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 /* Check if "c" is a direction that is independent of the previously found "n"
176 * If so, add it to the list, with the negative of the lower bound
177 * in the constant position, i.e., such that c corresponds to a bounding
178 * hyperplane (but not necessarily a facet).
179 * Assumes set "set" is bounded.
181 static int is_independent_bound(struct isl_set *set, isl_int *c,
182 struct isl_mat *dirs, int n)
187 isl_seq_cpy(dirs->row[n]+1, c+1, dirs->n_col-1);
189 int pos = isl_seq_first_non_zero(dirs->row[n]+1, dirs->n_col-1);
192 for (i = 0; i < n; ++i) {
194 pos_i = isl_seq_first_non_zero(dirs->row[i]+1, dirs->n_col-1);
199 isl_seq_elim(dirs->row[n]+1, dirs->row[i]+1, pos,
200 dirs->n_col-1, NULL);
201 pos = isl_seq_first_non_zero(dirs->row[n]+1, dirs->n_col-1);
207 is_bound = uset_is_bound(set, dirs->row[n], dirs->n_col);
210 isl_seq_normalize(set->ctx, dirs->row[n], dirs->n_col);
213 isl_int *t = dirs->row[n];
214 for (k = n; k > i; --k)
215 dirs->row[k] = dirs->row[k-1];
221 /* Compute and return a maximal set of linearly independent bounds
222 * on the set "set", based on the constraints of the basic sets
225 static struct isl_mat *independent_bounds(struct isl_set *set)
228 struct isl_mat *dirs = NULL;
229 unsigned dim = isl_set_n_dim(set);
231 dirs = isl_mat_alloc(set->ctx, dim, 1+dim);
236 for (i = 0; n < dim && i < set->n; ++i) {
238 struct isl_basic_set *bset = set->p[i];
240 for (j = 0; n < dim && j < bset->n_eq; ++j) {
241 f = is_independent_bound(set, bset->eq[j], dirs, n);
247 for (j = 0; n < dim && j < bset->n_ineq; ++j) {
248 f = is_independent_bound(set, bset->ineq[j], dirs, n);
262 struct isl_basic_set *isl_basic_set_set_rational(struct isl_basic_set *bset)
267 if (ISL_F_ISSET(bset, ISL_BASIC_MAP_RATIONAL))
270 bset = isl_basic_set_cow(bset);
274 ISL_F_SET(bset, ISL_BASIC_MAP_RATIONAL);
276 return isl_basic_set_finalize(bset);
279 static struct isl_set *isl_set_set_rational(struct isl_set *set)
283 set = isl_set_cow(set);
286 for (i = 0; i < set->n; ++i) {
287 set->p[i] = isl_basic_set_set_rational(set->p[i]);
297 static struct isl_basic_set *isl_basic_set_add_equality(
298 struct isl_basic_set *bset, isl_int *c)
303 if (ISL_F_ISSET(bset, ISL_BASIC_SET_EMPTY))
306 isl_assert(bset->ctx, isl_basic_set_n_param(bset) == 0, goto error);
307 isl_assert(bset->ctx, bset->n_div == 0, goto error);
308 dim = isl_basic_set_n_dim(bset);
309 bset = isl_basic_set_cow(bset);
310 bset = isl_basic_set_extend(bset, 0, dim, 0, 1, 0);
311 i = isl_basic_set_alloc_equality(bset);
314 isl_seq_cpy(bset->eq[i], c, 1 + dim);
317 isl_basic_set_free(bset);
321 static struct isl_set *isl_set_add_equality(struct isl_set *set, isl_int *c)
325 set = isl_set_cow(set);
328 for (i = 0; i < set->n; ++i) {
329 set->p[i] = isl_basic_set_add_equality(set->p[i], c);
339 /* Given a union of basic sets, construct the constraints for wrapping
340 * a facet around one of its ridges.
341 * In particular, if each of n the d-dimensional basic sets i in "set"
342 * contains the origin, satisfies the constraints x_1 >= 0 and x_2 >= 0
343 * and is defined by the constraints
347 * then the resulting set is of dimension n*(1+d) and has as constraints
356 static struct isl_basic_set *wrap_constraints(struct isl_set *set)
358 struct isl_basic_set *lp;
362 unsigned dim, lp_dim;
367 dim = 1 + isl_set_n_dim(set);
370 for (i = 0; i < set->n; ++i) {
371 n_eq += set->p[i]->n_eq;
372 n_ineq += set->p[i]->n_ineq;
374 lp = isl_basic_set_alloc(set->ctx, 0, dim * set->n, 0, n_eq, n_ineq);
377 lp_dim = isl_basic_set_n_dim(lp);
378 k = isl_basic_set_alloc_equality(lp);
379 isl_int_set_si(lp->eq[k][0], -1);
380 for (i = 0; i < set->n; ++i) {
381 isl_int_set_si(lp->eq[k][1+dim*i], 0);
382 isl_int_set_si(lp->eq[k][1+dim*i+1], 1);
383 isl_seq_clr(lp->eq[k]+1+dim*i+2, dim-2);
385 for (i = 0; i < set->n; ++i) {
386 k = isl_basic_set_alloc_inequality(lp);
387 isl_seq_clr(lp->ineq[k], 1+lp_dim);
388 isl_int_set_si(lp->ineq[k][1+dim*i], 1);
390 for (j = 0; j < set->p[i]->n_eq; ++j) {
391 k = isl_basic_set_alloc_equality(lp);
392 isl_seq_clr(lp->eq[k], 1+dim*i);
393 isl_seq_cpy(lp->eq[k]+1+dim*i, set->p[i]->eq[j], dim);
394 isl_seq_clr(lp->eq[k]+1+dim*(i+1), dim*(set->n-i-1));
397 for (j = 0; j < set->p[i]->n_ineq; ++j) {
398 k = isl_basic_set_alloc_inequality(lp);
399 isl_seq_clr(lp->ineq[k], 1+dim*i);
400 isl_seq_cpy(lp->ineq[k]+1+dim*i, set->p[i]->ineq[j], dim);
401 isl_seq_clr(lp->ineq[k]+1+dim*(i+1), dim*(set->n-i-1));
407 /* Given a facet "facet" of the convex hull of "set" and a facet "ridge"
408 * of that facet, compute the other facet of the convex hull that contains
411 * We first transform the set such that the facet constraint becomes
415 * I.e., the facet lies in
419 * and on that facet, the constraint that defines the ridge is
423 * (This transformation is not strictly needed, all that is needed is
424 * that the ridge contains the origin.)
426 * Since the ridge contains the origin, the cone of the convex hull
427 * will be of the form
432 * with this second constraint defining the new facet.
433 * The constant a is obtained by settting x_1 in the cone of the
434 * convex hull to 1 and minimizing x_2.
435 * Now, each element in the cone of the convex hull is the sum
436 * of elements in the cones of the basic sets.
437 * If a_i is the dilation factor of basic set i, then the problem
438 * we need to solve is
451 * the constraints of each (transformed) basic set.
452 * If a = n/d, then the constraint defining the new facet (in the transformed
455 * -n x_1 + d x_2 >= 0
457 * In the original space, we need to take the same combination of the
458 * corresponding constraints "facet" and "ridge".
460 * Note that a is always finite, since we only apply the wrapping
461 * technique to a union of polytopes.
463 static isl_int *wrap_facet(struct isl_set *set, isl_int *facet, isl_int *ridge)
466 struct isl_mat *T = NULL;
467 struct isl_basic_set *lp = NULL;
469 enum isl_lp_result res;
473 set = isl_set_copy(set);
475 dim = 1 + isl_set_n_dim(set);
476 T = isl_mat_alloc(set->ctx, 3, dim);
479 isl_int_set_si(T->row[0][0], 1);
480 isl_seq_clr(T->row[0]+1, dim - 1);
481 isl_seq_cpy(T->row[1], facet, dim);
482 isl_seq_cpy(T->row[2], ridge, dim);
483 T = isl_mat_right_inverse(T);
484 set = isl_set_preimage(set, T);
488 lp = wrap_constraints(set);
489 obj = isl_vec_alloc(set->ctx, 1 + dim*set->n);
492 isl_int_set_si(obj->block.data[0], 0);
493 for (i = 0; i < set->n; ++i) {
494 isl_seq_clr(obj->block.data + 1 + dim*i, 2);
495 isl_int_set_si(obj->block.data[1 + dim*i+2], 1);
496 isl_seq_clr(obj->block.data + 1 + dim*i+3, dim-3);
500 res = isl_basic_set_solve_lp(lp, 0,
501 obj->block.data, set->ctx->one, &num, &den, NULL);
502 if (res == isl_lp_ok) {
503 isl_int_neg(num, num);
504 isl_seq_combine(facet, num, facet, den, ridge, dim);
509 isl_basic_set_free(lp);
511 isl_assert(set->ctx, res == isl_lp_ok, return NULL);
514 isl_basic_set_free(lp);
520 /* Given a set of d linearly independent bounding constraints of the
521 * convex hull of "set", compute the constraint of a facet of "set".
523 * We first compute the intersection with the first bounding hyperplane
524 * and remove the component corresponding to this hyperplane from
525 * other bounds (in homogeneous space).
526 * We then wrap around one of the remaining bounding constraints
527 * and continue the process until all bounding constraints have been
528 * taken into account.
529 * The resulting linear combination of the bounding constraints will
530 * correspond to a facet of the convex hull.
532 static struct isl_mat *initial_facet_constraint(struct isl_set *set,
533 struct isl_mat *bounds)
535 struct isl_set *slice = NULL;
536 struct isl_basic_set *face = NULL;
537 struct isl_mat *m, *U, *Q;
539 unsigned dim = isl_set_n_dim(set);
541 isl_assert(set->ctx, set->n > 0, goto error);
542 isl_assert(set->ctx, bounds->n_row == dim, goto error);
544 while (bounds->n_row > 1) {
545 slice = isl_set_copy(set);
546 slice = isl_set_add_equality(slice, bounds->row[0]);
547 face = isl_set_affine_hull(slice);
550 if (face->n_eq == 1) {
551 isl_basic_set_free(face);
554 m = isl_mat_alloc(set->ctx, 1 + face->n_eq, 1 + dim);
557 isl_int_set_si(m->row[0][0], 1);
558 isl_seq_clr(m->row[0]+1, dim);
559 for (i = 0; i < face->n_eq; ++i)
560 isl_seq_cpy(m->row[1 + i], face->eq[i], 1 + dim);
561 U = isl_mat_right_inverse(m);
562 Q = isl_mat_right_inverse(isl_mat_copy(U));
563 U = isl_mat_drop_cols(U, 1 + face->n_eq, dim - face->n_eq);
564 Q = isl_mat_drop_rows(Q, 1 + face->n_eq, dim - face->n_eq);
565 U = isl_mat_drop_cols(U, 0, 1);
566 Q = isl_mat_drop_rows(Q, 0, 1);
567 bounds = isl_mat_product(bounds, U);
568 bounds = isl_mat_product(bounds, Q);
569 while (isl_seq_first_non_zero(bounds->row[bounds->n_row-1],
570 bounds->n_col) == -1) {
572 isl_assert(set->ctx, bounds->n_row > 1, goto error);
574 if (!wrap_facet(set, bounds->row[0],
575 bounds->row[bounds->n_row-1]))
577 isl_basic_set_free(face);
582 isl_basic_set_free(face);
583 isl_mat_free(bounds);
587 /* Given the bounding constraint "c" of a facet of the convex hull of "set",
588 * compute a hyperplane description of the facet, i.e., compute the facets
591 * We compute an affine transformation that transforms the constraint
600 * by computing the right inverse U of a matrix that starts with the rows
613 * Since z_1 is zero, we can drop this variable as well as the corresponding
614 * column of U to obtain
622 * with Q' equal to Q, but without the corresponding row.
623 * After computing the facets of the facet in the z' space,
624 * we convert them back to the x space through Q.
626 static struct isl_basic_set *compute_facet(struct isl_set *set, isl_int *c)
628 struct isl_mat *m, *U, *Q;
629 struct isl_basic_set *facet = NULL;
634 set = isl_set_copy(set);
635 dim = isl_set_n_dim(set);
636 m = isl_mat_alloc(set->ctx, 2, 1 + dim);
639 isl_int_set_si(m->row[0][0], 1);
640 isl_seq_clr(m->row[0]+1, dim);
641 isl_seq_cpy(m->row[1], c, 1+dim);
642 U = isl_mat_right_inverse(m);
643 Q = isl_mat_right_inverse(isl_mat_copy(U));
644 U = isl_mat_drop_cols(U, 1, 1);
645 Q = isl_mat_drop_rows(Q, 1, 1);
646 set = isl_set_preimage(set, U);
647 facet = uset_convex_hull_wrap_bounded(set);
648 facet = isl_basic_set_preimage(facet, Q);
649 isl_assert(ctx, facet->n_eq == 0, goto error);
652 isl_basic_set_free(facet);
657 /* Given an initial facet constraint, compute the remaining facets.
658 * We do this by running through all facets found so far and computing
659 * the adjacent facets through wrapping, adding those facets that we
660 * hadn't already found before.
662 * For each facet we have found so far, we first compute its facets
663 * in the resulting convex hull. That is, we compute the ridges
664 * of the resulting convex hull contained in the facet.
665 * We also compute the corresponding facet in the current approximation
666 * of the convex hull. There is no need to wrap around the ridges
667 * in this facet since that would result in a facet that is already
668 * present in the current approximation.
670 * This function can still be significantly optimized by checking which of
671 * the facets of the basic sets are also facets of the convex hull and
672 * using all the facets so far to help in constructing the facets of the
675 * using the technique in section "3.1 Ridge Generation" of
676 * "Extended Convex Hull" by Fukuda et al.
678 static struct isl_basic_set *extend(struct isl_basic_set *hull,
683 struct isl_basic_set *facet = NULL;
684 struct isl_basic_set *hull_facet = NULL;
690 isl_assert(set->ctx, set->n > 0, goto error);
692 dim = isl_set_n_dim(set);
694 for (i = 0; i < hull->n_ineq; ++i) {
695 facet = compute_facet(set, hull->ineq[i]);
696 facet = isl_basic_set_add_equality(facet, hull->ineq[i]);
697 facet = isl_basic_set_gauss(facet, NULL);
698 facet = isl_basic_set_normalize_constraints(facet);
699 hull_facet = isl_basic_set_copy(hull);
700 hull_facet = isl_basic_set_add_equality(hull_facet, hull->ineq[i]);
701 hull_facet = isl_basic_set_gauss(hull_facet, NULL);
702 hull_facet = isl_basic_set_normalize_constraints(hull_facet);
705 hull = isl_basic_set_cow(hull);
706 hull = isl_basic_set_extend_dim(hull,
707 isl_dim_copy(hull->dim), 0, 0, facet->n_ineq);
708 for (j = 0; j < facet->n_ineq; ++j) {
709 for (f = 0; f < hull_facet->n_ineq; ++f)
710 if (isl_seq_eq(facet->ineq[j],
711 hull_facet->ineq[f], 1 + dim))
713 if (f < hull_facet->n_ineq)
715 k = isl_basic_set_alloc_inequality(hull);
718 isl_seq_cpy(hull->ineq[k], hull->ineq[i], 1+dim);
719 if (!wrap_facet(set, hull->ineq[k], facet->ineq[j]))
722 isl_basic_set_free(hull_facet);
723 isl_basic_set_free(facet);
725 hull = isl_basic_set_simplify(hull);
726 hull = isl_basic_set_finalize(hull);
729 isl_basic_set_free(hull_facet);
730 isl_basic_set_free(facet);
731 isl_basic_set_free(hull);
735 /* Special case for computing the convex hull of a one dimensional set.
736 * We simply collect the lower and upper bounds of each basic set
737 * and the biggest of those.
739 static struct isl_basic_set *convex_hull_1d(struct isl_set *set)
741 struct isl_mat *c = NULL;
742 isl_int *lower = NULL;
743 isl_int *upper = NULL;
746 struct isl_basic_set *hull;
748 for (i = 0; i < set->n; ++i) {
749 set->p[i] = isl_basic_set_simplify(set->p[i]);
753 set = isl_set_remove_empty_parts(set);
756 isl_assert(set->ctx, set->n > 0, goto error);
757 c = isl_mat_alloc(set->ctx, 2, 2);
761 if (set->p[0]->n_eq > 0) {
762 isl_assert(set->ctx, set->p[0]->n_eq == 1, goto error);
765 if (isl_int_is_pos(set->p[0]->eq[0][1])) {
766 isl_seq_cpy(lower, set->p[0]->eq[0], 2);
767 isl_seq_neg(upper, set->p[0]->eq[0], 2);
769 isl_seq_neg(lower, set->p[0]->eq[0], 2);
770 isl_seq_cpy(upper, set->p[0]->eq[0], 2);
773 for (j = 0; j < set->p[0]->n_ineq; ++j) {
774 if (isl_int_is_pos(set->p[0]->ineq[j][1])) {
776 isl_seq_cpy(lower, set->p[0]->ineq[j], 2);
779 isl_seq_cpy(upper, set->p[0]->ineq[j], 2);
786 for (i = 0; i < set->n; ++i) {
787 struct isl_basic_set *bset = set->p[i];
791 for (j = 0; j < bset->n_eq; ++j) {
795 isl_int_mul(a, lower[0], bset->eq[j][1]);
796 isl_int_mul(b, lower[1], bset->eq[j][0]);
797 if (isl_int_lt(a, b) && isl_int_is_pos(bset->eq[j][1]))
798 isl_seq_cpy(lower, bset->eq[j], 2);
799 if (isl_int_gt(a, b) && isl_int_is_neg(bset->eq[j][1]))
800 isl_seq_neg(lower, bset->eq[j], 2);
803 isl_int_mul(a, upper[0], bset->eq[j][1]);
804 isl_int_mul(b, upper[1], bset->eq[j][0]);
805 if (isl_int_lt(a, b) && isl_int_is_pos(bset->eq[j][1]))
806 isl_seq_neg(upper, bset->eq[j], 2);
807 if (isl_int_gt(a, b) && isl_int_is_neg(bset->eq[j][1]))
808 isl_seq_cpy(upper, bset->eq[j], 2);
811 for (j = 0; j < bset->n_ineq; ++j) {
812 if (isl_int_is_pos(bset->ineq[j][1]))
814 if (isl_int_is_neg(bset->ineq[j][1]))
816 if (lower && isl_int_is_pos(bset->ineq[j][1])) {
817 isl_int_mul(a, lower[0], bset->ineq[j][1]);
818 isl_int_mul(b, lower[1], bset->ineq[j][0]);
819 if (isl_int_lt(a, b))
820 isl_seq_cpy(lower, bset->ineq[j], 2);
822 if (upper && isl_int_is_neg(bset->ineq[j][1])) {
823 isl_int_mul(a, upper[0], bset->ineq[j][1]);
824 isl_int_mul(b, upper[1], bset->ineq[j][0]);
825 if (isl_int_gt(a, b))
826 isl_seq_cpy(upper, bset->ineq[j], 2);
837 hull = isl_basic_set_alloc(set->ctx, 0, 1, 0, 0, 2);
838 hull = isl_basic_set_set_rational(hull);
842 k = isl_basic_set_alloc_inequality(hull);
843 isl_seq_cpy(hull->ineq[k], lower, 2);
846 k = isl_basic_set_alloc_inequality(hull);
847 isl_seq_cpy(hull->ineq[k], upper, 2);
849 hull = isl_basic_set_finalize(hull);
859 /* Project out final n dimensions using Fourier-Motzkin */
860 static struct isl_set *set_project_out(struct isl_ctx *ctx,
861 struct isl_set *set, unsigned n)
863 return isl_set_remove_dims(set, isl_set_n_dim(set) - n, n);
866 static struct isl_basic_set *convex_hull_0d(struct isl_set *set)
868 struct isl_basic_set *convex_hull;
873 if (isl_set_is_empty(set))
874 convex_hull = isl_basic_set_empty(isl_dim_copy(set->dim));
876 convex_hull = isl_basic_set_universe(isl_dim_copy(set->dim));
881 /* Compute the convex hull of a pair of basic sets without any parameters or
882 * integer divisions using Fourier-Motzkin elimination.
883 * The convex hull is the set of all points that can be written as
884 * the sum of points from both basic sets (in homogeneous coordinates).
885 * We set up the constraints in a space with dimensions for each of
886 * the three sets and then project out the dimensions corresponding
887 * to the two original basic sets, retaining only those corresponding
888 * to the convex hull.
890 static struct isl_basic_set *convex_hull_pair_elim(struct isl_basic_set *bset1,
891 struct isl_basic_set *bset2)
894 struct isl_basic_set *bset[2];
895 struct isl_basic_set *hull = NULL;
898 if (!bset1 || !bset2)
901 dim = isl_basic_set_n_dim(bset1);
902 hull = isl_basic_set_alloc(bset1->ctx, 0, 2 + 3 * dim, 0,
903 1 + dim + bset1->n_eq + bset2->n_eq,
904 2 + bset1->n_ineq + bset2->n_ineq);
907 for (i = 0; i < 2; ++i) {
908 for (j = 0; j < bset[i]->n_eq; ++j) {
909 k = isl_basic_set_alloc_equality(hull);
912 isl_seq_clr(hull->eq[k], (i+1) * (1+dim));
913 isl_seq_clr(hull->eq[k]+(i+2)*(1+dim), (1-i)*(1+dim));
914 isl_seq_cpy(hull->eq[k]+(i+1)*(1+dim), bset[i]->eq[j],
917 for (j = 0; j < bset[i]->n_ineq; ++j) {
918 k = isl_basic_set_alloc_inequality(hull);
921 isl_seq_clr(hull->ineq[k], (i+1) * (1+dim));
922 isl_seq_clr(hull->ineq[k]+(i+2)*(1+dim), (1-i)*(1+dim));
923 isl_seq_cpy(hull->ineq[k]+(i+1)*(1+dim),
924 bset[i]->ineq[j], 1+dim);
926 k = isl_basic_set_alloc_inequality(hull);
929 isl_seq_clr(hull->ineq[k], 1+2+3*dim);
930 isl_int_set_si(hull->ineq[k][(i+1)*(1+dim)], 1);
932 for (j = 0; j < 1+dim; ++j) {
933 k = isl_basic_set_alloc_equality(hull);
936 isl_seq_clr(hull->eq[k], 1+2+3*dim);
937 isl_int_set_si(hull->eq[k][j], -1);
938 isl_int_set_si(hull->eq[k][1+dim+j], 1);
939 isl_int_set_si(hull->eq[k][2*(1+dim)+j], 1);
941 hull = isl_basic_set_set_rational(hull);
942 hull = isl_basic_set_remove_dims(hull, dim, 2*(1+dim));
943 hull = isl_basic_set_convex_hull(hull);
944 isl_basic_set_free(bset1);
945 isl_basic_set_free(bset2);
948 isl_basic_set_free(bset1);
949 isl_basic_set_free(bset2);
950 isl_basic_set_free(hull);
954 static int isl_basic_set_is_bounded(struct isl_basic_set *bset)
959 tab = isl_tab_from_recession_cone(bset);
960 bounded = isl_tab_cone_is_bounded(tab);
965 static int isl_set_is_bounded(struct isl_set *set)
969 for (i = 0; i < set->n; ++i) {
970 int bounded = isl_basic_set_is_bounded(set->p[i]);
971 if (!bounded || bounded < 0)
977 /* Compute the lineality space of the convex hull of bset1 and bset2.
979 * We first compute the intersection of the recession cone of bset1
980 * with the negative of the recession cone of bset2 and then compute
981 * the linear hull of the resulting cone.
983 static struct isl_basic_set *induced_lineality_space(
984 struct isl_basic_set *bset1, struct isl_basic_set *bset2)
987 struct isl_basic_set *lin = NULL;
990 if (!bset1 || !bset2)
993 dim = isl_basic_set_total_dim(bset1);
994 lin = isl_basic_set_alloc_dim(isl_basic_set_get_dim(bset1), 0,
995 bset1->n_eq + bset2->n_eq,
996 bset1->n_ineq + bset2->n_ineq);
997 lin = isl_basic_set_set_rational(lin);
1000 for (i = 0; i < bset1->n_eq; ++i) {
1001 k = isl_basic_set_alloc_equality(lin);
1004 isl_int_set_si(lin->eq[k][0], 0);
1005 isl_seq_cpy(lin->eq[k] + 1, bset1->eq[i] + 1, dim);
1007 for (i = 0; i < bset1->n_ineq; ++i) {
1008 k = isl_basic_set_alloc_inequality(lin);
1011 isl_int_set_si(lin->ineq[k][0], 0);
1012 isl_seq_cpy(lin->ineq[k] + 1, bset1->ineq[i] + 1, dim);
1014 for (i = 0; i < bset2->n_eq; ++i) {
1015 k = isl_basic_set_alloc_equality(lin);
1018 isl_int_set_si(lin->eq[k][0], 0);
1019 isl_seq_neg(lin->eq[k] + 1, bset2->eq[i] + 1, dim);
1021 for (i = 0; i < bset2->n_ineq; ++i) {
1022 k = isl_basic_set_alloc_inequality(lin);
1025 isl_int_set_si(lin->ineq[k][0], 0);
1026 isl_seq_neg(lin->ineq[k] + 1, bset2->ineq[i] + 1, dim);
1029 isl_basic_set_free(bset1);
1030 isl_basic_set_free(bset2);
1031 return isl_basic_set_affine_hull(lin);
1033 isl_basic_set_free(lin);
1034 isl_basic_set_free(bset1);
1035 isl_basic_set_free(bset2);
1039 static struct isl_basic_set *uset_convex_hull(struct isl_set *set);
1041 /* Given a set and a linear space "lin" of dimension n > 0,
1042 * project the linear space from the set, compute the convex hull
1043 * and then map the set back to the original space.
1049 * describe the linear space. We first compute the Hermite normal
1050 * form H = M U of M = H Q, to obtain
1054 * The last n rows of H will be zero, so the last n variables of x' = Q x
1055 * are the one we want to project out. We do this by transforming each
1056 * basic set A x >= b to A U x' >= b and then removing the last n dimensions.
1057 * After computing the convex hull in x'_1, i.e., A' x'_1 >= b',
1058 * we transform the hull back to the original space as A' Q_1 x >= b',
1059 * with Q_1 all but the last n rows of Q.
1061 static struct isl_basic_set *modulo_lineality(struct isl_set *set,
1062 struct isl_basic_set *lin)
1064 unsigned total = isl_basic_set_total_dim(lin);
1066 struct isl_basic_set *hull;
1067 struct isl_mat *M, *U, *Q;
1071 lin_dim = total - lin->n_eq;
1072 M = isl_mat_sub_alloc(set->ctx, lin->eq, 0, lin->n_eq, 1, total);
1073 M = isl_mat_left_hermite(M, 0, &U, &Q);
1077 isl_basic_set_free(lin);
1079 Q = isl_mat_drop_rows(Q, Q->n_row - lin_dim, lin_dim);
1081 U = isl_mat_lin_to_aff(U);
1082 Q = isl_mat_lin_to_aff(Q);
1084 set = isl_set_preimage(set, U);
1085 set = isl_set_remove_dims(set, total - lin_dim, lin_dim);
1086 hull = uset_convex_hull(set);
1087 hull = isl_basic_set_preimage(hull, Q);
1091 isl_basic_set_free(lin);
1096 /* Given two polyhedra with as constraints h_{ij} x >= 0 in homegeneous space,
1097 * set up an LP for solving
1099 * \sum_j \alpha_{1j} h_{1j} = \sum_j \alpha_{2j} h_{2j}
1101 * \alpha{i0} corresponds to the (implicit) positivity constraint 1 >= 0
1102 * The next \alpha{ij} correspond to the equalities and come in pairs.
1103 * The final \alpha{ij} correspond to the inequalities.
1105 static struct isl_basic_set *valid_direction_lp(
1106 struct isl_basic_set *bset1, struct isl_basic_set *bset2)
1108 struct isl_dim *dim;
1109 struct isl_basic_set *lp;
1114 if (!bset1 || !bset2)
1116 d = 1 + isl_basic_set_total_dim(bset1);
1118 2 * bset1->n_eq + bset1->n_ineq + 2 * bset2->n_eq + bset2->n_ineq;
1119 dim = isl_dim_set_alloc(bset1->ctx, 0, n);
1120 lp = isl_basic_set_alloc_dim(dim, 0, d, n);
1123 for (i = 0; i < n; ++i) {
1124 k = isl_basic_set_alloc_inequality(lp);
1127 isl_seq_clr(lp->ineq[k] + 1, n);
1128 isl_int_set_si(lp->ineq[k][0], -1);
1129 isl_int_set_si(lp->ineq[k][1 + i], 1);
1131 for (i = 0; i < d; ++i) {
1132 k = isl_basic_set_alloc_equality(lp);
1136 isl_int_set_si(lp->eq[k][n++], 0);
1137 /* positivity constraint 1 >= 0 */
1138 isl_int_set_si(lp->eq[k][n++], i == 0);
1139 for (j = 0; j < bset1->n_eq; ++j) {
1140 isl_int_set(lp->eq[k][n++], bset1->eq[j][i]);
1141 isl_int_neg(lp->eq[k][n++], bset1->eq[j][i]);
1143 for (j = 0; j < bset1->n_ineq; ++j)
1144 isl_int_set(lp->eq[k][n++], bset1->ineq[j][i]);
1145 /* positivity constraint 1 >= 0 */
1146 isl_int_set_si(lp->eq[k][n++], -(i == 0));
1147 for (j = 0; j < bset2->n_eq; ++j) {
1148 isl_int_neg(lp->eq[k][n++], bset2->eq[j][i]);
1149 isl_int_set(lp->eq[k][n++], bset2->eq[j][i]);
1151 for (j = 0; j < bset2->n_ineq; ++j)
1152 isl_int_neg(lp->eq[k][n++], bset2->ineq[j][i]);
1154 lp = isl_basic_set_gauss(lp, NULL);
1155 isl_basic_set_free(bset1);
1156 isl_basic_set_free(bset2);
1159 isl_basic_set_free(bset1);
1160 isl_basic_set_free(bset2);
1164 /* Compute a vector s in the homogeneous space such that <s, r> > 0
1165 * for all rays in the homogeneous space of the two cones that correspond
1166 * to the input polyhedra bset1 and bset2.
1168 * We compute s as a vector that satisfies
1170 * s = \sum_j \alpha_{ij} h_{ij} for i = 1,2 (*)
1172 * with h_{ij} the normals of the facets of polyhedron i
1173 * (including the "positivity constraint" 1 >= 0) and \alpha_{ij}
1174 * strictly positive numbers. For simplicity we impose \alpha_{ij} >= 1.
1175 * We first set up an LP with as variables the \alpha{ij}.
1176 * In this formulateion, for each polyhedron i,
1177 * the first constraint is the positivity constraint, followed by pairs
1178 * of variables for the equalities, followed by variables for the inequalities.
1179 * We then simply pick a feasible solution and compute s using (*).
1181 * Note that we simply pick any valid direction and make no attempt
1182 * to pick a "good" or even the "best" valid direction.
1184 static struct isl_vec *valid_direction(
1185 struct isl_basic_set *bset1, struct isl_basic_set *bset2)
1187 struct isl_basic_set *lp;
1188 struct isl_tab *tab;
1189 struct isl_vec *sample = NULL;
1190 struct isl_vec *dir;
1195 if (!bset1 || !bset2)
1197 lp = valid_direction_lp(isl_basic_set_copy(bset1),
1198 isl_basic_set_copy(bset2));
1199 tab = isl_tab_from_basic_set(lp);
1200 sample = isl_tab_get_sample_value(tab);
1202 isl_basic_set_free(lp);
1205 d = isl_basic_set_total_dim(bset1);
1206 dir = isl_vec_alloc(bset1->ctx, 1 + d);
1209 isl_seq_clr(dir->block.data + 1, dir->size - 1);
1211 /* positivity constraint 1 >= 0 */
1212 isl_int_set(dir->block.data[0], sample->block.data[n++]);
1213 for (i = 0; i < bset1->n_eq; ++i) {
1214 isl_int_sub(sample->block.data[n],
1215 sample->block.data[n], sample->block.data[n+1]);
1216 isl_seq_combine(dir->block.data,
1217 bset1->ctx->one, dir->block.data,
1218 sample->block.data[n], bset1->eq[i], 1 + d);
1222 for (i = 0; i < bset1->n_ineq; ++i)
1223 isl_seq_combine(dir->block.data,
1224 bset1->ctx->one, dir->block.data,
1225 sample->block.data[n++], bset1->ineq[i], 1 + d);
1226 isl_vec_free(sample);
1227 isl_seq_normalize(bset1->ctx, dir->block.data + 1, dir->size - 1);
1228 isl_basic_set_free(bset1);
1229 isl_basic_set_free(bset2);
1232 isl_vec_free(sample);
1233 isl_basic_set_free(bset1);
1234 isl_basic_set_free(bset2);
1238 /* Given a polyhedron b_i + A_i x >= 0 and a map T = S^{-1},
1239 * compute b_i' + A_i' x' >= 0, with
1241 * [ b_i A_i ] [ y' ] [ y' ]
1242 * [ 1 0 ] S^{-1} [ x' ] >= 0 or [ b_i' A_i' ] [ x' ] >= 0
1244 * In particular, add the "positivity constraint" and then perform
1247 static struct isl_basic_set *homogeneous_map(struct isl_basic_set *bset,
1254 bset = isl_basic_set_extend_constraints(bset, 0, 1);
1255 k = isl_basic_set_alloc_inequality(bset);
1258 isl_seq_clr(bset->ineq[k] + 1, isl_basic_set_total_dim(bset));
1259 isl_int_set_si(bset->ineq[k][0], 1);
1260 bset = isl_basic_set_preimage(bset, T);
1264 isl_basic_set_free(bset);
1268 /* Compute the convex hull of a pair of basic sets without any parameters or
1269 * integer divisions, where the convex hull is known to be pointed,
1270 * but the basic sets may be unbounded.
1272 * We turn this problem into the computation of a convex hull of a pair
1273 * _bounded_ polyhedra by "changing the direction of the homogeneous
1274 * dimension". This idea is due to Matthias Koeppe.
1276 * Consider the cones in homogeneous space that correspond to the
1277 * input polyhedra. The rays of these cones are also rays of the
1278 * polyhedra if the coordinate that corresponds to the homogeneous
1279 * dimension is zero. That is, if the inner product of the rays
1280 * with the homogeneous direction is zero.
1281 * The cones in the homogeneous space can also be considered to
1282 * correspond to other pairs of polyhedra by chosing a different
1283 * homogeneous direction. To ensure that both of these polyhedra
1284 * are bounded, we need to make sure that all rays of the cones
1285 * correspond to vertices and not to rays.
1286 * Let s be a direction such that <s, r> > 0 for all rays r of both cones.
1287 * Then using s as a homogeneous direction, we obtain a pair of polytopes.
1288 * The vector s is computed in valid_direction.
1290 * Note that we need to consider _all_ rays of the cones and not just
1291 * the rays that correspond to rays in the polyhedra. If we were to
1292 * only consider those rays and turn them into vertices, then we
1293 * may inadvertently turn some vertices into rays.
1295 * The standard homogeneous direction is the unit vector in the 0th coordinate.
1296 * We therefore transform the two polyhedra such that the selected
1297 * direction is mapped onto this standard direction and then proceed
1298 * with the normal computation.
1299 * Let S be a non-singular square matrix with s as its first row,
1300 * then we want to map the polyhedra to the space
1302 * [ y' ] [ y ] [ y ] [ y' ]
1303 * [ x' ] = S [ x ] i.e., [ x ] = S^{-1} [ x' ]
1305 * We take S to be the unimodular completion of s to limit the growth
1306 * of the coefficients in the following computations.
1308 * Let b_i + A_i x >= 0 be the constraints of polyhedron i.
1309 * We first move to the homogeneous dimension
1311 * b_i y + A_i x >= 0 [ b_i A_i ] [ y ] [ 0 ]
1312 * y >= 0 or [ 1 0 ] [ x ] >= [ 0 ]
1314 * Then we change directoin
1316 * [ b_i A_i ] [ y' ] [ y' ]
1317 * [ 1 0 ] S^{-1} [ x' ] >= 0 or [ b_i' A_i' ] [ x' ] >= 0
1319 * Then we compute the convex hull of the polytopes b_i' + A_i' x' >= 0
1320 * resulting in b' + A' x' >= 0, which we then convert back
1323 * [ b' A' ] S [ x ] >= 0 or [ b A ] [ x ] >= 0
1325 * The polyhedron b + A x >= 0 is then the convex hull of the input polyhedra.
1327 static struct isl_basic_set *convex_hull_pair_pointed(
1328 struct isl_basic_set *bset1, struct isl_basic_set *bset2)
1330 struct isl_ctx *ctx = NULL;
1331 struct isl_vec *dir = NULL;
1332 struct isl_mat *T = NULL;
1333 struct isl_mat *T2 = NULL;
1334 struct isl_basic_set *hull;
1335 struct isl_set *set;
1337 if (!bset1 || !bset2)
1340 dir = valid_direction(isl_basic_set_copy(bset1),
1341 isl_basic_set_copy(bset2));
1344 T = isl_mat_alloc(bset1->ctx, dir->size, dir->size);
1347 isl_seq_cpy(T->row[0], dir->block.data, dir->size);
1348 T = isl_mat_unimodular_complete(T, 1);
1349 T2 = isl_mat_right_inverse(isl_mat_copy(T));
1351 bset1 = homogeneous_map(bset1, isl_mat_copy(T2));
1352 bset2 = homogeneous_map(bset2, T2);
1353 set = isl_set_alloc_dim(isl_basic_set_get_dim(bset1), 2, 0);
1354 set = isl_set_add(set, bset1);
1355 set = isl_set_add(set, bset2);
1356 hull = uset_convex_hull(set);
1357 hull = isl_basic_set_preimage(hull, T);
1364 isl_basic_set_free(bset1);
1365 isl_basic_set_free(bset2);
1369 /* Compute the convex hull of a pair of basic sets without any parameters or
1370 * integer divisions.
1372 * If the convex hull of the two basic sets would have a non-trivial
1373 * lineality space, we first project out this lineality space.
1375 static struct isl_basic_set *convex_hull_pair(struct isl_basic_set *bset1,
1376 struct isl_basic_set *bset2)
1378 struct isl_basic_set *lin;
1380 if (isl_basic_set_is_bounded(bset1) || isl_basic_set_is_bounded(bset2))
1381 return convex_hull_pair_pointed(bset1, bset2);
1383 lin = induced_lineality_space(isl_basic_set_copy(bset1),
1384 isl_basic_set_copy(bset2));
1387 if (isl_basic_set_is_universe(lin)) {
1388 isl_basic_set_free(bset1);
1389 isl_basic_set_free(bset2);
1392 if (lin->n_eq < isl_basic_set_total_dim(lin)) {
1393 struct isl_set *set;
1394 set = isl_set_alloc_dim(isl_basic_set_get_dim(bset1), 2, 0);
1395 set = isl_set_add(set, bset1);
1396 set = isl_set_add(set, bset2);
1397 return modulo_lineality(set, lin);
1399 isl_basic_set_free(lin);
1401 return convex_hull_pair_pointed(bset1, bset2);
1403 isl_basic_set_free(bset1);
1404 isl_basic_set_free(bset2);
1408 /* Compute the lineality space of a basic set.
1409 * We currently do not allow the basic set to have any divs.
1410 * We basically just drop the constants and turn every inequality
1413 struct isl_basic_set *isl_basic_set_lineality_space(struct isl_basic_set *bset)
1416 struct isl_basic_set *lin = NULL;
1421 isl_assert(bset->ctx, bset->n_div == 0, goto error);
1422 dim = isl_basic_set_total_dim(bset);
1424 lin = isl_basic_set_alloc_dim(isl_basic_set_get_dim(bset), 0, dim, 0);
1427 for (i = 0; i < bset->n_eq; ++i) {
1428 k = isl_basic_set_alloc_equality(lin);
1431 isl_int_set_si(lin->eq[k][0], 0);
1432 isl_seq_cpy(lin->eq[k] + 1, bset->eq[i] + 1, dim);
1434 lin = isl_basic_set_gauss(lin, NULL);
1437 for (i = 0; i < bset->n_ineq && lin->n_eq < dim; ++i) {
1438 k = isl_basic_set_alloc_equality(lin);
1441 isl_int_set_si(lin->eq[k][0], 0);
1442 isl_seq_cpy(lin->eq[k] + 1, bset->ineq[i] + 1, dim);
1443 lin = isl_basic_set_gauss(lin, NULL);
1447 isl_basic_set_free(bset);
1450 isl_basic_set_free(lin);
1451 isl_basic_set_free(bset);
1455 /* Compute the (linear) hull of the lineality spaces of the basic sets in the
1456 * "underlying" set "set".
1458 static struct isl_basic_set *uset_combined_lineality_space(struct isl_set *set)
1461 struct isl_set *lin = NULL;
1466 struct isl_dim *dim = isl_set_get_dim(set);
1468 return isl_basic_set_empty(dim);
1471 lin = isl_set_alloc_dim(isl_set_get_dim(set), set->n, 0);
1472 for (i = 0; i < set->n; ++i)
1473 lin = isl_set_add(lin,
1474 isl_basic_set_lineality_space(isl_basic_set_copy(set->p[i])));
1476 return isl_set_affine_hull(lin);
1479 /* Compute the convex hull of a set without any parameters or
1480 * integer divisions.
1481 * In each step, we combined two basic sets until only one
1482 * basic set is left.
1483 * The input basic sets are assumed not to have a non-trivial
1484 * lineality space. If any of the intermediate results has
1485 * a non-trivial lineality space, it is projected out.
1487 static struct isl_basic_set *uset_convex_hull_unbounded(struct isl_set *set)
1489 struct isl_basic_set *convex_hull = NULL;
1491 convex_hull = isl_set_copy_basic_set(set);
1492 set = isl_set_drop_basic_set(set, convex_hull);
1495 while (set->n > 0) {
1496 struct isl_basic_set *t;
1497 t = isl_set_copy_basic_set(set);
1500 set = isl_set_drop_basic_set(set, t);
1503 convex_hull = convex_hull_pair(convex_hull, t);
1506 t = isl_basic_set_lineality_space(isl_basic_set_copy(convex_hull));
1509 if (isl_basic_set_is_universe(t)) {
1510 isl_basic_set_free(convex_hull);
1514 if (t->n_eq < isl_basic_set_total_dim(t)) {
1515 set = isl_set_add(set, convex_hull);
1516 return modulo_lineality(set, t);
1518 isl_basic_set_free(t);
1524 isl_basic_set_free(convex_hull);
1528 /* Compute an initial hull for wrapping containing a single initial
1529 * facet by first computing bounds on the set and then using these
1530 * bounds to construct an initial facet.
1531 * This function is a remnant of an older implementation where the
1532 * bounds were also used to check whether the set was bounded.
1533 * Since this function will now only be called when we know the
1534 * set to be bounded, the initial facet should probably be constructed
1535 * by simply using the coordinate directions instead.
1537 static struct isl_basic_set *initial_hull(struct isl_basic_set *hull,
1538 struct isl_set *set)
1540 struct isl_mat *bounds = NULL;
1546 bounds = independent_bounds(set);
1549 isl_assert(set->ctx, bounds->n_row == isl_set_n_dim(set), goto error);
1550 bounds = initial_facet_constraint(set, bounds);
1553 k = isl_basic_set_alloc_inequality(hull);
1556 dim = isl_set_n_dim(set);
1557 isl_assert(set->ctx, 1 + dim == bounds->n_col, goto error);
1558 isl_seq_cpy(hull->ineq[k], bounds->row[0], bounds->n_col);
1559 isl_mat_free(bounds);
1563 isl_basic_set_free(hull);
1564 isl_mat_free(bounds);
1568 struct max_constraint {
1574 static int max_constraint_equal(const void *entry, const void *val)
1576 struct max_constraint *a = (struct max_constraint *)entry;
1577 isl_int *b = (isl_int *)val;
1579 return isl_seq_eq(a->c->row[0] + 1, b, a->c->n_col - 1);
1582 static void update_constraint(struct isl_ctx *ctx, struct isl_hash_table *table,
1583 isl_int *con, unsigned len, int n, int ineq)
1585 struct isl_hash_table_entry *entry;
1586 struct max_constraint *c;
1589 c_hash = isl_seq_get_hash(con + 1, len);
1590 entry = isl_hash_table_find(ctx, table, c_hash, max_constraint_equal,
1596 isl_hash_table_remove(ctx, table, entry);
1600 if (isl_int_gt(c->c->row[0][0], con[0]))
1602 if (isl_int_eq(c->c->row[0][0], con[0])) {
1607 c->c = isl_mat_cow(c->c);
1608 isl_int_set(c->c->row[0][0], con[0]);
1612 /* Check whether the constraint hash table "table" constains the constraint
1615 static int has_constraint(struct isl_ctx *ctx, struct isl_hash_table *table,
1616 isl_int *con, unsigned len, int n)
1618 struct isl_hash_table_entry *entry;
1619 struct max_constraint *c;
1622 c_hash = isl_seq_get_hash(con + 1, len);
1623 entry = isl_hash_table_find(ctx, table, c_hash, max_constraint_equal,
1630 return isl_int_eq(c->c->row[0][0], con[0]);
1633 /* Check for inequality constraints of a basic set without equalities
1634 * such that the same or more stringent copies of the constraint appear
1635 * in all of the basic sets. Such constraints are necessarily facet
1636 * constraints of the convex hull.
1638 * If the resulting basic set is by chance identical to one of
1639 * the basic sets in "set", then we know that this basic set contains
1640 * all other basic sets and is therefore the convex hull of set.
1641 * In this case we set *is_hull to 1.
1643 static struct isl_basic_set *common_constraints(struct isl_basic_set *hull,
1644 struct isl_set *set, int *is_hull)
1647 int min_constraints;
1649 struct max_constraint *constraints = NULL;
1650 struct isl_hash_table *table = NULL;
1655 for (i = 0; i < set->n; ++i)
1656 if (set->p[i]->n_eq == 0)
1660 min_constraints = set->p[i]->n_ineq;
1662 for (i = best + 1; i < set->n; ++i) {
1663 if (set->p[i]->n_eq != 0)
1665 if (set->p[i]->n_ineq >= min_constraints)
1667 min_constraints = set->p[i]->n_ineq;
1670 constraints = isl_calloc_array(hull->ctx, struct max_constraint,
1674 table = isl_alloc_type(hull->ctx, struct isl_hash_table);
1675 if (isl_hash_table_init(hull->ctx, table, min_constraints))
1678 total = isl_dim_total(set->dim);
1679 for (i = 0; i < set->p[best]->n_ineq; ++i) {
1680 constraints[i].c = isl_mat_sub_alloc(hull->ctx,
1681 set->p[best]->ineq + i, 0, 1, 0, 1 + total);
1682 if (!constraints[i].c)
1684 constraints[i].ineq = 1;
1686 for (i = 0; i < min_constraints; ++i) {
1687 struct isl_hash_table_entry *entry;
1689 c_hash = isl_seq_get_hash(constraints[i].c->row[0] + 1, total);
1690 entry = isl_hash_table_find(hull->ctx, table, c_hash,
1691 max_constraint_equal, constraints[i].c->row[0] + 1, 1);
1694 isl_assert(hull->ctx, !entry->data, goto error);
1695 entry->data = &constraints[i];
1699 for (s = 0; s < set->n; ++s) {
1703 for (i = 0; i < set->p[s]->n_eq; ++i) {
1704 isl_int *eq = set->p[s]->eq[i];
1705 for (j = 0; j < 2; ++j) {
1706 isl_seq_neg(eq, eq, 1 + total);
1707 update_constraint(hull->ctx, table,
1711 for (i = 0; i < set->p[s]->n_ineq; ++i) {
1712 isl_int *ineq = set->p[s]->ineq[i];
1713 update_constraint(hull->ctx, table, ineq, total, n,
1714 set->p[s]->n_eq == 0);
1719 for (i = 0; i < min_constraints; ++i) {
1720 if (constraints[i].count < n)
1722 if (!constraints[i].ineq)
1724 j = isl_basic_set_alloc_inequality(hull);
1727 isl_seq_cpy(hull->ineq[j], constraints[i].c->row[0], 1 + total);
1730 for (s = 0; s < set->n; ++s) {
1731 if (set->p[s]->n_eq)
1733 if (set->p[s]->n_ineq != hull->n_ineq)
1735 for (i = 0; i < set->p[s]->n_ineq; ++i) {
1736 isl_int *ineq = set->p[s]->ineq[i];
1737 if (!has_constraint(hull->ctx, table, ineq, total, n))
1740 if (i == set->p[s]->n_ineq)
1744 isl_hash_table_clear(table);
1745 for (i = 0; i < min_constraints; ++i)
1746 isl_mat_free(constraints[i].c);
1751 isl_hash_table_clear(table);
1754 for (i = 0; i < min_constraints; ++i)
1755 isl_mat_free(constraints[i].c);
1760 /* Create a template for the convex hull of "set" and fill it up
1761 * obvious facet constraints, if any. If the result happens to
1762 * be the convex hull of "set" then *is_hull is set to 1.
1764 static struct isl_basic_set *proto_hull(struct isl_set *set, int *is_hull)
1766 struct isl_basic_set *hull;
1771 for (i = 0; i < set->n; ++i) {
1772 n_ineq += set->p[i]->n_eq;
1773 n_ineq += set->p[i]->n_ineq;
1775 hull = isl_basic_set_alloc_dim(isl_dim_copy(set->dim), 0, 0, n_ineq);
1776 hull = isl_basic_set_set_rational(hull);
1779 return common_constraints(hull, set, is_hull);
1782 static struct isl_basic_set *uset_convex_hull_wrap(struct isl_set *set)
1784 struct isl_basic_set *hull;
1787 hull = proto_hull(set, &is_hull);
1788 if (hull && !is_hull) {
1789 if (hull->n_ineq == 0)
1790 hull = initial_hull(hull, set);
1791 hull = extend(hull, set);
1798 /* Compute the convex hull of a set without any parameters or
1799 * integer divisions. Depending on whether the set is bounded,
1800 * we pass control to the wrapping based convex hull or
1801 * the Fourier-Motzkin elimination based convex hull.
1802 * We also handle a few special cases before checking the boundedness.
1804 static struct isl_basic_set *uset_convex_hull(struct isl_set *set)
1806 struct isl_basic_set *convex_hull = NULL;
1807 struct isl_basic_set *lin;
1809 if (isl_set_n_dim(set) == 0)
1810 return convex_hull_0d(set);
1812 set = isl_set_coalesce(set);
1813 set = isl_set_set_rational(set);
1820 convex_hull = isl_basic_set_copy(set->p[0]);
1824 if (isl_set_n_dim(set) == 1)
1825 return convex_hull_1d(set);
1827 if (isl_set_is_bounded(set))
1828 return uset_convex_hull_wrap(set);
1830 lin = uset_combined_lineality_space(isl_set_copy(set));
1833 if (isl_basic_set_is_universe(lin)) {
1837 if (lin->n_eq < isl_basic_set_total_dim(lin))
1838 return modulo_lineality(set, lin);
1839 isl_basic_set_free(lin);
1841 return uset_convex_hull_unbounded(set);
1844 isl_basic_set_free(convex_hull);
1848 /* This is the core procedure, where "set" is a "pure" set, i.e.,
1849 * without parameters or divs and where the convex hull of set is
1850 * known to be full-dimensional.
1852 static struct isl_basic_set *uset_convex_hull_wrap_bounded(struct isl_set *set)
1854 struct isl_basic_set *convex_hull = NULL;
1856 if (isl_set_n_dim(set) == 0) {
1857 convex_hull = isl_basic_set_universe(isl_dim_copy(set->dim));
1859 convex_hull = isl_basic_set_set_rational(convex_hull);
1863 set = isl_set_set_rational(set);
1867 set = isl_set_coalesce(set);
1871 convex_hull = isl_basic_set_copy(set->p[0]);
1875 if (isl_set_n_dim(set) == 1)
1876 return convex_hull_1d(set);
1878 return uset_convex_hull_wrap(set);
1884 /* Compute the convex hull of set "set" with affine hull "affine_hull",
1885 * We first remove the equalities (transforming the set), compute the
1886 * convex hull of the transformed set and then add the equalities back
1887 * (after performing the inverse transformation.
1889 static struct isl_basic_set *modulo_affine_hull(struct isl_ctx *ctx,
1890 struct isl_set *set, struct isl_basic_set *affine_hull)
1894 struct isl_basic_set *dummy;
1895 struct isl_basic_set *convex_hull;
1897 dummy = isl_basic_set_remove_equalities(
1898 isl_basic_set_copy(affine_hull), &T, &T2);
1901 isl_basic_set_free(dummy);
1902 set = isl_set_preimage(set, T);
1903 convex_hull = uset_convex_hull(set);
1904 convex_hull = isl_basic_set_preimage(convex_hull, T2);
1905 convex_hull = isl_basic_set_intersect(convex_hull, affine_hull);
1908 isl_basic_set_free(affine_hull);
1913 /* Compute the convex hull of a map.
1915 * The implementation was inspired by "Extended Convex Hull" by Fukuda et al.,
1916 * specifically, the wrapping of facets to obtain new facets.
1918 struct isl_basic_map *isl_map_convex_hull(struct isl_map *map)
1920 struct isl_basic_set *bset;
1921 struct isl_basic_map *model = NULL;
1922 struct isl_basic_set *affine_hull = NULL;
1923 struct isl_basic_map *convex_hull = NULL;
1924 struct isl_set *set = NULL;
1925 struct isl_ctx *ctx;
1932 convex_hull = isl_basic_map_empty_like_map(map);
1937 map = isl_map_detect_equalities(map);
1938 map = isl_map_align_divs(map);
1939 model = isl_basic_map_copy(map->p[0]);
1940 set = isl_map_underlying_set(map);
1944 affine_hull = isl_set_affine_hull(isl_set_copy(set));
1947 if (affine_hull->n_eq != 0)
1948 bset = modulo_affine_hull(ctx, set, affine_hull);
1950 isl_basic_set_free(affine_hull);
1951 bset = uset_convex_hull(set);
1954 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 struct sh_data_entry {
1973 struct isl_hash_table *table;
1974 struct isl_tab *tab;
1977 /* Holds the data needed during the simple hull computation.
1979 * n the number of basic sets in the original set
1980 * hull_table a hash table of already computed constraints
1981 * in the simple hull
1982 * p for each basic set,
1983 * table a hash table of the constraints
1984 * tab the tableau corresponding to the basic set
1987 struct isl_ctx *ctx;
1989 struct isl_hash_table *hull_table;
1990 struct sh_data_entry p[1];
1993 static void sh_data_free(struct sh_data *data)
1999 isl_hash_table_free(data->ctx, data->hull_table);
2000 for (i = 0; i < data->n; ++i) {
2001 isl_hash_table_free(data->ctx, data->p[i].table);
2002 isl_tab_free(data->p[i].tab);
2007 struct ineq_cmp_data {
2012 static int has_ineq(const void *entry, const void *val)
2014 isl_int *row = (isl_int *)entry;
2015 struct ineq_cmp_data *v = (struct ineq_cmp_data *)val;
2017 return isl_seq_eq(row + 1, v->p + 1, v->len) ||
2018 isl_seq_is_neg(row + 1, v->p + 1, v->len);
2021 static int hash_ineq(struct isl_ctx *ctx, struct isl_hash_table *table,
2022 isl_int *ineq, unsigned len)
2025 struct ineq_cmp_data v;
2026 struct isl_hash_table_entry *entry;
2030 c_hash = isl_seq_get_hash(ineq + 1, len);
2031 entry = isl_hash_table_find(ctx, table, c_hash, has_ineq, &v, 1);
2038 /* Fill hash table "table" with the constraints of "bset".
2039 * Equalities are added as two inequalities.
2040 * The value in the hash table is a pointer to the (in)equality of "bset".
2042 static int hash_basic_set(struct isl_hash_table *table,
2043 struct isl_basic_set *bset)
2046 unsigned dim = isl_basic_set_total_dim(bset);
2048 for (i = 0; i < bset->n_eq; ++i) {
2049 for (j = 0; j < 2; ++j) {
2050 isl_seq_neg(bset->eq[i], bset->eq[i], 1 + dim);
2051 if (hash_ineq(bset->ctx, table, bset->eq[i], dim) < 0)
2055 for (i = 0; i < bset->n_ineq; ++i) {
2056 if (hash_ineq(bset->ctx, table, bset->ineq[i], dim) < 0)
2062 static struct sh_data *sh_data_alloc(struct isl_set *set, unsigned n_ineq)
2064 struct sh_data *data;
2067 data = isl_calloc(set->ctx, struct sh_data,
2068 sizeof(struct sh_data) +
2069 (set->n - 1) * sizeof(struct sh_data_entry));
2072 data->ctx = set->ctx;
2074 data->hull_table = isl_hash_table_alloc(set->ctx, n_ineq);
2075 if (!data->hull_table)
2077 for (i = 0; i < set->n; ++i) {
2078 data->p[i].table = isl_hash_table_alloc(set->ctx,
2079 2 * set->p[i]->n_eq + set->p[i]->n_ineq);
2080 if (!data->p[i].table)
2082 if (hash_basic_set(data->p[i].table, set->p[i]) < 0)
2091 /* Check if inequality "ineq" is a bound for basic set "j" or if
2092 * it can be relaxed (by increasing the constant term) to become
2093 * a bound for that basic set. In the latter case, the constant
2095 * Return 1 if "ineq" is a bound
2096 * 0 if "ineq" may attain arbitrarily small values on basic set "j"
2097 * -1 if some error occurred
2099 static int is_bound(struct sh_data *data, struct isl_set *set, int j,
2102 enum isl_lp_result res;
2105 if (!data->p[j].tab) {
2106 data->p[j].tab = isl_tab_from_basic_set(set->p[j]);
2107 if (!data->p[j].tab)
2113 res = isl_tab_min(data->p[j].tab, ineq, data->ctx->one,
2115 if (res == isl_lp_ok && isl_int_is_neg(opt))
2116 isl_int_sub(ineq[0], ineq[0], opt);
2120 return res == isl_lp_ok ? 1 :
2121 res == isl_lp_unbounded ? 0 : -1;
2124 /* Check if inequality "ineq" from basic set "i" can be relaxed to
2125 * become a bound on the whole set. If so, add the (relaxed) inequality
2128 * We first check if "hull" already contains a translate of the inequality.
2129 * If so, we are done.
2130 * Then, we check if any of the previous basic sets contains a translate
2131 * of the inequality. If so, then we have already considered this
2132 * inequality and we are done.
2133 * Otherwise, for each basic set other than "i", we check if the inequality
2134 * is a bound on the basic set.
2135 * For previous basic sets, we know that they do not contain a translate
2136 * of the inequality, so we directly call is_bound.
2137 * For following basic sets, we first check if a translate of the
2138 * inequality appears in its description and if so directly update
2139 * the inequality accordingly.
2141 static struct isl_basic_set *add_bound(struct isl_basic_set *hull,
2142 struct sh_data *data, struct isl_set *set, int i, isl_int *ineq)
2145 struct ineq_cmp_data v;
2146 struct isl_hash_table_entry *entry;
2152 v.len = isl_basic_set_total_dim(hull);
2154 c_hash = isl_seq_get_hash(ineq + 1, v.len);
2156 entry = isl_hash_table_find(hull->ctx, data->hull_table, c_hash,
2161 for (j = 0; j < i; ++j) {
2162 entry = isl_hash_table_find(hull->ctx, data->p[j].table,
2163 c_hash, has_ineq, &v, 0);
2170 k = isl_basic_set_alloc_inequality(hull);
2171 isl_seq_cpy(hull->ineq[k], ineq, 1 + v.len);
2175 for (j = 0; j < i; ++j) {
2177 bound = is_bound(data, set, j, hull->ineq[k]);
2184 isl_basic_set_free_inequality(hull, 1);
2188 for (j = i + 1; j < set->n; ++j) {
2191 entry = isl_hash_table_find(hull->ctx, data->p[j].table,
2192 c_hash, has_ineq, &v, 0);
2194 ineq_j = entry->data;
2195 neg = isl_seq_is_neg(ineq_j + 1,
2196 hull->ineq[k] + 1, v.len);
2198 isl_int_neg(ineq_j[0], ineq_j[0]);
2199 if (isl_int_gt(ineq_j[0], hull->ineq[k][0]))
2200 isl_int_set(hull->ineq[k][0], ineq_j[0]);
2202 isl_int_neg(ineq_j[0], ineq_j[0]);
2205 bound = is_bound(data, set, j, hull->ineq[k]);
2212 isl_basic_set_free_inequality(hull, 1);
2216 entry = isl_hash_table_find(hull->ctx, data->hull_table, c_hash,
2220 entry->data = hull->ineq[k];
2224 isl_basic_set_free(hull);
2228 /* Check if any inequality from basic set "i" can be relaxed to
2229 * become a bound on the whole set. If so, add the (relaxed) inequality
2232 static struct isl_basic_set *add_bounds(struct isl_basic_set *bset,
2233 struct sh_data *data, struct isl_set *set, int i)
2236 unsigned dim = isl_basic_set_total_dim(bset);
2238 for (j = 0; j < set->p[i]->n_eq; ++j) {
2239 for (k = 0; k < 2; ++k) {
2240 isl_seq_neg(set->p[i]->eq[j], set->p[i]->eq[j], 1+dim);
2241 add_bound(bset, data, set, i, set->p[i]->eq[j]);
2244 for (j = 0; j < set->p[i]->n_ineq; ++j)
2245 add_bound(bset, data, set, i, set->p[i]->ineq[j]);
2249 /* Compute a superset of the convex hull of set that is described
2250 * by only translates of the constraints in the constituents of set.
2252 static struct isl_basic_set *uset_simple_hull(struct isl_set *set)
2254 struct sh_data *data = NULL;
2255 struct isl_basic_set *hull = NULL;
2263 for (i = 0; i < set->n; ++i) {
2266 n_ineq += 2 * set->p[i]->n_eq + set->p[i]->n_ineq;
2269 hull = isl_basic_set_alloc_dim(isl_dim_copy(set->dim), 0, 0, n_ineq);
2273 data = sh_data_alloc(set, n_ineq);
2277 for (i = 0; i < set->n; ++i)
2278 hull = add_bounds(hull, data, set, i);
2286 isl_basic_set_free(hull);
2291 /* Compute a superset of the convex hull of map that is described
2292 * by only translates of the constraints in the constituents of map.
2294 struct isl_basic_map *isl_map_simple_hull(struct isl_map *map)
2296 struct isl_set *set = NULL;
2297 struct isl_basic_map *model = NULL;
2298 struct isl_basic_map *hull;
2299 struct isl_basic_map *affine_hull;
2300 struct isl_basic_set *bset = NULL;
2305 hull = isl_basic_map_empty_like_map(map);
2310 hull = isl_basic_map_copy(map->p[0]);
2315 map = isl_map_detect_equalities(map);
2316 affine_hull = isl_map_affine_hull(isl_map_copy(map));
2317 map = isl_map_align_divs(map);
2318 model = isl_basic_map_copy(map->p[0]);
2320 set = isl_map_underlying_set(map);
2322 bset = uset_simple_hull(set);
2324 hull = isl_basic_map_overlying_set(bset, model);
2326 hull = isl_basic_map_intersect(hull, affine_hull);
2327 hull = isl_basic_map_convex_hull(hull);
2328 ISL_F_SET(hull, ISL_BASIC_MAP_NO_IMPLICIT);
2329 ISL_F_SET(hull, ISL_BASIC_MAP_ALL_EQUALITIES);
2334 struct isl_basic_set *isl_set_simple_hull(struct isl_set *set)
2336 return (struct isl_basic_set *)
2337 isl_map_simple_hull((struct isl_map *)set);
2340 /* Given a set "set", return parametric bounds on the dimension "dim".
2342 static struct isl_basic_set *set_bounds(struct isl_set *set, int dim)
2344 unsigned set_dim = isl_set_dim(set, isl_dim_set);
2345 set = isl_set_copy(set);
2346 set = isl_set_eliminate_dims(set, dim + 1, set_dim - (dim + 1));
2347 set = isl_set_eliminate_dims(set, 0, dim);
2348 return isl_set_convex_hull(set);
2351 /* Computes a "simple hull" and then check if each dimension in the
2352 * resulting hull is bounded by a symbolic constant. If not, the
2353 * hull is intersected with the corresponding bounds on the whole set.
2355 struct isl_basic_set *isl_set_bounded_simple_hull(struct isl_set *set)
2358 struct isl_basic_set *hull;
2359 unsigned nparam, left;
2360 int removed_divs = 0;
2362 hull = isl_set_simple_hull(isl_set_copy(set));
2366 nparam = isl_basic_set_dim(hull, isl_dim_param);
2367 for (i = 0; i < isl_basic_set_dim(hull, isl_dim_set); ++i) {
2368 int lower = 0, upper = 0;
2369 struct isl_basic_set *bounds;
2371 left = isl_basic_set_total_dim(hull) - nparam - i - 1;
2372 for (j = 0; j < hull->n_eq; ++j) {
2373 if (isl_int_is_zero(hull->eq[j][1 + nparam + i]))
2375 if (isl_seq_first_non_zero(hull->eq[j]+1+nparam+i+1,
2382 for (j = 0; j < hull->n_ineq; ++j) {
2383 if (isl_int_is_zero(hull->ineq[j][1 + nparam + i]))
2385 if (isl_seq_first_non_zero(hull->ineq[j]+1+nparam+i+1,
2387 isl_seq_first_non_zero(hull->ineq[j]+1+nparam,
2390 if (isl_int_is_pos(hull->ineq[j][1 + nparam + i]))
2401 if (!removed_divs) {
2402 set = isl_set_remove_divs(set);
2407 bounds = set_bounds(set, i);
2408 hull = isl_basic_set_intersect(hull, bounds);