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_basic_map_solve_lp(*bmap, 0, c, (*bmap)->ctx->one,
55 if (res == isl_lp_unbounded)
57 if (res == isl_lp_error)
59 if (res == isl_lp_empty) {
60 *bmap = isl_basic_map_set_to_empty(*bmap);
63 return !isl_int_is_neg(*opt_n);
66 int isl_basic_set_constraint_is_redundant(struct isl_basic_set **bset,
67 isl_int *c, isl_int *opt_n, isl_int *opt_d)
69 return isl_basic_map_constraint_is_redundant(
70 (struct isl_basic_map **)bset, c, opt_n, opt_d);
73 /* Compute the convex hull of a basic map, by removing the redundant
74 * constraints. If the minimal value along the normal of a constraint
75 * is the same if the constraint is removed, then the constraint is redundant.
77 * Alternatively, we could have intersected the basic map with the
78 * corresponding equality and the checked if the dimension was that
81 struct isl_basic_map *isl_basic_map_convex_hull(struct isl_basic_map *bmap)
88 bmap = isl_basic_map_gauss(bmap, NULL);
89 if (ISL_F_ISSET(bmap, ISL_BASIC_MAP_EMPTY))
91 if (ISL_F_ISSET(bmap, ISL_BASIC_MAP_NO_REDUNDANT))
93 if (bmap->n_ineq <= 1)
96 tab = isl_tab_from_basic_map(bmap);
97 tab = isl_tab_detect_equalities(tab);
98 tab = isl_tab_detect_redundant(tab);
99 bmap = isl_basic_map_update_from_tab(bmap, tab);
101 ISL_F_SET(bmap, ISL_BASIC_MAP_NO_IMPLICIT);
102 ISL_F_SET(bmap, ISL_BASIC_MAP_NO_REDUNDANT);
106 struct isl_basic_set *isl_basic_set_convex_hull(struct isl_basic_set *bset)
108 return (struct isl_basic_set *)
109 isl_basic_map_convex_hull((struct isl_basic_map *)bset);
112 /* Check if the set set is bound in the direction of the affine
113 * constraint c and if so, set the constant term such that the
114 * resulting constraint is a bounding constraint for the set.
116 static int uset_is_bound(struct isl_set *set, 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_basic_set_solve_lp(set->p[j],
133 0, c, set->ctx->one, &opt, &opt_denom, NULL);
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_set *set, isl_int *c,
167 struct isl_mat *dirs, int n)
172 isl_seq_cpy(dirs->row[n]+1, c+1, dirs->n_col-1);
174 int pos = isl_seq_first_non_zero(dirs->row[n]+1, dirs->n_col-1);
177 for (i = 0; i < n; ++i) {
179 pos_i = isl_seq_first_non_zero(dirs->row[i]+1, dirs->n_col-1);
184 isl_seq_elim(dirs->row[n]+1, dirs->row[i]+1, pos,
185 dirs->n_col-1, NULL);
186 pos = isl_seq_first_non_zero(dirs->row[n]+1, dirs->n_col-1);
192 is_bound = uset_is_bound(set, dirs->row[n], dirs->n_col);
197 isl_int *t = dirs->row[n];
198 for (k = n; k > i; --k)
199 dirs->row[k] = dirs->row[k-1];
205 /* Compute and return a maximal set of linearly independent bounds
206 * on the set "set", based on the constraints of the basic sets
209 static struct isl_mat *independent_bounds(struct isl_set *set)
212 struct isl_mat *dirs = NULL;
213 unsigned dim = isl_set_n_dim(set);
215 dirs = isl_mat_alloc(set->ctx, dim, 1+dim);
220 for (i = 0; n < dim && i < set->n; ++i) {
222 struct isl_basic_set *bset = set->p[i];
224 for (j = 0; n < dim && j < bset->n_eq; ++j) {
225 f = is_independent_bound(set, bset->eq[j], dirs, n);
231 for (j = 0; n < dim && j < bset->n_ineq; ++j) {
232 f = is_independent_bound(set, bset->ineq[j], dirs, n);
246 struct isl_basic_set *isl_basic_set_set_rational(struct isl_basic_set *bset)
251 if (ISL_F_ISSET(bset, ISL_BASIC_MAP_RATIONAL))
254 bset = isl_basic_set_cow(bset);
258 ISL_F_SET(bset, ISL_BASIC_MAP_RATIONAL);
260 return isl_basic_set_finalize(bset);
263 static struct isl_set *isl_set_set_rational(struct isl_set *set)
267 set = isl_set_cow(set);
270 for (i = 0; i < set->n; ++i) {
271 set->p[i] = isl_basic_set_set_rational(set->p[i]);
281 static struct isl_basic_set *isl_basic_set_add_equality(
282 struct isl_basic_set *bset, isl_int *c)
288 if (ISL_F_ISSET(bset, ISL_BASIC_SET_EMPTY))
291 isl_assert(ctx, isl_basic_set_n_param(bset) == 0, goto error);
292 isl_assert(ctx, bset->n_div == 0, goto error);
293 dim = isl_basic_set_n_dim(bset);
294 bset = isl_basic_set_cow(bset);
295 bset = isl_basic_set_extend(bset, 0, dim, 0, 1, 0);
296 i = isl_basic_set_alloc_equality(bset);
299 isl_seq_cpy(bset->eq[i], c, 1 + dim);
302 isl_basic_set_free(bset);
306 static struct isl_set *isl_set_add_equality(struct isl_set *set, isl_int *c)
310 set = isl_set_cow(set);
313 for (i = 0; i < set->n; ++i) {
314 set->p[i] = isl_basic_set_add_equality(set->p[i], c);
324 /* Given a union of basic sets, construct the constraints for wrapping
325 * a facet around one of its ridges.
326 * In particular, if each of n the d-dimensional basic sets i in "set"
327 * contains the origin, satisfies the constraints x_1 >= 0 and x_2 >= 0
328 * and is defined by the constraints
332 * then the resulting set is of dimension n*(1+d) and has as constraints
341 static struct isl_basic_set *wrap_constraints(struct isl_set *set)
343 struct isl_basic_set *lp;
347 unsigned dim, lp_dim;
352 dim = 1 + isl_set_n_dim(set);
355 for (i = 0; i < set->n; ++i) {
356 n_eq += set->p[i]->n_eq;
357 n_ineq += set->p[i]->n_ineq;
359 lp = isl_basic_set_alloc(set->ctx, 0, dim * set->n, 0, n_eq, n_ineq);
362 lp_dim = isl_basic_set_n_dim(lp);
363 k = isl_basic_set_alloc_equality(lp);
364 isl_int_set_si(lp->eq[k][0], -1);
365 for (i = 0; i < set->n; ++i) {
366 isl_int_set_si(lp->eq[k][1+dim*i], 0);
367 isl_int_set_si(lp->eq[k][1+dim*i+1], 1);
368 isl_seq_clr(lp->eq[k]+1+dim*i+2, dim-2);
370 for (i = 0; i < set->n; ++i) {
371 k = isl_basic_set_alloc_inequality(lp);
372 isl_seq_clr(lp->ineq[k], 1+lp_dim);
373 isl_int_set_si(lp->ineq[k][1+dim*i], 1);
375 for (j = 0; j < set->p[i]->n_eq; ++j) {
376 k = isl_basic_set_alloc_equality(lp);
377 isl_seq_clr(lp->eq[k], 1+dim*i);
378 isl_seq_cpy(lp->eq[k]+1+dim*i, set->p[i]->eq[j], dim);
379 isl_seq_clr(lp->eq[k]+1+dim*(i+1), dim*(set->n-i-1));
382 for (j = 0; j < set->p[i]->n_ineq; ++j) {
383 k = isl_basic_set_alloc_inequality(lp);
384 isl_seq_clr(lp->ineq[k], 1+dim*i);
385 isl_seq_cpy(lp->ineq[k]+1+dim*i, set->p[i]->ineq[j], dim);
386 isl_seq_clr(lp->ineq[k]+1+dim*(i+1), dim*(set->n-i-1));
392 /* Given a facet "facet" of the convex hull of "set" and a facet "ridge"
393 * of that facet, compute the other facet of the convex hull that contains
396 * We first transform the set such that the facet constraint becomes
400 * I.e., the facet lies in
404 * and on that facet, the constraint that defines the ridge is
408 * (This transformation is not strictly needed, all that is needed is
409 * that the ridge contains the origin.)
411 * Since the ridge contains the origin, the cone of the convex hull
412 * will be of the form
417 * with this second constraint defining the new facet.
418 * The constant a is obtained by settting x_1 in the cone of the
419 * convex hull to 1 and minimizing x_2.
420 * Now, each element in the cone of the convex hull is the sum
421 * of elements in the cones of the basic sets.
422 * If a_i is the dilation factor of basic set i, then the problem
423 * we need to solve is
436 * the constraints of each (transformed) basic set.
437 * If a = n/d, then the constraint defining the new facet (in the transformed
440 * -n x_1 + d x_2 >= 0
442 * In the original space, we need to take the same combination of the
443 * corresponding constraints "facet" and "ridge".
445 * Note that a is always finite, since we only apply the wrapping
446 * technique to a union of polytopes.
448 static isl_int *wrap_facet(struct isl_set *set, isl_int *facet, isl_int *ridge)
451 struct isl_mat *T = NULL;
452 struct isl_basic_set *lp = NULL;
454 enum isl_lp_result res;
458 set = isl_set_copy(set);
460 dim = 1 + isl_set_n_dim(set);
461 T = isl_mat_alloc(set->ctx, 3, dim);
464 isl_int_set_si(T->row[0][0], 1);
465 isl_seq_clr(T->row[0]+1, dim - 1);
466 isl_seq_cpy(T->row[1], facet, dim);
467 isl_seq_cpy(T->row[2], ridge, dim);
468 T = isl_mat_right_inverse(T);
469 set = isl_set_preimage(set, T);
473 lp = wrap_constraints(set);
474 obj = isl_vec_alloc(set->ctx, 1 + dim*set->n);
477 isl_int_set_si(obj->block.data[0], 0);
478 for (i = 0; i < set->n; ++i) {
479 isl_seq_clr(obj->block.data + 1 + dim*i, 2);
480 isl_int_set_si(obj->block.data[1 + dim*i+2], 1);
481 isl_seq_clr(obj->block.data + 1 + dim*i+3, dim-3);
485 res = isl_basic_set_solve_lp(lp, 0,
486 obj->block.data, set->ctx->one, &num, &den, NULL);
487 if (res == isl_lp_ok) {
488 isl_int_neg(num, num);
489 isl_seq_combine(facet, num, facet, den, ridge, dim);
494 isl_basic_set_free(lp);
496 isl_assert(set->ctx, res == isl_lp_ok, return NULL);
499 isl_basic_set_free(lp);
505 /* Given a set of d linearly independent bounding constraints of the
506 * convex hull of "set", compute the constraint of a facet of "set".
508 * We first compute the intersection with the first bounding hyperplane
509 * and remove the component corresponding to this hyperplane from
510 * other bounds (in homogeneous space).
511 * We then wrap around one of the remaining bounding constraints
512 * and continue the process until all bounding constraints have been
513 * taken into account.
514 * The resulting linear combination of the bounding constraints will
515 * correspond to a facet of the convex hull.
517 static struct isl_mat *initial_facet_constraint(struct isl_set *set,
518 struct isl_mat *bounds)
520 struct isl_set *slice = NULL;
521 struct isl_basic_set *face = NULL;
522 struct isl_mat *m, *U, *Q;
524 unsigned dim = isl_set_n_dim(set);
526 isl_assert(ctx, set->n > 0, goto error);
527 isl_assert(ctx, bounds->n_row == dim, goto error);
529 while (bounds->n_row > 1) {
530 slice = isl_set_copy(set);
531 slice = isl_set_add_equality(slice, bounds->row[0]);
532 face = isl_set_affine_hull(slice);
535 if (face->n_eq == 1) {
536 isl_basic_set_free(face);
539 m = isl_mat_alloc(set->ctx, 1 + face->n_eq, 1 + dim);
542 isl_int_set_si(m->row[0][0], 1);
543 isl_seq_clr(m->row[0]+1, dim);
544 for (i = 0; i < face->n_eq; ++i)
545 isl_seq_cpy(m->row[1 + i], face->eq[i], 1 + dim);
546 U = isl_mat_right_inverse(m);
547 Q = isl_mat_right_inverse(isl_mat_copy(U));
548 U = isl_mat_drop_cols(U, 1 + face->n_eq, dim - face->n_eq);
549 Q = isl_mat_drop_rows(Q, 1 + face->n_eq, dim - face->n_eq);
550 U = isl_mat_drop_cols(U, 0, 1);
551 Q = isl_mat_drop_rows(Q, 0, 1);
552 bounds = isl_mat_product(bounds, U);
553 bounds = isl_mat_product(bounds, Q);
554 while (isl_seq_first_non_zero(bounds->row[bounds->n_row-1],
555 bounds->n_col) == -1) {
557 isl_assert(ctx, bounds->n_row > 1, goto error);
559 if (!wrap_facet(set, bounds->row[0],
560 bounds->row[bounds->n_row-1]))
562 isl_basic_set_free(face);
567 isl_basic_set_free(face);
568 isl_mat_free(bounds);
572 /* Given the bounding constraint "c" of a facet of the convex hull of "set",
573 * compute a hyperplane description of the facet, i.e., compute the facets
576 * We compute an affine transformation that transforms the constraint
585 * by computing the right inverse U of a matrix that starts with the rows
598 * Since z_1 is zero, we can drop this variable as well as the corresponding
599 * column of U to obtain
607 * with Q' equal to Q, but without the corresponding row.
608 * After computing the facets of the facet in the z' space,
609 * we convert them back to the x space through Q.
611 static struct isl_basic_set *compute_facet(struct isl_set *set, isl_int *c)
613 struct isl_mat *m, *U, *Q;
614 struct isl_basic_set *facet = NULL;
619 set = isl_set_copy(set);
620 dim = isl_set_n_dim(set);
621 m = isl_mat_alloc(set->ctx, 2, 1 + dim);
624 isl_int_set_si(m->row[0][0], 1);
625 isl_seq_clr(m->row[0]+1, dim);
626 isl_seq_cpy(m->row[1], c, 1+dim);
627 U = isl_mat_right_inverse(m);
628 Q = isl_mat_right_inverse(isl_mat_copy(U));
629 U = isl_mat_drop_cols(U, 1, 1);
630 Q = isl_mat_drop_rows(Q, 1, 1);
631 set = isl_set_preimage(set, U);
632 facet = uset_convex_hull_wrap_bounded(set);
633 facet = isl_basic_set_preimage(facet, Q);
634 isl_assert(ctx, facet->n_eq == 0, goto error);
637 isl_basic_set_free(facet);
642 /* Given an initial facet constraint, compute the remaining facets.
643 * We do this by running through all facets found so far and computing
644 * the adjacent facets through wrapping, adding those facets that we
645 * hadn't already found before.
647 * For each facet we have found so far, we first compute its facets
648 * in the resulting convex hull. That is, we compute the ridges
649 * of the resulting convex hull contained in the facet.
650 * We also compute the corresponding facet in the current approximation
651 * of the convex hull. There is no need to wrap around the ridges
652 * in this facet since that would result in a facet that is already
653 * present in the current approximation.
655 * This function can still be significantly optimized by checking which of
656 * the facets of the basic sets are also facets of the convex hull and
657 * using all the facets so far to help in constructing the facets of the
660 * using the technique in section "3.1 Ridge Generation" of
661 * "Extended Convex Hull" by Fukuda et al.
663 static struct isl_basic_set *extend(struct isl_basic_set *hull,
668 struct isl_basic_set *facet = NULL;
669 struct isl_basic_set *hull_facet = NULL;
673 isl_assert(set->ctx, set->n > 0, goto error);
675 dim = isl_set_n_dim(set);
677 for (i = 0; i < hull->n_ineq; ++i) {
678 facet = compute_facet(set, hull->ineq[i]);
679 facet = isl_basic_set_add_equality(facet, hull->ineq[i]);
680 facet = isl_basic_set_gauss(facet, NULL);
681 facet = isl_basic_set_normalize_constraints(facet);
682 hull_facet = isl_basic_set_copy(hull);
683 hull_facet = isl_basic_set_add_equality(hull_facet, hull->ineq[i]);
684 hull_facet = isl_basic_set_gauss(hull_facet, NULL);
685 hull_facet = isl_basic_set_normalize_constraints(hull_facet);
688 hull = isl_basic_set_cow(hull);
689 hull = isl_basic_set_extend_dim(hull,
690 isl_dim_copy(hull->dim), 0, 0, facet->n_ineq);
691 for (j = 0; j < facet->n_ineq; ++j) {
692 for (f = 0; f < hull_facet->n_ineq; ++f)
693 if (isl_seq_eq(facet->ineq[j],
694 hull_facet->ineq[f], 1 + dim))
696 if (f < hull_facet->n_ineq)
698 k = isl_basic_set_alloc_inequality(hull);
701 isl_seq_cpy(hull->ineq[k], hull->ineq[i], 1+dim);
702 if (!wrap_facet(set, hull->ineq[k], facet->ineq[j]))
705 isl_basic_set_free(hull_facet);
706 isl_basic_set_free(facet);
708 hull = isl_basic_set_simplify(hull);
709 hull = isl_basic_set_finalize(hull);
712 isl_basic_set_free(hull_facet);
713 isl_basic_set_free(facet);
714 isl_basic_set_free(hull);
718 /* Special case for computing the convex hull of a one dimensional set.
719 * We simply collect the lower and upper bounds of each basic set
720 * and the biggest of those.
722 static struct isl_basic_set *convex_hull_1d(struct isl_set *set)
724 struct isl_mat *c = NULL;
725 isl_int *lower = NULL;
726 isl_int *upper = NULL;
729 struct isl_basic_set *hull;
731 for (i = 0; i < set->n; ++i) {
732 set->p[i] = isl_basic_set_simplify(set->p[i]);
736 set = isl_set_remove_empty_parts(set);
739 isl_assert(set->ctx, set->n > 0, goto error);
740 c = isl_mat_alloc(set->ctx, 2, 2);
744 if (set->p[0]->n_eq > 0) {
745 isl_assert(set->ctx, set->p[0]->n_eq == 1, goto error);
748 if (isl_int_is_pos(set->p[0]->eq[0][1])) {
749 isl_seq_cpy(lower, set->p[0]->eq[0], 2);
750 isl_seq_neg(upper, set->p[0]->eq[0], 2);
752 isl_seq_neg(lower, set->p[0]->eq[0], 2);
753 isl_seq_cpy(upper, set->p[0]->eq[0], 2);
756 for (j = 0; j < set->p[0]->n_ineq; ++j) {
757 if (isl_int_is_pos(set->p[0]->ineq[j][1])) {
759 isl_seq_cpy(lower, set->p[0]->ineq[j], 2);
762 isl_seq_cpy(upper, set->p[0]->ineq[j], 2);
769 for (i = 0; i < set->n; ++i) {
770 struct isl_basic_set *bset = set->p[i];
774 for (j = 0; j < bset->n_eq; ++j) {
778 isl_int_mul(a, lower[0], bset->eq[j][1]);
779 isl_int_mul(b, lower[1], bset->eq[j][0]);
780 if (isl_int_lt(a, b) && isl_int_is_pos(bset->eq[j][1]))
781 isl_seq_cpy(lower, bset->eq[j], 2);
782 if (isl_int_gt(a, b) && isl_int_is_neg(bset->eq[j][1]))
783 isl_seq_neg(lower, bset->eq[j], 2);
786 isl_int_mul(a, upper[0], bset->eq[j][1]);
787 isl_int_mul(b, upper[1], bset->eq[j][0]);
788 if (isl_int_lt(a, b) && isl_int_is_pos(bset->eq[j][1]))
789 isl_seq_neg(upper, bset->eq[j], 2);
790 if (isl_int_gt(a, b) && isl_int_is_neg(bset->eq[j][1]))
791 isl_seq_cpy(upper, bset->eq[j], 2);
794 for (j = 0; j < bset->n_ineq; ++j) {
795 if (isl_int_is_pos(bset->ineq[j][1]))
797 if (isl_int_is_neg(bset->ineq[j][1]))
799 if (lower && isl_int_is_pos(bset->ineq[j][1])) {
800 isl_int_mul(a, lower[0], bset->ineq[j][1]);
801 isl_int_mul(b, lower[1], bset->ineq[j][0]);
802 if (isl_int_lt(a, b))
803 isl_seq_cpy(lower, bset->ineq[j], 2);
805 if (upper && isl_int_is_neg(bset->ineq[j][1])) {
806 isl_int_mul(a, upper[0], bset->ineq[j][1]);
807 isl_int_mul(b, upper[1], bset->ineq[j][0]);
808 if (isl_int_gt(a, b))
809 isl_seq_cpy(upper, bset->ineq[j], 2);
820 hull = isl_basic_set_alloc(set->ctx, 0, 1, 0, 0, 2);
821 hull = isl_basic_set_set_rational(hull);
825 k = isl_basic_set_alloc_inequality(hull);
826 isl_seq_cpy(hull->ineq[k], lower, 2);
829 k = isl_basic_set_alloc_inequality(hull);
830 isl_seq_cpy(hull->ineq[k], upper, 2);
832 hull = isl_basic_set_finalize(hull);
842 /* Project out final n dimensions using Fourier-Motzkin */
843 static struct isl_set *set_project_out(struct isl_ctx *ctx,
844 struct isl_set *set, unsigned n)
846 return isl_set_remove_dims(set, isl_set_n_dim(set) - n, n);
849 static struct isl_basic_set *convex_hull_0d(struct isl_set *set)
851 struct isl_basic_set *convex_hull;
856 if (isl_set_is_empty(set))
857 convex_hull = isl_basic_set_empty(isl_dim_copy(set->dim));
859 convex_hull = isl_basic_set_universe(isl_dim_copy(set->dim));
864 /* Compute the convex hull of a pair of basic sets without any parameters or
865 * integer divisions using Fourier-Motzkin elimination.
866 * The convex hull is the set of all points that can be written as
867 * the sum of points from both basic sets (in homogeneous coordinates).
868 * We set up the constraints in a space with dimensions for each of
869 * the three sets and then project out the dimensions corresponding
870 * to the two original basic sets, retaining only those corresponding
871 * to the convex hull.
873 static struct isl_basic_set *convex_hull_pair_elim(struct isl_basic_set *bset1,
874 struct isl_basic_set *bset2)
877 struct isl_basic_set *bset[2];
878 struct isl_basic_set *hull = NULL;
881 if (!bset1 || !bset2)
884 dim = isl_basic_set_n_dim(bset1);
885 hull = isl_basic_set_alloc(bset1->ctx, 0, 2 + 3 * dim, 0,
886 1 + dim + bset1->n_eq + bset2->n_eq,
887 2 + bset1->n_ineq + bset2->n_ineq);
890 for (i = 0; i < 2; ++i) {
891 for (j = 0; j < bset[i]->n_eq; ++j) {
892 k = isl_basic_set_alloc_equality(hull);
895 isl_seq_clr(hull->eq[k], (i+1) * (1+dim));
896 isl_seq_clr(hull->eq[k]+(i+2)*(1+dim), (1-i)*(1+dim));
897 isl_seq_cpy(hull->eq[k]+(i+1)*(1+dim), bset[i]->eq[j],
900 for (j = 0; j < bset[i]->n_ineq; ++j) {
901 k = isl_basic_set_alloc_inequality(hull);
904 isl_seq_clr(hull->ineq[k], (i+1) * (1+dim));
905 isl_seq_clr(hull->ineq[k]+(i+2)*(1+dim), (1-i)*(1+dim));
906 isl_seq_cpy(hull->ineq[k]+(i+1)*(1+dim),
907 bset[i]->ineq[j], 1+dim);
909 k = isl_basic_set_alloc_inequality(hull);
912 isl_seq_clr(hull->ineq[k], 1+2+3*dim);
913 isl_int_set_si(hull->ineq[k][(i+1)*(1+dim)], 1);
915 for (j = 0; j < 1+dim; ++j) {
916 k = isl_basic_set_alloc_equality(hull);
919 isl_seq_clr(hull->eq[k], 1+2+3*dim);
920 isl_int_set_si(hull->eq[k][j], -1);
921 isl_int_set_si(hull->eq[k][1+dim+j], 1);
922 isl_int_set_si(hull->eq[k][2*(1+dim)+j], 1);
924 hull = isl_basic_set_set_rational(hull);
925 hull = isl_basic_set_remove_dims(hull, dim, 2*(1+dim));
926 hull = isl_basic_set_convex_hull(hull);
927 isl_basic_set_free(bset1);
928 isl_basic_set_free(bset2);
931 isl_basic_set_free(bset1);
932 isl_basic_set_free(bset2);
933 isl_basic_set_free(hull);
937 static int isl_basic_set_is_bounded(struct isl_basic_set *bset)
942 tab = isl_tab_from_recession_cone((struct isl_basic_map *)bset);
943 bounded = isl_tab_cone_is_bounded(tab);
948 static int isl_set_is_bounded(struct isl_set *set)
952 for (i = 0; i < set->n; ++i) {
953 int bounded = isl_basic_set_is_bounded(set->p[i]);
954 if (!bounded || bounded < 0)
960 /* Compute the lineality space of the convex hull of bset1 and bset2.
962 * We first compute the intersection of the recession cone of bset1
963 * with the negative of the recession cone of bset2 and then compute
964 * the linear hull of the resulting cone.
966 static struct isl_basic_set *induced_lineality_space(
967 struct isl_basic_set *bset1, struct isl_basic_set *bset2)
970 struct isl_basic_set *lin = NULL;
973 if (!bset1 || !bset2)
976 dim = isl_basic_set_total_dim(bset1);
977 lin = isl_basic_set_alloc_dim(isl_basic_set_get_dim(bset1), 0,
978 bset1->n_eq + bset2->n_eq,
979 bset1->n_ineq + bset2->n_ineq);
980 lin = isl_basic_set_set_rational(lin);
983 for (i = 0; i < bset1->n_eq; ++i) {
984 k = isl_basic_set_alloc_equality(lin);
987 isl_int_set_si(lin->eq[k][0], 0);
988 isl_seq_cpy(lin->eq[k] + 1, bset1->eq[i] + 1, dim);
990 for (i = 0; i < bset1->n_ineq; ++i) {
991 k = isl_basic_set_alloc_inequality(lin);
994 isl_int_set_si(lin->ineq[k][0], 0);
995 isl_seq_cpy(lin->ineq[k] + 1, bset1->ineq[i] + 1, dim);
997 for (i = 0; i < bset2->n_eq; ++i) {
998 k = isl_basic_set_alloc_equality(lin);
1001 isl_int_set_si(lin->eq[k][0], 0);
1002 isl_seq_neg(lin->eq[k] + 1, bset2->eq[i] + 1, dim);
1004 for (i = 0; i < bset2->n_ineq; ++i) {
1005 k = isl_basic_set_alloc_inequality(lin);
1008 isl_int_set_si(lin->ineq[k][0], 0);
1009 isl_seq_neg(lin->ineq[k] + 1, bset2->ineq[i] + 1, dim);
1012 isl_basic_set_free(bset1);
1013 isl_basic_set_free(bset2);
1014 return isl_basic_set_affine_hull(lin);
1016 isl_basic_set_free(lin);
1017 isl_basic_set_free(bset1);
1018 isl_basic_set_free(bset2);
1022 static struct isl_basic_set *uset_convex_hull(struct isl_set *set);
1024 /* Given a set and a linear space "lin" of dimension n > 0,
1025 * project the linear space from the set, compute the convex hull
1026 * and then map the set back to the original space.
1032 * describe the linear space. We first compute the Hermite normal
1033 * form H = M U of M = H Q, to obtain
1037 * The last n rows of H will be zero, so the last n variables of x' = Q x
1038 * are the one we want to project out. We do this by transforming each
1039 * basic set A x >= b to A U x' >= b and then removing the last n dimensions.
1040 * After computing the convex hull in x'_1, i.e., A' x'_1 >= b',
1041 * we transform the hull back to the original space as A' Q_1 x >= b',
1042 * with Q_1 all but the last n rows of Q.
1044 static struct isl_basic_set *modulo_lineality(struct isl_set *set,
1045 struct isl_basic_set *lin)
1047 unsigned total = isl_basic_set_total_dim(lin);
1049 struct isl_basic_set *hull;
1050 struct isl_mat *M, *U, *Q;
1054 lin_dim = total - lin->n_eq;
1055 M = isl_mat_sub_alloc(set->ctx, lin->eq, 0, lin->n_eq, 1, total);
1056 M = isl_mat_left_hermite(M, 0, &U, &Q);
1060 isl_basic_set_free(lin);
1062 Q = isl_mat_drop_rows(Q, Q->n_row - lin_dim, lin_dim);
1064 U = isl_mat_lin_to_aff(U);
1065 Q = isl_mat_lin_to_aff(Q);
1067 set = isl_set_preimage(set, U);
1068 set = isl_set_remove_dims(set, total - lin_dim, lin_dim);
1069 hull = uset_convex_hull(set);
1070 hull = isl_basic_set_preimage(hull, Q);
1074 isl_basic_set_free(lin);
1079 /* Given two polyhedra with as constraints h_{ij} x >= 0 in homegeneous space,
1080 * set up an LP for solving
1082 * \sum_j \alpha_{1j} h_{1j} = \sum_j \alpha_{2j} h_{2j}
1084 * \alpha{i0} corresponds to the (implicit) positivity constraint 1 >= 0
1085 * The next \alpha{ij} correspond to the equalities and come in pairs.
1086 * The final \alpha{ij} correspond to the inequalities.
1088 static struct isl_basic_set *valid_direction_lp(
1089 struct isl_basic_set *bset1, struct isl_basic_set *bset2)
1091 struct isl_dim *dim;
1092 struct isl_basic_set *lp;
1097 if (!bset1 || !bset2)
1099 d = 1 + isl_basic_set_total_dim(bset1);
1101 2 * bset1->n_eq + bset1->n_ineq + 2 * bset2->n_eq + bset2->n_ineq;
1102 dim = isl_dim_set_alloc(bset1->ctx, 0, n);
1103 lp = isl_basic_set_alloc_dim(dim, 0, d, n);
1106 for (i = 0; i < n; ++i) {
1107 k = isl_basic_set_alloc_inequality(lp);
1110 isl_seq_clr(lp->ineq[k] + 1, n);
1111 isl_int_set_si(lp->ineq[k][0], -1);
1112 isl_int_set_si(lp->ineq[k][1 + i], 1);
1114 for (i = 0; i < d; ++i) {
1115 k = isl_basic_set_alloc_equality(lp);
1119 isl_int_set_si(lp->eq[k][n++], 0);
1120 /* positivity constraint 1 >= 0 */
1121 isl_int_set_si(lp->eq[k][n++], i == 0);
1122 for (j = 0; j < bset1->n_eq; ++j) {
1123 isl_int_set(lp->eq[k][n++], bset1->eq[j][i]);
1124 isl_int_neg(lp->eq[k][n++], bset1->eq[j][i]);
1126 for (j = 0; j < bset1->n_ineq; ++j)
1127 isl_int_set(lp->eq[k][n++], bset1->ineq[j][i]);
1128 /* positivity constraint 1 >= 0 */
1129 isl_int_set_si(lp->eq[k][n++], -(i == 0));
1130 for (j = 0; j < bset2->n_eq; ++j) {
1131 isl_int_neg(lp->eq[k][n++], bset2->eq[j][i]);
1132 isl_int_set(lp->eq[k][n++], bset2->eq[j][i]);
1134 for (j = 0; j < bset2->n_ineq; ++j)
1135 isl_int_neg(lp->eq[k][n++], bset2->ineq[j][i]);
1137 lp = isl_basic_set_gauss(lp, NULL);
1138 isl_basic_set_free(bset1);
1139 isl_basic_set_free(bset2);
1142 isl_basic_set_free(bset1);
1143 isl_basic_set_free(bset2);
1147 /* Compute a vector s in the homogeneous space such that <s, r> > 0
1148 * for all rays in the homogeneous space of the two cones that correspond
1149 * to the input polyhedra bset1 and bset2.
1151 * We compute s as a vector that satisfies
1153 * s = \sum_j \alpha_{ij} h_{ij} for i = 1,2 (*)
1155 * with h_{ij} the normals of the facets of polyhedron i
1156 * (including the "positivity constraint" 1 >= 0) and \alpha_{ij}
1157 * strictly positive numbers. For simplicity we impose \alpha_{ij} >= 1.
1158 * We first set up an LP with as variables the \alpha{ij}.
1159 * In this formulateion, for each polyhedron i,
1160 * the first constraint is the positivity constraint, followed by pairs
1161 * of variables for the equalities, followed by variables for the inequalities.
1162 * We then simply pick a feasible solution and compute s using (*).
1164 * Note that we simply pick any valid direction and make no attempt
1165 * to pick a "good" or even the "best" valid direction.
1167 static struct isl_vec *valid_direction(
1168 struct isl_basic_set *bset1, struct isl_basic_set *bset2)
1170 struct isl_basic_set *lp;
1171 struct isl_tab *tab;
1172 struct isl_vec *sample = NULL;
1173 struct isl_vec *dir;
1178 if (!bset1 || !bset2)
1180 lp = valid_direction_lp(isl_basic_set_copy(bset1),
1181 isl_basic_set_copy(bset2));
1182 tab = isl_tab_from_basic_set(lp);
1183 sample = isl_tab_get_sample_value(tab);
1185 isl_basic_set_free(lp);
1188 d = isl_basic_set_total_dim(bset1);
1189 dir = isl_vec_alloc(bset1->ctx, 1 + d);
1192 isl_seq_clr(dir->block.data + 1, dir->size - 1);
1194 /* positivity constraint 1 >= 0 */
1195 isl_int_set(dir->block.data[0], sample->block.data[n++]);
1196 for (i = 0; i < bset1->n_eq; ++i) {
1197 isl_int_sub(sample->block.data[n],
1198 sample->block.data[n], sample->block.data[n+1]);
1199 isl_seq_combine(dir->block.data,
1200 bset1->ctx->one, dir->block.data,
1201 sample->block.data[n], bset1->eq[i], 1 + d);
1205 for (i = 0; i < bset1->n_ineq; ++i)
1206 isl_seq_combine(dir->block.data,
1207 bset1->ctx->one, dir->block.data,
1208 sample->block.data[n++], bset1->ineq[i], 1 + d);
1209 isl_vec_free(sample);
1210 isl_seq_normalize(bset1->ctx, dir->block.data + 1, dir->size - 1);
1211 isl_basic_set_free(bset1);
1212 isl_basic_set_free(bset2);
1215 isl_vec_free(sample);
1216 isl_basic_set_free(bset1);
1217 isl_basic_set_free(bset2);
1221 /* Given a polyhedron b_i + A_i x >= 0 and a map T = S^{-1},
1222 * compute b_i' + A_i' x' >= 0, with
1224 * [ b_i A_i ] [ y' ] [ y' ]
1225 * [ 1 0 ] S^{-1} [ x' ] >= 0 or [ b_i' A_i' ] [ x' ] >= 0
1227 * In particular, add the "positivity constraint" and then perform
1230 static struct isl_basic_set *homogeneous_map(struct isl_basic_set *bset,
1237 bset = isl_basic_set_extend_constraints(bset, 0, 1);
1238 k = isl_basic_set_alloc_inequality(bset);
1241 isl_seq_clr(bset->ineq[k] + 1, isl_basic_set_total_dim(bset));
1242 isl_int_set_si(bset->ineq[k][0], 1);
1243 bset = isl_basic_set_preimage(bset, T);
1247 isl_basic_set_free(bset);
1251 /* Compute the convex hull of a pair of basic sets without any parameters or
1252 * integer divisions, where the convex hull is known to be pointed,
1253 * but the basic sets may be unbounded.
1255 * We turn this problem into the computation of a convex hull of a pair
1256 * _bounded_ polyhedra by "changing the direction of the homogeneous
1257 * dimension". This idea is due to Matthias Koeppe.
1259 * Consider the cones in homogeneous space that correspond to the
1260 * input polyhedra. The rays of these cones are also rays of the
1261 * polyhedra if the coordinate that corresponds to the homogeneous
1262 * dimension is zero. That is, if the inner product of the rays
1263 * with the homogeneous direction is zero.
1264 * The cones in the homogeneous space can also be considered to
1265 * correspond to other pairs of polyhedra by chosing a different
1266 * homogeneous direction. To ensure that both of these polyhedra
1267 * are bounded, we need to make sure that all rays of the cones
1268 * correspond to vertices and not to rays.
1269 * Let s be a direction such that <s, r> > 0 for all rays r of both cones.
1270 * Then using s as a homogeneous direction, we obtain a pair of polytopes.
1271 * The vector s is computed in valid_direction.
1273 * Note that we need to consider _all_ rays of the cones and not just
1274 * the rays that correspond to rays in the polyhedra. If we were to
1275 * only consider those rays and turn them into vertices, then we
1276 * may inadvertently turn some vertices into rays.
1278 * The standard homogeneous direction is the unit vector in the 0th coordinate.
1279 * We therefore transform the two polyhedra such that the selected
1280 * direction is mapped onto this standard direction and then proceed
1281 * with the normal computation.
1282 * Let S be a non-singular square matrix with s as its first row,
1283 * then we want to map the polyhedra to the space
1285 * [ y' ] [ y ] [ y ] [ y' ]
1286 * [ x' ] = S [ x ] i.e., [ x ] = S^{-1} [ x' ]
1288 * We take S to be the unimodular completion of s to limit the growth
1289 * of the coefficients in the following computations.
1291 * Let b_i + A_i x >= 0 be the constraints of polyhedron i.
1292 * We first move to the homogeneous dimension
1294 * b_i y + A_i x >= 0 [ b_i A_i ] [ y ] [ 0 ]
1295 * y >= 0 or [ 1 0 ] [ x ] >= [ 0 ]
1297 * Then we change directoin
1299 * [ b_i A_i ] [ y' ] [ y' ]
1300 * [ 1 0 ] S^{-1} [ x' ] >= 0 or [ b_i' A_i' ] [ x' ] >= 0
1302 * Then we compute the convex hull of the polytopes b_i' + A_i' x' >= 0
1303 * resulting in b' + A' x' >= 0, which we then convert back
1306 * [ b' A' ] S [ x ] >= 0 or [ b A ] [ x ] >= 0
1308 * The polyhedron b + A x >= 0 is then the convex hull of the input polyhedra.
1310 static struct isl_basic_set *convex_hull_pair_pointed(
1311 struct isl_basic_set *bset1, struct isl_basic_set *bset2)
1313 struct isl_ctx *ctx = NULL;
1314 struct isl_vec *dir = NULL;
1315 struct isl_mat *T = NULL;
1316 struct isl_mat *T2 = NULL;
1317 struct isl_basic_set *hull;
1318 struct isl_set *set;
1320 if (!bset1 || !bset2)
1323 dir = valid_direction(isl_basic_set_copy(bset1),
1324 isl_basic_set_copy(bset2));
1327 T = isl_mat_alloc(bset1->ctx, dir->size, dir->size);
1330 isl_seq_cpy(T->row[0], dir->block.data, dir->size);
1331 T = isl_mat_unimodular_complete(T, 1);
1332 T2 = isl_mat_right_inverse(isl_mat_copy(T));
1334 bset1 = homogeneous_map(bset1, isl_mat_copy(T2));
1335 bset2 = homogeneous_map(bset2, T2);
1336 set = isl_set_alloc_dim(isl_basic_set_get_dim(bset1), 2, 0);
1337 set = isl_set_add(set, bset1);
1338 set = isl_set_add(set, bset2);
1339 hull = uset_convex_hull(set);
1340 hull = isl_basic_set_preimage(hull, T);
1347 isl_basic_set_free(bset1);
1348 isl_basic_set_free(bset2);
1352 /* Compute the convex hull of a pair of basic sets without any parameters or
1353 * integer divisions.
1355 * If the convex hull of the two basic sets would have a non-trivial
1356 * lineality space, we first project out this lineality space.
1358 static struct isl_basic_set *convex_hull_pair(struct isl_basic_set *bset1,
1359 struct isl_basic_set *bset2)
1361 struct isl_basic_set *lin;
1363 if (isl_basic_set_is_bounded(bset1) || isl_basic_set_is_bounded(bset2))
1364 return convex_hull_pair_pointed(bset1, bset2);
1366 lin = induced_lineality_space(isl_basic_set_copy(bset1),
1367 isl_basic_set_copy(bset2));
1370 if (isl_basic_set_is_universe(lin)) {
1371 isl_basic_set_free(bset1);
1372 isl_basic_set_free(bset2);
1375 if (lin->n_eq < isl_basic_set_total_dim(lin)) {
1376 struct isl_set *set;
1377 set = isl_set_alloc_dim(isl_basic_set_get_dim(bset1), 2, 0);
1378 set = isl_set_add(set, bset1);
1379 set = isl_set_add(set, bset2);
1380 return modulo_lineality(set, lin);
1382 isl_basic_set_free(lin);
1384 return convex_hull_pair_pointed(bset1, bset2);
1386 isl_basic_set_free(bset1);
1387 isl_basic_set_free(bset2);
1391 /* Compute the lineality space of a basic set.
1392 * We currently do not allow the basic set to have any divs.
1393 * We basically just drop the constants and turn every inequality
1396 struct isl_basic_set *isl_basic_set_lineality_space(struct isl_basic_set *bset)
1399 struct isl_basic_set *lin = NULL;
1404 isl_assert(bset->ctx, bset->n_div == 0, goto error);
1405 dim = isl_basic_set_total_dim(bset);
1407 lin = isl_basic_set_alloc_dim(isl_basic_set_get_dim(bset), 0, dim, 0);
1410 for (i = 0; i < bset->n_eq; ++i) {
1411 k = isl_basic_set_alloc_equality(lin);
1414 isl_int_set_si(lin->eq[k][0], 0);
1415 isl_seq_cpy(lin->eq[k] + 1, bset->eq[i] + 1, dim);
1417 lin = isl_basic_set_gauss(lin, NULL);
1420 for (i = 0; i < bset->n_ineq && lin->n_eq < dim; ++i) {
1421 k = isl_basic_set_alloc_equality(lin);
1424 isl_int_set_si(lin->eq[k][0], 0);
1425 isl_seq_cpy(lin->eq[k] + 1, bset->ineq[i] + 1, dim);
1426 lin = isl_basic_set_gauss(lin, NULL);
1430 isl_basic_set_free(bset);
1433 isl_basic_set_free(lin);
1434 isl_basic_set_free(bset);
1438 /* Compute the (linear) hull of the lineality spaces of the basic sets in the
1439 * "underlying" set "set".
1441 static struct isl_basic_set *uset_combined_lineality_space(struct isl_set *set)
1444 struct isl_set *lin = NULL;
1449 struct isl_dim *dim = isl_set_get_dim(set);
1451 return isl_basic_set_empty(dim);
1454 lin = isl_set_alloc_dim(isl_set_get_dim(set), set->n, 0);
1455 for (i = 0; i < set->n; ++i)
1456 lin = isl_set_add(lin,
1457 isl_basic_set_lineality_space(isl_basic_set_copy(set->p[i])));
1459 return isl_set_affine_hull(lin);
1462 /* Compute the convex hull of a set without any parameters or
1463 * integer divisions.
1464 * In each step, we combined two basic sets until only one
1465 * basic set is left.
1466 * The input basic sets are assumed not to have a non-trivial
1467 * lineality space. If any of the intermediate results has
1468 * a non-trivial lineality space, it is projected out.
1470 static struct isl_basic_set *uset_convex_hull_unbounded(struct isl_set *set)
1472 struct isl_basic_set *convex_hull = NULL;
1474 convex_hull = isl_set_copy_basic_set(set);
1475 set = isl_set_drop_basic_set(set, convex_hull);
1478 while (set->n > 0) {
1479 struct isl_basic_set *t;
1480 t = isl_set_copy_basic_set(set);
1483 set = isl_set_drop_basic_set(set, t);
1486 convex_hull = convex_hull_pair(convex_hull, t);
1489 t = isl_basic_set_lineality_space(isl_basic_set_copy(convex_hull));
1492 if (isl_basic_set_is_universe(t)) {
1493 isl_basic_set_free(convex_hull);
1497 if (t->n_eq < isl_basic_set_total_dim(t)) {
1498 set = isl_set_add(set, convex_hull);
1499 return modulo_lineality(set, t);
1501 isl_basic_set_free(t);
1507 isl_basic_set_free(convex_hull);
1511 /* Compute an initial hull for wrapping containing a single initial
1512 * facet by first computing bounds on the set and then using these
1513 * bounds to construct an initial facet.
1514 * This function is a remnant of an older implementation where the
1515 * bounds were also used to check whether the set was bounded.
1516 * Since this function will now only be called when we know the
1517 * set to be bounded, the initial facet should probably be constructed
1518 * by simply using the coordinate directions instead.
1520 static struct isl_basic_set *initial_hull(struct isl_basic_set *hull,
1521 struct isl_set *set)
1523 struct isl_mat *bounds = NULL;
1529 bounds = independent_bounds(set);
1532 isl_assert(set->ctx, bounds->n_row == isl_set_n_dim(set), goto error);
1533 bounds = initial_facet_constraint(set, bounds);
1536 k = isl_basic_set_alloc_inequality(hull);
1539 dim = isl_set_n_dim(set);
1540 isl_assert(set->ctx, 1 + dim == bounds->n_col, goto error);
1541 isl_seq_cpy(hull->ineq[k], bounds->row[0], bounds->n_col);
1542 isl_mat_free(bounds);
1546 isl_basic_set_free(hull);
1547 isl_mat_free(bounds);
1551 struct max_constraint {
1557 static int max_constraint_equal(const void *entry, const void *val)
1559 struct max_constraint *a = (struct max_constraint *)entry;
1560 isl_int *b = (isl_int *)val;
1562 return isl_seq_eq(a->c->row[0] + 1, b, a->c->n_col - 1);
1565 static void update_constraint(struct isl_ctx *ctx, struct isl_hash_table *table,
1566 isl_int *con, unsigned len, int n, int ineq)
1568 struct isl_hash_table_entry *entry;
1569 struct max_constraint *c;
1572 c_hash = isl_seq_get_hash(con + 1, len);
1573 entry = isl_hash_table_find(ctx, table, c_hash, max_constraint_equal,
1579 isl_hash_table_remove(ctx, table, entry);
1583 if (isl_int_gt(c->c->row[0][0], con[0]))
1585 if (isl_int_eq(c->c->row[0][0], con[0])) {
1590 c->c = isl_mat_cow(c->c);
1591 isl_int_set(c->c->row[0][0], con[0]);
1595 /* Check whether the constraint hash table "table" constains the constraint
1598 static int has_constraint(struct isl_ctx *ctx, struct isl_hash_table *table,
1599 isl_int *con, unsigned len, int n)
1601 struct isl_hash_table_entry *entry;
1602 struct max_constraint *c;
1605 c_hash = isl_seq_get_hash(con + 1, len);
1606 entry = isl_hash_table_find(ctx, table, c_hash, max_constraint_equal,
1613 return isl_int_eq(c->c->row[0][0], con[0]);
1616 /* Check for inequality constraints of a basic set without equalities
1617 * such that the same or more stringent copies of the constraint appear
1618 * in all of the basic sets. Such constraints are necessarily facet
1619 * constraints of the convex hull.
1621 * If the resulting basic set is by chance identical to one of
1622 * the basic sets in "set", then we know that this basic set contains
1623 * all other basic sets and is therefore the convex hull of set.
1624 * In this case we set *is_hull to 1.
1626 static struct isl_basic_set *common_constraints(struct isl_basic_set *hull,
1627 struct isl_set *set, int *is_hull)
1630 int min_constraints;
1632 struct max_constraint *constraints = NULL;
1633 struct isl_hash_table *table = NULL;
1638 for (i = 0; i < set->n; ++i)
1639 if (set->p[i]->n_eq == 0)
1643 min_constraints = set->p[i]->n_ineq;
1645 for (i = best + 1; i < set->n; ++i) {
1646 if (set->p[i]->n_eq != 0)
1648 if (set->p[i]->n_ineq >= min_constraints)
1650 min_constraints = set->p[i]->n_ineq;
1653 constraints = isl_calloc_array(hull->ctx, struct max_constraint,
1657 table = isl_alloc_type(hull->ctx, struct isl_hash_table);
1658 if (isl_hash_table_init(hull->ctx, table, min_constraints))
1661 total = isl_dim_total(set->dim);
1662 for (i = 0; i < set->p[best]->n_ineq; ++i) {
1663 constraints[i].c = isl_mat_sub_alloc(hull->ctx,
1664 set->p[best]->ineq + i, 0, 1, 0, 1 + total);
1665 if (!constraints[i].c)
1667 constraints[i].ineq = 1;
1669 for (i = 0; i < min_constraints; ++i) {
1670 struct isl_hash_table_entry *entry;
1672 c_hash = isl_seq_get_hash(constraints[i].c->row[0] + 1, total);
1673 entry = isl_hash_table_find(hull->ctx, table, c_hash,
1674 max_constraint_equal, constraints[i].c->row[0] + 1, 1);
1677 isl_assert(hull->ctx, !entry->data, goto error);
1678 entry->data = &constraints[i];
1682 for (s = 0; s < set->n; ++s) {
1686 for (i = 0; i < set->p[s]->n_eq; ++i) {
1687 isl_int *eq = set->p[s]->eq[i];
1688 for (j = 0; j < 2; ++j) {
1689 isl_seq_neg(eq, eq, 1 + total);
1690 update_constraint(hull->ctx, table,
1694 for (i = 0; i < set->p[s]->n_ineq; ++i) {
1695 isl_int *ineq = set->p[s]->ineq[i];
1696 update_constraint(hull->ctx, table, ineq, total, n,
1697 set->p[s]->n_eq == 0);
1702 for (i = 0; i < min_constraints; ++i) {
1703 if (constraints[i].count < n)
1705 if (!constraints[i].ineq)
1707 j = isl_basic_set_alloc_inequality(hull);
1710 isl_seq_cpy(hull->ineq[j], constraints[i].c->row[0], 1 + total);
1713 for (s = 0; s < set->n; ++s) {
1714 if (set->p[s]->n_eq)
1716 if (set->p[s]->n_ineq != hull->n_ineq)
1718 for (i = 0; i < set->p[s]->n_ineq; ++i) {
1719 isl_int *ineq = set->p[s]->ineq[i];
1720 if (!has_constraint(hull->ctx, table, ineq, total, n))
1723 if (i == set->p[s]->n_ineq)
1727 isl_hash_table_clear(table);
1728 for (i = 0; i < min_constraints; ++i)
1729 isl_mat_free(constraints[i].c);
1734 isl_hash_table_clear(table);
1737 for (i = 0; i < min_constraints; ++i)
1738 isl_mat_free(constraints[i].c);
1743 /* Create a template for the convex hull of "set" and fill it up
1744 * obvious facet constraints, if any. If the result happens to
1745 * be the convex hull of "set" then *is_hull is set to 1.
1747 static struct isl_basic_set *proto_hull(struct isl_set *set, int *is_hull)
1749 struct isl_basic_set *hull;
1754 for (i = 0; i < set->n; ++i) {
1755 n_ineq += set->p[i]->n_eq;
1756 n_ineq += set->p[i]->n_ineq;
1758 hull = isl_basic_set_alloc_dim(isl_dim_copy(set->dim), 0, 0, n_ineq);
1759 hull = isl_basic_set_set_rational(hull);
1762 return common_constraints(hull, set, is_hull);
1765 static struct isl_basic_set *uset_convex_hull_wrap(struct isl_set *set)
1767 struct isl_basic_set *hull;
1770 hull = proto_hull(set, &is_hull);
1771 if (hull && !is_hull) {
1772 if (hull->n_ineq == 0)
1773 hull = initial_hull(hull, set);
1774 hull = extend(hull, set);
1781 /* Compute the convex hull of a set without any parameters or
1782 * integer divisions. Depending on whether the set is bounded,
1783 * we pass control to the wrapping based convex hull or
1784 * the Fourier-Motzkin elimination based convex hull.
1785 * We also handle a few special cases before checking the boundedness.
1787 static struct isl_basic_set *uset_convex_hull(struct isl_set *set)
1790 struct isl_basic_set *convex_hull = NULL;
1791 struct isl_basic_set *lin;
1793 if (isl_set_n_dim(set) == 0)
1794 return convex_hull_0d(set);
1796 set = isl_set_coalesce(set);
1797 set = isl_set_set_rational(set);
1804 convex_hull = isl_basic_set_copy(set->p[0]);
1808 if (isl_set_n_dim(set) == 1)
1809 return convex_hull_1d(set);
1811 if (isl_set_is_bounded(set))
1812 return uset_convex_hull_wrap(set);
1814 lin = uset_combined_lineality_space(isl_set_copy(set));
1817 if (isl_basic_set_is_universe(lin)) {
1821 if (lin->n_eq < isl_basic_set_total_dim(lin))
1822 return modulo_lineality(set, lin);
1823 isl_basic_set_free(lin);
1825 return uset_convex_hull_unbounded(set);
1828 isl_basic_set_free(convex_hull);
1832 /* This is the core procedure, where "set" is a "pure" set, i.e.,
1833 * without parameters or divs and where the convex hull of set is
1834 * known to be full-dimensional.
1836 static struct isl_basic_set *uset_convex_hull_wrap_bounded(struct isl_set *set)
1839 struct isl_basic_set *convex_hull = NULL;
1841 if (isl_set_n_dim(set) == 0) {
1842 convex_hull = isl_basic_set_universe(isl_dim_copy(set->dim));
1844 convex_hull = isl_basic_set_set_rational(convex_hull);
1848 set = isl_set_set_rational(set);
1852 set = isl_set_coalesce(set);
1856 convex_hull = isl_basic_set_copy(set->p[0]);
1860 if (isl_set_n_dim(set) == 1)
1861 return convex_hull_1d(set);
1863 return uset_convex_hull_wrap(set);
1869 /* Compute the convex hull of set "set" with affine hull "affine_hull",
1870 * We first remove the equalities (transforming the set), compute the
1871 * convex hull of the transformed set and then add the equalities back
1872 * (after performing the inverse transformation.
1874 static struct isl_basic_set *modulo_affine_hull(struct isl_ctx *ctx,
1875 struct isl_set *set, struct isl_basic_set *affine_hull)
1879 struct isl_basic_set *dummy;
1880 struct isl_basic_set *convex_hull;
1882 dummy = isl_basic_set_remove_equalities(
1883 isl_basic_set_copy(affine_hull), &T, &T2);
1886 isl_basic_set_free(dummy);
1887 set = isl_set_preimage(set, T);
1888 convex_hull = uset_convex_hull(set);
1889 convex_hull = isl_basic_set_preimage(convex_hull, T2);
1890 convex_hull = isl_basic_set_intersect(convex_hull, affine_hull);
1893 isl_basic_set_free(affine_hull);
1898 /* Compute the convex hull of a map.
1900 * The implementation was inspired by "Extended Convex Hull" by Fukuda et al.,
1901 * specifically, the wrapping of facets to obtain new facets.
1903 struct isl_basic_map *isl_map_convex_hull(struct isl_map *map)
1905 struct isl_basic_set *bset;
1906 struct isl_basic_map *model = NULL;
1907 struct isl_basic_set *affine_hull = NULL;
1908 struct isl_basic_map *convex_hull = NULL;
1909 struct isl_set *set = NULL;
1910 struct isl_ctx *ctx;
1917 convex_hull = isl_basic_map_empty_like_map(map);
1922 map = isl_map_detect_equalities(map);
1923 map = isl_map_align_divs(map);
1924 model = isl_basic_map_copy(map->p[0]);
1925 set = isl_map_underlying_set(map);
1929 affine_hull = isl_set_affine_hull(isl_set_copy(set));
1932 if (affine_hull->n_eq != 0)
1933 bset = modulo_affine_hull(ctx, set, affine_hull);
1935 isl_basic_set_free(affine_hull);
1936 bset = uset_convex_hull(set);
1939 convex_hull = isl_basic_map_overlying_set(bset, model);
1941 ISL_F_SET(convex_hull, ISL_BASIC_MAP_NO_IMPLICIT);
1942 ISL_F_SET(convex_hull, ISL_BASIC_MAP_ALL_EQUALITIES);
1943 ISL_F_CLR(convex_hull, ISL_BASIC_MAP_RATIONAL);
1947 isl_basic_map_free(model);
1951 struct isl_basic_set *isl_set_convex_hull(struct isl_set *set)
1953 return (struct isl_basic_set *)
1954 isl_map_convex_hull((struct isl_map *)set);
1957 struct sh_data_entry {
1958 struct isl_hash_table *table;
1959 struct isl_tab *tab;
1962 /* Holds the data needed during the simple hull computation.
1964 * n the number of basic sets in the original set
1965 * hull_table a hash table of already computed constraints
1966 * in the simple hull
1967 * p for each basic set,
1968 * table a hash table of the constraints
1969 * tab the tableau corresponding to the basic set
1972 struct isl_ctx *ctx;
1974 struct isl_hash_table *hull_table;
1975 struct sh_data_entry p[1];
1978 static void sh_data_free(struct sh_data *data)
1984 isl_hash_table_free(data->ctx, data->hull_table);
1985 for (i = 0; i < data->n; ++i) {
1986 isl_hash_table_free(data->ctx, data->p[i].table);
1987 isl_tab_free(data->p[i].tab);
1992 struct ineq_cmp_data {
1997 static int has_ineq(const void *entry, const void *val)
1999 isl_int *row = (isl_int *)entry;
2000 struct ineq_cmp_data *v = (struct ineq_cmp_data *)val;
2002 return isl_seq_eq(row + 1, v->p + 1, v->len) ||
2003 isl_seq_is_neg(row + 1, v->p + 1, v->len);
2006 static int hash_ineq(struct isl_ctx *ctx, struct isl_hash_table *table,
2007 isl_int *ineq, unsigned len)
2010 struct ineq_cmp_data v;
2011 struct isl_hash_table_entry *entry;
2015 c_hash = isl_seq_get_hash(ineq + 1, len);
2016 entry = isl_hash_table_find(ctx, table, c_hash, has_ineq, &v, 1);
2023 /* Fill hash table "table" with the constraints of "bset".
2024 * Equalities are added as two inequalities.
2025 * The value in the hash table is a pointer to the (in)equality of "bset".
2027 static int hash_basic_set(struct isl_hash_table *table,
2028 struct isl_basic_set *bset)
2031 unsigned dim = isl_basic_set_total_dim(bset);
2033 for (i = 0; i < bset->n_eq; ++i) {
2034 for (j = 0; j < 2; ++j) {
2035 isl_seq_neg(bset->eq[i], bset->eq[i], 1 + dim);
2036 if (hash_ineq(bset->ctx, table, bset->eq[i], dim) < 0)
2040 for (i = 0; i < bset->n_ineq; ++i) {
2041 if (hash_ineq(bset->ctx, table, bset->ineq[i], dim) < 0)
2047 static struct sh_data *sh_data_alloc(struct isl_set *set, unsigned n_ineq)
2049 struct sh_data *data;
2052 data = isl_calloc(set->ctx, struct sh_data,
2053 sizeof(struct sh_data) +
2054 (set->n - 1) * sizeof(struct sh_data_entry));
2057 data->ctx = set->ctx;
2059 data->hull_table = isl_hash_table_alloc(set->ctx, n_ineq);
2060 if (!data->hull_table)
2062 for (i = 0; i < set->n; ++i) {
2063 data->p[i].table = isl_hash_table_alloc(set->ctx,
2064 2 * set->p[i]->n_eq + set->p[i]->n_ineq);
2065 if (!data->p[i].table)
2067 if (hash_basic_set(data->p[i].table, set->p[i]) < 0)
2076 /* Check if inequality "ineq" is a bound for basic set "j" or if
2077 * it can be relaxed (by increasing the constant term) to become
2078 * a bound for that basic set. In the latter case, the constant
2080 * Return 1 if "ineq" is a bound
2081 * 0 if "ineq" may attain arbitrarily small values on basic set "j"
2082 * -1 if some error occurred
2084 static int is_bound(struct sh_data *data, struct isl_set *set, int j,
2087 enum isl_lp_result res;
2090 if (!data->p[j].tab) {
2091 data->p[j].tab = isl_tab_from_basic_set(set->p[j]);
2092 if (!data->p[j].tab)
2098 res = isl_tab_min(data->p[j].tab, ineq, data->ctx->one,
2100 if (res == isl_lp_ok && isl_int_is_neg(opt))
2101 isl_int_sub(ineq[0], ineq[0], opt);
2105 return res == isl_lp_ok ? 1 :
2106 res == isl_lp_unbounded ? 0 : -1;
2109 /* Check if inequality "ineq" from basic set "i" can be relaxed to
2110 * become a bound on the whole set. If so, add the (relaxed) inequality
2113 * We first check if "hull" already contains a translate of the inequality.
2114 * If so, we are done.
2115 * Then, we check if any of the previous basic sets contains a translate
2116 * of the inequality. If so, then we have already considered this
2117 * inequality and we are done.
2118 * Otherwise, for each basic set other than "i", we check if the inequality
2119 * is a bound on the basic set.
2120 * For previous basic sets, we know that they do not contain a translate
2121 * of the inequality, so we directly call is_bound.
2122 * For following basic sets, we first check if a translate of the
2123 * inequality appears in its description and if so directly update
2124 * the inequality accordingly.
2126 static struct isl_basic_set *add_bound(struct isl_basic_set *hull,
2127 struct sh_data *data, struct isl_set *set, int i, isl_int *ineq)
2130 struct ineq_cmp_data v;
2131 struct isl_hash_table_entry *entry;
2137 v.len = isl_basic_set_total_dim(hull);
2139 c_hash = isl_seq_get_hash(ineq + 1, v.len);
2141 entry = isl_hash_table_find(hull->ctx, data->hull_table, c_hash,
2146 for (j = 0; j < i; ++j) {
2147 entry = isl_hash_table_find(hull->ctx, data->p[j].table,
2148 c_hash, has_ineq, &v, 0);
2155 k = isl_basic_set_alloc_inequality(hull);
2156 isl_seq_cpy(hull->ineq[k], ineq, 1 + v.len);
2160 for (j = 0; j < i; ++j) {
2162 bound = is_bound(data, set, j, hull->ineq[k]);
2169 isl_basic_set_free_inequality(hull, 1);
2173 for (j = i + 1; j < set->n; ++j) {
2176 entry = isl_hash_table_find(hull->ctx, data->p[j].table,
2177 c_hash, has_ineq, &v, 0);
2179 ineq_j = entry->data;
2180 neg = isl_seq_is_neg(ineq_j + 1,
2181 hull->ineq[k] + 1, v.len);
2183 isl_int_neg(ineq_j[0], ineq_j[0]);
2184 if (isl_int_gt(ineq_j[0], hull->ineq[k][0]))
2185 isl_int_set(hull->ineq[k][0], ineq_j[0]);
2187 isl_int_neg(ineq_j[0], ineq_j[0]);
2190 bound = is_bound(data, set, j, hull->ineq[k]);
2197 isl_basic_set_free_inequality(hull, 1);
2201 entry = isl_hash_table_find(hull->ctx, data->hull_table, c_hash,
2205 entry->data = hull->ineq[k];
2209 isl_basic_set_free(hull);
2213 /* Check if any inequality from basic set "i" can be relaxed to
2214 * become a bound on the whole set. If so, add the (relaxed) inequality
2217 static struct isl_basic_set *add_bounds(struct isl_basic_set *bset,
2218 struct sh_data *data, struct isl_set *set, int i)
2221 unsigned dim = isl_basic_set_total_dim(bset);
2223 for (j = 0; j < set->p[i]->n_eq; ++j) {
2224 for (k = 0; k < 2; ++k) {
2225 isl_seq_neg(set->p[i]->eq[j], set->p[i]->eq[j], 1+dim);
2226 add_bound(bset, data, set, i, set->p[i]->eq[j]);
2229 for (j = 0; j < set->p[i]->n_ineq; ++j)
2230 add_bound(bset, data, set, i, set->p[i]->ineq[j]);
2234 /* Compute a superset of the convex hull of set that is described
2235 * by only translates of the constraints in the constituents of set.
2237 static struct isl_basic_set *uset_simple_hull(struct isl_set *set)
2239 struct sh_data *data = NULL;
2240 struct isl_basic_set *hull = NULL;
2248 for (i = 0; i < set->n; ++i) {
2251 n_ineq += 2 * set->p[i]->n_eq + set->p[i]->n_ineq;
2254 hull = isl_basic_set_alloc_dim(isl_dim_copy(set->dim), 0, 0, n_ineq);
2258 data = sh_data_alloc(set, n_ineq);
2262 for (i = 0; i < set->n; ++i)
2263 hull = add_bounds(hull, data, set, i);
2271 isl_basic_set_free(hull);
2276 /* Compute a superset of the convex hull of map that is described
2277 * by only translates of the constraints in the constituents of map.
2279 struct isl_basic_map *isl_map_simple_hull(struct isl_map *map)
2281 struct isl_set *set = NULL;
2282 struct isl_basic_map *model = NULL;
2283 struct isl_basic_map *hull;
2284 struct isl_basic_map *affine_hull;
2285 struct isl_basic_set *bset = NULL;
2290 hull = isl_basic_map_empty_like_map(map);
2295 hull = isl_basic_map_copy(map->p[0]);
2300 map = isl_map_detect_equalities(map);
2301 affine_hull = isl_map_affine_hull(isl_map_copy(map));
2302 map = isl_map_align_divs(map);
2303 model = isl_basic_map_copy(map->p[0]);
2305 set = isl_map_underlying_set(map);
2307 bset = uset_simple_hull(set);
2309 hull = isl_basic_map_overlying_set(bset, model);
2311 hull = isl_basic_map_intersect(hull, affine_hull);
2312 hull = isl_basic_map_convex_hull(hull);
2313 ISL_F_SET(hull, ISL_BASIC_MAP_NO_IMPLICIT);
2314 ISL_F_SET(hull, ISL_BASIC_MAP_ALL_EQUALITIES);
2319 struct isl_basic_set *isl_set_simple_hull(struct isl_set *set)
2321 return (struct isl_basic_set *)
2322 isl_map_simple_hull((struct isl_map *)set);
2325 /* Given a set "set", return parametric bounds on the dimension "dim".
2327 static struct isl_basic_set *set_bounds(struct isl_set *set, int dim)
2329 unsigned set_dim = isl_set_dim(set, isl_dim_set);
2330 set = isl_set_copy(set);
2331 set = isl_set_eliminate_dims(set, dim + 1, set_dim - (dim + 1));
2332 set = isl_set_eliminate_dims(set, 0, dim);
2333 return isl_set_convex_hull(set);
2336 /* Computes a "simple hull" and then check if each dimension in the
2337 * resulting hull is bounded by a symbolic constant. If not, the
2338 * hull is intersected with the corresponding bounds on the whole set.
2340 struct isl_basic_set *isl_set_bounded_simple_hull(struct isl_set *set)
2343 struct isl_basic_set *hull;
2344 unsigned nparam, left;
2345 int removed_divs = 0;
2347 hull = isl_set_simple_hull(isl_set_copy(set));
2351 nparam = isl_basic_set_dim(hull, isl_dim_param);
2352 for (i = 0; i < isl_basic_set_dim(hull, isl_dim_set); ++i) {
2353 int lower = 0, upper = 0;
2354 struct isl_basic_set *bounds;
2356 left = isl_basic_set_total_dim(hull) - nparam - i - 1;
2357 for (j = 0; j < hull->n_eq; ++j) {
2358 if (isl_int_is_zero(hull->eq[j][1 + nparam + i]))
2360 if (isl_seq_first_non_zero(hull->eq[j]+1+nparam+i+1,
2367 for (j = 0; j < hull->n_ineq; ++j) {
2368 if (isl_int_is_zero(hull->ineq[j][1 + nparam + i]))
2370 if (isl_seq_first_non_zero(hull->ineq[j]+1+nparam+i+1,
2372 isl_seq_first_non_zero(hull->ineq[j]+1+nparam,
2375 if (isl_int_is_pos(hull->ineq[j][1 + nparam + i]))
2386 if (!removed_divs) {
2387 set = isl_set_remove_divs(set);
2392 bounds = set_bounds(set, i);
2393 hull = isl_basic_set_intersect(hull, bounds);