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 tab = isl_tab_detect_implicit_equalities(tab);
107 if (isl_tab_detect_redundant(tab) < 0)
109 bmap = isl_basic_map_update_from_tab(bmap, tab);
111 ISL_F_SET(bmap, ISL_BASIC_MAP_NO_IMPLICIT);
112 ISL_F_SET(bmap, ISL_BASIC_MAP_NO_REDUNDANT);
116 isl_basic_map_free(bmap);
120 struct isl_basic_set *isl_basic_set_convex_hull(struct isl_basic_set *bset)
122 return (struct isl_basic_set *)
123 isl_basic_map_convex_hull((struct isl_basic_map *)bset);
126 /* Check if the set set is bound in the direction of the affine
127 * constraint c and if so, set the constant term such that the
128 * resulting constraint is a bounding constraint for the set.
130 static int uset_is_bound(struct isl_set *set, isl_int *c, unsigned len)
138 isl_int_init(opt_denom);
140 for (j = 0; j < set->n; ++j) {
141 enum isl_lp_result res;
143 if (ISL_F_ISSET(set->p[j], ISL_BASIC_SET_EMPTY))
146 res = isl_basic_set_solve_lp(set->p[j],
147 0, c, set->ctx->one, &opt, &opt_denom, NULL);
148 if (res == isl_lp_unbounded)
150 if (res == isl_lp_error)
152 if (res == isl_lp_empty) {
153 set->p[j] = isl_basic_set_set_to_empty(set->p[j]);
158 if (first || isl_int_is_neg(opt)) {
159 if (!isl_int_is_one(opt_denom))
160 isl_seq_scale(c, c, opt_denom, len);
161 isl_int_sub(c[0], c[0], opt);
166 isl_int_clear(opt_denom);
170 isl_int_clear(opt_denom);
174 struct isl_basic_set *isl_basic_set_set_rational(struct isl_basic_set *bset)
179 if (ISL_F_ISSET(bset, ISL_BASIC_MAP_RATIONAL))
182 bset = isl_basic_set_cow(bset);
186 ISL_F_SET(bset, ISL_BASIC_MAP_RATIONAL);
188 return isl_basic_set_finalize(bset);
191 static struct isl_set *isl_set_set_rational(struct isl_set *set)
195 set = isl_set_cow(set);
198 for (i = 0; i < set->n; ++i) {
199 set->p[i] = isl_basic_set_set_rational(set->p[i]);
209 static struct isl_basic_set *isl_basic_set_add_equality(
210 struct isl_basic_set *bset, isl_int *c)
215 if (ISL_F_ISSET(bset, ISL_BASIC_SET_EMPTY))
218 isl_assert(bset->ctx, isl_basic_set_n_param(bset) == 0, goto error);
219 isl_assert(bset->ctx, bset->n_div == 0, goto error);
220 dim = isl_basic_set_n_dim(bset);
221 bset = isl_basic_set_cow(bset);
222 bset = isl_basic_set_extend(bset, 0, dim, 0, 1, 0);
223 i = isl_basic_set_alloc_equality(bset);
226 isl_seq_cpy(bset->eq[i], c, 1 + dim);
229 isl_basic_set_free(bset);
233 static struct isl_set *isl_set_add_basic_set_equality(struct isl_set *set, isl_int *c)
237 set = isl_set_cow(set);
240 for (i = 0; i < set->n; ++i) {
241 set->p[i] = isl_basic_set_add_equality(set->p[i], c);
251 /* Given a union of basic sets, construct the constraints for wrapping
252 * a facet around one of its ridges.
253 * In particular, if each of n the d-dimensional basic sets i in "set"
254 * contains the origin, satisfies the constraints x_1 >= 0 and x_2 >= 0
255 * and is defined by the constraints
259 * then the resulting set is of dimension n*(1+d) and has as constraints
268 static struct isl_basic_set *wrap_constraints(struct isl_set *set)
270 struct isl_basic_set *lp;
274 unsigned dim, lp_dim;
279 dim = 1 + isl_set_n_dim(set);
282 for (i = 0; i < set->n; ++i) {
283 n_eq += set->p[i]->n_eq;
284 n_ineq += set->p[i]->n_ineq;
286 lp = isl_basic_set_alloc(set->ctx, 0, dim * set->n, 0, n_eq, n_ineq);
289 lp_dim = isl_basic_set_n_dim(lp);
290 k = isl_basic_set_alloc_equality(lp);
291 isl_int_set_si(lp->eq[k][0], -1);
292 for (i = 0; i < set->n; ++i) {
293 isl_int_set_si(lp->eq[k][1+dim*i], 0);
294 isl_int_set_si(lp->eq[k][1+dim*i+1], 1);
295 isl_seq_clr(lp->eq[k]+1+dim*i+2, dim-2);
297 for (i = 0; i < set->n; ++i) {
298 k = isl_basic_set_alloc_inequality(lp);
299 isl_seq_clr(lp->ineq[k], 1+lp_dim);
300 isl_int_set_si(lp->ineq[k][1+dim*i], 1);
302 for (j = 0; j < set->p[i]->n_eq; ++j) {
303 k = isl_basic_set_alloc_equality(lp);
304 isl_seq_clr(lp->eq[k], 1+dim*i);
305 isl_seq_cpy(lp->eq[k]+1+dim*i, set->p[i]->eq[j], dim);
306 isl_seq_clr(lp->eq[k]+1+dim*(i+1), dim*(set->n-i-1));
309 for (j = 0; j < set->p[i]->n_ineq; ++j) {
310 k = isl_basic_set_alloc_inequality(lp);
311 isl_seq_clr(lp->ineq[k], 1+dim*i);
312 isl_seq_cpy(lp->ineq[k]+1+dim*i, set->p[i]->ineq[j], dim);
313 isl_seq_clr(lp->ineq[k]+1+dim*(i+1), dim*(set->n-i-1));
319 /* Given a facet "facet" of the convex hull of "set" and a facet "ridge"
320 * of that facet, compute the other facet of the convex hull that contains
323 * We first transform the set such that the facet constraint becomes
327 * I.e., the facet lies in
331 * and on that facet, the constraint that defines the ridge is
335 * (This transformation is not strictly needed, all that is needed is
336 * that the ridge contains the origin.)
338 * Since the ridge contains the origin, the cone of the convex hull
339 * will be of the form
344 * with this second constraint defining the new facet.
345 * The constant a is obtained by settting x_1 in the cone of the
346 * convex hull to 1 and minimizing x_2.
347 * Now, each element in the cone of the convex hull is the sum
348 * of elements in the cones of the basic sets.
349 * If a_i is the dilation factor of basic set i, then the problem
350 * we need to solve is
363 * the constraints of each (transformed) basic set.
364 * If a = n/d, then the constraint defining the new facet (in the transformed
367 * -n x_1 + d x_2 >= 0
369 * In the original space, we need to take the same combination of the
370 * corresponding constraints "facet" and "ridge".
372 * If a = -infty = "-1/0", then we just return the original facet constraint.
373 * This means that the facet is unbounded, but has a bounded intersection
374 * with the union of sets.
376 isl_int *isl_set_wrap_facet(__isl_keep isl_set *set,
377 isl_int *facet, isl_int *ridge)
380 struct isl_mat *T = NULL;
381 struct isl_basic_set *lp = NULL;
383 enum isl_lp_result res;
387 set = isl_set_copy(set);
388 set = isl_set_set_rational(set);
390 dim = 1 + isl_set_n_dim(set);
391 T = isl_mat_alloc(set->ctx, 3, dim);
394 isl_int_set_si(T->row[0][0], 1);
395 isl_seq_clr(T->row[0]+1, dim - 1);
396 isl_seq_cpy(T->row[1], facet, dim);
397 isl_seq_cpy(T->row[2], ridge, dim);
398 T = isl_mat_right_inverse(T);
399 set = isl_set_preimage(set, T);
403 lp = wrap_constraints(set);
404 obj = isl_vec_alloc(set->ctx, 1 + dim*set->n);
407 isl_int_set_si(obj->block.data[0], 0);
408 for (i = 0; i < set->n; ++i) {
409 isl_seq_clr(obj->block.data + 1 + dim*i, 2);
410 isl_int_set_si(obj->block.data[1 + dim*i+2], 1);
411 isl_seq_clr(obj->block.data + 1 + dim*i+3, dim-3);
415 res = isl_basic_set_solve_lp(lp, 0,
416 obj->block.data, set->ctx->one, &num, &den, NULL);
417 if (res == isl_lp_ok) {
418 isl_int_neg(num, num);
419 isl_seq_combine(facet, num, facet, den, ridge, dim);
424 isl_basic_set_free(lp);
426 isl_assert(set->ctx, res == isl_lp_ok || res == isl_lp_unbounded,
430 isl_basic_set_free(lp);
436 /* Compute the constraint of a facet of "set".
438 * We first compute the intersection with a bounding constraint
439 * that is orthogonal to one of the coordinate axes.
440 * If the affine hull of this intersection has only one equality,
441 * we have found a facet.
442 * Otherwise, we wrap the current bounding constraint around
443 * one of the equalities of the face (one that is not equal to
444 * the current bounding constraint).
445 * This process continues until we have found a facet.
446 * The dimension of the intersection increases by at least
447 * one on each iteration, so termination is guaranteed.
449 static __isl_give isl_mat *initial_facet_constraint(__isl_keep isl_set *set)
451 struct isl_set *slice = NULL;
452 struct isl_basic_set *face = NULL;
453 struct isl_mat *m, *U, *Q;
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 static int isl_basic_set_is_bounded(struct isl_basic_set *bset)
872 tab = isl_tab_from_recession_cone(bset);
873 bounded = isl_tab_cone_is_bounded(tab);
878 static int isl_set_is_bounded(struct isl_set *set)
882 for (i = 0; i < set->n; ++i) {
883 int bounded = isl_basic_set_is_bounded(set->p[i]);
884 if (!bounded || bounded < 0)
890 /* Compute the lineality space of the convex hull of bset1 and bset2.
892 * We first compute the intersection of the recession cone of bset1
893 * with the negative of the recession cone of bset2 and then compute
894 * the linear hull of the resulting cone.
896 static struct isl_basic_set *induced_lineality_space(
897 struct isl_basic_set *bset1, struct isl_basic_set *bset2)
900 struct isl_basic_set *lin = NULL;
903 if (!bset1 || !bset2)
906 dim = isl_basic_set_total_dim(bset1);
907 lin = isl_basic_set_alloc_dim(isl_basic_set_get_dim(bset1), 0,
908 bset1->n_eq + bset2->n_eq,
909 bset1->n_ineq + bset2->n_ineq);
910 lin = isl_basic_set_set_rational(lin);
913 for (i = 0; i < bset1->n_eq; ++i) {
914 k = isl_basic_set_alloc_equality(lin);
917 isl_int_set_si(lin->eq[k][0], 0);
918 isl_seq_cpy(lin->eq[k] + 1, bset1->eq[i] + 1, dim);
920 for (i = 0; i < bset1->n_ineq; ++i) {
921 k = isl_basic_set_alloc_inequality(lin);
924 isl_int_set_si(lin->ineq[k][0], 0);
925 isl_seq_cpy(lin->ineq[k] + 1, bset1->ineq[i] + 1, dim);
927 for (i = 0; i < bset2->n_eq; ++i) {
928 k = isl_basic_set_alloc_equality(lin);
931 isl_int_set_si(lin->eq[k][0], 0);
932 isl_seq_neg(lin->eq[k] + 1, bset2->eq[i] + 1, dim);
934 for (i = 0; i < bset2->n_ineq; ++i) {
935 k = isl_basic_set_alloc_inequality(lin);
938 isl_int_set_si(lin->ineq[k][0], 0);
939 isl_seq_neg(lin->ineq[k] + 1, bset2->ineq[i] + 1, dim);
942 isl_basic_set_free(bset1);
943 isl_basic_set_free(bset2);
944 return isl_basic_set_affine_hull(lin);
946 isl_basic_set_free(lin);
947 isl_basic_set_free(bset1);
948 isl_basic_set_free(bset2);
952 static struct isl_basic_set *uset_convex_hull(struct isl_set *set);
954 /* Given a set and a linear space "lin" of dimension n > 0,
955 * project the linear space from the set, compute the convex hull
956 * and then map the set back to the original space.
962 * describe the linear space. We first compute the Hermite normal
963 * form H = M U of M = H Q, to obtain
967 * The last n rows of H will be zero, so the last n variables of x' = Q x
968 * are the one we want to project out. We do this by transforming each
969 * basic set A x >= b to A U x' >= b and then removing the last n dimensions.
970 * After computing the convex hull in x'_1, i.e., A' x'_1 >= b',
971 * we transform the hull back to the original space as A' Q_1 x >= b',
972 * with Q_1 all but the last n rows of Q.
974 static struct isl_basic_set *modulo_lineality(struct isl_set *set,
975 struct isl_basic_set *lin)
977 unsigned total = isl_basic_set_total_dim(lin);
979 struct isl_basic_set *hull;
980 struct isl_mat *M, *U, *Q;
984 lin_dim = total - lin->n_eq;
985 M = isl_mat_sub_alloc(set->ctx, lin->eq, 0, lin->n_eq, 1, total);
986 M = isl_mat_left_hermite(M, 0, &U, &Q);
990 isl_basic_set_free(lin);
992 Q = isl_mat_drop_rows(Q, Q->n_row - lin_dim, lin_dim);
994 U = isl_mat_lin_to_aff(U);
995 Q = isl_mat_lin_to_aff(Q);
997 set = isl_set_preimage(set, U);
998 set = isl_set_remove_dims(set, total - lin_dim, lin_dim);
999 hull = uset_convex_hull(set);
1000 hull = isl_basic_set_preimage(hull, Q);
1004 isl_basic_set_free(lin);
1009 /* Given two polyhedra with as constraints h_{ij} x >= 0 in homegeneous space,
1010 * set up an LP for solving
1012 * \sum_j \alpha_{1j} h_{1j} = \sum_j \alpha_{2j} h_{2j}
1014 * \alpha{i0} corresponds to the (implicit) positivity constraint 1 >= 0
1015 * The next \alpha{ij} correspond to the equalities and come in pairs.
1016 * The final \alpha{ij} correspond to the inequalities.
1018 static struct isl_basic_set *valid_direction_lp(
1019 struct isl_basic_set *bset1, struct isl_basic_set *bset2)
1021 struct isl_dim *dim;
1022 struct isl_basic_set *lp;
1027 if (!bset1 || !bset2)
1029 d = 1 + isl_basic_set_total_dim(bset1);
1031 2 * bset1->n_eq + bset1->n_ineq + 2 * bset2->n_eq + bset2->n_ineq;
1032 dim = isl_dim_set_alloc(bset1->ctx, 0, n);
1033 lp = isl_basic_set_alloc_dim(dim, 0, d, n);
1036 for (i = 0; i < n; ++i) {
1037 k = isl_basic_set_alloc_inequality(lp);
1040 isl_seq_clr(lp->ineq[k] + 1, n);
1041 isl_int_set_si(lp->ineq[k][0], -1);
1042 isl_int_set_si(lp->ineq[k][1 + i], 1);
1044 for (i = 0; i < d; ++i) {
1045 k = isl_basic_set_alloc_equality(lp);
1049 isl_int_set_si(lp->eq[k][n++], 0);
1050 /* positivity constraint 1 >= 0 */
1051 isl_int_set_si(lp->eq[k][n++], i == 0);
1052 for (j = 0; j < bset1->n_eq; ++j) {
1053 isl_int_set(lp->eq[k][n++], bset1->eq[j][i]);
1054 isl_int_neg(lp->eq[k][n++], bset1->eq[j][i]);
1056 for (j = 0; j < bset1->n_ineq; ++j)
1057 isl_int_set(lp->eq[k][n++], bset1->ineq[j][i]);
1058 /* positivity constraint 1 >= 0 */
1059 isl_int_set_si(lp->eq[k][n++], -(i == 0));
1060 for (j = 0; j < bset2->n_eq; ++j) {
1061 isl_int_neg(lp->eq[k][n++], bset2->eq[j][i]);
1062 isl_int_set(lp->eq[k][n++], bset2->eq[j][i]);
1064 for (j = 0; j < bset2->n_ineq; ++j)
1065 isl_int_neg(lp->eq[k][n++], bset2->ineq[j][i]);
1067 lp = isl_basic_set_gauss(lp, NULL);
1068 isl_basic_set_free(bset1);
1069 isl_basic_set_free(bset2);
1072 isl_basic_set_free(bset1);
1073 isl_basic_set_free(bset2);
1077 /* Compute a vector s in the homogeneous space such that <s, r> > 0
1078 * for all rays in the homogeneous space of the two cones that correspond
1079 * to the input polyhedra bset1 and bset2.
1081 * We compute s as a vector that satisfies
1083 * s = \sum_j \alpha_{ij} h_{ij} for i = 1,2 (*)
1085 * with h_{ij} the normals of the facets of polyhedron i
1086 * (including the "positivity constraint" 1 >= 0) and \alpha_{ij}
1087 * strictly positive numbers. For simplicity we impose \alpha_{ij} >= 1.
1088 * We first set up an LP with as variables the \alpha{ij}.
1089 * In this formulation, for each polyhedron i,
1090 * the first constraint is the positivity constraint, followed by pairs
1091 * of variables for the equalities, followed by variables for the inequalities.
1092 * We then simply pick a feasible solution and compute s using (*).
1094 * Note that we simply pick any valid direction and make no attempt
1095 * to pick a "good" or even the "best" valid direction.
1097 static struct isl_vec *valid_direction(
1098 struct isl_basic_set *bset1, struct isl_basic_set *bset2)
1100 struct isl_basic_set *lp;
1101 struct isl_tab *tab;
1102 struct isl_vec *sample = NULL;
1103 struct isl_vec *dir;
1108 if (!bset1 || !bset2)
1110 lp = valid_direction_lp(isl_basic_set_copy(bset1),
1111 isl_basic_set_copy(bset2));
1112 tab = isl_tab_from_basic_set(lp);
1113 sample = isl_tab_get_sample_value(tab);
1115 isl_basic_set_free(lp);
1118 d = isl_basic_set_total_dim(bset1);
1119 dir = isl_vec_alloc(bset1->ctx, 1 + d);
1122 isl_seq_clr(dir->block.data + 1, dir->size - 1);
1124 /* positivity constraint 1 >= 0 */
1125 isl_int_set(dir->block.data[0], sample->block.data[n++]);
1126 for (i = 0; i < bset1->n_eq; ++i) {
1127 isl_int_sub(sample->block.data[n],
1128 sample->block.data[n], sample->block.data[n+1]);
1129 isl_seq_combine(dir->block.data,
1130 bset1->ctx->one, dir->block.data,
1131 sample->block.data[n], bset1->eq[i], 1 + d);
1135 for (i = 0; i < bset1->n_ineq; ++i)
1136 isl_seq_combine(dir->block.data,
1137 bset1->ctx->one, dir->block.data,
1138 sample->block.data[n++], bset1->ineq[i], 1 + d);
1139 isl_vec_free(sample);
1140 isl_seq_normalize(bset1->ctx, dir->el, dir->size);
1141 isl_basic_set_free(bset1);
1142 isl_basic_set_free(bset2);
1145 isl_vec_free(sample);
1146 isl_basic_set_free(bset1);
1147 isl_basic_set_free(bset2);
1151 /* Given a polyhedron b_i + A_i x >= 0 and a map T = S^{-1},
1152 * compute b_i' + A_i' x' >= 0, with
1154 * [ b_i A_i ] [ y' ] [ y' ]
1155 * [ 1 0 ] S^{-1} [ x' ] >= 0 or [ b_i' A_i' ] [ x' ] >= 0
1157 * In particular, add the "positivity constraint" and then perform
1160 static struct isl_basic_set *homogeneous_map(struct isl_basic_set *bset,
1167 bset = isl_basic_set_extend_constraints(bset, 0, 1);
1168 k = isl_basic_set_alloc_inequality(bset);
1171 isl_seq_clr(bset->ineq[k] + 1, isl_basic_set_total_dim(bset));
1172 isl_int_set_si(bset->ineq[k][0], 1);
1173 bset = isl_basic_set_preimage(bset, T);
1177 isl_basic_set_free(bset);
1181 /* Compute the convex hull of a pair of basic sets without any parameters or
1182 * integer divisions, where the convex hull is known to be pointed,
1183 * but the basic sets may be unbounded.
1185 * We turn this problem into the computation of a convex hull of a pair
1186 * _bounded_ polyhedra by "changing the direction of the homogeneous
1187 * dimension". This idea is due to Matthias Koeppe.
1189 * Consider the cones in homogeneous space that correspond to the
1190 * input polyhedra. The rays of these cones are also rays of the
1191 * polyhedra if the coordinate that corresponds to the homogeneous
1192 * dimension is zero. That is, if the inner product of the rays
1193 * with the homogeneous direction is zero.
1194 * The cones in the homogeneous space can also be considered to
1195 * correspond to other pairs of polyhedra by chosing a different
1196 * homogeneous direction. To ensure that both of these polyhedra
1197 * are bounded, we need to make sure that all rays of the cones
1198 * correspond to vertices and not to rays.
1199 * Let s be a direction such that <s, r> > 0 for all rays r of both cones.
1200 * Then using s as a homogeneous direction, we obtain a pair of polytopes.
1201 * The vector s is computed in valid_direction.
1203 * Note that we need to consider _all_ rays of the cones and not just
1204 * the rays that correspond to rays in the polyhedra. If we were to
1205 * only consider those rays and turn them into vertices, then we
1206 * may inadvertently turn some vertices into rays.
1208 * The standard homogeneous direction is the unit vector in the 0th coordinate.
1209 * We therefore transform the two polyhedra such that the selected
1210 * direction is mapped onto this standard direction and then proceed
1211 * with the normal computation.
1212 * Let S be a non-singular square matrix with s as its first row,
1213 * then we want to map the polyhedra to the space
1215 * [ y' ] [ y ] [ y ] [ y' ]
1216 * [ x' ] = S [ x ] i.e., [ x ] = S^{-1} [ x' ]
1218 * We take S to be the unimodular completion of s to limit the growth
1219 * of the coefficients in the following computations.
1221 * Let b_i + A_i x >= 0 be the constraints of polyhedron i.
1222 * We first move to the homogeneous dimension
1224 * b_i y + A_i x >= 0 [ b_i A_i ] [ y ] [ 0 ]
1225 * y >= 0 or [ 1 0 ] [ x ] >= [ 0 ]
1227 * Then we change directoin
1229 * [ b_i A_i ] [ y' ] [ y' ]
1230 * [ 1 0 ] S^{-1} [ x' ] >= 0 or [ b_i' A_i' ] [ x' ] >= 0
1232 * Then we compute the convex hull of the polytopes b_i' + A_i' x' >= 0
1233 * resulting in b' + A' x' >= 0, which we then convert back
1236 * [ b' A' ] S [ x ] >= 0 or [ b A ] [ x ] >= 0
1238 * The polyhedron b + A x >= 0 is then the convex hull of the input polyhedra.
1240 static struct isl_basic_set *convex_hull_pair_pointed(
1241 struct isl_basic_set *bset1, struct isl_basic_set *bset2)
1243 struct isl_ctx *ctx = NULL;
1244 struct isl_vec *dir = NULL;
1245 struct isl_mat *T = NULL;
1246 struct isl_mat *T2 = NULL;
1247 struct isl_basic_set *hull;
1248 struct isl_set *set;
1250 if (!bset1 || !bset2)
1253 dir = valid_direction(isl_basic_set_copy(bset1),
1254 isl_basic_set_copy(bset2));
1257 T = isl_mat_alloc(bset1->ctx, dir->size, dir->size);
1260 isl_seq_cpy(T->row[0], dir->block.data, dir->size);
1261 T = isl_mat_unimodular_complete(T, 1);
1262 T2 = isl_mat_right_inverse(isl_mat_copy(T));
1264 bset1 = homogeneous_map(bset1, isl_mat_copy(T2));
1265 bset2 = homogeneous_map(bset2, T2);
1266 set = isl_set_alloc_dim(isl_basic_set_get_dim(bset1), 2, 0);
1267 set = isl_set_add_basic_set(set, bset1);
1268 set = isl_set_add_basic_set(set, bset2);
1269 hull = uset_convex_hull(set);
1270 hull = isl_basic_set_preimage(hull, T);
1277 isl_basic_set_free(bset1);
1278 isl_basic_set_free(bset2);
1282 /* Compute the convex hull of a pair of basic sets without any parameters or
1283 * integer divisions.
1285 * If the convex hull of the two basic sets would have a non-trivial
1286 * lineality space, we first project out this lineality space.
1288 static struct isl_basic_set *convex_hull_pair(struct isl_basic_set *bset1,
1289 struct isl_basic_set *bset2)
1291 struct isl_basic_set *lin;
1293 if (isl_basic_set_is_bounded(bset1) || isl_basic_set_is_bounded(bset2))
1294 return convex_hull_pair_pointed(bset1, bset2);
1296 lin = induced_lineality_space(isl_basic_set_copy(bset1),
1297 isl_basic_set_copy(bset2));
1300 if (isl_basic_set_is_universe(lin)) {
1301 isl_basic_set_free(bset1);
1302 isl_basic_set_free(bset2);
1305 if (lin->n_eq < isl_basic_set_total_dim(lin)) {
1306 struct isl_set *set;
1307 set = isl_set_alloc_dim(isl_basic_set_get_dim(bset1), 2, 0);
1308 set = isl_set_add_basic_set(set, bset1);
1309 set = isl_set_add_basic_set(set, bset2);
1310 return modulo_lineality(set, lin);
1312 isl_basic_set_free(lin);
1314 return convex_hull_pair_pointed(bset1, bset2);
1316 isl_basic_set_free(bset1);
1317 isl_basic_set_free(bset2);
1321 /* Compute the lineality space of a basic set.
1322 * We currently do not allow the basic set to have any divs.
1323 * We basically just drop the constants and turn every inequality
1326 struct isl_basic_set *isl_basic_set_lineality_space(struct isl_basic_set *bset)
1329 struct isl_basic_set *lin = NULL;
1334 isl_assert(bset->ctx, bset->n_div == 0, goto error);
1335 dim = isl_basic_set_total_dim(bset);
1337 lin = isl_basic_set_alloc_dim(isl_basic_set_get_dim(bset), 0, dim, 0);
1340 for (i = 0; i < bset->n_eq; ++i) {
1341 k = isl_basic_set_alloc_equality(lin);
1344 isl_int_set_si(lin->eq[k][0], 0);
1345 isl_seq_cpy(lin->eq[k] + 1, bset->eq[i] + 1, dim);
1347 lin = isl_basic_set_gauss(lin, NULL);
1350 for (i = 0; i < bset->n_ineq && lin->n_eq < dim; ++i) {
1351 k = isl_basic_set_alloc_equality(lin);
1354 isl_int_set_si(lin->eq[k][0], 0);
1355 isl_seq_cpy(lin->eq[k] + 1, bset->ineq[i] + 1, dim);
1356 lin = isl_basic_set_gauss(lin, NULL);
1360 isl_basic_set_free(bset);
1363 isl_basic_set_free(lin);
1364 isl_basic_set_free(bset);
1368 /* Compute the (linear) hull of the lineality spaces of the basic sets in the
1369 * "underlying" set "set".
1371 static struct isl_basic_set *uset_combined_lineality_space(struct isl_set *set)
1374 struct isl_set *lin = NULL;
1379 struct isl_dim *dim = isl_set_get_dim(set);
1381 return isl_basic_set_empty(dim);
1384 lin = isl_set_alloc_dim(isl_set_get_dim(set), set->n, 0);
1385 for (i = 0; i < set->n; ++i)
1386 lin = isl_set_add_basic_set(lin,
1387 isl_basic_set_lineality_space(isl_basic_set_copy(set->p[i])));
1389 return isl_set_affine_hull(lin);
1392 /* Compute the convex hull of a set without any parameters or
1393 * integer divisions.
1394 * In each step, we combined two basic sets until only one
1395 * basic set is left.
1396 * The input basic sets are assumed not to have a non-trivial
1397 * lineality space. If any of the intermediate results has
1398 * a non-trivial lineality space, it is projected out.
1400 static struct isl_basic_set *uset_convex_hull_unbounded(struct isl_set *set)
1402 struct isl_basic_set *convex_hull = NULL;
1404 convex_hull = isl_set_copy_basic_set(set);
1405 set = isl_set_drop_basic_set(set, convex_hull);
1408 while (set->n > 0) {
1409 struct isl_basic_set *t;
1410 t = isl_set_copy_basic_set(set);
1413 set = isl_set_drop_basic_set(set, t);
1416 convex_hull = convex_hull_pair(convex_hull, t);
1419 t = isl_basic_set_lineality_space(isl_basic_set_copy(convex_hull));
1422 if (isl_basic_set_is_universe(t)) {
1423 isl_basic_set_free(convex_hull);
1427 if (t->n_eq < isl_basic_set_total_dim(t)) {
1428 set = isl_set_add_basic_set(set, convex_hull);
1429 return modulo_lineality(set, t);
1431 isl_basic_set_free(t);
1437 isl_basic_set_free(convex_hull);
1441 /* Compute an initial hull for wrapping containing a single initial
1443 * This function assumes that the given set is bounded.
1445 static struct isl_basic_set *initial_hull(struct isl_basic_set *hull,
1446 struct isl_set *set)
1448 struct isl_mat *bounds = NULL;
1454 bounds = initial_facet_constraint(set);
1457 k = isl_basic_set_alloc_inequality(hull);
1460 dim = isl_set_n_dim(set);
1461 isl_assert(set->ctx, 1 + dim == bounds->n_col, goto error);
1462 isl_seq_cpy(hull->ineq[k], bounds->row[0], bounds->n_col);
1463 isl_mat_free(bounds);
1467 isl_basic_set_free(hull);
1468 isl_mat_free(bounds);
1472 struct max_constraint {
1478 static int max_constraint_equal(const void *entry, const void *val)
1480 struct max_constraint *a = (struct max_constraint *)entry;
1481 isl_int *b = (isl_int *)val;
1483 return isl_seq_eq(a->c->row[0] + 1, b, a->c->n_col - 1);
1486 static void update_constraint(struct isl_ctx *ctx, struct isl_hash_table *table,
1487 isl_int *con, unsigned len, int n, int ineq)
1489 struct isl_hash_table_entry *entry;
1490 struct max_constraint *c;
1493 c_hash = isl_seq_get_hash(con + 1, len);
1494 entry = isl_hash_table_find(ctx, table, c_hash, max_constraint_equal,
1500 isl_hash_table_remove(ctx, table, entry);
1504 if (isl_int_gt(c->c->row[0][0], con[0]))
1506 if (isl_int_eq(c->c->row[0][0], con[0])) {
1511 c->c = isl_mat_cow(c->c);
1512 isl_int_set(c->c->row[0][0], con[0]);
1516 /* Check whether the constraint hash table "table" constains the constraint
1519 static int has_constraint(struct isl_ctx *ctx, struct isl_hash_table *table,
1520 isl_int *con, unsigned len, int n)
1522 struct isl_hash_table_entry *entry;
1523 struct max_constraint *c;
1526 c_hash = isl_seq_get_hash(con + 1, len);
1527 entry = isl_hash_table_find(ctx, table, c_hash, max_constraint_equal,
1534 return isl_int_eq(c->c->row[0][0], con[0]);
1537 /* Check for inequality constraints of a basic set without equalities
1538 * such that the same or more stringent copies of the constraint appear
1539 * in all of the basic sets. Such constraints are necessarily facet
1540 * constraints of the convex hull.
1542 * If the resulting basic set is by chance identical to one of
1543 * the basic sets in "set", then we know that this basic set contains
1544 * all other basic sets and is therefore the convex hull of set.
1545 * In this case we set *is_hull to 1.
1547 static struct isl_basic_set *common_constraints(struct isl_basic_set *hull,
1548 struct isl_set *set, int *is_hull)
1551 int min_constraints;
1553 struct max_constraint *constraints = NULL;
1554 struct isl_hash_table *table = NULL;
1559 for (i = 0; i < set->n; ++i)
1560 if (set->p[i]->n_eq == 0)
1564 min_constraints = set->p[i]->n_ineq;
1566 for (i = best + 1; i < set->n; ++i) {
1567 if (set->p[i]->n_eq != 0)
1569 if (set->p[i]->n_ineq >= min_constraints)
1571 min_constraints = set->p[i]->n_ineq;
1574 constraints = isl_calloc_array(hull->ctx, struct max_constraint,
1578 table = isl_alloc_type(hull->ctx, struct isl_hash_table);
1579 if (isl_hash_table_init(hull->ctx, table, min_constraints))
1582 total = isl_dim_total(set->dim);
1583 for (i = 0; i < set->p[best]->n_ineq; ++i) {
1584 constraints[i].c = isl_mat_sub_alloc(hull->ctx,
1585 set->p[best]->ineq + i, 0, 1, 0, 1 + total);
1586 if (!constraints[i].c)
1588 constraints[i].ineq = 1;
1590 for (i = 0; i < min_constraints; ++i) {
1591 struct isl_hash_table_entry *entry;
1593 c_hash = isl_seq_get_hash(constraints[i].c->row[0] + 1, total);
1594 entry = isl_hash_table_find(hull->ctx, table, c_hash,
1595 max_constraint_equal, constraints[i].c->row[0] + 1, 1);
1598 isl_assert(hull->ctx, !entry->data, goto error);
1599 entry->data = &constraints[i];
1603 for (s = 0; s < set->n; ++s) {
1607 for (i = 0; i < set->p[s]->n_eq; ++i) {
1608 isl_int *eq = set->p[s]->eq[i];
1609 for (j = 0; j < 2; ++j) {
1610 isl_seq_neg(eq, eq, 1 + total);
1611 update_constraint(hull->ctx, table,
1615 for (i = 0; i < set->p[s]->n_ineq; ++i) {
1616 isl_int *ineq = set->p[s]->ineq[i];
1617 update_constraint(hull->ctx, table, ineq, total, n,
1618 set->p[s]->n_eq == 0);
1623 for (i = 0; i < min_constraints; ++i) {
1624 if (constraints[i].count < n)
1626 if (!constraints[i].ineq)
1628 j = isl_basic_set_alloc_inequality(hull);
1631 isl_seq_cpy(hull->ineq[j], constraints[i].c->row[0], 1 + total);
1634 for (s = 0; s < set->n; ++s) {
1635 if (set->p[s]->n_eq)
1637 if (set->p[s]->n_ineq != hull->n_ineq)
1639 for (i = 0; i < set->p[s]->n_ineq; ++i) {
1640 isl_int *ineq = set->p[s]->ineq[i];
1641 if (!has_constraint(hull->ctx, table, ineq, total, n))
1644 if (i == set->p[s]->n_ineq)
1648 isl_hash_table_clear(table);
1649 for (i = 0; i < min_constraints; ++i)
1650 isl_mat_free(constraints[i].c);
1655 isl_hash_table_clear(table);
1658 for (i = 0; i < min_constraints; ++i)
1659 isl_mat_free(constraints[i].c);
1664 /* Create a template for the convex hull of "set" and fill it up
1665 * obvious facet constraints, if any. If the result happens to
1666 * be the convex hull of "set" then *is_hull is set to 1.
1668 static struct isl_basic_set *proto_hull(struct isl_set *set, int *is_hull)
1670 struct isl_basic_set *hull;
1675 for (i = 0; i < set->n; ++i) {
1676 n_ineq += set->p[i]->n_eq;
1677 n_ineq += set->p[i]->n_ineq;
1679 hull = isl_basic_set_alloc_dim(isl_dim_copy(set->dim), 0, 0, n_ineq);
1680 hull = isl_basic_set_set_rational(hull);
1683 return common_constraints(hull, set, is_hull);
1686 static struct isl_basic_set *uset_convex_hull_wrap(struct isl_set *set)
1688 struct isl_basic_set *hull;
1691 hull = proto_hull(set, &is_hull);
1692 if (hull && !is_hull) {
1693 if (hull->n_ineq == 0)
1694 hull = initial_hull(hull, set);
1695 hull = extend(hull, set);
1702 /* Compute the convex hull of a set without any parameters or
1703 * integer divisions. Depending on whether the set is bounded,
1704 * we pass control to the wrapping based convex hull or
1705 * the Fourier-Motzkin elimination based convex hull.
1706 * We also handle a few special cases before checking the boundedness.
1708 static struct isl_basic_set *uset_convex_hull(struct isl_set *set)
1710 struct isl_basic_set *convex_hull = NULL;
1711 struct isl_basic_set *lin;
1713 if (isl_set_n_dim(set) == 0)
1714 return convex_hull_0d(set);
1716 set = isl_set_coalesce(set);
1717 set = isl_set_set_rational(set);
1724 convex_hull = isl_basic_set_copy(set->p[0]);
1728 if (isl_set_n_dim(set) == 1)
1729 return convex_hull_1d(set);
1731 if (isl_set_is_bounded(set))
1732 return uset_convex_hull_wrap(set);
1734 lin = uset_combined_lineality_space(isl_set_copy(set));
1737 if (isl_basic_set_is_universe(lin)) {
1741 if (lin->n_eq < isl_basic_set_total_dim(lin))
1742 return modulo_lineality(set, lin);
1743 isl_basic_set_free(lin);
1745 return uset_convex_hull_unbounded(set);
1748 isl_basic_set_free(convex_hull);
1752 /* This is the core procedure, where "set" is a "pure" set, i.e.,
1753 * without parameters or divs and where the convex hull of set is
1754 * known to be full-dimensional.
1756 static struct isl_basic_set *uset_convex_hull_wrap_bounded(struct isl_set *set)
1758 struct isl_basic_set *convex_hull = NULL;
1760 if (isl_set_n_dim(set) == 0) {
1761 convex_hull = isl_basic_set_universe(isl_dim_copy(set->dim));
1763 convex_hull = isl_basic_set_set_rational(convex_hull);
1767 set = isl_set_set_rational(set);
1771 set = isl_set_coalesce(set);
1775 convex_hull = isl_basic_set_copy(set->p[0]);
1779 if (isl_set_n_dim(set) == 1)
1780 return convex_hull_1d(set);
1782 return uset_convex_hull_wrap(set);
1788 /* Compute the convex hull of set "set" with affine hull "affine_hull",
1789 * We first remove the equalities (transforming the set), compute the
1790 * convex hull of the transformed set and then add the equalities back
1791 * (after performing the inverse transformation.
1793 static struct isl_basic_set *modulo_affine_hull(struct isl_ctx *ctx,
1794 struct isl_set *set, struct isl_basic_set *affine_hull)
1798 struct isl_basic_set *dummy;
1799 struct isl_basic_set *convex_hull;
1801 dummy = isl_basic_set_remove_equalities(
1802 isl_basic_set_copy(affine_hull), &T, &T2);
1805 isl_basic_set_free(dummy);
1806 set = isl_set_preimage(set, T);
1807 convex_hull = uset_convex_hull(set);
1808 convex_hull = isl_basic_set_preimage(convex_hull, T2);
1809 convex_hull = isl_basic_set_intersect(convex_hull, affine_hull);
1812 isl_basic_set_free(affine_hull);
1817 /* Compute the convex hull of a map.
1819 * The implementation was inspired by "Extended Convex Hull" by Fukuda et al.,
1820 * specifically, the wrapping of facets to obtain new facets.
1822 struct isl_basic_map *isl_map_convex_hull(struct isl_map *map)
1824 struct isl_basic_set *bset;
1825 struct isl_basic_map *model = NULL;
1826 struct isl_basic_set *affine_hull = NULL;
1827 struct isl_basic_map *convex_hull = NULL;
1828 struct isl_set *set = NULL;
1829 struct isl_ctx *ctx;
1836 convex_hull = isl_basic_map_empty_like_map(map);
1841 map = isl_map_detect_equalities(map);
1842 map = isl_map_align_divs(map);
1843 model = isl_basic_map_copy(map->p[0]);
1844 set = isl_map_underlying_set(map);
1848 affine_hull = isl_set_affine_hull(isl_set_copy(set));
1851 if (affine_hull->n_eq != 0)
1852 bset = modulo_affine_hull(ctx, set, affine_hull);
1854 isl_basic_set_free(affine_hull);
1855 bset = uset_convex_hull(set);
1858 convex_hull = isl_basic_map_overlying_set(bset, model);
1860 ISL_F_SET(convex_hull, ISL_BASIC_MAP_NO_IMPLICIT);
1861 ISL_F_SET(convex_hull, ISL_BASIC_MAP_ALL_EQUALITIES);
1862 ISL_F_CLR(convex_hull, ISL_BASIC_MAP_RATIONAL);
1866 isl_basic_map_free(model);
1870 struct isl_basic_set *isl_set_convex_hull(struct isl_set *set)
1872 return (struct isl_basic_set *)
1873 isl_map_convex_hull((struct isl_map *)set);
1876 struct sh_data_entry {
1877 struct isl_hash_table *table;
1878 struct isl_tab *tab;
1881 /* Holds the data needed during the simple hull computation.
1883 * n the number of basic sets in the original set
1884 * hull_table a hash table of already computed constraints
1885 * in the simple hull
1886 * p for each basic set,
1887 * table a hash table of the constraints
1888 * tab the tableau corresponding to the basic set
1891 struct isl_ctx *ctx;
1893 struct isl_hash_table *hull_table;
1894 struct sh_data_entry p[1];
1897 static void sh_data_free(struct sh_data *data)
1903 isl_hash_table_free(data->ctx, data->hull_table);
1904 for (i = 0; i < data->n; ++i) {
1905 isl_hash_table_free(data->ctx, data->p[i].table);
1906 isl_tab_free(data->p[i].tab);
1911 struct ineq_cmp_data {
1916 static int has_ineq(const void *entry, const void *val)
1918 isl_int *row = (isl_int *)entry;
1919 struct ineq_cmp_data *v = (struct ineq_cmp_data *)val;
1921 return isl_seq_eq(row + 1, v->p + 1, v->len) ||
1922 isl_seq_is_neg(row + 1, v->p + 1, v->len);
1925 static int hash_ineq(struct isl_ctx *ctx, struct isl_hash_table *table,
1926 isl_int *ineq, unsigned len)
1929 struct ineq_cmp_data v;
1930 struct isl_hash_table_entry *entry;
1934 c_hash = isl_seq_get_hash(ineq + 1, len);
1935 entry = isl_hash_table_find(ctx, table, c_hash, has_ineq, &v, 1);
1942 /* Fill hash table "table" with the constraints of "bset".
1943 * Equalities are added as two inequalities.
1944 * The value in the hash table is a pointer to the (in)equality of "bset".
1946 static int hash_basic_set(struct isl_hash_table *table,
1947 struct isl_basic_set *bset)
1950 unsigned dim = isl_basic_set_total_dim(bset);
1952 for (i = 0; i < bset->n_eq; ++i) {
1953 for (j = 0; j < 2; ++j) {
1954 isl_seq_neg(bset->eq[i], bset->eq[i], 1 + dim);
1955 if (hash_ineq(bset->ctx, table, bset->eq[i], dim) < 0)
1959 for (i = 0; i < bset->n_ineq; ++i) {
1960 if (hash_ineq(bset->ctx, table, bset->ineq[i], dim) < 0)
1966 static struct sh_data *sh_data_alloc(struct isl_set *set, unsigned n_ineq)
1968 struct sh_data *data;
1971 data = isl_calloc(set->ctx, struct sh_data,
1972 sizeof(struct sh_data) +
1973 (set->n - 1) * sizeof(struct sh_data_entry));
1976 data->ctx = set->ctx;
1978 data->hull_table = isl_hash_table_alloc(set->ctx, n_ineq);
1979 if (!data->hull_table)
1981 for (i = 0; i < set->n; ++i) {
1982 data->p[i].table = isl_hash_table_alloc(set->ctx,
1983 2 * set->p[i]->n_eq + set->p[i]->n_ineq);
1984 if (!data->p[i].table)
1986 if (hash_basic_set(data->p[i].table, set->p[i]) < 0)
1995 /* Check if inequality "ineq" is a bound for basic set "j" or if
1996 * it can be relaxed (by increasing the constant term) to become
1997 * a bound for that basic set. In the latter case, the constant
1999 * Return 1 if "ineq" is a bound
2000 * 0 if "ineq" may attain arbitrarily small values on basic set "j"
2001 * -1 if some error occurred
2003 static int is_bound(struct sh_data *data, struct isl_set *set, int j,
2006 enum isl_lp_result res;
2009 if (!data->p[j].tab) {
2010 data->p[j].tab = isl_tab_from_basic_set(set->p[j]);
2011 if (!data->p[j].tab)
2017 res = isl_tab_min(data->p[j].tab, ineq, data->ctx->one,
2019 if (res == isl_lp_ok && isl_int_is_neg(opt))
2020 isl_int_sub(ineq[0], ineq[0], opt);
2024 return res == isl_lp_ok ? 1 :
2025 res == isl_lp_unbounded ? 0 : -1;
2028 /* Check if inequality "ineq" from basic set "i" can be relaxed to
2029 * become a bound on the whole set. If so, add the (relaxed) inequality
2032 * We first check if "hull" already contains a translate of the inequality.
2033 * If so, we are done.
2034 * Then, we check if any of the previous basic sets contains a translate
2035 * of the inequality. If so, then we have already considered this
2036 * inequality and we are done.
2037 * Otherwise, for each basic set other than "i", we check if the inequality
2038 * is a bound on the basic set.
2039 * For previous basic sets, we know that they do not contain a translate
2040 * of the inequality, so we directly call is_bound.
2041 * For following basic sets, we first check if a translate of the
2042 * inequality appears in its description and if so directly update
2043 * the inequality accordingly.
2045 static struct isl_basic_set *add_bound(struct isl_basic_set *hull,
2046 struct sh_data *data, struct isl_set *set, int i, isl_int *ineq)
2049 struct ineq_cmp_data v;
2050 struct isl_hash_table_entry *entry;
2056 v.len = isl_basic_set_total_dim(hull);
2058 c_hash = isl_seq_get_hash(ineq + 1, v.len);
2060 entry = isl_hash_table_find(hull->ctx, data->hull_table, c_hash,
2065 for (j = 0; j < i; ++j) {
2066 entry = isl_hash_table_find(hull->ctx, data->p[j].table,
2067 c_hash, has_ineq, &v, 0);
2074 k = isl_basic_set_alloc_inequality(hull);
2075 isl_seq_cpy(hull->ineq[k], ineq, 1 + v.len);
2079 for (j = 0; j < i; ++j) {
2081 bound = is_bound(data, set, j, hull->ineq[k]);
2088 isl_basic_set_free_inequality(hull, 1);
2092 for (j = i + 1; j < set->n; ++j) {
2095 entry = isl_hash_table_find(hull->ctx, data->p[j].table,
2096 c_hash, has_ineq, &v, 0);
2098 ineq_j = entry->data;
2099 neg = isl_seq_is_neg(ineq_j + 1,
2100 hull->ineq[k] + 1, v.len);
2102 isl_int_neg(ineq_j[0], ineq_j[0]);
2103 if (isl_int_gt(ineq_j[0], hull->ineq[k][0]))
2104 isl_int_set(hull->ineq[k][0], ineq_j[0]);
2106 isl_int_neg(ineq_j[0], ineq_j[0]);
2109 bound = is_bound(data, set, j, hull->ineq[k]);
2116 isl_basic_set_free_inequality(hull, 1);
2120 entry = isl_hash_table_find(hull->ctx, data->hull_table, c_hash,
2124 entry->data = hull->ineq[k];
2128 isl_basic_set_free(hull);
2132 /* Check if any inequality from basic set "i" can be relaxed to
2133 * become a bound on the whole set. If so, add the (relaxed) inequality
2136 static struct isl_basic_set *add_bounds(struct isl_basic_set *bset,
2137 struct sh_data *data, struct isl_set *set, int i)
2140 unsigned dim = isl_basic_set_total_dim(bset);
2142 for (j = 0; j < set->p[i]->n_eq; ++j) {
2143 for (k = 0; k < 2; ++k) {
2144 isl_seq_neg(set->p[i]->eq[j], set->p[i]->eq[j], 1+dim);
2145 add_bound(bset, data, set, i, set->p[i]->eq[j]);
2148 for (j = 0; j < set->p[i]->n_ineq; ++j)
2149 add_bound(bset, data, set, i, set->p[i]->ineq[j]);
2153 /* Compute a superset of the convex hull of set that is described
2154 * by only translates of the constraints in the constituents of set.
2156 static struct isl_basic_set *uset_simple_hull(struct isl_set *set)
2158 struct sh_data *data = NULL;
2159 struct isl_basic_set *hull = NULL;
2167 for (i = 0; i < set->n; ++i) {
2170 n_ineq += 2 * set->p[i]->n_eq + set->p[i]->n_ineq;
2173 hull = isl_basic_set_alloc_dim(isl_dim_copy(set->dim), 0, 0, n_ineq);
2177 data = sh_data_alloc(set, n_ineq);
2181 for (i = 0; i < set->n; ++i)
2182 hull = add_bounds(hull, data, set, i);
2190 isl_basic_set_free(hull);
2195 /* Compute a superset of the convex hull of map that is described
2196 * by only translates of the constraints in the constituents of map.
2198 struct isl_basic_map *isl_map_simple_hull(struct isl_map *map)
2200 struct isl_set *set = NULL;
2201 struct isl_basic_map *model = NULL;
2202 struct isl_basic_map *hull;
2203 struct isl_basic_map *affine_hull;
2204 struct isl_basic_set *bset = NULL;
2209 hull = isl_basic_map_empty_like_map(map);
2214 hull = isl_basic_map_copy(map->p[0]);
2219 map = isl_map_detect_equalities(map);
2220 affine_hull = isl_map_affine_hull(isl_map_copy(map));
2221 map = isl_map_align_divs(map);
2222 model = isl_basic_map_copy(map->p[0]);
2224 set = isl_map_underlying_set(map);
2226 bset = uset_simple_hull(set);
2228 hull = isl_basic_map_overlying_set(bset, model);
2230 hull = isl_basic_map_intersect(hull, affine_hull);
2231 hull = isl_basic_map_convex_hull(hull);
2232 ISL_F_SET(hull, ISL_BASIC_MAP_NO_IMPLICIT);
2233 ISL_F_SET(hull, ISL_BASIC_MAP_ALL_EQUALITIES);
2238 struct isl_basic_set *isl_set_simple_hull(struct isl_set *set)
2240 return (struct isl_basic_set *)
2241 isl_map_simple_hull((struct isl_map *)set);
2244 /* Given a set "set", return parametric bounds on the dimension "dim".
2246 static struct isl_basic_set *set_bounds(struct isl_set *set, int dim)
2248 unsigned set_dim = isl_set_dim(set, isl_dim_set);
2249 set = isl_set_copy(set);
2250 set = isl_set_eliminate_dims(set, dim + 1, set_dim - (dim + 1));
2251 set = isl_set_eliminate_dims(set, 0, dim);
2252 return isl_set_convex_hull(set);
2255 /* Computes a "simple hull" and then check if each dimension in the
2256 * resulting hull is bounded by a symbolic constant. If not, the
2257 * hull is intersected with the corresponding bounds on the whole set.
2259 struct isl_basic_set *isl_set_bounded_simple_hull(struct isl_set *set)
2262 struct isl_basic_set *hull;
2263 unsigned nparam, left;
2264 int removed_divs = 0;
2266 hull = isl_set_simple_hull(isl_set_copy(set));
2270 nparam = isl_basic_set_dim(hull, isl_dim_param);
2271 for (i = 0; i < isl_basic_set_dim(hull, isl_dim_set); ++i) {
2272 int lower = 0, upper = 0;
2273 struct isl_basic_set *bounds;
2275 left = isl_basic_set_total_dim(hull) - nparam - i - 1;
2276 for (j = 0; j < hull->n_eq; ++j) {
2277 if (isl_int_is_zero(hull->eq[j][1 + nparam + i]))
2279 if (isl_seq_first_non_zero(hull->eq[j]+1+nparam+i+1,
2286 for (j = 0; j < hull->n_ineq; ++j) {
2287 if (isl_int_is_zero(hull->ineq[j][1 + nparam + i]))
2289 if (isl_seq_first_non_zero(hull->ineq[j]+1+nparam+i+1,
2291 isl_seq_first_non_zero(hull->ineq[j]+1+nparam,
2294 if (isl_int_is_pos(hull->ineq[j][1 + nparam + i]))
2305 if (!removed_divs) {
2306 set = isl_set_remove_divs(set);
2311 bounds = set_bounds(set, i);
2312 hull = isl_basic_set_intersect(hull, bounds);