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
10 #include <isl_ctx_private.h>
11 #include <isl_map_private.h>
14 #include <isl_mat_private.h>
17 #include "isl_equalities.h"
20 static struct isl_basic_set *uset_convex_hull_wrap_bounded(struct isl_set *set);
22 static void swap_ineq(struct isl_basic_map *bmap, unsigned i, unsigned j)
28 bmap->ineq[i] = bmap->ineq[j];
33 /* Return 1 if constraint c is redundant with respect to the constraints
34 * in bmap. If c is a lower [upper] bound in some variable and bmap
35 * does not have a lower [upper] bound in that variable, then c cannot
36 * be redundant and we do not need solve any lp.
38 int isl_basic_map_constraint_is_redundant(struct isl_basic_map **bmap,
39 isl_int *c, isl_int *opt_n, isl_int *opt_d)
41 enum isl_lp_result res;
48 total = isl_basic_map_total_dim(*bmap);
49 for (i = 0; i < total; ++i) {
51 if (isl_int_is_zero(c[1+i]))
53 sign = isl_int_sgn(c[1+i]);
54 for (j = 0; j < (*bmap)->n_ineq; ++j)
55 if (sign == isl_int_sgn((*bmap)->ineq[j][1+i]))
57 if (j == (*bmap)->n_ineq)
63 res = isl_basic_map_solve_lp(*bmap, 0, c, (*bmap)->ctx->one,
65 if (res == isl_lp_unbounded)
67 if (res == isl_lp_error)
69 if (res == isl_lp_empty) {
70 *bmap = isl_basic_map_set_to_empty(*bmap);
73 return !isl_int_is_neg(*opt_n);
76 int isl_basic_set_constraint_is_redundant(struct isl_basic_set **bset,
77 isl_int *c, isl_int *opt_n, isl_int *opt_d)
79 return isl_basic_map_constraint_is_redundant(
80 (struct isl_basic_map **)bset, c, opt_n, opt_d);
84 * constraints. If the minimal value along the normal of a constraint
85 * is the same if the constraint is removed, then the constraint is redundant.
87 * Alternatively, we could have intersected the basic map with the
88 * corresponding equality and the checked if the dimension was that
91 __isl_give isl_basic_map *isl_basic_map_remove_redundancies(
92 __isl_take isl_basic_map *bmap)
99 bmap = isl_basic_map_gauss(bmap, NULL);
100 if (ISL_F_ISSET(bmap, ISL_BASIC_MAP_EMPTY))
102 if (ISL_F_ISSET(bmap, ISL_BASIC_MAP_NO_REDUNDANT))
104 if (bmap->n_ineq <= 1)
107 tab = isl_tab_from_basic_map(bmap);
108 if (isl_tab_detect_implicit_equalities(tab) < 0)
110 if (isl_tab_detect_redundant(tab) < 0)
112 bmap = isl_basic_map_update_from_tab(bmap, tab);
114 ISL_F_SET(bmap, ISL_BASIC_MAP_NO_IMPLICIT);
115 ISL_F_SET(bmap, ISL_BASIC_MAP_NO_REDUNDANT);
119 isl_basic_map_free(bmap);
123 __isl_give isl_basic_set *isl_basic_set_remove_redundancies(
124 __isl_take isl_basic_set *bset)
126 return (struct isl_basic_set *)
127 isl_basic_map_remove_redundancies((struct isl_basic_map *)bset);
130 /* Check if the set set is bound in the direction of the affine
131 * constraint c and if so, set the constant term such that the
132 * resulting constraint is a bounding constraint for the set.
134 static int uset_is_bound(struct isl_set *set, isl_int *c, unsigned len)
142 isl_int_init(opt_denom);
144 for (j = 0; j < set->n; ++j) {
145 enum isl_lp_result res;
147 if (ISL_F_ISSET(set->p[j], ISL_BASIC_SET_EMPTY))
150 res = isl_basic_set_solve_lp(set->p[j],
151 0, c, set->ctx->one, &opt, &opt_denom, NULL);
152 if (res == isl_lp_unbounded)
154 if (res == isl_lp_error)
156 if (res == isl_lp_empty) {
157 set->p[j] = isl_basic_set_set_to_empty(set->p[j]);
162 if (first || isl_int_is_neg(opt)) {
163 if (!isl_int_is_one(opt_denom))
164 isl_seq_scale(c, c, opt_denom, len);
165 isl_int_sub(c[0], c[0], opt);
170 isl_int_clear(opt_denom);
174 isl_int_clear(opt_denom);
178 __isl_give isl_basic_map *isl_basic_map_set_rational(
179 __isl_take isl_basic_set *bmap)
184 if (ISL_F_ISSET(bmap, ISL_BASIC_MAP_RATIONAL))
187 bmap = isl_basic_map_cow(bmap);
191 ISL_F_SET(bmap, ISL_BASIC_MAP_RATIONAL);
193 return isl_basic_map_finalize(bmap);
196 __isl_give isl_basic_set *isl_basic_set_set_rational(
197 __isl_take isl_basic_set *bset)
199 return isl_basic_map_set_rational(bset);
202 static struct isl_set *isl_set_set_rational(struct isl_set *set)
206 set = isl_set_cow(set);
209 for (i = 0; i < set->n; ++i) {
210 set->p[i] = isl_basic_set_set_rational(set->p[i]);
220 static struct isl_basic_set *isl_basic_set_add_equality(
221 struct isl_basic_set *bset, isl_int *c)
229 if (ISL_F_ISSET(bset, ISL_BASIC_SET_EMPTY))
232 isl_assert(bset->ctx, isl_basic_set_n_param(bset) == 0, goto error);
233 isl_assert(bset->ctx, bset->n_div == 0, goto error);
234 dim = isl_basic_set_n_dim(bset);
235 bset = isl_basic_set_cow(bset);
236 bset = isl_basic_set_extend(bset, 0, dim, 0, 1, 0);
237 i = isl_basic_set_alloc_equality(bset);
240 isl_seq_cpy(bset->eq[i], c, 1 + dim);
243 isl_basic_set_free(bset);
247 static struct isl_set *isl_set_add_basic_set_equality(struct isl_set *set, isl_int *c)
251 set = isl_set_cow(set);
254 for (i = 0; i < set->n; ++i) {
255 set->p[i] = isl_basic_set_add_equality(set->p[i], c);
265 /* Given a union of basic sets, construct the constraints for wrapping
266 * a facet around one of its ridges.
267 * In particular, if each of n the d-dimensional basic sets i in "set"
268 * contains the origin, satisfies the constraints x_1 >= 0 and x_2 >= 0
269 * and is defined by the constraints
273 * then the resulting set is of dimension n*(1+d) and has as constraints
282 static struct isl_basic_set *wrap_constraints(struct isl_set *set)
284 struct isl_basic_set *lp;
288 unsigned dim, lp_dim;
293 dim = 1 + isl_set_n_dim(set);
296 for (i = 0; i < set->n; ++i) {
297 n_eq += set->p[i]->n_eq;
298 n_ineq += set->p[i]->n_ineq;
300 lp = isl_basic_set_alloc(set->ctx, 0, dim * set->n, 0, n_eq, n_ineq);
301 lp = isl_basic_set_set_rational(lp);
304 lp_dim = isl_basic_set_n_dim(lp);
305 k = isl_basic_set_alloc_equality(lp);
306 isl_int_set_si(lp->eq[k][0], -1);
307 for (i = 0; i < set->n; ++i) {
308 isl_int_set_si(lp->eq[k][1+dim*i], 0);
309 isl_int_set_si(lp->eq[k][1+dim*i+1], 1);
310 isl_seq_clr(lp->eq[k]+1+dim*i+2, dim-2);
312 for (i = 0; i < set->n; ++i) {
313 k = isl_basic_set_alloc_inequality(lp);
314 isl_seq_clr(lp->ineq[k], 1+lp_dim);
315 isl_int_set_si(lp->ineq[k][1+dim*i], 1);
317 for (j = 0; j < set->p[i]->n_eq; ++j) {
318 k = isl_basic_set_alloc_equality(lp);
319 isl_seq_clr(lp->eq[k], 1+dim*i);
320 isl_seq_cpy(lp->eq[k]+1+dim*i, set->p[i]->eq[j], dim);
321 isl_seq_clr(lp->eq[k]+1+dim*(i+1), dim*(set->n-i-1));
324 for (j = 0; j < set->p[i]->n_ineq; ++j) {
325 k = isl_basic_set_alloc_inequality(lp);
326 isl_seq_clr(lp->ineq[k], 1+dim*i);
327 isl_seq_cpy(lp->ineq[k]+1+dim*i, set->p[i]->ineq[j], dim);
328 isl_seq_clr(lp->ineq[k]+1+dim*(i+1), dim*(set->n-i-1));
334 /* Given a facet "facet" of the convex hull of "set" and a facet "ridge"
335 * of that facet, compute the other facet of the convex hull that contains
338 * We first transform the set such that the facet constraint becomes
342 * I.e., the facet lies in
346 * and on that facet, the constraint that defines the ridge is
350 * (This transformation is not strictly needed, all that is needed is
351 * that the ridge contains the origin.)
353 * Since the ridge contains the origin, the cone of the convex hull
354 * will be of the form
359 * with this second constraint defining the new facet.
360 * The constant a is obtained by settting x_1 in the cone of the
361 * convex hull to 1 and minimizing x_2.
362 * Now, each element in the cone of the convex hull is the sum
363 * of elements in the cones of the basic sets.
364 * If a_i is the dilation factor of basic set i, then the problem
365 * we need to solve is
378 * the constraints of each (transformed) basic set.
379 * If a = n/d, then the constraint defining the new facet (in the transformed
382 * -n x_1 + d x_2 >= 0
384 * In the original space, we need to take the same combination of the
385 * corresponding constraints "facet" and "ridge".
387 * If a = -infty = "-1/0", then we just return the original facet constraint.
388 * This means that the facet is unbounded, but has a bounded intersection
389 * with the union of sets.
391 isl_int *isl_set_wrap_facet(__isl_keep isl_set *set,
392 isl_int *facet, isl_int *ridge)
396 struct isl_mat *T = NULL;
397 struct isl_basic_set *lp = NULL;
399 enum isl_lp_result res;
406 set = isl_set_copy(set);
407 set = isl_set_set_rational(set);
409 dim = 1 + isl_set_n_dim(set);
410 T = isl_mat_alloc(ctx, 3, dim);
413 isl_int_set_si(T->row[0][0], 1);
414 isl_seq_clr(T->row[0]+1, dim - 1);
415 isl_seq_cpy(T->row[1], facet, dim);
416 isl_seq_cpy(T->row[2], ridge, dim);
417 T = isl_mat_right_inverse(T);
418 set = isl_set_preimage(set, T);
422 lp = wrap_constraints(set);
423 obj = isl_vec_alloc(ctx, 1 + dim*set->n);
426 isl_int_set_si(obj->block.data[0], 0);
427 for (i = 0; i < set->n; ++i) {
428 isl_seq_clr(obj->block.data + 1 + dim*i, 2);
429 isl_int_set_si(obj->block.data[1 + dim*i+2], 1);
430 isl_seq_clr(obj->block.data + 1 + dim*i+3, dim-3);
434 res = isl_basic_set_solve_lp(lp, 0,
435 obj->block.data, ctx->one, &num, &den, NULL);
436 if (res == isl_lp_ok) {
437 isl_int_neg(num, num);
438 isl_seq_combine(facet, num, facet, den, ridge, dim);
439 isl_seq_normalize(ctx, facet, dim);
444 isl_basic_set_free(lp);
446 if (res == isl_lp_error)
448 isl_assert(ctx, res == isl_lp_ok || res == isl_lp_unbounded,
452 isl_basic_set_free(lp);
458 /* Compute the constraint of a facet of "set".
460 * We first compute the intersection with a bounding constraint
461 * that is orthogonal to one of the coordinate axes.
462 * If the affine hull of this intersection has only one equality,
463 * we have found a facet.
464 * Otherwise, we wrap the current bounding constraint around
465 * one of the equalities of the face (one that is not equal to
466 * the current bounding constraint).
467 * This process continues until we have found a facet.
468 * The dimension of the intersection increases by at least
469 * one on each iteration, so termination is guaranteed.
471 static __isl_give isl_mat *initial_facet_constraint(__isl_keep isl_set *set)
473 struct isl_set *slice = NULL;
474 struct isl_basic_set *face = NULL;
476 unsigned dim = isl_set_n_dim(set);
480 isl_assert(set->ctx, set->n > 0, goto error);
481 bounds = isl_mat_alloc(set->ctx, 1, 1 + dim);
485 isl_seq_clr(bounds->row[0], dim);
486 isl_int_set_si(bounds->row[0][1 + dim - 1], 1);
487 is_bound = uset_is_bound(set, bounds->row[0], 1 + dim);
490 isl_assert(set->ctx, is_bound, goto error);
491 isl_seq_normalize(set->ctx, bounds->row[0], 1 + dim);
495 slice = isl_set_copy(set);
496 slice = isl_set_add_basic_set_equality(slice, bounds->row[0]);
497 face = isl_set_affine_hull(slice);
500 if (face->n_eq == 1) {
501 isl_basic_set_free(face);
504 for (i = 0; i < face->n_eq; ++i)
505 if (!isl_seq_eq(bounds->row[0], face->eq[i], 1 + dim) &&
506 !isl_seq_is_neg(bounds->row[0],
507 face->eq[i], 1 + dim))
509 isl_assert(set->ctx, i < face->n_eq, goto error);
510 if (!isl_set_wrap_facet(set, bounds->row[0], face->eq[i]))
512 isl_seq_normalize(set->ctx, bounds->row[0], bounds->n_col);
513 isl_basic_set_free(face);
518 isl_basic_set_free(face);
519 isl_mat_free(bounds);
523 /* Given the bounding constraint "c" of a facet of the convex hull of "set",
524 * compute a hyperplane description of the facet, i.e., compute the facets
527 * We compute an affine transformation that transforms the constraint
536 * by computing the right inverse U of a matrix that starts with the rows
549 * Since z_1 is zero, we can drop this variable as well as the corresponding
550 * column of U to obtain
558 * with Q' equal to Q, but without the corresponding row.
559 * After computing the facets of the facet in the z' space,
560 * we convert them back to the x space through Q.
562 static struct isl_basic_set *compute_facet(struct isl_set *set, isl_int *c)
564 struct isl_mat *m, *U, *Q;
565 struct isl_basic_set *facet = NULL;
570 set = isl_set_copy(set);
571 dim = isl_set_n_dim(set);
572 m = isl_mat_alloc(set->ctx, 2, 1 + dim);
575 isl_int_set_si(m->row[0][0], 1);
576 isl_seq_clr(m->row[0]+1, dim);
577 isl_seq_cpy(m->row[1], c, 1+dim);
578 U = isl_mat_right_inverse(m);
579 Q = isl_mat_right_inverse(isl_mat_copy(U));
580 U = isl_mat_drop_cols(U, 1, 1);
581 Q = isl_mat_drop_rows(Q, 1, 1);
582 set = isl_set_preimage(set, U);
583 facet = uset_convex_hull_wrap_bounded(set);
584 facet = isl_basic_set_preimage(facet, Q);
586 isl_assert(ctx, facet->n_eq == 0, goto error);
589 isl_basic_set_free(facet);
594 /* Given an initial facet constraint, compute the remaining facets.
595 * We do this by running through all facets found so far and computing
596 * the adjacent facets through wrapping, adding those facets that we
597 * hadn't already found before.
599 * For each facet we have found so far, we first compute its facets
600 * in the resulting convex hull. That is, we compute the ridges
601 * of the resulting convex hull contained in the facet.
602 * We also compute the corresponding facet in the current approximation
603 * of the convex hull. There is no need to wrap around the ridges
604 * in this facet since that would result in a facet that is already
605 * present in the current approximation.
607 * This function can still be significantly optimized by checking which of
608 * the facets of the basic sets are also facets of the convex hull and
609 * using all the facets so far to help in constructing the facets of the
612 * using the technique in section "3.1 Ridge Generation" of
613 * "Extended Convex Hull" by Fukuda et al.
615 static struct isl_basic_set *extend(struct isl_basic_set *hull,
620 struct isl_basic_set *facet = NULL;
621 struct isl_basic_set *hull_facet = NULL;
627 isl_assert(set->ctx, set->n > 0, goto error);
629 dim = isl_set_n_dim(set);
631 for (i = 0; i < hull->n_ineq; ++i) {
632 facet = compute_facet(set, hull->ineq[i]);
633 facet = isl_basic_set_add_equality(facet, hull->ineq[i]);
634 facet = isl_basic_set_gauss(facet, NULL);
635 facet = isl_basic_set_normalize_constraints(facet);
636 hull_facet = isl_basic_set_copy(hull);
637 hull_facet = isl_basic_set_add_equality(hull_facet, hull->ineq[i]);
638 hull_facet = isl_basic_set_gauss(hull_facet, NULL);
639 hull_facet = isl_basic_set_normalize_constraints(hull_facet);
640 if (!facet || !hull_facet)
642 hull = isl_basic_set_cow(hull);
643 hull = isl_basic_set_extend_dim(hull,
644 isl_dim_copy(hull->dim), 0, 0, facet->n_ineq);
647 for (j = 0; j < facet->n_ineq; ++j) {
648 for (f = 0; f < hull_facet->n_ineq; ++f)
649 if (isl_seq_eq(facet->ineq[j],
650 hull_facet->ineq[f], 1 + dim))
652 if (f < hull_facet->n_ineq)
654 k = isl_basic_set_alloc_inequality(hull);
657 isl_seq_cpy(hull->ineq[k], hull->ineq[i], 1+dim);
658 if (!isl_set_wrap_facet(set, hull->ineq[k], facet->ineq[j]))
661 isl_basic_set_free(hull_facet);
662 isl_basic_set_free(facet);
664 hull = isl_basic_set_simplify(hull);
665 hull = isl_basic_set_finalize(hull);
668 isl_basic_set_free(hull_facet);
669 isl_basic_set_free(facet);
670 isl_basic_set_free(hull);
674 /* Special case for computing the convex hull of a one dimensional set.
675 * We simply collect the lower and upper bounds of each basic set
676 * and the biggest of those.
678 static struct isl_basic_set *convex_hull_1d(struct isl_set *set)
680 struct isl_mat *c = NULL;
681 isl_int *lower = NULL;
682 isl_int *upper = NULL;
685 struct isl_basic_set *hull;
687 for (i = 0; i < set->n; ++i) {
688 set->p[i] = isl_basic_set_simplify(set->p[i]);
692 set = isl_set_remove_empty_parts(set);
695 isl_assert(set->ctx, set->n > 0, goto error);
696 c = isl_mat_alloc(set->ctx, 2, 2);
700 if (set->p[0]->n_eq > 0) {
701 isl_assert(set->ctx, set->p[0]->n_eq == 1, goto error);
704 if (isl_int_is_pos(set->p[0]->eq[0][1])) {
705 isl_seq_cpy(lower, set->p[0]->eq[0], 2);
706 isl_seq_neg(upper, set->p[0]->eq[0], 2);
708 isl_seq_neg(lower, set->p[0]->eq[0], 2);
709 isl_seq_cpy(upper, set->p[0]->eq[0], 2);
712 for (j = 0; j < set->p[0]->n_ineq; ++j) {
713 if (isl_int_is_pos(set->p[0]->ineq[j][1])) {
715 isl_seq_cpy(lower, set->p[0]->ineq[j], 2);
718 isl_seq_cpy(upper, set->p[0]->ineq[j], 2);
725 for (i = 0; i < set->n; ++i) {
726 struct isl_basic_set *bset = set->p[i];
730 for (j = 0; j < bset->n_eq; ++j) {
734 isl_int_mul(a, lower[0], bset->eq[j][1]);
735 isl_int_mul(b, lower[1], bset->eq[j][0]);
736 if (isl_int_lt(a, b) && isl_int_is_pos(bset->eq[j][1]))
737 isl_seq_cpy(lower, bset->eq[j], 2);
738 if (isl_int_gt(a, b) && isl_int_is_neg(bset->eq[j][1]))
739 isl_seq_neg(lower, bset->eq[j], 2);
742 isl_int_mul(a, upper[0], bset->eq[j][1]);
743 isl_int_mul(b, upper[1], bset->eq[j][0]);
744 if (isl_int_lt(a, b) && isl_int_is_pos(bset->eq[j][1]))
745 isl_seq_neg(upper, bset->eq[j], 2);
746 if (isl_int_gt(a, b) && isl_int_is_neg(bset->eq[j][1]))
747 isl_seq_cpy(upper, bset->eq[j], 2);
750 for (j = 0; j < bset->n_ineq; ++j) {
751 if (isl_int_is_pos(bset->ineq[j][1]))
753 if (isl_int_is_neg(bset->ineq[j][1]))
755 if (lower && isl_int_is_pos(bset->ineq[j][1])) {
756 isl_int_mul(a, lower[0], bset->ineq[j][1]);
757 isl_int_mul(b, lower[1], bset->ineq[j][0]);
758 if (isl_int_lt(a, b))
759 isl_seq_cpy(lower, bset->ineq[j], 2);
761 if (upper && isl_int_is_neg(bset->ineq[j][1])) {
762 isl_int_mul(a, upper[0], bset->ineq[j][1]);
763 isl_int_mul(b, upper[1], bset->ineq[j][0]);
764 if (isl_int_gt(a, b))
765 isl_seq_cpy(upper, bset->ineq[j], 2);
776 hull = isl_basic_set_alloc(set->ctx, 0, 1, 0, 0, 2);
777 hull = isl_basic_set_set_rational(hull);
781 k = isl_basic_set_alloc_inequality(hull);
782 isl_seq_cpy(hull->ineq[k], lower, 2);
785 k = isl_basic_set_alloc_inequality(hull);
786 isl_seq_cpy(hull->ineq[k], upper, 2);
788 hull = isl_basic_set_finalize(hull);
798 /* Project out final n dimensions using Fourier-Motzkin */
799 static struct isl_set *set_project_out(struct isl_ctx *ctx,
800 struct isl_set *set, unsigned n)
802 return isl_set_remove_dims(set, isl_dim_set, isl_set_n_dim(set) - n, n);
805 static struct isl_basic_set *convex_hull_0d(struct isl_set *set)
807 struct isl_basic_set *convex_hull;
812 if (isl_set_is_empty(set))
813 convex_hull = isl_basic_set_empty(isl_dim_copy(set->dim));
815 convex_hull = isl_basic_set_universe(isl_dim_copy(set->dim));
820 /* Compute the convex hull of a pair of basic sets without any parameters or
821 * integer divisions using Fourier-Motzkin elimination.
822 * The convex hull is the set of all points that can be written as
823 * the sum of points from both basic sets (in homogeneous coordinates).
824 * We set up the constraints in a space with dimensions for each of
825 * the three sets and then project out the dimensions corresponding
826 * to the two original basic sets, retaining only those corresponding
827 * to the convex hull.
829 static struct isl_basic_set *convex_hull_pair_elim(struct isl_basic_set *bset1,
830 struct isl_basic_set *bset2)
833 struct isl_basic_set *bset[2];
834 struct isl_basic_set *hull = NULL;
837 if (!bset1 || !bset2)
840 dim = isl_basic_set_n_dim(bset1);
841 hull = isl_basic_set_alloc(bset1->ctx, 0, 2 + 3 * dim, 0,
842 1 + dim + bset1->n_eq + bset2->n_eq,
843 2 + bset1->n_ineq + bset2->n_ineq);
846 for (i = 0; i < 2; ++i) {
847 for (j = 0; j < bset[i]->n_eq; ++j) {
848 k = isl_basic_set_alloc_equality(hull);
851 isl_seq_clr(hull->eq[k], (i+1) * (1+dim));
852 isl_seq_clr(hull->eq[k]+(i+2)*(1+dim), (1-i)*(1+dim));
853 isl_seq_cpy(hull->eq[k]+(i+1)*(1+dim), bset[i]->eq[j],
856 for (j = 0; j < bset[i]->n_ineq; ++j) {
857 k = isl_basic_set_alloc_inequality(hull);
860 isl_seq_clr(hull->ineq[k], (i+1) * (1+dim));
861 isl_seq_clr(hull->ineq[k]+(i+2)*(1+dim), (1-i)*(1+dim));
862 isl_seq_cpy(hull->ineq[k]+(i+1)*(1+dim),
863 bset[i]->ineq[j], 1+dim);
865 k = isl_basic_set_alloc_inequality(hull);
868 isl_seq_clr(hull->ineq[k], 1+2+3*dim);
869 isl_int_set_si(hull->ineq[k][(i+1)*(1+dim)], 1);
871 for (j = 0; j < 1+dim; ++j) {
872 k = isl_basic_set_alloc_equality(hull);
875 isl_seq_clr(hull->eq[k], 1+2+3*dim);
876 isl_int_set_si(hull->eq[k][j], -1);
877 isl_int_set_si(hull->eq[k][1+dim+j], 1);
878 isl_int_set_si(hull->eq[k][2*(1+dim)+j], 1);
880 hull = isl_basic_set_set_rational(hull);
881 hull = isl_basic_set_remove_dims(hull, isl_dim_set, dim, 2*(1+dim));
882 hull = isl_basic_set_remove_redundancies(hull);
883 isl_basic_set_free(bset1);
884 isl_basic_set_free(bset2);
887 isl_basic_set_free(bset1);
888 isl_basic_set_free(bset2);
889 isl_basic_set_free(hull);
893 /* Is the set bounded for each value of the parameters?
895 int isl_basic_set_is_bounded(__isl_keep isl_basic_set *bset)
902 if (isl_basic_set_plain_is_empty(bset))
905 tab = isl_tab_from_recession_cone(bset, 1);
906 bounded = isl_tab_cone_is_bounded(tab);
911 /* Is the image bounded for each value of the parameters and
912 * the domain variables?
914 int isl_basic_map_image_is_bounded(__isl_keep isl_basic_map *bmap)
916 unsigned nparam = isl_basic_map_dim(bmap, isl_dim_param);
917 unsigned n_in = isl_basic_map_dim(bmap, isl_dim_in);
920 bmap = isl_basic_map_copy(bmap);
921 bmap = isl_basic_map_cow(bmap);
922 bmap = isl_basic_map_move_dims(bmap, isl_dim_param, nparam,
923 isl_dim_in, 0, n_in);
924 bounded = isl_basic_set_is_bounded((isl_basic_set *)bmap);
925 isl_basic_map_free(bmap);
930 /* Is the set bounded for each value of the parameters?
932 int isl_set_is_bounded(__isl_keep isl_set *set)
939 for (i = 0; i < set->n; ++i) {
940 int bounded = isl_basic_set_is_bounded(set->p[i]);
941 if (!bounded || bounded < 0)
947 /* Compute the lineality space of the convex hull of bset1 and bset2.
949 * We first compute the intersection of the recession cone of bset1
950 * with the negative of the recession cone of bset2 and then compute
951 * the linear hull of the resulting cone.
953 static struct isl_basic_set *induced_lineality_space(
954 struct isl_basic_set *bset1, struct isl_basic_set *bset2)
957 struct isl_basic_set *lin = NULL;
960 if (!bset1 || !bset2)
963 dim = isl_basic_set_total_dim(bset1);
964 lin = isl_basic_set_alloc_dim(isl_basic_set_get_dim(bset1), 0,
965 bset1->n_eq + bset2->n_eq,
966 bset1->n_ineq + bset2->n_ineq);
967 lin = isl_basic_set_set_rational(lin);
970 for (i = 0; i < bset1->n_eq; ++i) {
971 k = isl_basic_set_alloc_equality(lin);
974 isl_int_set_si(lin->eq[k][0], 0);
975 isl_seq_cpy(lin->eq[k] + 1, bset1->eq[i] + 1, dim);
977 for (i = 0; i < bset1->n_ineq; ++i) {
978 k = isl_basic_set_alloc_inequality(lin);
981 isl_int_set_si(lin->ineq[k][0], 0);
982 isl_seq_cpy(lin->ineq[k] + 1, bset1->ineq[i] + 1, dim);
984 for (i = 0; i < bset2->n_eq; ++i) {
985 k = isl_basic_set_alloc_equality(lin);
988 isl_int_set_si(lin->eq[k][0], 0);
989 isl_seq_neg(lin->eq[k] + 1, bset2->eq[i] + 1, dim);
991 for (i = 0; i < bset2->n_ineq; ++i) {
992 k = isl_basic_set_alloc_inequality(lin);
995 isl_int_set_si(lin->ineq[k][0], 0);
996 isl_seq_neg(lin->ineq[k] + 1, bset2->ineq[i] + 1, dim);
999 isl_basic_set_free(bset1);
1000 isl_basic_set_free(bset2);
1001 return isl_basic_set_affine_hull(lin);
1003 isl_basic_set_free(lin);
1004 isl_basic_set_free(bset1);
1005 isl_basic_set_free(bset2);
1009 static struct isl_basic_set *uset_convex_hull(struct isl_set *set);
1011 /* Given a set and a linear space "lin" of dimension n > 0,
1012 * project the linear space from the set, compute the convex hull
1013 * and then map the set back to the original space.
1019 * describe the linear space. We first compute the Hermite normal
1020 * form H = M U of M = H Q, to obtain
1024 * The last n rows of H will be zero, so the last n variables of x' = Q x
1025 * are the one we want to project out. We do this by transforming each
1026 * basic set A x >= b to A U x' >= b and then removing the last n dimensions.
1027 * After computing the convex hull in x'_1, i.e., A' x'_1 >= b',
1028 * we transform the hull back to the original space as A' Q_1 x >= b',
1029 * with Q_1 all but the last n rows of Q.
1031 static struct isl_basic_set *modulo_lineality(struct isl_set *set,
1032 struct isl_basic_set *lin)
1034 unsigned total = isl_basic_set_total_dim(lin);
1036 struct isl_basic_set *hull;
1037 struct isl_mat *M, *U, *Q;
1041 lin_dim = total - lin->n_eq;
1042 M = isl_mat_sub_alloc6(set->ctx, lin->eq, 0, lin->n_eq, 1, total);
1043 M = isl_mat_left_hermite(M, 0, &U, &Q);
1047 isl_basic_set_free(lin);
1049 Q = isl_mat_drop_rows(Q, Q->n_row - lin_dim, lin_dim);
1051 U = isl_mat_lin_to_aff(U);
1052 Q = isl_mat_lin_to_aff(Q);
1054 set = isl_set_preimage(set, U);
1055 set = isl_set_remove_dims(set, isl_dim_set, total - lin_dim, lin_dim);
1056 hull = uset_convex_hull(set);
1057 hull = isl_basic_set_preimage(hull, Q);
1061 isl_basic_set_free(lin);
1066 /* Given two polyhedra with as constraints h_{ij} x >= 0 in homegeneous space,
1067 * set up an LP for solving
1069 * \sum_j \alpha_{1j} h_{1j} = \sum_j \alpha_{2j} h_{2j}
1071 * \alpha{i0} corresponds to the (implicit) positivity constraint 1 >= 0
1072 * The next \alpha{ij} correspond to the equalities and come in pairs.
1073 * The final \alpha{ij} correspond to the inequalities.
1075 static struct isl_basic_set *valid_direction_lp(
1076 struct isl_basic_set *bset1, struct isl_basic_set *bset2)
1078 struct isl_dim *dim;
1079 struct isl_basic_set *lp;
1084 if (!bset1 || !bset2)
1086 d = 1 + isl_basic_set_total_dim(bset1);
1088 2 * bset1->n_eq + bset1->n_ineq + 2 * bset2->n_eq + bset2->n_ineq;
1089 dim = isl_dim_set_alloc(bset1->ctx, 0, n);
1090 lp = isl_basic_set_alloc_dim(dim, 0, d, n);
1093 for (i = 0; i < n; ++i) {
1094 k = isl_basic_set_alloc_inequality(lp);
1097 isl_seq_clr(lp->ineq[k] + 1, n);
1098 isl_int_set_si(lp->ineq[k][0], -1);
1099 isl_int_set_si(lp->ineq[k][1 + i], 1);
1101 for (i = 0; i < d; ++i) {
1102 k = isl_basic_set_alloc_equality(lp);
1106 isl_int_set_si(lp->eq[k][n], 0); n++;
1107 /* positivity constraint 1 >= 0 */
1108 isl_int_set_si(lp->eq[k][n], i == 0); n++;
1109 for (j = 0; j < bset1->n_eq; ++j) {
1110 isl_int_set(lp->eq[k][n], bset1->eq[j][i]); n++;
1111 isl_int_neg(lp->eq[k][n], bset1->eq[j][i]); n++;
1113 for (j = 0; j < bset1->n_ineq; ++j) {
1114 isl_int_set(lp->eq[k][n], bset1->ineq[j][i]); n++;
1116 /* positivity constraint 1 >= 0 */
1117 isl_int_set_si(lp->eq[k][n], -(i == 0)); n++;
1118 for (j = 0; j < bset2->n_eq; ++j) {
1119 isl_int_neg(lp->eq[k][n], bset2->eq[j][i]); n++;
1120 isl_int_set(lp->eq[k][n], bset2->eq[j][i]); n++;
1122 for (j = 0; j < bset2->n_ineq; ++j) {
1123 isl_int_neg(lp->eq[k][n], bset2->ineq[j][i]); n++;
1126 lp = isl_basic_set_gauss(lp, NULL);
1127 isl_basic_set_free(bset1);
1128 isl_basic_set_free(bset2);
1131 isl_basic_set_free(bset1);
1132 isl_basic_set_free(bset2);
1136 /* Compute a vector s in the homogeneous space such that <s, r> > 0
1137 * for all rays in the homogeneous space of the two cones that correspond
1138 * to the input polyhedra bset1 and bset2.
1140 * We compute s as a vector that satisfies
1142 * s = \sum_j \alpha_{ij} h_{ij} for i = 1,2 (*)
1144 * with h_{ij} the normals of the facets of polyhedron i
1145 * (including the "positivity constraint" 1 >= 0) and \alpha_{ij}
1146 * strictly positive numbers. For simplicity we impose \alpha_{ij} >= 1.
1147 * We first set up an LP with as variables the \alpha{ij}.
1148 * In this formulation, for each polyhedron i,
1149 * the first constraint is the positivity constraint, followed by pairs
1150 * of variables for the equalities, followed by variables for the inequalities.
1151 * We then simply pick a feasible solution and compute s using (*).
1153 * Note that we simply pick any valid direction and make no attempt
1154 * to pick a "good" or even the "best" valid direction.
1156 static struct isl_vec *valid_direction(
1157 struct isl_basic_set *bset1, struct isl_basic_set *bset2)
1159 struct isl_basic_set *lp;
1160 struct isl_tab *tab;
1161 struct isl_vec *sample = NULL;
1162 struct isl_vec *dir;
1167 if (!bset1 || !bset2)
1169 lp = valid_direction_lp(isl_basic_set_copy(bset1),
1170 isl_basic_set_copy(bset2));
1171 tab = isl_tab_from_basic_set(lp);
1172 sample = isl_tab_get_sample_value(tab);
1174 isl_basic_set_free(lp);
1177 d = isl_basic_set_total_dim(bset1);
1178 dir = isl_vec_alloc(bset1->ctx, 1 + d);
1181 isl_seq_clr(dir->block.data + 1, dir->size - 1);
1183 /* positivity constraint 1 >= 0 */
1184 isl_int_set(dir->block.data[0], sample->block.data[n]); n++;
1185 for (i = 0; i < bset1->n_eq; ++i) {
1186 isl_int_sub(sample->block.data[n],
1187 sample->block.data[n], sample->block.data[n+1]);
1188 isl_seq_combine(dir->block.data,
1189 bset1->ctx->one, dir->block.data,
1190 sample->block.data[n], bset1->eq[i], 1 + d);
1194 for (i = 0; i < bset1->n_ineq; ++i)
1195 isl_seq_combine(dir->block.data,
1196 bset1->ctx->one, dir->block.data,
1197 sample->block.data[n++], bset1->ineq[i], 1 + d);
1198 isl_vec_free(sample);
1199 isl_seq_normalize(bset1->ctx, dir->el, dir->size);
1200 isl_basic_set_free(bset1);
1201 isl_basic_set_free(bset2);
1204 isl_vec_free(sample);
1205 isl_basic_set_free(bset1);
1206 isl_basic_set_free(bset2);
1210 /* Given a polyhedron b_i + A_i x >= 0 and a map T = S^{-1},
1211 * compute b_i' + A_i' x' >= 0, with
1213 * [ b_i A_i ] [ y' ] [ y' ]
1214 * [ 1 0 ] S^{-1} [ x' ] >= 0 or [ b_i' A_i' ] [ x' ] >= 0
1216 * In particular, add the "positivity constraint" and then perform
1219 static struct isl_basic_set *homogeneous_map(struct isl_basic_set *bset,
1226 bset = isl_basic_set_extend_constraints(bset, 0, 1);
1227 k = isl_basic_set_alloc_inequality(bset);
1230 isl_seq_clr(bset->ineq[k] + 1, isl_basic_set_total_dim(bset));
1231 isl_int_set_si(bset->ineq[k][0], 1);
1232 bset = isl_basic_set_preimage(bset, T);
1236 isl_basic_set_free(bset);
1240 /* Compute the convex hull of a pair of basic sets without any parameters or
1241 * integer divisions, where the convex hull is known to be pointed,
1242 * but the basic sets may be unbounded.
1244 * We turn this problem into the computation of a convex hull of a pair
1245 * _bounded_ polyhedra by "changing the direction of the homogeneous
1246 * dimension". This idea is due to Matthias Koeppe.
1248 * Consider the cones in homogeneous space that correspond to the
1249 * input polyhedra. The rays of these cones are also rays of the
1250 * polyhedra if the coordinate that corresponds to the homogeneous
1251 * dimension is zero. That is, if the inner product of the rays
1252 * with the homogeneous direction is zero.
1253 * The cones in the homogeneous space can also be considered to
1254 * correspond to other pairs of polyhedra by chosing a different
1255 * homogeneous direction. To ensure that both of these polyhedra
1256 * are bounded, we need to make sure that all rays of the cones
1257 * correspond to vertices and not to rays.
1258 * Let s be a direction such that <s, r> > 0 for all rays r of both cones.
1259 * Then using s as a homogeneous direction, we obtain a pair of polytopes.
1260 * The vector s is computed in valid_direction.
1262 * Note that we need to consider _all_ rays of the cones and not just
1263 * the rays that correspond to rays in the polyhedra. If we were to
1264 * only consider those rays and turn them into vertices, then we
1265 * may inadvertently turn some vertices into rays.
1267 * The standard homogeneous direction is the unit vector in the 0th coordinate.
1268 * We therefore transform the two polyhedra such that the selected
1269 * direction is mapped onto this standard direction and then proceed
1270 * with the normal computation.
1271 * Let S be a non-singular square matrix with s as its first row,
1272 * then we want to map the polyhedra to the space
1274 * [ y' ] [ y ] [ y ] [ y' ]
1275 * [ x' ] = S [ x ] i.e., [ x ] = S^{-1} [ x' ]
1277 * We take S to be the unimodular completion of s to limit the growth
1278 * of the coefficients in the following computations.
1280 * Let b_i + A_i x >= 0 be the constraints of polyhedron i.
1281 * We first move to the homogeneous dimension
1283 * b_i y + A_i x >= 0 [ b_i A_i ] [ y ] [ 0 ]
1284 * y >= 0 or [ 1 0 ] [ x ] >= [ 0 ]
1286 * Then we change directoin
1288 * [ b_i A_i ] [ y' ] [ y' ]
1289 * [ 1 0 ] S^{-1} [ x' ] >= 0 or [ b_i' A_i' ] [ x' ] >= 0
1291 * Then we compute the convex hull of the polytopes b_i' + A_i' x' >= 0
1292 * resulting in b' + A' x' >= 0, which we then convert back
1295 * [ b' A' ] S [ x ] >= 0 or [ b A ] [ x ] >= 0
1297 * The polyhedron b + A x >= 0 is then the convex hull of the input polyhedra.
1299 static struct isl_basic_set *convex_hull_pair_pointed(
1300 struct isl_basic_set *bset1, struct isl_basic_set *bset2)
1302 struct isl_ctx *ctx = NULL;
1303 struct isl_vec *dir = NULL;
1304 struct isl_mat *T = NULL;
1305 struct isl_mat *T2 = NULL;
1306 struct isl_basic_set *hull;
1307 struct isl_set *set;
1309 if (!bset1 || !bset2)
1312 dir = valid_direction(isl_basic_set_copy(bset1),
1313 isl_basic_set_copy(bset2));
1316 T = isl_mat_alloc(bset1->ctx, dir->size, dir->size);
1319 isl_seq_cpy(T->row[0], dir->block.data, dir->size);
1320 T = isl_mat_unimodular_complete(T, 1);
1321 T2 = isl_mat_right_inverse(isl_mat_copy(T));
1323 bset1 = homogeneous_map(bset1, isl_mat_copy(T2));
1324 bset2 = homogeneous_map(bset2, T2);
1325 set = isl_set_alloc_dim(isl_basic_set_get_dim(bset1), 2, 0);
1326 set = isl_set_add_basic_set(set, bset1);
1327 set = isl_set_add_basic_set(set, bset2);
1328 hull = uset_convex_hull(set);
1329 hull = isl_basic_set_preimage(hull, T);
1336 isl_basic_set_free(bset1);
1337 isl_basic_set_free(bset2);
1341 static struct isl_basic_set *uset_convex_hull_wrap(struct isl_set *set);
1342 static struct isl_basic_set *modulo_affine_hull(
1343 struct isl_set *set, struct isl_basic_set *affine_hull);
1345 /* Compute the convex hull of a pair of basic sets without any parameters or
1346 * integer divisions.
1348 * This function is called from uset_convex_hull_unbounded, which
1349 * means that the complete convex hull is unbounded. Some pairs
1350 * of basic sets may still be bounded, though.
1351 * They may even lie inside a lower dimensional space, in which
1352 * case they need to be handled inside their affine hull since
1353 * the main algorithm assumes that the result is full-dimensional.
1355 * If the convex hull of the two basic sets would have a non-trivial
1356 * lineality space, we first project out this lineality space.
1358 static struct isl_basic_set *convex_hull_pair(struct isl_basic_set *bset1,
1359 struct isl_basic_set *bset2)
1361 isl_basic_set *lin, *aff;
1362 int bounded1, bounded2;
1364 if (bset1->ctx->opt->convex == ISL_CONVEX_HULL_FM)
1365 return convex_hull_pair_elim(bset1, bset2);
1367 aff = isl_set_affine_hull(isl_basic_set_union(isl_basic_set_copy(bset1),
1368 isl_basic_set_copy(bset2)));
1372 return modulo_affine_hull(isl_basic_set_union(bset1, bset2), aff);
1373 isl_basic_set_free(aff);
1375 bounded1 = isl_basic_set_is_bounded(bset1);
1376 bounded2 = isl_basic_set_is_bounded(bset2);
1378 if (bounded1 < 0 || bounded2 < 0)
1381 if (bounded1 && bounded2)
1382 uset_convex_hull_wrap(isl_basic_set_union(bset1, bset2));
1384 if (bounded1 || bounded2)
1385 return convex_hull_pair_pointed(bset1, bset2);
1387 lin = induced_lineality_space(isl_basic_set_copy(bset1),
1388 isl_basic_set_copy(bset2));
1391 if (isl_basic_set_is_universe(lin)) {
1392 isl_basic_set_free(bset1);
1393 isl_basic_set_free(bset2);
1396 if (lin->n_eq < isl_basic_set_total_dim(lin)) {
1397 struct isl_set *set;
1398 set = isl_set_alloc_dim(isl_basic_set_get_dim(bset1), 2, 0);
1399 set = isl_set_add_basic_set(set, bset1);
1400 set = isl_set_add_basic_set(set, bset2);
1401 return modulo_lineality(set, lin);
1403 isl_basic_set_free(lin);
1405 return convex_hull_pair_pointed(bset1, bset2);
1407 isl_basic_set_free(bset1);
1408 isl_basic_set_free(bset2);
1412 /* Compute the lineality space of a basic set.
1413 * We currently do not allow the basic set to have any divs.
1414 * We basically just drop the constants and turn every inequality
1417 struct isl_basic_set *isl_basic_set_lineality_space(struct isl_basic_set *bset)
1420 struct isl_basic_set *lin = NULL;
1425 isl_assert(bset->ctx, bset->n_div == 0, goto error);
1426 dim = isl_basic_set_total_dim(bset);
1428 lin = isl_basic_set_alloc_dim(isl_basic_set_get_dim(bset), 0, dim, 0);
1431 for (i = 0; i < bset->n_eq; ++i) {
1432 k = isl_basic_set_alloc_equality(lin);
1435 isl_int_set_si(lin->eq[k][0], 0);
1436 isl_seq_cpy(lin->eq[k] + 1, bset->eq[i] + 1, dim);
1438 lin = isl_basic_set_gauss(lin, NULL);
1441 for (i = 0; i < bset->n_ineq && lin->n_eq < dim; ++i) {
1442 k = isl_basic_set_alloc_equality(lin);
1445 isl_int_set_si(lin->eq[k][0], 0);
1446 isl_seq_cpy(lin->eq[k] + 1, bset->ineq[i] + 1, dim);
1447 lin = isl_basic_set_gauss(lin, NULL);
1451 isl_basic_set_free(bset);
1454 isl_basic_set_free(lin);
1455 isl_basic_set_free(bset);
1459 /* Compute the (linear) hull of the lineality spaces of the basic sets in the
1460 * "underlying" set "set".
1462 static struct isl_basic_set *uset_combined_lineality_space(struct isl_set *set)
1465 struct isl_set *lin = NULL;
1470 struct isl_dim *dim = isl_set_get_dim(set);
1472 return isl_basic_set_empty(dim);
1475 lin = isl_set_alloc_dim(isl_set_get_dim(set), set->n, 0);
1476 for (i = 0; i < set->n; ++i)
1477 lin = isl_set_add_basic_set(lin,
1478 isl_basic_set_lineality_space(isl_basic_set_copy(set->p[i])));
1480 return isl_set_affine_hull(lin);
1483 /* Compute the convex hull of a set without any parameters or
1484 * integer divisions.
1485 * In each step, we combined two basic sets until only one
1486 * basic set is left.
1487 * The input basic sets are assumed not to have a non-trivial
1488 * lineality space. If any of the intermediate results has
1489 * a non-trivial lineality space, it is projected out.
1491 static struct isl_basic_set *uset_convex_hull_unbounded(struct isl_set *set)
1493 struct isl_basic_set *convex_hull = NULL;
1495 convex_hull = isl_set_copy_basic_set(set);
1496 set = isl_set_drop_basic_set(set, convex_hull);
1499 while (set->n > 0) {
1500 struct isl_basic_set *t;
1501 t = isl_set_copy_basic_set(set);
1504 set = isl_set_drop_basic_set(set, t);
1507 convex_hull = convex_hull_pair(convex_hull, t);
1510 t = isl_basic_set_lineality_space(isl_basic_set_copy(convex_hull));
1513 if (isl_basic_set_is_universe(t)) {
1514 isl_basic_set_free(convex_hull);
1518 if (t->n_eq < isl_basic_set_total_dim(t)) {
1519 set = isl_set_add_basic_set(set, convex_hull);
1520 return modulo_lineality(set, t);
1522 isl_basic_set_free(t);
1528 isl_basic_set_free(convex_hull);
1532 /* Compute an initial hull for wrapping containing a single initial
1534 * This function assumes that the given set is bounded.
1536 static struct isl_basic_set *initial_hull(struct isl_basic_set *hull,
1537 struct isl_set *set)
1539 struct isl_mat *bounds = NULL;
1545 bounds = initial_facet_constraint(set);
1548 k = isl_basic_set_alloc_inequality(hull);
1551 dim = isl_set_n_dim(set);
1552 isl_assert(set->ctx, 1 + dim == bounds->n_col, goto error);
1553 isl_seq_cpy(hull->ineq[k], bounds->row[0], bounds->n_col);
1554 isl_mat_free(bounds);
1558 isl_basic_set_free(hull);
1559 isl_mat_free(bounds);
1563 struct max_constraint {
1569 static int max_constraint_equal(const void *entry, const void *val)
1571 struct max_constraint *a = (struct max_constraint *)entry;
1572 isl_int *b = (isl_int *)val;
1574 return isl_seq_eq(a->c->row[0] + 1, b, a->c->n_col - 1);
1577 static void update_constraint(struct isl_ctx *ctx, struct isl_hash_table *table,
1578 isl_int *con, unsigned len, int n, int ineq)
1580 struct isl_hash_table_entry *entry;
1581 struct max_constraint *c;
1584 c_hash = isl_seq_get_hash(con + 1, len);
1585 entry = isl_hash_table_find(ctx, table, c_hash, max_constraint_equal,
1591 isl_hash_table_remove(ctx, table, entry);
1595 if (isl_int_gt(c->c->row[0][0], con[0]))
1597 if (isl_int_eq(c->c->row[0][0], con[0])) {
1602 c->c = isl_mat_cow(c->c);
1603 isl_int_set(c->c->row[0][0], con[0]);
1607 /* Check whether the constraint hash table "table" constains the constraint
1610 static int has_constraint(struct isl_ctx *ctx, struct isl_hash_table *table,
1611 isl_int *con, unsigned len, int n)
1613 struct isl_hash_table_entry *entry;
1614 struct max_constraint *c;
1617 c_hash = isl_seq_get_hash(con + 1, len);
1618 entry = isl_hash_table_find(ctx, table, c_hash, max_constraint_equal,
1625 return isl_int_eq(c->c->row[0][0], con[0]);
1628 /* Check for inequality constraints of a basic set without equalities
1629 * such that the same or more stringent copies of the constraint appear
1630 * in all of the basic sets. Such constraints are necessarily facet
1631 * constraints of the convex hull.
1633 * If the resulting basic set is by chance identical to one of
1634 * the basic sets in "set", then we know that this basic set contains
1635 * all other basic sets and is therefore the convex hull of set.
1636 * In this case we set *is_hull to 1.
1638 static struct isl_basic_set *common_constraints(struct isl_basic_set *hull,
1639 struct isl_set *set, int *is_hull)
1642 int min_constraints;
1644 struct max_constraint *constraints = NULL;
1645 struct isl_hash_table *table = NULL;
1650 for (i = 0; i < set->n; ++i)
1651 if (set->p[i]->n_eq == 0)
1655 min_constraints = set->p[i]->n_ineq;
1657 for (i = best + 1; i < set->n; ++i) {
1658 if (set->p[i]->n_eq != 0)
1660 if (set->p[i]->n_ineq >= min_constraints)
1662 min_constraints = set->p[i]->n_ineq;
1665 constraints = isl_calloc_array(hull->ctx, struct max_constraint,
1669 table = isl_alloc_type(hull->ctx, struct isl_hash_table);
1670 if (isl_hash_table_init(hull->ctx, table, min_constraints))
1673 total = isl_dim_total(set->dim);
1674 for (i = 0; i < set->p[best]->n_ineq; ++i) {
1675 constraints[i].c = isl_mat_sub_alloc6(hull->ctx,
1676 set->p[best]->ineq + i, 0, 1, 0, 1 + total);
1677 if (!constraints[i].c)
1679 constraints[i].ineq = 1;
1681 for (i = 0; i < min_constraints; ++i) {
1682 struct isl_hash_table_entry *entry;
1684 c_hash = isl_seq_get_hash(constraints[i].c->row[0] + 1, total);
1685 entry = isl_hash_table_find(hull->ctx, table, c_hash,
1686 max_constraint_equal, constraints[i].c->row[0] + 1, 1);
1689 isl_assert(hull->ctx, !entry->data, goto error);
1690 entry->data = &constraints[i];
1694 for (s = 0; s < set->n; ++s) {
1698 for (i = 0; i < set->p[s]->n_eq; ++i) {
1699 isl_int *eq = set->p[s]->eq[i];
1700 for (j = 0; j < 2; ++j) {
1701 isl_seq_neg(eq, eq, 1 + total);
1702 update_constraint(hull->ctx, table,
1706 for (i = 0; i < set->p[s]->n_ineq; ++i) {
1707 isl_int *ineq = set->p[s]->ineq[i];
1708 update_constraint(hull->ctx, table, ineq, total, n,
1709 set->p[s]->n_eq == 0);
1714 for (i = 0; i < min_constraints; ++i) {
1715 if (constraints[i].count < n)
1717 if (!constraints[i].ineq)
1719 j = isl_basic_set_alloc_inequality(hull);
1722 isl_seq_cpy(hull->ineq[j], constraints[i].c->row[0], 1 + total);
1725 for (s = 0; s < set->n; ++s) {
1726 if (set->p[s]->n_eq)
1728 if (set->p[s]->n_ineq != hull->n_ineq)
1730 for (i = 0; i < set->p[s]->n_ineq; ++i) {
1731 isl_int *ineq = set->p[s]->ineq[i];
1732 if (!has_constraint(hull->ctx, table, ineq, total, n))
1735 if (i == set->p[s]->n_ineq)
1739 isl_hash_table_clear(table);
1740 for (i = 0; i < min_constraints; ++i)
1741 isl_mat_free(constraints[i].c);
1746 isl_hash_table_clear(table);
1749 for (i = 0; i < min_constraints; ++i)
1750 isl_mat_free(constraints[i].c);
1755 /* Create a template for the convex hull of "set" and fill it up
1756 * obvious facet constraints, if any. If the result happens to
1757 * be the convex hull of "set" then *is_hull is set to 1.
1759 static struct isl_basic_set *proto_hull(struct isl_set *set, int *is_hull)
1761 struct isl_basic_set *hull;
1766 for (i = 0; i < set->n; ++i) {
1767 n_ineq += set->p[i]->n_eq;
1768 n_ineq += set->p[i]->n_ineq;
1770 hull = isl_basic_set_alloc_dim(isl_dim_copy(set->dim), 0, 0, n_ineq);
1771 hull = isl_basic_set_set_rational(hull);
1774 return common_constraints(hull, set, is_hull);
1777 static struct isl_basic_set *uset_convex_hull_wrap(struct isl_set *set)
1779 struct isl_basic_set *hull;
1782 hull = proto_hull(set, &is_hull);
1783 if (hull && !is_hull) {
1784 if (hull->n_ineq == 0)
1785 hull = initial_hull(hull, set);
1786 hull = extend(hull, set);
1793 /* Compute the convex hull of a set without any parameters or
1794 * integer divisions. Depending on whether the set is bounded,
1795 * we pass control to the wrapping based convex hull or
1796 * the Fourier-Motzkin elimination based convex hull.
1797 * We also handle a few special cases before checking the boundedness.
1799 static struct isl_basic_set *uset_convex_hull(struct isl_set *set)
1801 struct isl_basic_set *convex_hull = NULL;
1802 struct isl_basic_set *lin;
1804 if (isl_set_n_dim(set) == 0)
1805 return convex_hull_0d(set);
1807 set = isl_set_coalesce(set);
1808 set = isl_set_set_rational(set);
1815 convex_hull = isl_basic_set_copy(set->p[0]);
1819 if (isl_set_n_dim(set) == 1)
1820 return convex_hull_1d(set);
1822 if (isl_set_is_bounded(set) &&
1823 set->ctx->opt->convex == ISL_CONVEX_HULL_WRAP)
1824 return uset_convex_hull_wrap(set);
1826 lin = uset_combined_lineality_space(isl_set_copy(set));
1829 if (isl_basic_set_is_universe(lin)) {
1833 if (lin->n_eq < isl_basic_set_total_dim(lin))
1834 return modulo_lineality(set, lin);
1835 isl_basic_set_free(lin);
1837 return uset_convex_hull_unbounded(set);
1840 isl_basic_set_free(convex_hull);
1844 /* This is the core procedure, where "set" is a "pure" set, i.e.,
1845 * without parameters or divs and where the convex hull of set is
1846 * known to be full-dimensional.
1848 static struct isl_basic_set *uset_convex_hull_wrap_bounded(struct isl_set *set)
1850 struct isl_basic_set *convex_hull = NULL;
1855 if (isl_set_n_dim(set) == 0) {
1856 convex_hull = isl_basic_set_universe(isl_dim_copy(set->dim));
1858 convex_hull = isl_basic_set_set_rational(convex_hull);
1862 set = isl_set_set_rational(set);
1863 set = isl_set_coalesce(set);
1867 convex_hull = isl_basic_set_copy(set->p[0]);
1871 if (isl_set_n_dim(set) == 1)
1872 return convex_hull_1d(set);
1874 return uset_convex_hull_wrap(set);
1880 /* Compute the convex hull of set "set" with affine hull "affine_hull",
1881 * We first remove the equalities (transforming the set), compute the
1882 * convex hull of the transformed set and then add the equalities back
1883 * (after performing the inverse transformation.
1885 static struct isl_basic_set *modulo_affine_hull(
1886 struct isl_set *set, struct isl_basic_set *affine_hull)
1890 struct isl_basic_set *dummy;
1891 struct isl_basic_set *convex_hull;
1893 dummy = isl_basic_set_remove_equalities(
1894 isl_basic_set_copy(affine_hull), &T, &T2);
1897 isl_basic_set_free(dummy);
1898 set = isl_set_preimage(set, T);
1899 convex_hull = uset_convex_hull(set);
1900 convex_hull = isl_basic_set_preimage(convex_hull, T2);
1901 convex_hull = isl_basic_set_intersect(convex_hull, affine_hull);
1904 isl_basic_set_free(affine_hull);
1909 /* Compute the convex hull of a map.
1911 * The implementation was inspired by "Extended Convex Hull" by Fukuda et al.,
1912 * specifically, the wrapping of facets to obtain new facets.
1914 struct isl_basic_map *isl_map_convex_hull(struct isl_map *map)
1916 struct isl_basic_set *bset;
1917 struct isl_basic_map *model = NULL;
1918 struct isl_basic_set *affine_hull = NULL;
1919 struct isl_basic_map *convex_hull = NULL;
1920 struct isl_set *set = NULL;
1921 struct isl_ctx *ctx;
1928 convex_hull = isl_basic_map_empty_like_map(map);
1933 map = isl_map_detect_equalities(map);
1934 map = isl_map_align_divs(map);
1937 model = isl_basic_map_copy(map->p[0]);
1938 set = isl_map_underlying_set(map);
1942 affine_hull = isl_set_affine_hull(isl_set_copy(set));
1945 if (affine_hull->n_eq != 0)
1946 bset = modulo_affine_hull(set, affine_hull);
1948 isl_basic_set_free(affine_hull);
1949 bset = uset_convex_hull(set);
1952 convex_hull = isl_basic_map_overlying_set(bset, model);
1956 ISL_F_SET(convex_hull, ISL_BASIC_MAP_NO_IMPLICIT);
1957 ISL_F_SET(convex_hull, ISL_BASIC_MAP_ALL_EQUALITIES);
1958 ISL_F_CLR(convex_hull, ISL_BASIC_MAP_RATIONAL);
1962 isl_basic_map_free(model);
1966 struct isl_basic_set *isl_set_convex_hull(struct isl_set *set)
1968 return (struct isl_basic_set *)
1969 isl_map_convex_hull((struct isl_map *)set);
1972 __isl_give isl_basic_map *isl_map_polyhedral_hull(__isl_take isl_map *map)
1974 isl_basic_map *hull;
1976 hull = isl_map_convex_hull(map);
1977 return isl_basic_map_remove_divs(hull);
1980 __isl_give isl_basic_set *isl_set_polyhedral_hull(__isl_take isl_set *set)
1982 return (isl_basic_set *)isl_map_polyhedral_hull((isl_map *)set);
1985 struct sh_data_entry {
1986 struct isl_hash_table *table;
1987 struct isl_tab *tab;
1990 /* Holds the data needed during the simple hull computation.
1992 * n the number of basic sets in the original set
1993 * hull_table a hash table of already computed constraints
1994 * in the simple hull
1995 * p for each basic set,
1996 * table a hash table of the constraints
1997 * tab the tableau corresponding to the basic set
2000 struct isl_ctx *ctx;
2002 struct isl_hash_table *hull_table;
2003 struct sh_data_entry p[1];
2006 static void sh_data_free(struct sh_data *data)
2012 isl_hash_table_free(data->ctx, data->hull_table);
2013 for (i = 0; i < data->n; ++i) {
2014 isl_hash_table_free(data->ctx, data->p[i].table);
2015 isl_tab_free(data->p[i].tab);
2020 struct ineq_cmp_data {
2025 static int has_ineq(const void *entry, const void *val)
2027 isl_int *row = (isl_int *)entry;
2028 struct ineq_cmp_data *v = (struct ineq_cmp_data *)val;
2030 return isl_seq_eq(row + 1, v->p + 1, v->len) ||
2031 isl_seq_is_neg(row + 1, v->p + 1, v->len);
2034 static int hash_ineq(struct isl_ctx *ctx, struct isl_hash_table *table,
2035 isl_int *ineq, unsigned len)
2038 struct ineq_cmp_data v;
2039 struct isl_hash_table_entry *entry;
2043 c_hash = isl_seq_get_hash(ineq + 1, len);
2044 entry = isl_hash_table_find(ctx, table, c_hash, has_ineq, &v, 1);
2051 /* Fill hash table "table" with the constraints of "bset".
2052 * Equalities are added as two inequalities.
2053 * The value in the hash table is a pointer to the (in)equality of "bset".
2055 static int hash_basic_set(struct isl_hash_table *table,
2056 struct isl_basic_set *bset)
2059 unsigned dim = isl_basic_set_total_dim(bset);
2061 for (i = 0; i < bset->n_eq; ++i) {
2062 for (j = 0; j < 2; ++j) {
2063 isl_seq_neg(bset->eq[i], bset->eq[i], 1 + dim);
2064 if (hash_ineq(bset->ctx, table, bset->eq[i], dim) < 0)
2068 for (i = 0; i < bset->n_ineq; ++i) {
2069 if (hash_ineq(bset->ctx, table, bset->ineq[i], dim) < 0)
2075 static struct sh_data *sh_data_alloc(struct isl_set *set, unsigned n_ineq)
2077 struct sh_data *data;
2080 data = isl_calloc(set->ctx, struct sh_data,
2081 sizeof(struct sh_data) +
2082 (set->n - 1) * sizeof(struct sh_data_entry));
2085 data->ctx = set->ctx;
2087 data->hull_table = isl_hash_table_alloc(set->ctx, n_ineq);
2088 if (!data->hull_table)
2090 for (i = 0; i < set->n; ++i) {
2091 data->p[i].table = isl_hash_table_alloc(set->ctx,
2092 2 * set->p[i]->n_eq + set->p[i]->n_ineq);
2093 if (!data->p[i].table)
2095 if (hash_basic_set(data->p[i].table, set->p[i]) < 0)
2104 /* Check if inequality "ineq" is a bound for basic set "j" or if
2105 * it can be relaxed (by increasing the constant term) to become
2106 * a bound for that basic set. In the latter case, the constant
2108 * Return 1 if "ineq" is a bound
2109 * 0 if "ineq" may attain arbitrarily small values on basic set "j"
2110 * -1 if some error occurred
2112 static int is_bound(struct sh_data *data, struct isl_set *set, int j,
2115 enum isl_lp_result res;
2118 if (!data->p[j].tab) {
2119 data->p[j].tab = isl_tab_from_basic_set(set->p[j]);
2120 if (!data->p[j].tab)
2126 res = isl_tab_min(data->p[j].tab, ineq, data->ctx->one,
2128 if (res == isl_lp_ok && isl_int_is_neg(opt))
2129 isl_int_sub(ineq[0], ineq[0], opt);
2133 return (res == isl_lp_ok || res == isl_lp_empty) ? 1 :
2134 res == isl_lp_unbounded ? 0 : -1;
2137 /* Check if inequality "ineq" from basic set "i" can be relaxed to
2138 * become a bound on the whole set. If so, add the (relaxed) inequality
2141 * We first check if "hull" already contains a translate of the inequality.
2142 * If so, we are done.
2143 * Then, we check if any of the previous basic sets contains a translate
2144 * of the inequality. If so, then we have already considered this
2145 * inequality and we are done.
2146 * Otherwise, for each basic set other than "i", we check if the inequality
2147 * is a bound on the basic set.
2148 * For previous basic sets, we know that they do not contain a translate
2149 * of the inequality, so we directly call is_bound.
2150 * For following basic sets, we first check if a translate of the
2151 * inequality appears in its description and if so directly update
2152 * the inequality accordingly.
2154 static struct isl_basic_set *add_bound(struct isl_basic_set *hull,
2155 struct sh_data *data, struct isl_set *set, int i, isl_int *ineq)
2158 struct ineq_cmp_data v;
2159 struct isl_hash_table_entry *entry;
2165 v.len = isl_basic_set_total_dim(hull);
2167 c_hash = isl_seq_get_hash(ineq + 1, v.len);
2169 entry = isl_hash_table_find(hull->ctx, data->hull_table, c_hash,
2174 for (j = 0; j < i; ++j) {
2175 entry = isl_hash_table_find(hull->ctx, data->p[j].table,
2176 c_hash, has_ineq, &v, 0);
2183 k = isl_basic_set_alloc_inequality(hull);
2184 isl_seq_cpy(hull->ineq[k], ineq, 1 + v.len);
2188 for (j = 0; j < i; ++j) {
2190 bound = is_bound(data, set, j, hull->ineq[k]);
2197 isl_basic_set_free_inequality(hull, 1);
2201 for (j = i + 1; j < set->n; ++j) {
2204 entry = isl_hash_table_find(hull->ctx, data->p[j].table,
2205 c_hash, has_ineq, &v, 0);
2207 ineq_j = entry->data;
2208 neg = isl_seq_is_neg(ineq_j + 1,
2209 hull->ineq[k] + 1, v.len);
2211 isl_int_neg(ineq_j[0], ineq_j[0]);
2212 if (isl_int_gt(ineq_j[0], hull->ineq[k][0]))
2213 isl_int_set(hull->ineq[k][0], ineq_j[0]);
2215 isl_int_neg(ineq_j[0], ineq_j[0]);
2218 bound = is_bound(data, set, j, hull->ineq[k]);
2225 isl_basic_set_free_inequality(hull, 1);
2229 entry = isl_hash_table_find(hull->ctx, data->hull_table, c_hash,
2233 entry->data = hull->ineq[k];
2237 isl_basic_set_free(hull);
2241 /* Check if any inequality from basic set "i" can be relaxed to
2242 * become a bound on the whole set. If so, add the (relaxed) inequality
2245 static struct isl_basic_set *add_bounds(struct isl_basic_set *bset,
2246 struct sh_data *data, struct isl_set *set, int i)
2249 unsigned dim = isl_basic_set_total_dim(bset);
2251 for (j = 0; j < set->p[i]->n_eq; ++j) {
2252 for (k = 0; k < 2; ++k) {
2253 isl_seq_neg(set->p[i]->eq[j], set->p[i]->eq[j], 1+dim);
2254 bset = add_bound(bset, data, set, i, set->p[i]->eq[j]);
2257 for (j = 0; j < set->p[i]->n_ineq; ++j)
2258 bset = add_bound(bset, data, set, i, set->p[i]->ineq[j]);
2262 /* Compute a superset of the convex hull of set that is described
2263 * by only translates of the constraints in the constituents of set.
2265 static struct isl_basic_set *uset_simple_hull(struct isl_set *set)
2267 struct sh_data *data = NULL;
2268 struct isl_basic_set *hull = NULL;
2276 for (i = 0; i < set->n; ++i) {
2279 n_ineq += 2 * set->p[i]->n_eq + set->p[i]->n_ineq;
2282 hull = isl_basic_set_alloc_dim(isl_dim_copy(set->dim), 0, 0, n_ineq);
2286 data = sh_data_alloc(set, n_ineq);
2290 for (i = 0; i < set->n; ++i)
2291 hull = add_bounds(hull, data, set, i);
2299 isl_basic_set_free(hull);
2304 /* Compute a superset of the convex hull of map that is described
2305 * by only translates of the constraints in the constituents of map.
2307 struct isl_basic_map *isl_map_simple_hull(struct isl_map *map)
2309 struct isl_set *set = NULL;
2310 struct isl_basic_map *model = NULL;
2311 struct isl_basic_map *hull;
2312 struct isl_basic_map *affine_hull;
2313 struct isl_basic_set *bset = NULL;
2318 hull = isl_basic_map_empty_like_map(map);
2323 hull = isl_basic_map_copy(map->p[0]);
2328 map = isl_map_detect_equalities(map);
2329 affine_hull = isl_map_affine_hull(isl_map_copy(map));
2330 map = isl_map_align_divs(map);
2331 model = isl_basic_map_copy(map->p[0]);
2333 set = isl_map_underlying_set(map);
2335 bset = uset_simple_hull(set);
2337 hull = isl_basic_map_overlying_set(bset, model);
2339 hull = isl_basic_map_intersect(hull, affine_hull);
2340 hull = isl_basic_map_remove_redundancies(hull);
2341 ISL_F_SET(hull, ISL_BASIC_MAP_NO_IMPLICIT);
2342 ISL_F_SET(hull, ISL_BASIC_MAP_ALL_EQUALITIES);
2347 struct isl_basic_set *isl_set_simple_hull(struct isl_set *set)
2349 return (struct isl_basic_set *)
2350 isl_map_simple_hull((struct isl_map *)set);
2353 /* Given a set "set", return parametric bounds on the dimension "dim".
2355 static struct isl_basic_set *set_bounds(struct isl_set *set, int dim)
2357 unsigned set_dim = isl_set_dim(set, isl_dim_set);
2358 set = isl_set_copy(set);
2359 set = isl_set_eliminate_dims(set, dim + 1, set_dim - (dim + 1));
2360 set = isl_set_eliminate_dims(set, 0, dim);
2361 return isl_set_convex_hull(set);
2364 /* Computes a "simple hull" and then check if each dimension in the
2365 * resulting hull is bounded by a symbolic constant. If not, the
2366 * hull is intersected with the corresponding bounds on the whole set.
2368 struct isl_basic_set *isl_set_bounded_simple_hull(struct isl_set *set)
2371 struct isl_basic_set *hull;
2372 unsigned nparam, left;
2373 int removed_divs = 0;
2375 hull = isl_set_simple_hull(isl_set_copy(set));
2379 nparam = isl_basic_set_dim(hull, isl_dim_param);
2380 for (i = 0; i < isl_basic_set_dim(hull, isl_dim_set); ++i) {
2381 int lower = 0, upper = 0;
2382 struct isl_basic_set *bounds;
2384 left = isl_basic_set_total_dim(hull) - nparam - i - 1;
2385 for (j = 0; j < hull->n_eq; ++j) {
2386 if (isl_int_is_zero(hull->eq[j][1 + nparam + i]))
2388 if (isl_seq_first_non_zero(hull->eq[j]+1+nparam+i+1,
2395 for (j = 0; j < hull->n_ineq; ++j) {
2396 if (isl_int_is_zero(hull->ineq[j][1 + nparam + i]))
2398 if (isl_seq_first_non_zero(hull->ineq[j]+1+nparam+i+1,
2400 isl_seq_first_non_zero(hull->ineq[j]+1+nparam,
2403 if (isl_int_is_pos(hull->ineq[j][1 + nparam + i]))
2414 if (!removed_divs) {
2415 set = isl_set_remove_divs(set);
2420 bounds = set_bounds(set, i);
2421 hull = isl_basic_set_intersect(hull, bounds);