->p[0]->ineq[j], 2);
692 isl_seq_cpy(upper, set->p[0]->ineq[j], 2);
699 for (i = 0; i < set->n; ++i) {
700 struct isl_basic_set *bset = set->p[i];
704 for (j = 0; j < bset->n_eq; ++j) {
708 isl_int_mul(a, lower[0], bset->eq[j][1]);
709 isl_int_mul(b, lower[1], bset->eq[j][0]);
710 if (isl_int_lt(a, b) && isl_int_is_pos(bset->eq[j][1]))
711 isl_seq_cpy(lower, bset->eq[j], 2);
712 if (isl_int_gt(a, b) && isl_int_is_neg(bset->eq[j][1]))
713 isl_seq_neg(lower, bset->eq[j], 2);
716 isl_int_mul(a, upper[0], bset->eq[j][1]);
717 isl_int_mul(b, upper[1], bset->eq[j][0]);
718 if (isl_int_lt(a, b) && isl_int_is_pos(bset->eq[j][1]))
719 isl_seq_neg(upper, bset->eq[j], 2);
720 if (isl_int_gt(a, b) && isl_int_is_neg(bset->eq[j][1]))
721 isl_seq_cpy(upper, bset->eq[j], 2);
724 for (j = 0; j < bset->n_ineq; ++j) {
725 if (isl_int_is_pos(bset->ineq[j][1]))
727 if (isl_int_is_neg(bset->ineq[j][1]))
729 if (lower && isl_int_is_pos(bset->ineq[j][1])) {
730 isl_int_mul(a, lower[0], bset->ineq[j][1]);
731 isl_int_mul(b, lower[1], bset->ineq[j][0]);
732 if (isl_int_lt(a, b))
733 isl_seq_cpy(lower, bset->ineq[j], 2);
735 if (upper && isl_int_is_neg(bset->ineq[j][1])) {
736 isl_int_mul(a, upper[0], bset->ineq[j][1]);
737 isl_int_mul(b, upper[1], bset->ineq[j][0]);
738 if (isl_int_gt(a, b))
739 isl_seq_cpy(upper, bset->ineq[j], 2);
750 hull = isl_basic_set_alloc(set->ctx, 0, 1, 0, 0, 2);
751 hull = isl_basic_set_set_rational(hull);
755 k = isl_basic_set_alloc_inequality(hull);
756 isl_seq_cpy(hull->ineq[k], lower, 2);
759 k = isl_basic_set_alloc_inequality(hull);
760 isl_seq_cpy(hull->ineq[k], upper, 2);
762 hull = isl_basic_set_finalize(hull);
772 /* Project out final n dimensions using Fourier-Motzkin */
773 static struct isl_set *set_project_out(struct isl_ctx *ctx,
774 struct isl_set *set, unsigned n)
776 return isl_set_remove_dims(set, isl_set_n_dim(set) - n, n);
779 static struct isl_basic_set *convex_hull_0d(struct isl_set *set)
781 struct isl_basic_set *convex_hull;
786 if (isl_set_is_empty(set))
787 convex_hull = isl_basic_set_empty(isl_dim_copy(set->dim));
789 convex_hull = isl_basic_set_universe(isl_dim_copy(set->dim));
794 /* Compute the convex hull of a pair of basic sets without any parameters or
795 * integer divisions using Fourier-Motzkin elimination.
796 * The convex hull is the set of all points that can be written as
797 * the sum of points from both basic sets (in homogeneous coordinates).
798 * We set up the constraints in a space with dimensions for each of
799 * the three sets and then project out the dimensions corresponding
800 * to the two original basic sets, retaining only those corresponding
801 * to the convex hull.
803 static struct isl_basic_set *convex_hull_pair_elim(struct isl_basic_set *bset1,
804 struct isl_basic_set *bset2)
807 struct isl_basic_set *bset[2];
808 struct isl_basic_set *hull = NULL;
811 if (!bset1 || !bset2)
814 dim = isl_basic_set_n_dim(bset1);
815 hull = isl_basic_set_alloc(bset1->ctx, 0, 2 + 3 * dim, 0,
816 1 + dim + bset1->n_eq + bset2->n_eq,
817 2 + bset1->n_ineq + bset2->n_ineq);
820 for (i = 0; i < 2; ++i) {
821 for (j = 0; j < bset[i]->n_eq; ++j) {
822 k = isl_basic_set_alloc_equality(hull);
825 isl_seq_clr(hull->eq[k], (i+1) * (1+dim));
826 isl_seq_clr(hull->eq[k]+(i+2)*(1+dim), (1-i)*(1+dim));
827 isl_seq_cpy(hull->eq[k]+(i+1)*(1+dim), bset[i]->eq[j],
830 for (j = 0; j < bset[i]->n_ineq; ++j) {
831 k = isl_basic_set_alloc_inequality(hull);
834 isl_seq_clr(hull->ineq[k], (i+1) * (1+dim));
835 isl_seq_clr(hull->ineq[k]+(i+2)*(1+dim), (1-i)*(1+dim));
836 isl_seq_cpy(hull->ineq[k]+(i+1)*(1+dim),
837 bset[i]->ineq[j], 1+dim);
839 k = isl_basic_set_alloc_inequality(hull);
842 isl_seq_clr(hull->ineq[k], 1+2+3*dim);
843 isl_int_set_si(hull->ineq[k][(i+1)*(1+dim)], 1);
845 for (j = 0; j < 1+dim; ++j) {
846 k = isl_basic_set_alloc_equality(hull);
849 isl_seq_clr(hull->eq[k], 1+2+3*dim);
850 isl_int_set_si(hull->eq[k][j], -1);
851 isl_int_set_si(hull->eq[k][1+dim+j], 1);
852 isl_int_set_si(hull->eq[k][2*(1+dim)+j], 1);
854 hull = isl_basic_set_set_rational(hull);
855 hull = isl_basic_set_remove_dims(hull, dim, 2*(1+dim));
856 hull = isl_basic_set_convex_hull(hull);
857 isl_basic_set_free(bset1);
858 isl_basic_set_free(bset2);
861 isl_basic_set_free(bset1);
862 isl_basic_set_free(bset2);
863 isl_basic_set_free(hull);
867 /* Is the set bounded for each value of the parameters?
869 int isl_basic_set_is_bounded(__isl_keep isl_basic_set *bset)
876 if (isl_basic_set_fast_is_empty(bset))
879 tab = isl_tab_from_recession_cone(bset, 1);
880 bounded = isl_tab_cone_is_bounded(tab);
885 /* Is the set bounded for each value of the parameters?
887 int isl_set_is_bounded(__isl_keep isl_set *set)
894 for (i = 0; i < set->n; ++i) {
895 int bounded = isl_basic_set_is_bounded(set->p[i]);
896 if (!bounded || bounded < 0)
902 /* Compute the lineality space of the convex hull of bset1 and bset2.
904 * We first compute the intersection of the recession cone of bset1
905 * with the negative of the recession cone of bset2 and then compute
906 * the linear hull of the resulting cone.
908 static struct isl_basic_set *induced_lineality_space(
909 struct isl_basic_set *bset1, struct isl_basic_set *bset2)
912 struct isl_basic_set *lin = NULL;
915 if (!bset1 || !bset2)
918 dim = isl_basic_set_total_dim(bset1);
919 lin = isl_basic_set_alloc_dim(isl_basic_set_get_dim(bset1), 0,
920 bset1->n_eq + bset2->n_eq,
921 bset1->n_ineq + bset2->n_ineq);
922 lin = isl_basic_set_set_rational(lin);
925 for (i = 0; i < bset1->n_eq; ++i) {
926 k = isl_basic_set_alloc_equality(lin);
929 isl_int_set_si(lin->eq[k][0], 0);
930 isl_seq_cpy(lin->eq[k] + 1, bset1->eq[i] + 1, dim);
932 for (i = 0; i < bset1->n_ineq; ++i) {
933 k = isl_basic_set_alloc_inequality(lin);
936 isl_int_set_si(lin->ineq[k][0], 0);
937 isl_seq_cpy(lin->ineq[k] + 1, bset1->ineq[i] + 1, dim);
939 for (i = 0; i < bset2->n_eq; ++i) {
940 k = isl_basic_set_alloc_equality(lin);
943 isl_int_set_si(lin->eq[k][0], 0);
944 isl_seq_neg(lin->eq[k] + 1, bset2->eq[i] + 1, dim);
946 for (i = 0; i < bset2->n_ineq; ++i) {
947 k = isl_basic_set_alloc_inequality(lin);
950 isl_int_set_si(lin->ineq[k][0], 0);
951 isl_seq_neg(lin->ineq[k] + 1, bset2->ineq[i] + 1, dim);
954 isl_basic_set_free(bset1);
955 isl_basic_set_free(bset2);
956 return isl_basic_set_affine_hull(lin);
958 isl_basic_set_free(lin);
959 isl_basic_set_free(bset1);
960 isl_basic_set_free(bset2);
964 static struct isl_basic_set *uset_convex_hull(struct isl_set *set);
966 /* Given a set and a linear space "lin" of dimension n > 0,
967 * project the linear space from the set, compute the convex hull
968 * and then map the set back to the original space.
974 * describe the linear space. We first compute the Hermite normal
975 * form H = M U of M = H Q, to obtain
979 * The last n rows of H will be zero, so the last n variables of x' = Q x
980 * are the one we want to project out. We do this by transforming each
981 * basic set A x >= b to A U x' >= b and then removing the last n dimensions.
982 * After computing the convex hull in x'_1, i.e., A' x'_1 >= b',
983 * we transform the hull back to the original space as A' Q_1 x >= b',
984 * with Q_1 all but the last n rows of Q.
986 static struct isl_basic_set *modulo_lineality(struct isl_set *set,
987 struct isl_basic_set *lin)
989 unsigned total = isl_basic_set_total_dim(lin);
991 struct isl_basic_set *hull;
992 struct isl_mat *M, *U, *Q;
996 lin_dim = total - lin->n_eq;
997 M = isl_mat_sub_alloc(set->ctx, lin->eq, 0, lin->n_eq, 1, total);
998 M = isl_mat_left_hermite(M, 0, &U, &Q);
1002 isl_basic_set_free(lin);
1004 Q = isl_mat_drop_rows(Q, Q->n_row - lin_dim, lin_dim);
1006 U = isl_mat_lin_to_aff(U);
1007 Q = isl_mat_lin_to_aff(Q);
1009 set = isl_set_preimage(set, U);
1010 set = isl_set_remove_dims(set, total - lin_dim, lin_dim);
1011 hull = uset_convex_hull(set);
1012 hull = isl_basic_set_preimage(hull, Q);
1016 isl_basic_set_free(lin);
1021 /* Given two polyhedra with as constraints h_{ij} x >= 0 in homegeneous space,
1022 * set up an LP for solving
1024 * \sum_j \alpha_{1j} h_{1j} = \sum_j \alpha_{2j} h_{2j}
1026 * \alpha{i0} corresponds to the (implicit) positivity constraint 1 >= 0
1027 * The next \alpha{ij} correspond to the equalities and come in pairs.
1028 * The final \alpha{ij} correspond to the inequalities.
1030 static struct isl_basic_set *valid_direction_lp(
1031 struct isl_basic_set *bset1, struct isl_basic_set *bset2)
1033 struct isl_dim *dim;
1034 struct isl_basic_set *lp;
1039 if (!bset1 || !bset2)
1041 d = 1 + isl_basic_set_total_dim(bset1);
1043 2 * bset1->n_eq + bset1->n_ineq + 2 * bset2->n_eq + bset2->n_ineq;
1044 dim = isl_dim_set_alloc(bset1->ctx, 0, n);
1045 lp = isl_basic_set_alloc_dim(dim, 0, d, n);
1048 for (i = 0; i < n; ++i) {
1049 k = isl_basic_set_alloc_inequality(lp);
1052 isl_seq_clr(lp->ineq[k] + 1, n);
1053 isl_int_set_si(lp->ineq[k][0], -1);
1054 isl_int_set_si(lp->ineq[k][1 + i], 1);
1056 for (i = 0; i < d; ++i) {
1057 k = isl_basic_set_alloc_equality(lp);
1061 isl_int_set_si(lp->eq[k][n++], 0);
1062 /* positivity constraint 1 >= 0 */
1063 isl_int_set_si(lp->eq[k][n++], i == 0);
1064 for (j = 0; j < bset1->n_eq; ++j) {
1065 isl_int_set(lp->eq[k][n++], bset1->eq[j][i]);
1066 isl_int_neg(lp->eq[k][n++], bset1->eq[j][i]);
1068 for (j = 0; j < bset1->n_ineq; ++j)
1069 isl_int_set(lp->eq[k][n++], bset1->ineq[j][i]);
1070 /* positivity constraint 1 >= 0 */
1071 isl_int_set_si(lp->eq[k][n++], -(i == 0));
1072 for (j = 0; j < bset2->n_eq; ++j) {
1073 isl_int_neg(lp->eq[k][n++], bset2->eq[j][i]);
1074 isl_int_set(lp->eq[k][n++], bset2->eq[j][i]);
1076 for (j = 0; j < bset2->n_ineq; ++j)
1077 isl_int_neg(lp->eq[k][n++], bset2->ineq[j][i]);
1079 lp = isl_basic_set_gauss(lp, NULL);
1080 isl_basic_set_free(bset1);
1081 isl_basic_set_free(bset2);
1084 isl_basic_set_free(bset1);
1085 isl_basic_set_free(bset2);
1089 /* Compute a vector s in the homogeneous space such that <s, r> > 0
1090 * for all rays in the homogeneous space of the two cones that correspond
1091 * to the input polyhedra bset1 and bset2.
1093 * We compute s as a vector that satisfies
1095 * s = \sum_j \alpha_{ij} h_{ij} for i = 1,2 (*)
1097 * with h_{ij} the normals of the facets of polyhedron i
1098 * (including the "positivity constraint" 1 >= 0) and \alpha_{ij}
1099 * strictly positive numbers. For simplicity we impose \alpha_{ij} >= 1.
1100 * We first set up an LP with as variables the \alpha{ij}.
1101 * In this formulation, for each polyhedron i,
1102 * the first constraint is the positivity constraint, followed by pairs
1103 * of variables for the equalities, followed by variables for the inequalities.
1104 * We then simply pick a feasible solution and compute s using (*).
1106 * Note that we simply pick any valid direction and make no attempt
1107 * to pick a "good" or even the "best" valid direction.
1109 static struct isl_vec *valid_direction(
1110 struct isl_basic_set *bset1, struct isl_basic_set *bset2)
1112 struct isl_basic_set *lp;
1113 struct isl_tab *tab;
1114 struct isl_vec *sample = NULL;
1115 struct isl_vec *dir;
1120 if (!bset1 || !bset2)
1122 lp = valid_direction_lp(isl_basic_set_copy(bset1),
1123 isl_basic_set_copy(bset2));
1124 tab = isl_tab_from_basic_set(lp);
1125 sample = isl_tab_get_sample_value(tab);
1127 isl_basic_set_free(lp);
1130 d = isl_basic_set_total_dim(bset1);
1131 dir = isl_vec_alloc(bset1->ctx, 1 + d);
1134 isl_seq_clr(dir->block.data + 1, dir->size - 1);
1136 /* positivity constraint 1 >= 0 */
1137 isl_int_set(dir->block.data[0], sample->block.data[n++]);
1138 for (i = 0; i < bset1->n_eq; ++i) {
1139 isl_int_sub(sample->block.data[n],
1140 sample->block.data[n], sample->block.data[n+1]);
1141 isl_seq_combine(dir->block.data,
1142 bset1->ctx->one, dir->block.data,
1143 sample->block.data[n], bset1->eq[i], 1 + d);
1147 for (i = 0; i < bset1->n_ineq; ++i)
1148 isl_seq_combine(dir->block.data,
1149 bset1->ctx->one, dir->block.data,
1150 sample->block.data[n++], bset1->ineq[i], 1 + d);
1151 isl_vec_free(sample);
1152 isl_seq_normalize(bset1->ctx, dir->el, dir->size);
1153 isl_basic_set_free(bset1);
1154 isl_basic_set_free(bset2);
1157 isl_vec_free(sample);
1158 isl_basic_set_free(bset1);
1159 isl_basic_set_free(bset2);
1163 /* Given a polyhedron b_i + A_i x >= 0 and a map T = S^{-1},
1164 * compute b_i' + A_i' x' >= 0, with
1166 * [ b_i A_i ] [ y' ] [ y' ]
1167 * [ 1 0 ] S^{-1} [ x' ] >= 0 or [ b_i' A_i' ] [ x' ] >= 0
1169 * In particular, add the "positivity constraint" and then perform
1172 static struct isl_basic_set *homogeneous_map(struct isl_basic_set *bset,
1179 bset = isl_basic_set_extend_constraints(bset, 0, 1);
1180 k = isl_basic_set_alloc_inequality(bset);
1183 isl_seq_clr(bset->ineq[k] + 1, isl_basic_set_total_dim(bset));
1184 isl_int_set_si(bset->ineq[k][0], 1);
1185 bset = isl_basic_set_preimage(bset, T);
1189 isl_basic_set_free(bset);
1193 /* Compute the convex hull of a pair of basic sets without any parameters or
1194 * integer divisions, where the convex hull is known to be pointed,
1195 * but the basic sets may be unbounded.
1197 * We turn this problem into the computation of a convex hull of a pair
1198 * _bounded_ polyhedra by "changing the direction of the homogeneous
1199 * dimension". This idea is due to Matthias Koeppe.
1201 * Consider the cones in homogeneous space that correspond to the
1202 * input polyhedra. The rays of these cones are also rays of the
1203 * polyhedra if the coordinate that corresponds to the homogeneous
1204 * dimension is zero. That is, if the inner product of the rays
1205 * with the homogeneous direction is zero.
1206 * The cones in the homogeneous space can also be considered to
1207 * correspond to other pairs of polyhedra by chosing a different
1208 * homogeneous direction. To ensure that both of these polyhedra
1209 * are bounded, we need to make sure that all rays of the cones
1210 * correspond to vertices and not to rays.
1211 * Let s be a direction such that <s, r> > 0 for all rays r of both cones.
1212 * Then using s as a homogeneous direction, we obtain a pair of polytopes.
1213 * The vector s is computed in valid_direction.
1215 * Note that we need to consider _all_ rays of the cones and not just
1216 * the rays that correspond to rays in the polyhedra. If we were to
1217 * only consider those rays and turn them into vertices, then we
1218 * may inadvertently turn some vertices into rays.
1220 * The standard homogeneous direction is the unit vector in the 0th coordinate.
1221 * We therefore transform the two polyhedra such that the selected
1222 * direction is mapped onto this standard direction and then proceed
1223 * with the normal computation.
1224 * Let S be a non-singular square matrix with s as its first row,
1225 * then we want to map the polyhedra to the space
1227 * [ y' ] [ y ] [ y ] [ y' ]
1228 * [ x' ] = S [ x ] i.e., [ x ] = S^{-1} [ x' ]
1230 * We take S to be the unimodular completion of s to limit the growth
1231 * of the coefficients in the following computations.
1233 * Let b_i + A_i x >= 0 be the constraints of polyhedron i.
1234 * We first move to the homogeneous dimension
1236 * b_i y + A_i x >= 0 [ b_i A_i ] [ y ] [ 0 ]
1237 * y >= 0 or [ 1 0 ] [ x ] >= [ 0 ]
1239 * Then we change directoin
1241 * [ b_i A_i ] [ y' ] [ y' ]
1242 * [ 1 0 ] S^{-1} [ x' ] >= 0 or [ b_i' A_i' ] [ x' ] >= 0
1244 * Then we compute the convex hull of the polytopes b_i' + A_i' x' >= 0
1245 * resulting in b' + A' x' >= 0, which we then convert back
1248 * [ b' A' ] S [ x ] >= 0 or [ b A ] [ x ] >= 0
1250 * The polyhedron b + A x >= 0 is then the convex hull of the input polyhedra.
1252 static struct isl_basic_set *convex_hull_pair_pointed(
1253 struct isl_basic_set *bset1, struct isl_basic_set *bset2)
1255 struct isl_ctx *ctx = NULL;
1256 struct isl_vec *dir = NULL;
1257 struct isl_mat *T = NULL;
1258 struct isl_mat *T2 = NULL;
1259 struct isl_basic_set *hull;
1260 struct isl_set *set;
1262 if (!bset1 || !bset2)
1265 dir = valid_direction(isl_basic_set_copy(bset1),
1266 isl_basic_set_copy(bset2));
1269 T = isl_mat_alloc(bset1->ctx, dir->size, dir->size);
1272 isl_seq_cpy(T->row[0], dir->block.data, dir->size);
1273 T = isl_mat_unimodular_complete(T, 1);
1274 T2 = isl_mat_right_inverse(isl_mat_copy(T));
1276 bset1 = homogeneous_map(bset1, isl_mat_copy(T2));
1277 bset2 = homogeneous_map(bset2, T2);
1278 set = isl_set_alloc_dim(isl_basic_set_get_dim(bset1), 2, 0);
1279 set = isl_set_add_basic_set(set, bset1);
1280 set = isl_set_add_basic_set(set, bset2);
1281 hull = uset_convex_hull(set);
1282 hull = isl_basic_set_preimage(hull, T);
1289 isl_basic_set_free(bset1);
1290 isl_basic_set_free(bset2);
1294 static struct isl_basic_set *uset_convex_hull_wrap(struct isl_set *set);
1295 static struct isl_basic_set *modulo_affine_hull(
1296 struct isl_set *set, struct isl_basic_set *affine_hull);
1298 /* Compute the convex hull of a pair of basic sets without any parameters or
1299 * integer divisions.
1301 * This function is called from uset_convex_hull_unbounded, which
1302 * means that the complete convex hull is unbounded. Some pairs
1303 * of basic sets may still be bounded, though.
1304 * They may even lie inside a lower dimensional space, in which
1305 * case they need to be handled inside their affine hull since
1306 * the main algorithm assumes that the result is full-dimensional.
1308 * If the convex hull of the two basic sets would have a non-trivial
1309 * lineality space, we first project out this lineality space.
1311 static struct isl_basic_set *convex_hull_pair(struct isl_basic_set *bset1,
1312 struct isl_basic_set *bset2)
1314 isl_basic_set *lin, *aff;
1315 int bounded1, bounded2;
1317 aff = isl_set_affine_hull(isl_basic_set_union(isl_basic_set_copy(bset1),
1318 isl_basic_set_copy(bset2)));
1322 return modulo_affine_hull(isl_basic_set_union(bset1, bset2), aff);
1323 isl_basic_set_free(aff);
1325 bounded1 = isl_basic_set_is_bounded(bset1);
1326 bounded2 = isl_basic_set_is_bounded(bset2);
1328 if (bounded1 < 0 || bounded2 < 0)
1331 if (bounded1 && bounded2)
1332 uset_convex_hull_wrap(isl_basic_set_union(bset1, bset2));
1334 if (bounded1 || bounded2)
1335 return convex_hull_pair_pointed(bset1, bset2);
1337 lin = induced_lineality_space(isl_basic_set_copy(bset1),
1338 isl_basic_set_copy(bset2));
1341 if (isl_basic_set_is_universe(lin)) {
1342 isl_basic_set_free(bset1);
1343 isl_basic_set_free(bset2);
1346 if (lin->n_eq < isl_basic_set_total_dim(lin)) {
1347 struct isl_set *set;
1348 set = isl_set_alloc_dim(isl_basic_set_get_dim(bset1), 2, 0);
1349 set = isl_set_add_basic_set(set, bset1);
1350 set = isl_set_add_basic_set(set, bset2);
1351 return modulo_lineality(set, lin);
1353 isl_basic_set_free(lin);
1355 return convex_hull_pair_pointed(bset1, bset2);
1357 isl_basic_set_free(bset1);
1358 isl_basic_set_free(bset2);
1362 /* Compute the lineality space of a basic set.
1363 * We currently do not allow the basic set to have any divs.
1364 * We basically just drop the constants and turn every inequality
1367 struct isl_basic_set *isl_basic_set_lineality_space(struct isl_basic_set *bset)
1370 struct isl_basic_set *lin = NULL;
1375 isl_assert(bset->ctx, bset->n_div == 0, goto error);
1376 dim = isl_basic_set_total_dim(bset);
1378 lin = isl_basic_set_alloc_dim(isl_basic_set_get_dim(bset), 0, dim, 0);
1381 for (i = 0; i < bset->n_eq; ++i) {
1382 k = isl_basic_set_alloc_equality(lin);
1385 isl_int_set_si(lin->eq[k][0], 0);
1386 isl_seq_cpy(lin->eq[k] + 1, bset->eq[i] + 1, dim);
1388 lin = isl_basic_set_gauss(lin, NULL);
1391 for (i = 0; i < bset->n_ineq && lin->n_eq < dim; ++i) {
1392 k = isl_basic_set_alloc_equality(lin);
1395 isl_int_set_si(lin->eq[k][0], 0);
1396 isl_seq_cpy(lin->eq[k] + 1, bset->ineq[i] + 1, dim);
1397 lin = isl_basic_set_gauss(lin, NULL);
1401 isl_basic_set_free(bset);
1404 isl_basic_set_free(lin);
1405 isl_basic_set_free(bset);
1409 /* Compute the (linear) hull of the lineality spaces of the basic sets in the
1410 * "underlying" set "set".
1412 static struct isl_basic_set *uset_combined_lineality_space(struct isl_set *set)
1415 struct isl_set *lin = NULL;
1420 struct isl_dim *dim = isl_set_get_dim(set);
1422 return isl_basic_set_empty(dim);
1425 lin = isl_set_alloc_dim(isl_set_get_dim(set), set->n, 0);
1426 for (i = 0; i < set->n; ++i)
1427 lin = isl_set_add_basic_set(lin,
1428 isl_basic_set_lineality_space(isl_basic_set_copy(set->p[i])));
1430 return isl_set_affine_hull(lin);
1433 /* Compute the convex hull of a set without any parameters or
1434 * integer divisions.
1435 * In each step, we combined two basic sets until only one
1436 * basic set is left.
1437 * The input basic sets are assumed not to have a non-trivial
1438 * lineality space. If any of the intermediate results has
1439 * a non-trivial lineality space, it is projected out.
1441 static struct isl_basic_set *uset_convex_hull_unbounded(struct isl_set *set)
1443 struct isl_basic_set *convex_hull = NULL;
1445 convex_hull = isl_set_copy_basic_set(set);
1446 set = isl_set_drop_basic_set(set, convex_hull);
1449 while (set->n > 0) {
1450 struct isl_basic_set *t;
1451 t = isl_set_copy_basic_set(set);
1454 set = isl_set_drop_basic_set(set, t);
1457 convex_hull = convex_hull_pair(convex_hull, t);
1460 t = isl_basic_set_lineality_space(isl_basic_set_copy(convex_hull));
1463 if (isl_basic_set_is_universe(t)) {
1464 isl_basic_set_free(convex_hull);
1468 if (t->n_eq < isl_basic_set_total_dim(t)) {
1469 set = isl_set_add_basic_set(set, convex_hull);
1470 return modulo_lineality(set, t);
1472 isl_basic_set_free(t);
1478 isl_basic_set_free(convex_hull);
1482 /* Compute an initial hull for wrapping containing a single initial
1484 * This function assumes that the given set is bounded.
1486 static struct isl_basic_set *initial_hull(struct isl_basic_set *hull,
1487 struct isl_set *set)
1489 struct isl_mat *bounds = NULL;
1495 bounds = initial_facet_constraint(set);
1498 k = isl_basic_set_alloc_inequality(hull);
1501 dim = isl_set_n_dim(set);
1502 isl_assert(set->ctx, 1 + dim == bounds->n_col, goto error);
1503 isl_seq_cpy(hull->ineq[k], bounds->row[0], bounds->n_col);
1504 isl_mat_free(bounds);
1508 isl_basic_set_free(hull);
1509 isl_mat_free(bounds);
1513 struct max_constraint {
1519 static int max_constraint_equal(const void *entry, const void *val)
1521 struct max_constraint *a = (struct max_constraint *)entry;
1522 isl_int *b = (isl_int *)val;
1524 return isl_seq_eq(a->c->row[0] + 1, b, a->c->n_col - 1);
1527 static void update_constraint(struct isl_ctx *ctx, struct isl_hash_table *table,
1528 isl_int *con, unsigned len, int n, int ineq)
1530 struct isl_hash_table_entry *entry;
1531 struct max_constraint *c;
1534 c_hash = isl_seq_get_hash(con + 1, len);
1535 entry = isl_hash_table_find(ctx, table, c_hash, max_constraint_equal,
1541 isl_hash_table_remove(ctx, table, entry);
1545 if (isl_int_gt(c->c->row[0][0], con[0]))
1547 if (isl_int_eq(c->c->row[0][0], con[0])) {
1552 c->c = isl_mat_cow(c->c);
1553 isl_int_set(c->c->row[0][0], con[0]);
1557 /* Check whether the constraint hash table "table" constains the constraint
1560 static int has_constraint(struct isl_ctx *ctx, struct isl_hash_table *table,
1561 isl_int *con, unsigned len, int n)
1563 struct isl_hash_table_entry *entry;
1564 struct max_constraint *c;
1567 c_hash = isl_seq_get_hash(con + 1, len);
1568 entry = isl_hash_table_find(ctx, table, c_hash, max_constraint_equal,
1575 return isl_int_eq(c->c->row[0][0], con[0]);
1578 /* Check for inequality constraints of a basic set without equalities
1579 * such that the same or more stringent copies of the constraint appear
1580 * in all of the basic sets. Such constraints are necessarily facet
1581 * constraints of the convex hull.
1583 * If the resulting basic set is by chance identical to one of
1584 * the basic sets in "set", then we know that this basic set contains
1585 * all other basic sets and is therefore the convex hull of set.
1586 * In this case we set *is_hull to 1.
1588 static struct isl_basic_set *common_constraints(struct isl_basic_set *hull,
1589 struct isl_set *set, int *is_hull)
1592 int min_constraints;
1594 struct max_constraint *constraints = NULL;
1595 struct isl_hash_table *table = NULL;
1600 for (i = 0; i < set->n; ++i)
1601 if (set->p[i]->n_eq == 0)
1605 min_constraints = set->p[i]->n_ineq;
1607 for (i = best + 1; i < set->n; ++i) {
1608 if (set->p[i]->n_eq != 0)
1610 if (set->p[i]->n_ineq >= min_constraints)
1612 min_constraints = set->p[i]->n_ineq;
1615 constraints = isl_calloc_array(hull->ctx, struct max_constraint,
1619 table = isl_alloc_type(hull->ctx, struct isl_hash_table);
1620 if (isl_hash_table_init(hull->ctx, table, min_constraints))
1623 total = isl_dim_total(set->dim);
1624 for (i = 0; i < set->p[best]->n_ineq; ++i) {
1625 constraints[i].c = isl_mat_sub_alloc(hull->ctx,
1626 set->p[best]->ineq + i, 0, 1, 0, 1 + total);
1627 if (!constraints[i].c)
1629 constraints[i].ineq = 1;
1631 for (i = 0; i < min_constraints; ++i) {
1632 struct isl_hash_table_entry *entry;
1634 c_hash = isl_seq_get_hash(constraints[i].c->row[0] + 1, total);
1635 entry = isl_hash_table_find(hull->ctx, table, c_hash,
1636 max_constraint_equal, constraints[i].c->row[0] + 1, 1);
1639 isl_assert(hull->ctx, !entry->data, goto error);
1640 entry->data = &constraints[i];
1644 for (s = 0; s < set->n; ++s) {
1648 for (i = 0; i < set->p[s]->n_eq; ++i) {
1649 isl_int *eq = set->p[s]->eq[i];
1650 for (j = 0; j < 2; ++j) {
1651 isl_seq_neg(eq, eq, 1 + total);
1652 update_constraint(hull->ctx, table,
1656 for (i = 0; i < set->p[s]->n_ineq; ++i) {
1657 isl_int *ineq = set->p[s]->ineq[i];
1658 update_constraint(hull->ctx, table, ineq, total, n,
1659 set->p[s]->n_eq == 0);
1664 for (i = 0; i < min_constraints; ++i) {
1665 if (constraints[i].count < n)
1667 if (!constraints[i].ineq)
1669 j = isl_basic_set_alloc_inequality(hull);
1672 isl_seq_cpy(hull->ineq[j], constraints[i].c->row[0], 1 + total);
1675 for (s = 0; s < set->n; ++s) {
1676 if (set->p[s]->n_eq)
1678 if (set->p[s]->n_ineq != hull->n_ineq)
1680 for (i = 0; i < set->p[s]->n_ineq; ++i) {
1681 isl_int *ineq = set->p[s]->ineq[i];
1682 if (!has_constraint(hull->ctx, table, ineq, total, n))
1685 if (i == set->p[s]->n_ineq)
1689 isl_hash_table_clear(table);
1690 for (i = 0; i < min_constraints; ++i)
1691 isl_mat_free(constraints[i].c);
1696 isl_hash_table_clear(table);
1699 for (i = 0; i < min_constraints; ++i)
1700 isl_mat_free(constraints[i].c);
1705 /* Create a template for the convex hull of "set" and fill it up
1706 * obvious facet constraints, if any. If the result happens to
1707 * be the convex hull of "set" then *is_hull is set to 1.
1709 static struct isl_basic_set *proto_hull(struct isl_set *set, int *is_hull)
1711 struct isl_basic_set *hull;
1716 for (i = 0; i < set->n; ++i) {
1717 n_ineq += set->p[i]->n_eq;
1718 n_ineq += set->p[i]->n_ineq;
1720 hull = isl_basic_set_alloc_dim(isl_dim_copy(set->dim), 0, 0, n_ineq);
1721 hull = isl_basic_set_set_rational(hull);
1724 return common_constraints(hull, set, is_hull);
1727 static struct isl_basic_set *uset_convex_hull_wrap(struct isl_set *set)
1729 struct isl_basic_set *hull;
1732 hull = proto_hull(set, &is_hull);
1733 if (hull && !is_hull) {
1734 if (hull->n_ineq == 0)
1735 hull = initial_hull(hull, set);
1736 hull = extend(hull, set);
1743 /* Compute the convex hull of a set without any parameters or
1744 * integer divisions. Depending on whether the set is bounded,
1745 * we pass control to the wrapping based convex hull or
1746 * the Fourier-Motzkin elimination based convex hull.
1747 * We also handle a few special cases before checking the boundedness.
1749 static struct isl_basic_set *uset_convex_hull(struct isl_set *set)
1751 struct isl_basic_set *convex_hull = NULL;
1752 struct isl_basic_set *lin;
1754 if (isl_set_n_dim(set) == 0)
1755 return convex_hull_0d(set);
1757 set = isl_set_coalesce(set);
1758 set = isl_set_set_rational(set);
1765 convex_hull = isl_basic_set_copy(set->p[0]);
1769 if (isl_set_n_dim(set) == 1)
1770 return convex_hull_1d(set);
1772 if (isl_set_is_bounded(set))
1773 return uset_convex_hull_wrap(set);
1775 lin = uset_combined_lineality_space(isl_set_copy(set));
1778 if (isl_basic_set_is_universe(lin)) {
1782 if (lin->n_eq < isl_basic_set_total_dim(lin))
1783 return modulo_lineality(set, lin);
1784 isl_basic_set_free(lin);
1786 return uset_convex_hull_unbounded(set);
1789 isl_basic_set_free(convex_hull);
1793 /* This is the core procedure, where "set" is a "pure" set, i.e.,
1794 * without parameters or divs and where the convex hull of set is
1795 * known to be full-dimensional.
1797 static struct isl_basic_set *uset_convex_hull_wrap_bounded(struct isl_set *set)
1799 struct isl_basic_set *convex_hull = NULL;
1801 if (isl_set_n_dim(set) == 0) {
1802 convex_hull = isl_basic_set_universe(isl_dim_copy(set->dim));
1804 convex_hull = isl_basic_set_set_rational(convex_hull);
1808 set = isl_set_set_rational(set);
1812 set = isl_set_coalesce(set);
1816 convex_hull = isl_basic_set_copy(set->p[0]);
1820 if (isl_set_n_dim(set) == 1)
1821 return convex_hull_1d(set);
1823 return uset_convex_hull_wrap(set);
1829 /* Compute the convex hull of set "set" with affine hull "affine_hull",
1830 * We first remove the equalities (transforming the set), compute the
1831 * convex hull of the transformed set and then add the equalities back
1832 * (after performing the inverse transformation.
1834 static struct isl_basic_set *modulo_affine_hull(
1835 struct isl_set *set, struct isl_basic_set *affine_hull)
1839 struct isl_basic_set *dummy;
1840 struct isl_basic_set *convex_hull;
1842 dummy = isl_basic_set_remove_equalities(
1843 isl_basic_set_copy(affine_hull), &T, &T2);
1846 isl_basic_set_free(dummy);
1847 set = isl_set_preimage(set, T);
1848 convex_hull = uset_convex_hull(set);
1849 convex_hull = isl_basic_set_preimage(convex_hull, T2);
1850 convex_hull = isl_basic_set_intersect(convex_hull, affine_hull);
1853 isl_basic_set_free(affine_hull);
1858 /* Compute the convex hull of a map.
1860 * The implementation was inspired by "Extended Convex Hull" by Fukuda et al.,
1861 * specifically, the wrapping of facets to obtain new facets.
1863 struct isl_basic_map *isl_map_convex_hull(struct isl_map *map)
1865 struct isl_basic_set *bset;
1866 struct isl_basic_map *model = NULL;
1867 struct isl_basic_set *affine_hull = NULL;
1868 struct isl_basic_map *convex_hull = NULL;
1869 struct isl_set *set = NULL;
1870 struct isl_ctx *ctx;
1877 convex_hull = isl_basic_map_empty_like_map(map);
1882 map = isl_map_detect_equalities(map);
1883 map = isl_map_align_divs(map);
1884 model = isl_basic_map_copy(map->p[0]);
1885 set = isl_map_underlying_set(map);
1889 affine_hull = isl_set_affine_hull(isl_set_copy(set));
1892 if (affine_hull->n_eq != 0)
1893 bset = modulo_affine_hull(set, affine_hull);
1895 isl_basic_set_free(affine_hull);
1896 bset = uset_convex_hull(set);
1899 convex_hull = isl_basic_map_overlying_set(bset, model);
1901 ISL_F_SET(convex_hull, ISL_BASIC_MAP_NO_IMPLICIT);
1902 ISL_F_SET(convex_hull, ISL_BASIC_MAP_ALL_EQUALITIES);
1903 ISL_F_CLR(convex_hull, ISL_BASIC_MAP_RATIONAL);
1907 isl_basic_map_free(model);
1911 struct isl_basic_set *isl_set_convex_hull(struct isl_set *set)
1913 return (struct isl_basic_set *)
1914 isl_map_convex_hull((struct isl_map *)set);
1917 struct sh_data_entry {
1918 struct isl_hash_table *table;
1919 struct isl_tab *tab;
1922 /* Holds the data needed during the simple hull computation.
1924 * n the number of basic sets in the original set
1925 * hull_table a hash table of already computed constraints
1926 * in the simple hull
1927 * p for each basic set,
1928 * table a hash table of the constraints
1929 * tab the tableau corresponding to the basic set
1932 struct isl_ctx *ctx;
1934 struct isl_hash_table *hull_table;
1935 struct sh_data_entry p[1];
1938 static void sh_data_free(struct sh_data *data)
1944 isl_hash_table_free(data->ctx, data->hull_table);
1945 for (i = 0; i < data->n; ++i) {
1946 isl_hash_table_free(data->ctx, data->p[i].table);
1947 isl_tab_free(data->p[i].tab);
1952 struct ineq_cmp_data {
1957 static int has_ineq(const void *entry, const void *val)
1959 isl_int *row = (isl_int *)entry;
1960 struct ineq_cmp_data *v = (struct ineq_cmp_data *)val;
1962 return isl_seq_eq(row + 1, v->p + 1, v->len) ||
1963 isl_seq_is_neg(row + 1, v->p + 1, v->len);
1966 static int hash_ineq(struct isl_ctx *ctx, struct isl_hash_table *table,
1967 isl_int *ineq, unsigned len)
1970 struct ineq_cmp_data v;
1971 struct isl_hash_table_entry *entry;
1975 c_hash = isl_seq_get_hash(ineq + 1, len);
1976 entry = isl_hash_table_find(ctx, table, c_hash, has_ineq, &v, 1);
1983 /* Fill hash table "table" with the constraints of "bset".
1984 * Equalities are added as two inequalities.
1985 * The value in the hash table is a pointer to the (in)equality of "bset".
1987 static int hash_basic_set(struct isl_hash_table *table,
1988 struct isl_basic_set *bset)
1991 unsigned dim = isl_basic_set_total_dim(bset);
1993 for (i = 0; i < bset->n_eq; ++i) {
1994 for (j = 0; j < 2; ++j) {
1995 isl_seq_neg(bset->eq[i], bset->eq[i], 1 + dim);
1996 if (hash_ineq(bset->ctx, table, bset->eq[i], dim) < 0)
2000 for (i = 0; i < bset->n_ineq; ++i) {
2001 if (hash_ineq(bset->ctx, table, bset->ineq[i], dim) < 0)
2007 static struct sh_data *sh_data_alloc(struct isl_set *set, unsigned n_ineq)
2009 struct sh_data *data;
2012 data = isl_calloc(set->ctx, struct sh_data,
2013 sizeof(struct sh_data) +
2014 (set->n - 1) * sizeof(struct sh_data_entry));
2017 data->ctx = set->ctx;
2019 data->hull_table = isl_hash_table_alloc(set->ctx, n_ineq);
2020 if (!data->hull_table)
2022 for (i = 0; i < set->n; ++i) {
2023 data->p[i].table = isl_hash_table_alloc(set->ctx,
2024 2 * set->p[i]->n_eq + set->p[i]->n_ineq);
2025 if (!data->p[i].table)
2027 if (hash_basic_set(data->p[i].table, set->p[i]) < 0)
2036 /* Check if inequality "ineq" is a bound for basic set "j" or if
2037 * it can be relaxed (by increasing the constant term) to become
2038 * a bound for that basic set. In the latter case, the constant
2040 * Return 1 if "ineq" is a bound
2041 * 0 if "ineq" may attain arbitrarily small values on basic set "j"
2042 * -1 if some error occurred
2044 static int is_bound(struct sh_data *data, struct isl_set *set, int j,
2047 enum isl_lp_result res;
2050 if (!data->p[j].tab) {
2051 data->p[j].tab = isl_tab_from_basic_set(set->p[j]);
2052 if (!data->p[j].tab)
2058 res = isl_tab_min(data->p[j].tab, ineq, data->ctx->one,
2060 if (res == isl_lp_ok && isl_int_is_neg(opt))
2061 isl_int_sub(ineq[0], ineq[0], opt);
2065 return (res == isl_lp_ok || res == isl_lp_empty) ? 1 :
2066 res == isl_lp_unbounded ? 0 : -1;
2069 /* Check if inequality "ineq" from basic set "i" can be relaxed to
2070 * become a bound on the whole set. If so, add the (relaxed) inequality
2073 * We first check if "hull" already contains a translate of the inequality.
2074 * If so, we are done.
2075 * Then, we check if any of the previous basic sets contains a translate
2076 * of the inequality. If so, then we have already considered this
2077 * inequality and we are done.
2078 * Otherwise, for each basic set other than "i", we check if the inequality
2079 * is a bound on the basic set.
2080 * For previous basic sets, we know that they do not contain a translate
2081 * of the inequality, so we directly call is_bound.
2082 * For following basic sets, we first check if a translate of the
2083 * inequality appears in its description and if so directly update
2084 * the inequality accordingly.
2086 static struct isl_basic_set *add_bound(struct isl_basic_set *hull,
2087 struct sh_data *data, struct isl_set *set, int i, isl_int *ineq)
2090 struct ineq_cmp_data v;
2091 struct isl_hash_table_entry *entry;
2097 v.len = isl_basic_set_total_dim(hull);
2099 c_hash = isl_seq_get_hash(ineq + 1, v.len);
2101 entry = isl_hash_table_find(hull->ctx, data->hull_table, c_hash,
2106 for (j = 0; j < i; ++j) {
2107 entry = isl_hash_table_find(hull->ctx, data->p[j].table,
2108 c_hash, has_ineq, &v, 0);
2115 k = isl_basic_set_alloc_inequality(hull);
2116 isl_seq_cpy(hull->ineq[k], ineq, 1 + v.len);
2120 for (j = 0; j < i; ++j) {
2122 bound = is_bound(data, set, j, hull->ineq[k]);
2129 isl_basic_set_free_inequality(hull, 1);
2133 for (j = i + 1; j < set->n; ++j) {
2136 entry = isl_hash_table_find(hull->ctx, data->p[j].table,
2137 c_hash, has_ineq, &v, 0);
2139 ineq_j = entry->data;
2140 neg = isl_seq_is_neg(ineq_j + 1,
2141 hull->ineq[k] + 1, v.len);
2143 isl_int_neg(ineq_j[0], ineq_j[0]);
2144 if (isl_int_gt(ineq_j[0], hull->ineq[k][0]))
2145 isl_int_set(hull->ineq[k][0], ineq_j[0]);
2147 isl_int_neg(ineq_j[0], ineq_j[0]);
2150 bound = is_bound(data, set, j, hull->ineq[k]);
2157 isl_basic_set_free_inequality(hull, 1);
2161 entry = isl_hash_table_find(hull->ctx, data->hull_table, c_hash,
2165 entry->data = hull->ineq[k];
2169 isl_basic_set_free(hull);
2173 /* Check if any inequality from basic set "i" can be relaxed to
2174 * become a bound on the whole set. If so, add the (relaxed) inequality
2177 static struct isl_basic_set *add_bounds(struct isl_basic_set *bset,
2178 struct sh_data *data, struct isl_set *set, int i)
2181 unsigned dim = isl_basic_set_total_dim(bset);
2183 for (j = 0; j < set->p[i]->n_eq; ++j) {
2184 for (k = 0; k < 2; ++k) {
2185 isl_seq_neg(set->p[i]->eq[j], set->p[i]->eq[j], 1+dim);
2186 bset = add_bound(bset, data, set, i, set->p[i]->eq[j]);
2189 for (j = 0; j < set->p[i]->n_ineq; ++j)
2190 bset = add_bound(bset, data, set, i, set->p[i]->ineq[j]);
2194 /* Compute a superset of the convex hull of set that is described
2195 * by only translates of the constraints in the constituents of set.
2197 static struct isl_basic_set *uset_simple_hull(struct isl_set *set)
2199 struct sh_data *data = NULL;
2200 struct isl_basic_set *hull = NULL;
2208 for (i = 0; i < set->n; ++i) {
2211 n_ineq += 2 * set->p[i]->n_eq + set->p[i]->n_ineq;
2214 hull = isl_basic_set_alloc_dim(isl_dim_copy(set->dim), 0, 0, n_ineq);
2218 data = sh_data_alloc(set, n_ineq);
2222 for (i = 0; i < set->n; ++i)
2223 hull = add_bounds(hull, data, set, i);
2231 isl_basic_set_free(hull);
2236 /* Compute a superset of the convex hull of map that is described
2237 * by only translates of the constraints in the constituents of map.
2239 struct isl_basic_map *isl_map_simple_hull(struct isl_map *map)
2241 struct isl_set *set = NULL;
2242 struct isl_basic_map *model = NULL;
2243 struct isl_basic_map *hull;
2244 struct isl_basic_map *affine_hull;
2245 struct isl_basic_set *bset = NULL;
2250 hull = isl_basic_map_empty_like_map(map);
2255 hull = isl_basic_map_copy(map->p[0]);
2260 map = isl_map_detect_equalities(map);
2261 affine_hull = isl_map_affine_hull(isl_map_copy(map));
2262 map = isl_map_align_divs(map);
2263 model = isl_basic_map_copy(map->p[0]);
2265 set = isl_map_underlying_set(map);
2267 bset = uset_simple_hull(set);
2269 hull = isl_basic_map_overlying_set(bset, model);
2271 hull = isl_basic_map_intersect(hull, affine_hull);
2272 hull = isl_basic_map_convex_hull(hull);
2273 ISL_F_SET(hull, ISL_BASIC_MAP_NO_IMPLICIT);
2274 ISL_F_SET(hull, ISL_BASIC_MAP_ALL_EQUALITIES);
2279 struct isl_basic_set *isl_set_simple_hull(struct isl_set *set)
2281 return (struct isl_basic_set *)
2282 isl_map_simple_hull((struct isl_map *)set);
2285 /* Given a set "set", return parametric bounds on the dimension "dim".
2287 static struct isl_basic_set *set_bounds(struct isl_set *set, int dim)
2289 unsigned set_dim = isl_set_dim(set, isl_dim_set);
2290 set = isl_set_copy(set);
2291 set = isl_set_eliminate_dims(set, dim + 1, set_dim - (dim + 1));
2292 set = isl_set_eliminate_dims(set, 0, dim);
2293 return isl_set_convex_hull(set);
2296 /* Computes a "simple hull" and then check if each dimension in the
2297 * resulting hull is bounded by a symbolic constant. If not, the
2298 * hull is intersected with the corresponding bounds on the whole set.
2300 struct isl_basic_set *isl_set_bounded_simple_hull(struct isl_set *set)
2303 struct isl_basic_set *hull;
2304 unsigned nparam, left;
2305 int removed_divs = 0;
2307 hull = isl_set_simple_hull(isl_set_copy(set));
2311 nparam = isl_basic_set_dim(hull, isl_dim_param);
2312 for (i = 0; i < isl_basic_set_dim(hull, isl_dim_set); ++i) {
2313 int lower = 0, upper = 0;
2314 struct isl_basic_set *bounds;
2316 left = isl_basic_set_total_dim(hull) - nparam - i - 1;
2317 for (j = 0; j < hull->n_eq; ++j) {
2318 if (isl_int_is_zero(hull->eq[j][1 + nparam + i]))
2320 if (isl_seq_first_non_zero(hull->eq[j]+1+nparam+i+1,
2327 for (j = 0; j < hull->n_ineq; ++j) {
2328 if (isl_int_is_zero(hull->ineq[j][1 + nparam + i]))
2330 if (isl_seq_first_non_zero(hull->ineq[j]+1+nparam+i+1,
2332 isl_seq_first_non_zero(hull->ineq[j]+1+nparam,
2335 if (isl_int_is_pos(hull->ineq[j][1 + nparam + i]))
2346 if (!removed_divs) {
2347 set = isl_set_remove_divs(set);
2352 bounds = set_bounds(set, i);
2353 hull = isl_basic_set_intersect(hull, bounds);