3 #include "isl_map_private.h"
7 #include "isl_equalities.h"
10 static struct isl_basic_set *uset_convex_hull_wrap_bounded(struct isl_set *set);
12 static void swap_ineq(struct isl_basic_map *bmap, unsigned i, unsigned j)
18 bmap->ineq[i] = bmap->ineq[j];
23 /* Return 1 if constraint c is redundant with respect to the constraints
24 * in bmap. If c is a lower [upper] bound in some variable and bmap
25 * does not have a lower [upper] bound in that variable, then c cannot
26 * be redundant and we do not need solve any lp.
28 int isl_basic_map_constraint_is_redundant(struct isl_basic_map **bmap,
29 isl_int *c, isl_int *opt_n, isl_int *opt_d)
31 enum isl_lp_result res;
38 total = isl_basic_map_total_dim(*bmap);
39 for (i = 0; i < total; ++i) {
41 if (isl_int_is_zero(c[1+i]))
43 sign = isl_int_sgn(c[1+i]);
44 for (j = 0; j < (*bmap)->n_ineq; ++j)
45 if (sign == isl_int_sgn((*bmap)->ineq[j][1+i]))
47 if (j == (*bmap)->n_ineq)
53 res = isl_solve_lp(*bmap, 0, c, (*bmap)->ctx->one, opt_n, opt_d);
54 if (res == isl_lp_unbounded)
56 if (res == isl_lp_error)
58 if (res == isl_lp_empty) {
59 *bmap = isl_basic_map_set_to_empty(*bmap);
62 return !isl_int_is_neg(*opt_n);
65 int isl_basic_set_constraint_is_redundant(struct isl_basic_set **bset,
66 isl_int *c, isl_int *opt_n, isl_int *opt_d)
68 return isl_basic_map_constraint_is_redundant(
69 (struct isl_basic_map **)bset, c, opt_n, opt_d);
72 /* Compute the convex hull of a basic map, by removing the redundant
73 * constraints. If the minimal value along the normal of a constraint
74 * is the same if the constraint is removed, then the constraint is redundant.
76 * Alternatively, we could have intersected the basic map with the
77 * corresponding equality and the checked if the dimension was that
80 struct isl_basic_map *isl_basic_map_convex_hull(struct isl_basic_map *bmap)
87 bmap = isl_basic_map_gauss(bmap, NULL);
88 if (ISL_F_ISSET(bmap, ISL_BASIC_MAP_EMPTY))
90 if (ISL_F_ISSET(bmap, ISL_BASIC_MAP_NO_REDUNDANT))
92 if (bmap->n_ineq <= 1)
95 tab = isl_tab_from_basic_map(bmap);
96 tab = isl_tab_detect_equalities(bmap->ctx, tab);
97 tab = isl_tab_detect_redundant(bmap->ctx, tab);
98 bmap = isl_basic_map_update_from_tab(bmap, tab);
99 isl_tab_free(bmap->ctx, tab);
100 ISL_F_SET(bmap, ISL_BASIC_MAP_NO_IMPLICIT);
101 ISL_F_SET(bmap, ISL_BASIC_MAP_NO_REDUNDANT);
105 struct isl_basic_set *isl_basic_set_convex_hull(struct isl_basic_set *bset)
107 return (struct isl_basic_set *)
108 isl_basic_map_convex_hull((struct isl_basic_map *)bset);
111 /* Check if the set set is bound in the direction of the affine
112 * constraint c and if so, set the constant term such that the
113 * resulting constraint is a bounding constraint for the set.
115 static int uset_is_bound(struct isl_ctx *ctx, struct isl_set *set,
116 isl_int *c, unsigned len)
124 isl_int_init(opt_denom);
126 for (j = 0; j < set->n; ++j) {
127 enum isl_lp_result res;
129 if (ISL_F_ISSET(set->p[j], ISL_BASIC_SET_EMPTY))
132 res = isl_solve_lp((struct isl_basic_map*)set->p[j],
133 0, c, ctx->one, &opt, &opt_denom);
134 if (res == isl_lp_unbounded)
136 if (res == isl_lp_error)
138 if (res == isl_lp_empty) {
139 set->p[j] = isl_basic_set_set_to_empty(set->p[j]);
144 if (!isl_int_is_one(opt_denom))
145 isl_seq_scale(c, c, opt_denom, len);
146 if (first || isl_int_is_neg(opt))
147 isl_int_sub(c[0], c[0], opt);
151 isl_int_clear(opt_denom);
155 isl_int_clear(opt_denom);
159 /* Check if "c" is a direction that is independent of the previously found "n"
161 * If so, add it to the list, with the negative of the lower bound
162 * in the constant position, i.e., such that c corresponds to a bounding
163 * hyperplane (but not necessarily a facet).
164 * Assumes set "set" is bounded.
166 static int is_independent_bound(struct isl_ctx *ctx,
167 struct isl_set *set, isl_int *c,
168 struct isl_mat *dirs, int n)
173 isl_seq_cpy(dirs->row[n]+1, c+1, dirs->n_col-1);
175 int pos = isl_seq_first_non_zero(dirs->row[n]+1, dirs->n_col-1);
178 for (i = 0; i < n; ++i) {
180 pos_i = isl_seq_first_non_zero(dirs->row[i]+1, dirs->n_col-1);
185 isl_seq_elim(dirs->row[n]+1, dirs->row[i]+1, pos,
186 dirs->n_col-1, NULL);
187 pos = isl_seq_first_non_zero(dirs->row[n]+1, dirs->n_col-1);
193 is_bound = uset_is_bound(ctx, set, dirs->row[n], dirs->n_col);
198 isl_int *t = dirs->row[n];
199 for (k = n; k > i; --k)
200 dirs->row[k] = dirs->row[k-1];
206 /* Compute and return a maximal set of linearly independent bounds
207 * on the set "set", based on the constraints of the basic sets
210 static struct isl_mat *independent_bounds(struct isl_ctx *ctx,
214 struct isl_mat *dirs = NULL;
215 unsigned dim = isl_set_n_dim(set);
217 dirs = isl_mat_alloc(ctx, dim, 1+dim);
222 for (i = 0; n < dim && i < set->n; ++i) {
224 struct isl_basic_set *bset = set->p[i];
226 for (j = 0; n < dim && j < bset->n_eq; ++j) {
227 f = is_independent_bound(ctx, set, bset->eq[j],
234 for (j = 0; n < dim && j < bset->n_ineq; ++j) {
235 f = is_independent_bound(ctx, set, bset->ineq[j],
246 isl_mat_free(ctx, dirs);
250 static struct isl_basic_set *isl_basic_set_set_rational(
251 struct isl_basic_set *bset)
256 if (ISL_F_ISSET(bset, ISL_BASIC_MAP_RATIONAL))
259 bset = isl_basic_set_cow(bset);
263 ISL_F_SET(bset, ISL_BASIC_MAP_RATIONAL);
265 return isl_basic_set_finalize(bset);
268 static struct isl_set *isl_set_set_rational(struct isl_set *set)
272 set = isl_set_cow(set);
275 for (i = 0; i < set->n; ++i) {
276 set->p[i] = isl_basic_set_set_rational(set->p[i]);
286 static struct isl_basic_set *isl_basic_set_add_equality(struct isl_ctx *ctx,
287 struct isl_basic_set *bset, isl_int *c)
293 if (ISL_F_ISSET(bset, ISL_BASIC_SET_EMPTY))
296 isl_assert(ctx, isl_basic_set_n_param(bset) == 0, goto error);
297 isl_assert(ctx, bset->n_div == 0, goto error);
298 dim = isl_basic_set_n_dim(bset);
299 bset = isl_basic_set_cow(bset);
300 bset = isl_basic_set_extend(bset, 0, dim, 0, 1, 0);
301 i = isl_basic_set_alloc_equality(bset);
304 isl_seq_cpy(bset->eq[i], c, 1 + dim);
307 isl_basic_set_free(bset);
311 static struct isl_set *isl_set_add_equality(struct isl_ctx *ctx,
312 struct isl_set *set, isl_int *c)
316 set = isl_set_cow(set);
319 for (i = 0; i < set->n; ++i) {
320 set->p[i] = isl_basic_set_add_equality(ctx, set->p[i], c);
330 /* Given a union of basic sets, construct the constraints for wrapping
331 * a facet around one of its ridges.
332 * In particular, if each of n the d-dimensional basic sets i in "set"
333 * contains the origin, satisfies the constraints x_1 >= 0 and x_2 >= 0
334 * and is defined by the constraints
338 * then the resulting set is of dimension n*(1+d) and has as constraints
347 static struct isl_basic_set *wrap_constraints(struct isl_set *set)
349 struct isl_basic_set *lp;
353 unsigned dim, lp_dim;
358 dim = 1 + isl_set_n_dim(set);
361 for (i = 0; i < set->n; ++i) {
362 n_eq += set->p[i]->n_eq;
363 n_ineq += set->p[i]->n_ineq;
365 lp = isl_basic_set_alloc(set->ctx, 0, dim * set->n, 0, n_eq, n_ineq);
368 lp_dim = isl_basic_set_n_dim(lp);
369 k = isl_basic_set_alloc_equality(lp);
370 isl_int_set_si(lp->eq[k][0], -1);
371 for (i = 0; i < set->n; ++i) {
372 isl_int_set_si(lp->eq[k][1+dim*i], 0);
373 isl_int_set_si(lp->eq[k][1+dim*i+1], 1);
374 isl_seq_clr(lp->eq[k]+1+dim*i+2, dim-2);
376 for (i = 0; i < set->n; ++i) {
377 k = isl_basic_set_alloc_inequality(lp);
378 isl_seq_clr(lp->ineq[k], 1+lp_dim);
379 isl_int_set_si(lp->ineq[k][1+dim*i], 1);
381 for (j = 0; j < set->p[i]->n_eq; ++j) {
382 k = isl_basic_set_alloc_equality(lp);
383 isl_seq_clr(lp->eq[k], 1+dim*i);
384 isl_seq_cpy(lp->eq[k]+1+dim*i, set->p[i]->eq[j], dim);
385 isl_seq_clr(lp->eq[k]+1+dim*(i+1), dim*(set->n-i-1));
388 for (j = 0; j < set->p[i]->n_ineq; ++j) {
389 k = isl_basic_set_alloc_inequality(lp);
390 isl_seq_clr(lp->ineq[k], 1+dim*i);
391 isl_seq_cpy(lp->ineq[k]+1+dim*i, set->p[i]->ineq[j], dim);
392 isl_seq_clr(lp->ineq[k]+1+dim*(i+1), dim*(set->n-i-1));
398 /* Given a facet "facet" of the convex hull of "set" and a facet "ridge"
399 * of that facet, compute the other facet of the convex hull that contains
402 * We first transform the set such that the facet constraint becomes
406 * I.e., the facet lies in
410 * and on that facet, the constraint that defines the ridge is
414 * (This transformation is not strictly needed, all that is needed is
415 * that the ridge contains the origin.)
417 * Since the ridge contains the origin, the cone of the convex hull
418 * will be of the form
423 * with this second constraint defining the new facet.
424 * The constant a is obtained by settting x_1 in the cone of the
425 * convex hull to 1 and minimizing x_2.
426 * Now, each element in the cone of the convex hull is the sum
427 * of elements in the cones of the basic sets.
428 * If a_i is the dilation factor of basic set i, then the problem
429 * we need to solve is
442 * the constraints of each (transformed) basic set.
443 * If a = n/d, then the constraint defining the new facet (in the transformed
446 * -n x_1 + d x_2 >= 0
448 * In the original space, we need to take the same combination of the
449 * corresponding constraints "facet" and "ridge".
451 * Note that a is always finite, since we only apply the wrapping
452 * technique to a union of polytopes.
454 static isl_int *wrap_facet(struct isl_set *set, isl_int *facet, isl_int *ridge)
457 struct isl_mat *T = NULL;
458 struct isl_basic_set *lp = NULL;
460 enum isl_lp_result res;
464 set = isl_set_copy(set);
466 dim = 1 + isl_set_n_dim(set);
467 T = isl_mat_alloc(set->ctx, 3, dim);
470 isl_int_set_si(T->row[0][0], 1);
471 isl_seq_clr(T->row[0]+1, dim - 1);
472 isl_seq_cpy(T->row[1], facet, dim);
473 isl_seq_cpy(T->row[2], ridge, dim);
474 T = isl_mat_right_inverse(set->ctx, T);
475 set = isl_set_preimage(set, T);
479 lp = wrap_constraints(set);
480 obj = isl_vec_alloc(set->ctx, 1 + dim*set->n);
483 isl_int_set_si(obj->block.data[0], 0);
484 for (i = 0; i < set->n; ++i) {
485 isl_seq_clr(obj->block.data + 1 + dim*i, 2);
486 isl_int_set_si(obj->block.data[1 + dim*i+2], 1);
487 isl_seq_clr(obj->block.data + 1 + dim*i+3, dim-3);
491 res = isl_solve_lp((struct isl_basic_map *)lp, 0,
492 obj->block.data, set->ctx->one, &num, &den);
493 if (res == isl_lp_ok) {
494 isl_int_neg(num, num);
495 isl_seq_combine(facet, num, facet, den, ridge, dim);
500 isl_basic_set_free(lp);
502 isl_assert(set->ctx, res == isl_lp_ok, return NULL);
505 isl_basic_set_free(lp);
506 isl_mat_free(set->ctx, T);
511 /* Given a set of d linearly independent bounding constraints of the
512 * convex hull of "set", compute the constraint of a facet of "set".
514 * We first compute the intersection with the first bounding hyperplane
515 * and remove the component corresponding to this hyperplane from
516 * other bounds (in homogeneous space).
517 * We then wrap around one of the remaining bounding constraints
518 * and continue the process until all bounding constraints have been
519 * taken into account.
520 * The resulting linear combination of the bounding constraints will
521 * correspond to a facet of the convex hull.
523 static struct isl_mat *initial_facet_constraint(struct isl_ctx *ctx,
524 struct isl_set *set, struct isl_mat *bounds)
526 struct isl_set *slice = NULL;
527 struct isl_basic_set *face = NULL;
528 struct isl_mat *m, *U, *Q;
530 unsigned dim = isl_set_n_dim(set);
532 isl_assert(ctx, set->n > 0, goto error);
533 isl_assert(ctx, bounds->n_row == dim, goto error);
535 while (bounds->n_row > 1) {
536 slice = isl_set_copy(set);
537 slice = isl_set_add_equality(ctx, slice, bounds->row[0]);
538 face = isl_set_affine_hull(slice);
541 if (face->n_eq == 1) {
542 isl_basic_set_free(face);
545 m = isl_mat_alloc(ctx, 1 + face->n_eq, 1 + dim);
548 isl_int_set_si(m->row[0][0], 1);
549 isl_seq_clr(m->row[0]+1, dim);
550 for (i = 0; i < face->n_eq; ++i)
551 isl_seq_cpy(m->row[1 + i], face->eq[i], 1 + dim);
552 U = isl_mat_right_inverse(ctx, m);
553 Q = isl_mat_right_inverse(ctx, isl_mat_copy(ctx, U));
554 U = isl_mat_drop_cols(ctx, U, 1 + face->n_eq,
556 Q = isl_mat_drop_rows(ctx, Q, 1 + face->n_eq,
558 U = isl_mat_drop_cols(ctx, U, 0, 1);
559 Q = isl_mat_drop_rows(ctx, Q, 0, 1);
560 bounds = isl_mat_product(ctx, bounds, U);
561 bounds = isl_mat_product(ctx, bounds, Q);
562 while (isl_seq_first_non_zero(bounds->row[bounds->n_row-1],
563 bounds->n_col) == -1) {
565 isl_assert(ctx, bounds->n_row > 1, goto error);
567 if (!wrap_facet(set, bounds->row[0],
568 bounds->row[bounds->n_row-1]))
570 isl_basic_set_free(face);
575 isl_basic_set_free(face);
576 isl_mat_free(ctx, bounds);
580 /* Given the bounding constraint "c" of a facet of the convex hull of "set",
581 * compute a hyperplane description of the facet, i.e., compute the facets
584 * We compute an affine transformation that transforms the constraint
593 * by computing the right inverse U of a matrix that starts with the rows
606 * Since z_1 is zero, we can drop this variable as well as the corresponding
607 * column of U to obtain
615 * with Q' equal to Q, but without the corresponding row.
616 * After computing the facets of the facet in the z' space,
617 * we convert them back to the x space through Q.
619 static struct isl_basic_set *compute_facet(struct isl_set *set, isl_int *c)
621 struct isl_mat *m, *U, *Q;
622 struct isl_basic_set *facet = NULL;
627 set = isl_set_copy(set);
628 dim = isl_set_n_dim(set);
629 m = isl_mat_alloc(set->ctx, 2, 1 + dim);
632 isl_int_set_si(m->row[0][0], 1);
633 isl_seq_clr(m->row[0]+1, dim);
634 isl_seq_cpy(m->row[1], c, 1+dim);
635 U = isl_mat_right_inverse(set->ctx, m);
636 Q = isl_mat_right_inverse(set->ctx, isl_mat_copy(set->ctx, U));
637 U = isl_mat_drop_cols(set->ctx, U, 1, 1);
638 Q = isl_mat_drop_rows(set->ctx, Q, 1, 1);
639 set = isl_set_preimage(set, U);
640 facet = uset_convex_hull_wrap_bounded(set);
641 facet = isl_basic_set_preimage(facet, Q);
642 isl_assert(ctx, facet->n_eq == 0, goto error);
645 isl_basic_set_free(facet);
650 /* Given an initial facet constraint, compute the remaining facets.
651 * We do this by running through all facets found so far and computing
652 * the adjacent facets through wrapping, adding those facets that we
653 * hadn't already found before.
655 * For each facet we have found so far, we first compute its facets
656 * in the resulting convex hull. That is, we compute the ridges
657 * of the resulting convex hull contained in the facet.
658 * We also compute the corresponding facet in the current approximation
659 * of the convex hull. There is no need to wrap around the ridges
660 * in this facet since that would result in a facet that is already
661 * present in the current approximation.
663 * This function can still be significantly optimized by checking which of
664 * the facets of the basic sets are also facets of the convex hull and
665 * using all the facets so far to help in constructing the facets of the
668 * using the technique in section "3.1 Ridge Generation" of
669 * "Extended Convex Hull" by Fukuda et al.
671 static struct isl_basic_set *extend(struct isl_basic_set *hull,
676 struct isl_basic_set *facet = NULL;
677 struct isl_basic_set *hull_facet = NULL;
681 isl_assert(set->ctx, set->n > 0, goto error);
683 dim = isl_set_n_dim(set);
685 for (i = 0; i < hull->n_ineq; ++i) {
686 facet = compute_facet(set, hull->ineq[i]);
687 facet = isl_basic_set_add_equality(facet->ctx, facet, hull->ineq[i]);
688 facet = isl_basic_set_gauss(facet, NULL);
689 facet = isl_basic_set_normalize_constraints(facet);
690 hull_facet = isl_basic_set_copy(hull);
691 hull_facet = isl_basic_set_add_equality(hull_facet->ctx, hull_facet, hull->ineq[i]);
692 hull_facet = isl_basic_set_gauss(hull_facet, NULL);
693 hull_facet = isl_basic_set_normalize_constraints(hull_facet);
696 hull = isl_basic_set_cow(hull);
697 hull = isl_basic_set_extend_dim(hull,
698 isl_dim_copy(hull->dim), 0, 0, facet->n_ineq);
699 for (j = 0; j < facet->n_ineq; ++j) {
700 for (f = 0; f < hull_facet->n_ineq; ++f)
701 if (isl_seq_eq(facet->ineq[j],
702 hull_facet->ineq[f], 1 + dim))
704 if (f < hull_facet->n_ineq)
706 k = isl_basic_set_alloc_inequality(hull);
709 isl_seq_cpy(hull->ineq[k], hull->ineq[i], 1+dim);
710 if (!wrap_facet(set, hull->ineq[k], facet->ineq[j]))
713 isl_basic_set_free(hull_facet);
714 isl_basic_set_free(facet);
716 hull = isl_basic_set_simplify(hull);
717 hull = isl_basic_set_finalize(hull);
720 isl_basic_set_free(hull_facet);
721 isl_basic_set_free(facet);
722 isl_basic_set_free(hull);
726 /* Special case for computing the convex hull of a one dimensional set.
727 * We simply collect the lower and upper bounds of each basic set
728 * and the biggest of those.
730 static struct isl_basic_set *convex_hull_1d(struct isl_ctx *ctx,
733 struct isl_mat *c = NULL;
734 isl_int *lower = NULL;
735 isl_int *upper = NULL;
738 struct isl_basic_set *hull;
740 for (i = 0; i < set->n; ++i) {
741 set->p[i] = isl_basic_set_simplify(set->p[i]);
745 set = isl_set_remove_empty_parts(set);
748 isl_assert(ctx, set->n > 0, goto error);
749 c = isl_mat_alloc(ctx, 2, 2);
753 if (set->p[0]->n_eq > 0) {
754 isl_assert(ctx, set->p[0]->n_eq == 1, goto error);
757 if (isl_int_is_pos(set->p[0]->eq[0][1])) {
758 isl_seq_cpy(lower, set->p[0]->eq[0], 2);
759 isl_seq_neg(upper, set->p[0]->eq[0], 2);
761 isl_seq_neg(lower, set->p[0]->eq[0], 2);
762 isl_seq_cpy(upper, set->p[0]->eq[0], 2);
765 for (j = 0; j < set->p[0]->n_ineq; ++j) {
766 if (isl_int_is_pos(set->p[0]->ineq[j][1])) {
768 isl_seq_cpy(lower, set->p[0]->ineq[j], 2);
771 isl_seq_cpy(upper, set->p[0]->ineq[j], 2);
778 for (i = 0; i < set->n; ++i) {
779 struct isl_basic_set *bset = set->p[i];
783 for (j = 0; j < bset->n_eq; ++j) {
787 isl_int_mul(a, lower[0], bset->eq[j][1]);
788 isl_int_mul(b, lower[1], bset->eq[j][0]);
789 if (isl_int_lt(a, b) && isl_int_is_pos(bset->eq[j][1]))
790 isl_seq_cpy(lower, bset->eq[j], 2);
791 if (isl_int_gt(a, b) && isl_int_is_neg(bset->eq[j][1]))
792 isl_seq_neg(lower, bset->eq[j], 2);
795 isl_int_mul(a, upper[0], bset->eq[j][1]);
796 isl_int_mul(b, upper[1], bset->eq[j][0]);
797 if (isl_int_lt(a, b) && isl_int_is_pos(bset->eq[j][1]))
798 isl_seq_neg(upper, bset->eq[j], 2);
799 if (isl_int_gt(a, b) && isl_int_is_neg(bset->eq[j][1]))
800 isl_seq_cpy(upper, bset->eq[j], 2);
803 for (j = 0; j < bset->n_ineq; ++j) {
804 if (isl_int_is_pos(bset->ineq[j][1]))
806 if (isl_int_is_neg(bset->ineq[j][1]))
808 if (lower && isl_int_is_pos(bset->ineq[j][1])) {
809 isl_int_mul(a, lower[0], bset->ineq[j][1]);
810 isl_int_mul(b, lower[1], bset->ineq[j][0]);
811 if (isl_int_lt(a, b))
812 isl_seq_cpy(lower, bset->ineq[j], 2);
814 if (upper && isl_int_is_neg(bset->ineq[j][1])) {
815 isl_int_mul(a, upper[0], bset->ineq[j][1]);
816 isl_int_mul(b, upper[1], bset->ineq[j][0]);
817 if (isl_int_gt(a, b))
818 isl_seq_cpy(upper, bset->ineq[j], 2);
829 hull = isl_basic_set_alloc(ctx, 0, 1, 0, 0, 2);
830 hull = isl_basic_set_set_rational(hull);
834 k = isl_basic_set_alloc_inequality(hull);
835 isl_seq_cpy(hull->ineq[k], lower, 2);
838 k = isl_basic_set_alloc_inequality(hull);
839 isl_seq_cpy(hull->ineq[k], upper, 2);
841 hull = isl_basic_set_finalize(hull);
843 isl_mat_free(ctx, c);
847 isl_mat_free(ctx, c);
851 /* Project out final n dimensions using Fourier-Motzkin */
852 static struct isl_set *set_project_out(struct isl_ctx *ctx,
853 struct isl_set *set, unsigned n)
855 return isl_set_remove_dims(set, isl_set_n_dim(set) - n, n);
858 static struct isl_basic_set *convex_hull_0d(struct isl_set *set)
860 struct isl_basic_set *convex_hull;
865 if (isl_set_is_empty(set))
866 convex_hull = isl_basic_set_empty(isl_dim_copy(set->dim));
868 convex_hull = isl_basic_set_universe(isl_dim_copy(set->dim));
873 /* Compute the convex hull of a pair of basic sets without any parameters or
874 * integer divisions using Fourier-Motzkin elimination.
875 * The convex hull is the set of all points that can be written as
876 * the sum of points from both basic sets (in homogeneous coordinates).
877 * We set up the constraints in a space with dimensions for each of
878 * the three sets and then project out the dimensions corresponding
879 * to the two original basic sets, retaining only those corresponding
880 * to the convex hull.
882 static struct isl_basic_set *convex_hull_pair_elim(struct isl_basic_set *bset1,
883 struct isl_basic_set *bset2)
886 struct isl_basic_set *bset[2];
887 struct isl_basic_set *hull = NULL;
890 if (!bset1 || !bset2)
893 dim = isl_basic_set_n_dim(bset1);
894 hull = isl_basic_set_alloc(bset1->ctx, 0, 2 + 3 * dim, 0,
895 1 + dim + bset1->n_eq + bset2->n_eq,
896 2 + bset1->n_ineq + bset2->n_ineq);
899 for (i = 0; i < 2; ++i) {
900 for (j = 0; j < bset[i]->n_eq; ++j) {
901 k = isl_basic_set_alloc_equality(hull);
904 isl_seq_clr(hull->eq[k], (i+1) * (1+dim));
905 isl_seq_clr(hull->eq[k]+(i+2)*(1+dim), (1-i)*(1+dim));
906 isl_seq_cpy(hull->eq[k]+(i+1)*(1+dim), bset[i]->eq[j],
909 for (j = 0; j < bset[i]->n_ineq; ++j) {
910 k = isl_basic_set_alloc_inequality(hull);
913 isl_seq_clr(hull->ineq[k], (i+1) * (1+dim));
914 isl_seq_clr(hull->ineq[k]+(i+2)*(1+dim), (1-i)*(1+dim));
915 isl_seq_cpy(hull->ineq[k]+(i+1)*(1+dim),
916 bset[i]->ineq[j], 1+dim);
918 k = isl_basic_set_alloc_inequality(hull);
921 isl_seq_clr(hull->ineq[k], 1+2+3*dim);
922 isl_int_set_si(hull->ineq[k][(i+1)*(1+dim)], 1);
924 for (j = 0; j < 1+dim; ++j) {
925 k = isl_basic_set_alloc_equality(hull);
928 isl_seq_clr(hull->eq[k], 1+2+3*dim);
929 isl_int_set_si(hull->eq[k][j], -1);
930 isl_int_set_si(hull->eq[k][1+dim+j], 1);
931 isl_int_set_si(hull->eq[k][2*(1+dim)+j], 1);
933 hull = isl_basic_set_set_rational(hull);
934 hull = isl_basic_set_remove_dims(hull, dim, 2*(1+dim));
935 hull = isl_basic_set_convex_hull(hull);
936 isl_basic_set_free(bset1);
937 isl_basic_set_free(bset2);
940 isl_basic_set_free(bset1);
941 isl_basic_set_free(bset2);
942 isl_basic_set_free(hull);
946 static int isl_basic_set_is_bounded(struct isl_basic_set *bset)
951 tab = isl_tab_from_recession_cone((struct isl_basic_map *)bset);
952 bounded = isl_tab_cone_is_bounded(bset->ctx, tab);
953 isl_tab_free(bset->ctx, tab);
957 static int isl_set_is_bounded(struct isl_set *set)
961 for (i = 0; i < set->n; ++i) {
962 int bounded = isl_basic_set_is_bounded(set->p[i]);
963 if (!bounded || bounded < 0)
969 /* Compute the lineality space of the convex hull of bset1 and bset2.
971 * We first compute the intersection of the recession cone of bset1
972 * with the negative of the recession cone of bset2 and then compute
973 * the linear hull of the resulting cone.
975 static struct isl_basic_set *induced_lineality_space(
976 struct isl_basic_set *bset1, struct isl_basic_set *bset2)
979 struct isl_basic_set *lin = NULL;
982 if (!bset1 || !bset2)
985 dim = isl_basic_set_total_dim(bset1);
986 lin = isl_basic_set_alloc_dim(isl_basic_set_get_dim(bset1), 0,
987 bset1->n_eq + bset2->n_eq,
988 bset1->n_ineq + bset2->n_ineq);
989 lin = isl_basic_set_set_rational(lin);
992 for (i = 0; i < bset1->n_eq; ++i) {
993 k = isl_basic_set_alloc_equality(lin);
996 isl_int_set_si(lin->eq[k][0], 0);
997 isl_seq_cpy(lin->eq[k] + 1, bset1->eq[i] + 1, dim);
999 for (i = 0; i < bset1->n_ineq; ++i) {
1000 k = isl_basic_set_alloc_inequality(lin);
1003 isl_int_set_si(lin->ineq[k][0], 0);
1004 isl_seq_cpy(lin->ineq[k] + 1, bset1->ineq[i] + 1, dim);
1006 for (i = 0; i < bset2->n_eq; ++i) {
1007 k = isl_basic_set_alloc_equality(lin);
1010 isl_int_set_si(lin->eq[k][0], 0);
1011 isl_seq_neg(lin->eq[k] + 1, bset2->eq[i] + 1, dim);
1013 for (i = 0; i < bset2->n_ineq; ++i) {
1014 k = isl_basic_set_alloc_inequality(lin);
1017 isl_int_set_si(lin->ineq[k][0], 0);
1018 isl_seq_neg(lin->ineq[k] + 1, bset2->ineq[i] + 1, dim);
1021 isl_basic_set_free(bset1);
1022 isl_basic_set_free(bset2);
1023 return isl_basic_set_affine_hull(lin);
1025 isl_basic_set_free(lin);
1026 isl_basic_set_free(bset1);
1027 isl_basic_set_free(bset2);
1031 static struct isl_basic_set *uset_convex_hull(struct isl_set *set);
1033 /* Given a set and a linear space "lin" of dimension n > 0,
1034 * project the linear space from the set, compute the convex hull
1035 * and then map the set back to the original space.
1041 * describe the linear space. We first compute the Hermite normal
1042 * form H = M U of M = H Q, to obtain
1046 * The last n rows of H will be zero, so the last n variables of x' = Q x
1047 * are the one we want to project out. We do this by transforming each
1048 * basic set A x >= b to A U x' >= b and then removing the last n dimensions.
1049 * After computing the convex hull in x'_1, i.e., A' x'_1 >= b',
1050 * we transform the hull back to the original space as A' Q_1 x >= b',
1051 * with Q_1 all but the last n rows of Q.
1053 static struct isl_basic_set *modulo_lineality(struct isl_set *set,
1054 struct isl_basic_set *lin)
1056 unsigned total = isl_basic_set_total_dim(lin);
1058 struct isl_basic_set *hull;
1059 struct isl_mat *M, *U, *Q;
1063 lin_dim = total - lin->n_eq;
1064 M = isl_mat_sub_alloc(set->ctx, lin->eq, 0, lin->n_eq, 1, total);
1065 M = isl_mat_left_hermite(set->ctx, M, 0, &U, &Q);
1068 isl_mat_free(set->ctx, M);
1069 isl_basic_set_free(lin);
1071 isl_mat_drop_rows(set->ctx, Q, Q->n_row - lin_dim, lin_dim);
1073 U = isl_mat_lin_to_aff(set->ctx, U);
1074 Q = isl_mat_lin_to_aff(set->ctx, Q);
1076 set = isl_set_preimage(set, U);
1077 set = isl_set_remove_dims(set, total - lin_dim, lin_dim);
1078 hull = uset_convex_hull(set);
1079 hull = isl_basic_set_preimage(hull, Q);
1083 isl_basic_set_free(lin);
1088 /* Given two polyhedra with as constraints h_{ij} x >= 0 in homegeneous space,
1089 * set up an LP for solving
1091 * \sum_j \alpha_{1j} h_{1j} = \sum_j \alpha_{2j} h_{2j}
1093 * \alpha{i0} corresponds to the (implicit) positivity constraint 1 >= 0
1094 * The next \alpha{ij} correspond to the equalities and come in pairs.
1095 * The final \alpha{ij} correspond to the inequalities.
1097 static struct isl_basic_set *valid_direction_lp(
1098 struct isl_basic_set *bset1, struct isl_basic_set *bset2)
1100 struct isl_dim *dim;
1101 struct isl_basic_set *lp;
1106 if (!bset1 || !bset2)
1108 d = 1 + isl_basic_set_total_dim(bset1);
1110 2 * bset1->n_eq + bset1->n_ineq + 2 * bset2->n_eq + bset2->n_ineq;
1111 dim = isl_dim_set_alloc(bset1->ctx, 0, n);
1112 lp = isl_basic_set_alloc_dim(dim, 0, d, n);
1115 for (i = 0; i < n; ++i) {
1116 k = isl_basic_set_alloc_inequality(lp);
1119 isl_seq_clr(lp->ineq[k] + 1, n);
1120 isl_int_set_si(lp->ineq[k][0], -1);
1121 isl_int_set_si(lp->ineq[k][1 + i], 1);
1123 for (i = 0; i < d; ++i) {
1124 k = isl_basic_set_alloc_equality(lp);
1128 isl_int_set_si(lp->eq[k][n++], 0);
1129 /* positivity constraint 1 >= 0 */
1130 isl_int_set_si(lp->eq[k][n++], i == 0);
1131 for (j = 0; j < bset1->n_eq; ++j) {
1132 isl_int_set(lp->eq[k][n++], bset1->eq[j][i]);
1133 isl_int_neg(lp->eq[k][n++], bset1->eq[j][i]);
1135 for (j = 0; j < bset1->n_ineq; ++j)
1136 isl_int_set(lp->eq[k][n++], bset1->ineq[j][i]);
1137 /* positivity constraint 1 >= 0 */
1138 isl_int_set_si(lp->eq[k][n++], -(i == 0));
1139 for (j = 0; j < bset2->n_eq; ++j) {
1140 isl_int_neg(lp->eq[k][n++], bset2->eq[j][i]);
1141 isl_int_set(lp->eq[k][n++], bset2->eq[j][i]);
1143 for (j = 0; j < bset2->n_ineq; ++j)
1144 isl_int_neg(lp->eq[k][n++], bset2->ineq[j][i]);
1146 lp = isl_basic_set_gauss(lp, NULL);
1147 isl_basic_set_free(bset1);
1148 isl_basic_set_free(bset2);
1151 isl_basic_set_free(bset1);
1152 isl_basic_set_free(bset2);
1156 /* Compute a vector s in the homogeneous space such that <s, r> > 0
1157 * for all rays in the homogeneous space of the two cones that correspond
1158 * to the input polyhedra bset1 and bset2.
1160 * We compute s as a vector that satisfies
1162 * s = \sum_j \alpha_{ij} h_{ij} for i = 1,2 (*)
1164 * with h_{ij} the normals of the facets of polyhedron i
1165 * (including the "positivity constraint" 1 >= 0) and \alpha_{ij}
1166 * strictly positive numbers. For simplicity we impose \alpha_{ij} >= 1.
1167 * We first set up an LP with as variables the \alpha{ij}.
1168 * In this formulateion, for each polyhedron i,
1169 * the first constraint is the positivity constraint, followed by pairs
1170 * of variables for the equalities, followed by variables for the inequalities.
1171 * We then simply pick a feasible solution and compute s using (*).
1173 * Note that we simply pick any valid direction and make no attempt
1174 * to pick a "good" or even the "best" valid direction.
1176 static struct isl_vec *valid_direction(
1177 struct isl_basic_set *bset1, struct isl_basic_set *bset2)
1179 struct isl_ctx *ctx = NULL;
1180 struct isl_basic_set *lp;
1181 struct isl_tab *tab;
1182 struct isl_vec *sample = NULL;
1183 struct isl_vec *dir;
1188 if (!bset1 || !bset2)
1191 lp = valid_direction_lp(isl_basic_set_copy(bset1),
1192 isl_basic_set_copy(bset2));
1193 tab = isl_tab_from_basic_set(lp);
1194 sample = isl_tab_get_sample_value(ctx, tab);
1195 isl_tab_free(ctx, tab);
1196 isl_basic_set_free(lp);
1199 d = isl_basic_set_total_dim(bset1);
1200 dir = isl_vec_alloc(ctx, 1 + d);
1203 isl_seq_clr(dir->block.data + 1, dir->size - 1);
1205 /* positivity constraint 1 >= 0 */
1206 isl_int_set(dir->block.data[0], sample->block.data[n++]);
1207 for (i = 0; i < bset1->n_eq; ++i) {
1208 isl_int_sub(sample->block.data[n],
1209 sample->block.data[n], sample->block.data[n+1]);
1210 isl_seq_combine(dir->block.data,
1211 bset1->ctx->one, dir->block.data,
1212 sample->block.data[n], bset1->eq[i], 1 + d);
1216 for (i = 0; i < bset1->n_ineq; ++i)
1217 isl_seq_combine(dir->block.data,
1218 bset1->ctx->one, dir->block.data,
1219 sample->block.data[n++], bset1->ineq[i], 1 + d);
1220 isl_vec_free(sample);
1221 isl_basic_set_free(bset1);
1222 isl_basic_set_free(bset2);
1223 isl_seq_normalize(dir->block.data + 1, dir->size - 1);
1226 isl_vec_free(sample);
1227 isl_basic_set_free(bset1);
1228 isl_basic_set_free(bset2);
1232 /* Given a polyhedron b_i + A_i x >= 0 and a map T = S^{-1},
1233 * compute b_i' + A_i' x' >= 0, with
1235 * [ b_i A_i ] [ y' ] [ y' ]
1236 * [ 1 0 ] S^{-1} [ x' ] >= 0 or [ b_i' A_i' ] [ x' ] >= 0
1238 * In particular, add the "positivity constraint" and then perform
1241 static struct isl_basic_set *homogeneous_map(struct isl_basic_set *bset,
1244 struct isl_ctx *ctx = NULL;
1250 bset = isl_basic_set_extend_constraints(bset, 0, 1);
1251 k = isl_basic_set_alloc_inequality(bset);
1254 isl_seq_clr(bset->ineq[k] + 1, isl_basic_set_total_dim(bset));
1255 isl_int_set_si(bset->ineq[k][0], 1);
1256 bset = isl_basic_set_preimage(bset, T);
1259 isl_mat_free(ctx, T);
1260 isl_basic_set_free(bset);
1264 /* Compute the convex hull of a pair of basic sets without any parameters or
1265 * integer divisions, where the convex hull is known to be pointed,
1266 * but the basic sets may be unbounded.
1268 * We turn this problem into the computation of a convex hull of a pair
1269 * _bounded_ polyhedra by "changing the direction of the homogeneous
1270 * dimension". This idea is due to Matthias Koeppe.
1272 * Consider the cones in homogeneous space that correspond to the
1273 * input polyhedra. The rays of these cones are also rays of the
1274 * polyhedra if the coordinate that corresponds to the homogeneous
1275 * dimension is zero. That is, if the inner product of the rays
1276 * with the homogeneous direction is zero.
1277 * The cones in the homogeneous space can also be considered to
1278 * correspond to other pairs of polyhedra by chosing a different
1279 * homogeneous direction. To ensure that both of these polyhedra
1280 * are bounded, we need to make sure that all rays of the cones
1281 * correspond to vertices and not to rays.
1282 * Let s be a direction such that <s, r> > 0 for all rays r of both cones.
1283 * Then using s as a homogeneous direction, we obtain a pair of polytopes.
1284 * The vector s is computed in valid_direction.
1286 * Note that we need to consider _all_ rays of the cones and not just
1287 * the rays that correspond to rays in the polyhedra. If we were to
1288 * only consider those rays and turn them into vertices, then we
1289 * may inadvertently turn some vertices into rays.
1291 * The standard homogeneous direction is the unit vector in the 0th coordinate.
1292 * We therefore transform the two polyhedra such that the selected
1293 * direction is mapped onto this standard direction and then proceed
1294 * with the normal computation.
1295 * Let S be a non-singular square matrix with s as its first row,
1296 * then we want to map the polyhedra to the space
1298 * [ y' ] [ y ] [ y ] [ y' ]
1299 * [ x' ] = S [ x ] i.e., [ x ] = S^{-1} [ x' ]
1301 * We take S to be the unimodular completion of s to limit the growth
1302 * of the coefficients in the following computations.
1304 * Let b_i + A_i x >= 0 be the constraints of polyhedron i.
1305 * We first move to the homogeneous dimension
1307 * b_i y + A_i x >= 0 [ b_i A_i ] [ y ] [ 0 ]
1308 * y >= 0 or [ 1 0 ] [ x ] >= [ 0 ]
1310 * Then we change directoin
1312 * [ b_i A_i ] [ y' ] [ y' ]
1313 * [ 1 0 ] S^{-1} [ x' ] >= 0 or [ b_i' A_i' ] [ x' ] >= 0
1315 * Then we compute the convex hull of the polytopes b_i' + A_i' x' >= 0
1316 * resulting in b' + A' x' >= 0, which we then convert back
1319 * [ b' A' ] S [ x ] >= 0 or [ b A ] [ x ] >= 0
1321 * The polyhedron b + A x >= 0 is then the convex hull of the input polyhedra.
1323 static struct isl_basic_set *convex_hull_pair_pointed(
1324 struct isl_basic_set *bset1, struct isl_basic_set *bset2)
1326 struct isl_ctx *ctx = NULL;
1327 struct isl_vec *dir = NULL;
1328 struct isl_mat *T = NULL;
1329 struct isl_mat *T2 = NULL;
1330 struct isl_basic_set *hull;
1331 struct isl_set *set;
1333 if (!bset1 || !bset2)
1336 dir = valid_direction(isl_basic_set_copy(bset1),
1337 isl_basic_set_copy(bset2));
1340 T = isl_mat_alloc(bset1->ctx, dir->size, dir->size);
1343 isl_seq_cpy(T->row[0], dir->block.data, dir->size);
1344 T = isl_mat_unimodular_complete(ctx, T, 1);
1345 T2 = isl_mat_right_inverse(ctx, isl_mat_copy(ctx, T));
1347 bset1 = homogeneous_map(bset1, isl_mat_copy(ctx, T2));
1348 bset2 = homogeneous_map(bset2, T2);
1349 set = isl_set_alloc_dim(isl_basic_set_get_dim(bset1), 2, 0);
1350 set = isl_set_add(set, bset1);
1351 set = isl_set_add(set, bset2);
1352 hull = uset_convex_hull(set);
1353 hull = isl_basic_set_preimage(hull, T);
1360 isl_basic_set_free(bset1);
1361 isl_basic_set_free(bset2);
1365 /* Compute the convex hull of a pair of basic sets without any parameters or
1366 * integer divisions.
1368 * If the convex hull of the two basic sets would have a non-trivial
1369 * lineality space, we first project out this lineality space.
1371 static struct isl_basic_set *convex_hull_pair(struct isl_basic_set *bset1,
1372 struct isl_basic_set *bset2)
1374 struct isl_basic_set *lin;
1376 if (isl_basic_set_is_bounded(bset1) || isl_basic_set_is_bounded(bset2))
1377 return convex_hull_pair_pointed(bset1, bset2);
1379 lin = induced_lineality_space(isl_basic_set_copy(bset1),
1380 isl_basic_set_copy(bset2));
1383 if (isl_basic_set_is_universe(lin)) {
1384 isl_basic_set_free(bset1);
1385 isl_basic_set_free(bset2);
1388 if (lin->n_eq < isl_basic_set_total_dim(lin)) {
1389 struct isl_set *set;
1390 set = isl_set_alloc_dim(isl_basic_set_get_dim(bset1), 2, 0);
1391 set = isl_set_add(set, bset1);
1392 set = isl_set_add(set, bset2);
1393 return modulo_lineality(set, lin);
1395 isl_basic_set_free(lin);
1397 return convex_hull_pair_pointed(bset1, bset2);
1399 isl_basic_set_free(bset1);
1400 isl_basic_set_free(bset2);
1404 /* Compute the lineality space of an "underlying" basic set.
1405 * We basically just drop the constants and turn every inequality
1408 static struct isl_basic_set *ubasic_set_lineality_space(
1409 struct isl_basic_set *bset)
1412 struct isl_basic_set *lin = NULL;
1417 dim = isl_basic_set_total_dim(bset);
1419 lin = isl_basic_set_alloc_dim(isl_basic_set_get_dim(bset), 0, dim, 0);
1422 for (i = 0; i < bset->n_eq; ++i) {
1423 k = isl_basic_set_alloc_equality(lin);
1426 isl_int_set_si(lin->eq[k][0], 0);
1427 isl_seq_cpy(lin->eq[k] + 1, bset->eq[i] + 1, dim);
1429 lin = isl_basic_set_gauss(lin, NULL);
1432 for (i = 0; i < bset->n_ineq && lin->n_eq < dim; ++i) {
1433 k = isl_basic_set_alloc_equality(lin);
1436 isl_int_set_si(lin->eq[k][0], 0);
1437 isl_seq_cpy(lin->eq[k] + 1, bset->ineq[i] + 1, dim);
1438 lin = isl_basic_set_gauss(lin, NULL);
1442 isl_basic_set_free(bset);
1445 isl_basic_set_free(lin);
1446 isl_basic_set_free(bset);
1450 /* Compute the (linear) hull of the lineality spaces of the basic sets in the
1451 * "underlying" set "set".
1453 static struct isl_basic_set *uset_combined_lineality_space(struct isl_set *set)
1456 struct isl_set *lin = NULL;
1461 struct isl_dim *dim = isl_set_get_dim(set);
1463 return isl_basic_set_empty(dim);
1466 lin = isl_set_alloc_dim(isl_set_get_dim(set), set->n, 0);
1467 for (i = 0; i < set->n; ++i)
1468 lin = isl_set_add(lin,
1469 ubasic_set_lineality_space(isl_basic_set_copy(set->p[i])));
1471 return isl_set_affine_hull(lin);
1474 /* Compute the convex hull of a set without any parameters or
1475 * integer divisions.
1476 * In each step, we combined two basic sets until only one
1477 * basic set is left.
1478 * The input basic sets are assumed not to have a non-trivial
1479 * lineality space. If any of the intermediate results has
1480 * a non-trivial lineality space, it is projected out.
1482 static struct isl_basic_set *uset_convex_hull_unbounded(struct isl_set *set)
1484 struct isl_basic_set *convex_hull = NULL;
1486 convex_hull = isl_set_copy_basic_set(set);
1487 set = isl_set_drop_basic_set(set, convex_hull);
1490 while (set->n > 0) {
1491 struct isl_basic_set *t;
1492 t = isl_set_copy_basic_set(set);
1495 set = isl_set_drop_basic_set(set, t);
1498 convex_hull = convex_hull_pair(convex_hull, t);
1501 t = ubasic_set_lineality_space(isl_basic_set_copy(convex_hull));
1504 if (isl_basic_set_is_universe(t)) {
1505 isl_basic_set_free(convex_hull);
1509 if (t->n_eq < isl_basic_set_total_dim(t)) {
1510 set = isl_set_add(set, convex_hull);
1511 return modulo_lineality(set, t);
1513 isl_basic_set_free(t);
1519 isl_basic_set_free(convex_hull);
1523 /* Compute an initial hull for wrapping containing a single initial
1524 * facet by first computing bounds on the set and then using these
1525 * bounds to construct an initial facet.
1526 * This function is a remnant of an older implementation where the
1527 * bounds were also used to check whether the set was bounded.
1528 * Since this function will now only be called when we know the
1529 * set to be bounded, the initial facet should probably be constructed
1530 * by simply using the coordinate directions instead.
1532 static struct isl_basic_set *initial_hull(struct isl_basic_set *hull,
1533 struct isl_set *set)
1535 struct isl_mat *bounds = NULL;
1541 bounds = independent_bounds(set->ctx, set);
1544 isl_assert(set->ctx, bounds->n_row == isl_set_n_dim(set), goto error);
1545 bounds = initial_facet_constraint(set->ctx, set, bounds);
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(set->ctx, bounds);
1558 isl_basic_set_free(hull);
1559 isl_mat_free(set->ctx, 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_hash(con + 1, len, isl_hash_init());
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(ctx, 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_hash(con + 1, len, isl_hash_init());
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_alloc(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_hash(constraints[i].c->row[0] + 1, total,
1686 entry = isl_hash_table_find(hull->ctx, table, c_hash,
1687 max_constraint_equal, constraints[i].c->row[0] + 1, 1);
1690 isl_assert(hull->ctx, !entry->data, goto error);
1691 entry->data = &constraints[i];
1695 for (s = 0; s < set->n; ++s) {
1699 for (i = 0; i < set->p[s]->n_eq; ++i) {
1700 isl_int *eq = set->p[s]->eq[i];
1701 for (j = 0; j < 2; ++j) {
1702 isl_seq_neg(eq, eq, 1 + total);
1703 update_constraint(hull->ctx, table,
1707 for (i = 0; i < set->p[s]->n_ineq; ++i) {
1708 isl_int *ineq = set->p[s]->ineq[i];
1709 update_constraint(hull->ctx, table, ineq, total, n,
1710 set->p[s]->n_eq == 0);
1715 for (i = 0; i < min_constraints; ++i) {
1716 if (constraints[i].count < n)
1718 if (!constraints[i].ineq)
1720 j = isl_basic_set_alloc_inequality(hull);
1723 isl_seq_cpy(hull->ineq[j], constraints[i].c->row[0], 1 + total);
1726 for (s = 0; s < set->n; ++s) {
1727 if (set->p[s]->n_eq)
1729 if (set->p[s]->n_ineq != hull->n_ineq)
1731 for (i = 0; i < set->p[s]->n_ineq; ++i) {
1732 isl_int *ineq = set->p[s]->ineq[i];
1733 if (!has_constraint(hull->ctx, table, ineq, total, n))
1736 if (i == set->p[s]->n_ineq)
1740 isl_hash_table_clear(table);
1741 for (i = 0; i < min_constraints; ++i)
1742 isl_mat_free(hull->ctx, constraints[i].c);
1747 isl_hash_table_clear(table);
1750 for (i = 0; i < min_constraints; ++i)
1751 isl_mat_free(hull->ctx, constraints[i].c);
1756 /* Create a template for the convex hull of "set" and fill it up
1757 * obvious facet constraints, if any. If the result happens to
1758 * be the convex hull of "set" then *is_hull is set to 1.
1760 static struct isl_basic_set *proto_hull(struct isl_set *set, int *is_hull)
1762 struct isl_basic_set *hull;
1767 for (i = 0; i < set->n; ++i) {
1768 n_ineq += set->p[i]->n_eq;
1769 n_ineq += set->p[i]->n_ineq;
1771 hull = isl_basic_set_alloc_dim(isl_dim_copy(set->dim), 0, 0, n_ineq);
1772 hull = isl_basic_set_set_rational(hull);
1775 return common_constraints(hull, set, is_hull);
1778 static struct isl_basic_set *uset_convex_hull_wrap(struct isl_set *set)
1780 struct isl_basic_set *hull;
1783 hull = proto_hull(set, &is_hull);
1784 if (hull && !is_hull) {
1785 if (hull->n_ineq == 0)
1786 hull = initial_hull(hull, set);
1787 hull = extend(hull, set);
1794 /* Compute the convex hull of a set without any parameters or
1795 * integer divisions. Depending on whether the set is bounded,
1796 * we pass control to the wrapping based convex hull or
1797 * the Fourier-Motzkin elimination based convex hull.
1798 * We also handle a few special cases before checking the boundedness.
1800 static struct isl_basic_set *uset_convex_hull(struct isl_set *set)
1803 struct isl_basic_set *convex_hull = NULL;
1804 struct isl_basic_set *lin;
1806 if (isl_set_n_dim(set) == 0)
1807 return convex_hull_0d(set);
1809 set = isl_set_coalesce(set);
1810 set = isl_set_set_rational(set);
1817 convex_hull = isl_basic_set_copy(set->p[0]);
1821 if (isl_set_n_dim(set) == 1)
1822 return convex_hull_1d(set->ctx, set);
1824 if (isl_set_is_bounded(set))
1825 return uset_convex_hull_wrap(set);
1827 lin = uset_combined_lineality_space(isl_set_copy(set));
1830 if (isl_basic_set_is_universe(lin)) {
1834 if (lin->n_eq < isl_basic_set_total_dim(lin))
1835 return modulo_lineality(set, lin);
1836 isl_basic_set_free(lin);
1838 return uset_convex_hull_unbounded(set);
1841 isl_basic_set_free(convex_hull);
1845 /* This is the core procedure, where "set" is a "pure" set, i.e.,
1846 * without parameters or divs and where the convex hull of set is
1847 * known to be full-dimensional.
1849 static struct isl_basic_set *uset_convex_hull_wrap_bounded(struct isl_set *set)
1852 struct isl_basic_set *convex_hull = NULL;
1854 if (isl_set_n_dim(set) == 0) {
1855 convex_hull = isl_basic_set_universe(isl_dim_copy(set->dim));
1857 convex_hull = isl_basic_set_set_rational(convex_hull);
1861 set = isl_set_set_rational(set);
1865 set = isl_set_normalize(set);
1869 convex_hull = isl_basic_set_copy(set->p[0]);
1873 if (isl_set_n_dim(set) == 1)
1874 return convex_hull_1d(set->ctx, set);
1876 return uset_convex_hull_wrap(set);
1882 /* Compute the convex hull of set "set" with affine hull "affine_hull",
1883 * We first remove the equalities (transforming the set), compute the
1884 * convex hull of the transformed set and then add the equalities back
1885 * (after performing the inverse transformation.
1887 static struct isl_basic_set *modulo_affine_hull(struct isl_ctx *ctx,
1888 struct isl_set *set, struct isl_basic_set *affine_hull)
1892 struct isl_basic_set *dummy;
1893 struct isl_basic_set *convex_hull;
1895 dummy = isl_basic_set_remove_equalities(
1896 isl_basic_set_copy(affine_hull), &T, &T2);
1899 isl_basic_set_free(dummy);
1900 set = isl_set_preimage(set, T);
1901 convex_hull = uset_convex_hull(set);
1902 convex_hull = isl_basic_set_preimage(convex_hull, T2);
1903 convex_hull = isl_basic_set_intersect(convex_hull, affine_hull);
1906 isl_basic_set_free(affine_hull);
1911 /* Compute the convex hull of a map.
1913 * The implementation was inspired by "Extended Convex Hull" by Fukuda et al.,
1914 * specifically, the wrapping of facets to obtain new facets.
1916 struct isl_basic_map *isl_map_convex_hull(struct isl_map *map)
1918 struct isl_basic_set *bset;
1919 struct isl_basic_map *model = NULL;
1920 struct isl_basic_set *affine_hull = NULL;
1921 struct isl_basic_map *convex_hull = NULL;
1922 struct isl_set *set = NULL;
1923 struct isl_ctx *ctx;
1930 convex_hull = isl_basic_map_empty_like_map(map);
1935 map = isl_map_detect_equalities(map);
1936 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(ctx, 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);
1954 ISL_F_SET(convex_hull, ISL_BASIC_MAP_NO_IMPLICIT);
1955 ISL_F_SET(convex_hull, ISL_BASIC_MAP_ALL_EQUALITIES);
1956 ISL_F_CLR(convex_hull, ISL_BASIC_MAP_RATIONAL);
1960 isl_basic_map_free(model);
1964 struct isl_basic_set *isl_set_convex_hull(struct isl_set *set)
1966 return (struct isl_basic_set *)
1967 isl_map_convex_hull((struct isl_map *)set);
1970 struct sh_data_entry {
1971 struct isl_hash_table *table;
1972 struct isl_tab *tab;
1975 /* Holds the data needed during the simple hull computation.
1977 * n the number of basic sets in the original set
1978 * hull_table a hash table of already computed constraints
1979 * in the simple hull
1980 * p for each basic set,
1981 * table a hash table of the constraints
1982 * tab the tableau corresponding to the basic set
1985 struct isl_ctx *ctx;
1987 struct isl_hash_table *hull_table;
1988 struct sh_data_entry p[0];
1991 static void sh_data_free(struct sh_data *data)
1997 isl_hash_table_free(data->ctx, data->hull_table);
1998 for (i = 0; i < data->n; ++i) {
1999 isl_hash_table_free(data->ctx, data->p[i].table);
2000 isl_tab_free(data->ctx, data->p[i].tab);
2005 struct ineq_cmp_data {
2010 static int has_ineq(const void *entry, const void *val)
2012 isl_int *row = (isl_int *)entry;
2013 struct ineq_cmp_data *v = (struct ineq_cmp_data *)val;
2015 return isl_seq_eq(row + 1, v->p + 1, v->len) ||
2016 isl_seq_is_neg(row + 1, v->p + 1, v->len);
2019 static int hash_ineq(struct isl_ctx *ctx, struct isl_hash_table *table,
2020 isl_int *ineq, unsigned len)
2023 struct ineq_cmp_data v;
2024 struct isl_hash_table_entry *entry;
2028 c_hash = isl_seq_hash(ineq + 1, len, isl_hash_init());
2029 entry = isl_hash_table_find(ctx, table, c_hash, has_ineq, &v, 1);
2036 /* Fill hash table "table" with the constraints of "bset".
2037 * Equalities are added as two inequalities.
2038 * The value in the hash table is a pointer to the (in)equality of "bset".
2040 static int hash_basic_set(struct isl_hash_table *table,
2041 struct isl_basic_set *bset)
2044 unsigned dim = isl_basic_set_total_dim(bset);
2046 for (i = 0; i < bset->n_eq; ++i) {
2047 for (j = 0; j < 2; ++j) {
2048 isl_seq_neg(bset->eq[i], bset->eq[i], 1 + dim);
2049 if (hash_ineq(bset->ctx, table, bset->eq[i], dim) < 0)
2053 for (i = 0; i < bset->n_ineq; ++i) {
2054 if (hash_ineq(bset->ctx, table, bset->ineq[i], dim) < 0)
2060 static struct sh_data *sh_data_alloc(struct isl_set *set, unsigned n_ineq)
2062 struct sh_data *data;
2065 data = isl_calloc(set->ctx, struct sh_data,
2066 sizeof(struct sh_data) + set->n * sizeof(struct sh_data_entry));
2069 data->ctx = set->ctx;
2071 data->hull_table = isl_hash_table_alloc(set->ctx, n_ineq);
2072 if (!data->hull_table)
2074 for (i = 0; i < set->n; ++i) {
2075 data->p[i].table = isl_hash_table_alloc(set->ctx,
2076 2 * set->p[i]->n_eq + set->p[i]->n_ineq);
2077 if (!data->p[i].table)
2079 if (hash_basic_set(data->p[i].table, set->p[i]) < 0)
2088 /* Check if inequality "ineq" is a bound for basic set "j" or if
2089 * it can be relaxed (by increasing the constant term) to become
2090 * a bound for that basic set. In the latter case, the constant
2092 * Return 1 if "ineq" is a bound
2093 * 0 if "ineq" may attain arbitrarily small values on basic set "j"
2094 * -1 if some error occurred
2096 static int is_bound(struct sh_data *data, struct isl_set *set, int j,
2099 enum isl_lp_result res;
2102 if (!data->p[j].tab) {
2103 data->p[j].tab = isl_tab_from_basic_set(set->p[j]);
2104 if (!data->p[j].tab)
2110 res = isl_tab_min(data->ctx, data->p[j].tab, ineq, data->ctx->one,
2112 if (res == isl_lp_ok && isl_int_is_neg(opt))
2113 isl_int_sub(ineq[0], ineq[0], opt);
2117 return res == isl_lp_ok ? 1 :
2118 res == isl_lp_unbounded ? 0 : -1;
2121 /* Check if inequality "ineq" from basic set "i" can be relaxed to
2122 * become a bound on the whole set. If so, add the (relaxed) inequality
2125 * We first check if "hull" already contains a translate of the inequality.
2126 * If so, we are done.
2127 * Then, we check if any of the previous basic sets contains a translate
2128 * of the inequality. If so, then we have already considered this
2129 * inequality and we are done.
2130 * Otherwise, for each basic set other than "i", we check if the inequality
2131 * is a bound on the basic set.
2132 * For previous basic sets, we know that they do not contain a translate
2133 * of the inequality, so we directly call is_bound.
2134 * For following basic sets, we first check if a translate of the
2135 * inequality appears in its description and if so directly update
2136 * the inequality accordingly.
2138 static struct isl_basic_set *add_bound(struct isl_basic_set *hull,
2139 struct sh_data *data, struct isl_set *set, int i, isl_int *ineq)
2142 struct ineq_cmp_data v;
2143 struct isl_hash_table_entry *entry;
2149 v.len = isl_basic_set_total_dim(hull);
2151 c_hash = isl_seq_hash(ineq + 1, v.len, isl_hash_init());
2153 entry = isl_hash_table_find(hull->ctx, data->hull_table, c_hash,
2158 for (j = 0; j < i; ++j) {
2159 entry = isl_hash_table_find(hull->ctx, data->p[j].table,
2160 c_hash, has_ineq, &v, 0);
2167 k = isl_basic_set_alloc_inequality(hull);
2168 isl_seq_cpy(hull->ineq[k], ineq, 1 + v.len);
2172 for (j = 0; j < i; ++j) {
2174 bound = is_bound(data, set, j, hull->ineq[k]);
2181 isl_basic_set_free_inequality(hull, 1);
2185 for (j = i + 1; j < set->n; ++j) {
2188 entry = isl_hash_table_find(hull->ctx, data->p[j].table,
2189 c_hash, has_ineq, &v, 0);
2191 ineq_j = entry->data;
2192 neg = isl_seq_is_neg(ineq_j + 1,
2193 hull->ineq[k] + 1, v.len);
2195 isl_int_neg(ineq_j[0], ineq_j[0]);
2196 if (isl_int_gt(ineq_j[0], hull->ineq[k][0]))
2197 isl_int_set(hull->ineq[k][0], ineq_j[0]);
2199 isl_int_neg(ineq_j[0], ineq_j[0]);
2202 bound = is_bound(data, set, j, hull->ineq[k]);
2209 isl_basic_set_free_inequality(hull, 1);
2213 entry = isl_hash_table_find(hull->ctx, data->hull_table, c_hash,
2217 entry->data = hull->ineq[k];
2221 isl_basic_set_free(hull);
2225 /* Check if any inequality from basic set "i" can be relaxed to
2226 * become a bound on the whole set. If so, add the (relaxed) inequality
2229 static struct isl_basic_set *add_bounds(struct isl_basic_set *bset,
2230 struct sh_data *data, struct isl_set *set, int i)
2233 unsigned dim = isl_basic_set_total_dim(bset);
2235 for (j = 0; j < set->p[i]->n_eq; ++j) {
2236 for (k = 0; k < 2; ++k) {
2237 isl_seq_neg(set->p[i]->eq[j], set->p[i]->eq[j], 1+dim);
2238 add_bound(bset, data, set, i, set->p[i]->eq[j]);
2241 for (j = 0; j < set->p[i]->n_ineq; ++j)
2242 add_bound(bset, data, set, i, set->p[i]->ineq[j]);
2246 /* Compute a superset of the convex hull of set that is described
2247 * by only translates of the constraints in the constituents of set.
2249 static struct isl_basic_set *uset_simple_hull(struct isl_set *set)
2251 struct sh_data *data = NULL;
2252 struct isl_basic_set *hull = NULL;
2260 for (i = 0; i < set->n; ++i) {
2263 n_ineq += 2 * set->p[i]->n_eq + set->p[i]->n_ineq;
2266 hull = isl_basic_set_alloc_dim(isl_dim_copy(set->dim), 0, 0, n_ineq);
2270 data = sh_data_alloc(set, n_ineq);
2274 for (i = 0; i < set->n; ++i)
2275 hull = add_bounds(hull, data, set, i);
2283 isl_basic_set_free(hull);
2288 /* Compute a superset of the convex hull of map that is described
2289 * by only translates of the constraints in the constituents of map.
2291 struct isl_basic_map *isl_map_simple_hull(struct isl_map *map)
2293 struct isl_set *set = NULL;
2294 struct isl_basic_map *model = NULL;
2295 struct isl_basic_map *hull;
2296 struct isl_basic_map *affine_hull;
2297 struct isl_basic_set *bset = NULL;
2302 hull = isl_basic_map_empty_like_map(map);
2307 hull = isl_basic_map_copy(map->p[0]);
2312 map = isl_map_detect_equalities(map);
2313 affine_hull = isl_map_affine_hull(isl_map_copy(map));
2314 map = isl_map_align_divs(map);
2315 model = isl_basic_map_copy(map->p[0]);
2317 set = isl_map_underlying_set(map);
2319 bset = uset_simple_hull(set);
2321 hull = isl_basic_map_overlying_set(bset, model);
2323 hull = isl_basic_map_intersect(hull, affine_hull);
2324 hull = isl_basic_map_convex_hull(hull);
2325 ISL_F_SET(hull, ISL_BASIC_MAP_NO_IMPLICIT);
2326 ISL_F_SET(hull, ISL_BASIC_MAP_ALL_EQUALITIES);
2331 struct isl_basic_set *isl_set_simple_hull(struct isl_set *set)
2333 return (struct isl_basic_set *)
2334 isl_map_simple_hull((struct isl_map *)set);
2337 /* Given a set "set", return parametric bounds on the dimension "dim".
2339 static struct isl_basic_set *set_bounds(struct isl_set *set, int dim)
2341 unsigned set_dim = isl_set_dim(set, isl_dim_set);
2342 set = isl_set_copy(set);
2343 set = isl_set_eliminate_dims(set, dim + 1, set_dim - (dim + 1));
2344 set = isl_set_eliminate_dims(set, 0, dim);
2345 return isl_set_convex_hull(set);
2348 /* Computes a "simple hull" and then check if each dimension in the
2349 * resulting hull is bounded by a symbolic constant. If not, the
2350 * hull is intersected with the corresponding bounds on the whole set.
2352 struct isl_basic_set *isl_set_bounded_simple_hull(struct isl_set *set)
2355 struct isl_basic_set *hull;
2356 unsigned nparam, left;
2357 int removed_divs = 0;
2359 hull = isl_set_simple_hull(isl_set_copy(set));
2363 nparam = isl_basic_set_dim(hull, isl_dim_param);
2364 for (i = 0; i < isl_basic_set_dim(hull, isl_dim_set); ++i) {
2365 int lower = 0, upper = 0;
2366 struct isl_basic_set *bounds;
2368 left = isl_basic_set_total_dim(hull) - nparam - i - 1;
2369 for (j = 0; j < hull->n_eq; ++j) {
2370 if (isl_int_is_zero(hull->eq[j][1 + nparam + i]))
2372 if (isl_seq_first_non_zero(hull->eq[j]+1+nparam+i+1,
2379 for (j = 0; j < hull->n_ineq; ++j) {
2380 if (isl_int_is_zero(hull->ineq[j][1 + nparam + i]))
2382 if (isl_seq_first_non_zero(hull->ineq[j]+1+nparam+i+1,
2384 isl_seq_first_non_zero(hull->ineq[j]+1+nparam,
2387 if (isl_int_is_pos(hull->ineq[j][1 + nparam + i]))
2398 if (!removed_divs) {
2399 set = isl_set_remove_divs(set);
2404 bounds = set_bounds(set, i);
2405 hull = isl_basic_set_intersect(hull, bounds);