3 #include "isl_map_private.h"
7 #include "isl_equalities.h"
10 static struct isl_basic_set *uset_convex_hull_wrap(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+1, (*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);
63 isl_int_addmul(*opt_n, *opt_d, c[0]);
65 isl_int_add(*opt_n, *opt_n, c[0]);
66 return !isl_int_is_neg(*opt_n);
69 int isl_basic_set_constraint_is_redundant(struct isl_basic_set **bset,
70 isl_int *c, isl_int *opt_n, isl_int *opt_d)
72 return isl_basic_map_constraint_is_redundant(
73 (struct isl_basic_map **)bset, c, opt_n, opt_d);
76 /* Compute the convex hull of a basic map, by removing the redundant
77 * constraints. If the minimal value along the normal of a constraint
78 * is the same if the constraint is removed, then the constraint is redundant.
80 * Alternatively, we could have intersected the basic map with the
81 * corresponding equality and the checked if the dimension was that
84 struct isl_basic_map *isl_basic_map_convex_hull(struct isl_basic_map *bmap)
91 bmap = isl_basic_map_gauss(bmap, NULL);
92 if (ISL_F_ISSET(bmap, ISL_BASIC_MAP_EMPTY))
94 if (ISL_F_ISSET(bmap, ISL_BASIC_MAP_NO_REDUNDANT))
96 if (bmap->n_ineq <= 1)
99 tab = isl_tab_from_basic_map(bmap);
100 tab = isl_tab_detect_equalities(bmap->ctx, tab);
101 tab = isl_tab_detect_redundant(bmap->ctx, tab);
102 bmap = isl_basic_map_update_from_tab(bmap, tab);
103 isl_tab_free(bmap->ctx, tab);
104 ISL_F_SET(bmap, ISL_BASIC_MAP_NO_IMPLICIT);
105 ISL_F_SET(bmap, ISL_BASIC_MAP_NO_REDUNDANT);
109 struct isl_basic_set *isl_basic_set_convex_hull(struct isl_basic_set *bset)
111 return (struct isl_basic_set *)
112 isl_basic_map_convex_hull((struct isl_basic_map *)bset);
115 /* Check if the set set is bound in the direction of the affine
116 * constraint c and if so, set the constant term such that the
117 * resulting constraint is a bounding constraint for the set.
119 static int uset_is_bound(struct isl_ctx *ctx, struct isl_set *set,
120 isl_int *c, unsigned len)
128 isl_int_init(opt_denom);
130 for (j = 0; j < set->n; ++j) {
131 enum isl_lp_result res;
133 if (ISL_F_ISSET(set->p[j], ISL_BASIC_SET_EMPTY))
136 res = isl_solve_lp((struct isl_basic_map*)set->p[j],
137 0, c+1, ctx->one, &opt, &opt_denom);
138 if (res == isl_lp_unbounded)
140 if (res == isl_lp_error)
142 if (res == isl_lp_empty) {
143 set->p[j] = isl_basic_set_set_to_empty(set->p[j]);
148 if (!isl_int_is_one(opt_denom))
149 isl_seq_scale(c, c, opt_denom, len);
150 if (first || isl_int_lt(opt, c[0]))
151 isl_int_set(c[0], opt);
155 isl_int_clear(opt_denom);
156 isl_int_neg(c[0], c[0]);
160 isl_int_clear(opt_denom);
164 /* Check if "c" is a direction that is independent of the previously found "n"
166 * If so, add it to the list, with the negative of the lower bound
167 * in the constant position, i.e., such that c corresponds to a bounding
168 * hyperplane (but not necessarily a facet).
169 * Assumes set "set" is bounded.
171 static int is_independent_bound(struct isl_ctx *ctx,
172 struct isl_set *set, isl_int *c,
173 struct isl_mat *dirs, int n)
178 isl_seq_cpy(dirs->row[n]+1, c+1, dirs->n_col-1);
180 int pos = isl_seq_first_non_zero(dirs->row[n]+1, dirs->n_col-1);
183 for (i = 0; i < n; ++i) {
185 pos_i = isl_seq_first_non_zero(dirs->row[i]+1, dirs->n_col-1);
190 isl_seq_elim(dirs->row[n]+1, dirs->row[i]+1, pos,
191 dirs->n_col-1, NULL);
192 pos = isl_seq_first_non_zero(dirs->row[n]+1, dirs->n_col-1);
198 is_bound = uset_is_bound(ctx, set, dirs->row[n], dirs->n_col);
203 isl_int *t = dirs->row[n];
204 for (k = n; k > i; --k)
205 dirs->row[k] = dirs->row[k-1];
211 /* Compute and return a maximal set of linearly independent bounds
212 * on the set "set", based on the constraints of the basic sets
215 static struct isl_mat *independent_bounds(struct isl_ctx *ctx,
219 struct isl_mat *dirs = NULL;
220 unsigned dim = isl_set_n_dim(set);
222 dirs = isl_mat_alloc(ctx, dim, 1+dim);
227 for (i = 0; n < dim && i < set->n; ++i) {
229 struct isl_basic_set *bset = set->p[i];
231 for (j = 0; n < dim && j < bset->n_eq; ++j) {
232 f = is_independent_bound(ctx, set, bset->eq[j],
239 for (j = 0; n < dim && j < bset->n_ineq; ++j) {
240 f = is_independent_bound(ctx, set, bset->ineq[j],
251 isl_mat_free(ctx, dirs);
255 static struct isl_basic_set *isl_basic_set_set_rational(
256 struct isl_basic_set *bset)
261 if (ISL_F_ISSET(bset, ISL_BASIC_MAP_RATIONAL))
264 bset = isl_basic_set_cow(bset);
268 ISL_F_SET(bset, ISL_BASIC_MAP_RATIONAL);
270 return isl_basic_set_finalize(bset);
273 static struct isl_set *isl_set_set_rational(struct isl_set *set)
277 set = isl_set_cow(set);
280 for (i = 0; i < set->n; ++i) {
281 set->p[i] = isl_basic_set_set_rational(set->p[i]);
291 static struct isl_basic_set *isl_basic_set_add_equality(struct isl_ctx *ctx,
292 struct isl_basic_set *bset, isl_int *c)
298 if (ISL_F_ISSET(bset, ISL_BASIC_SET_EMPTY))
301 isl_assert(ctx, isl_basic_set_n_param(bset) == 0, goto error);
302 isl_assert(ctx, bset->n_div == 0, goto error);
303 dim = isl_basic_set_n_dim(bset);
304 bset = isl_basic_set_extend(bset, 0, dim, 0, 1, 0);
305 i = isl_basic_set_alloc_equality(bset);
308 isl_seq_cpy(bset->eq[i], c, 1 + dim);
311 isl_basic_set_free(bset);
315 static struct isl_set *isl_set_add_equality(struct isl_ctx *ctx,
316 struct isl_set *set, isl_int *c)
320 set = isl_set_cow(set);
323 for (i = 0; i < set->n; ++i) {
324 set->p[i] = isl_basic_set_add_equality(ctx, set->p[i], c);
334 /* Given a union of basic sets, construct the constraints for wrapping
335 * a facet around one of its ridges.
336 * In particular, if each of n the d-dimensional basic sets i in "set"
337 * contains the origin, satisfies the constraints x_1 >= 0 and x_2 >= 0
338 * and is defined by the constraints
342 * then the resulting set is of dimension n*(1+d) and has as contraints
351 static struct isl_basic_set *wrap_constraints(struct isl_ctx *ctx,
354 struct isl_basic_set *lp;
358 unsigned dim, lp_dim;
363 dim = 1 + isl_set_n_dim(set);
366 for (i = 0; i < set->n; ++i) {
367 n_eq += set->p[i]->n_eq;
368 n_ineq += set->p[i]->n_ineq;
370 lp = isl_basic_set_alloc(ctx, 0, dim * set->n, 0, n_eq, n_ineq);
373 lp_dim = isl_basic_set_n_dim(lp);
374 k = isl_basic_set_alloc_equality(lp);
375 isl_int_set_si(lp->eq[k][0], -1);
376 for (i = 0; i < set->n; ++i) {
377 isl_int_set_si(lp->eq[k][1+dim*i], 0);
378 isl_int_set_si(lp->eq[k][1+dim*i+1], 1);
379 isl_seq_clr(lp->eq[k]+1+dim*i+2, dim-2);
381 for (i = 0; i < set->n; ++i) {
382 k = isl_basic_set_alloc_inequality(lp);
383 isl_seq_clr(lp->ineq[k], 1+lp_dim);
384 isl_int_set_si(lp->ineq[k][1+dim*i], 1);
386 for (j = 0; j < set->p[i]->n_eq; ++j) {
387 k = isl_basic_set_alloc_equality(lp);
388 isl_seq_clr(lp->eq[k], 1+dim*i);
389 isl_seq_cpy(lp->eq[k]+1+dim*i, set->p[i]->eq[j], dim);
390 isl_seq_clr(lp->eq[k]+1+dim*(i+1), dim*(set->n-i-1));
393 for (j = 0; j < set->p[i]->n_ineq; ++j) {
394 k = isl_basic_set_alloc_inequality(lp);
395 isl_seq_clr(lp->ineq[k], 1+dim*i);
396 isl_seq_cpy(lp->ineq[k]+1+dim*i, set->p[i]->ineq[j], dim);
397 isl_seq_clr(lp->ineq[k]+1+dim*(i+1), dim*(set->n-i-1));
403 /* Given a facet "facet" of the convex hull of "set" and a facet "ridge"
404 * of that facet, compute the other facet of the convex hull that contains
407 * We first transform the set such that the facet constraint becomes
411 * I.e., the facet lies in
415 * and on that facet, the constraint that defines the ridge is
419 * (This transformation is not strictly needed, all that is needed is
420 * that the ridge contains the origin.)
422 * Since the ridge contains the origin, the cone of the convex hull
423 * will be of the form
428 * with this second constraint defining the new facet.
429 * The constant a is obtained by settting x_1 in the cone of the
430 * convex hull to 1 and minimizing x_2.
431 * Now, each element in the cone of the convex hull is the sum
432 * of elements in the cones of the basic sets.
433 * If a_i is the dilation factor of basic set i, then the problem
434 * we need to solve is
447 * the constraints of each (transformed) basic set.
448 * If a = n/d, then the constraint defining the new facet (in the transformed
451 * -n x_1 + d x_2 >= 0
453 * In the original space, we need to take the same combination of the
454 * corresponding constraints "facet" and "ridge".
456 * If a = -infty = "-1/0", then we just return the original facet constraint.
457 * This means that the facet is unbounded, but has a bounded intersection
458 * with the union of sets.
460 static isl_int *wrap_facet(struct isl_ctx *ctx, struct isl_set *set,
461 isl_int *facet, isl_int *ridge)
464 struct isl_mat *T = NULL;
465 struct isl_basic_set *lp = NULL;
467 enum isl_lp_result res;
471 set = isl_set_copy(set);
473 dim = 1 + isl_set_n_dim(set);
474 T = isl_mat_alloc(ctx, 3, dim);
477 isl_int_set_si(T->row[0][0], 1);
478 isl_seq_clr(T->row[0]+1, dim - 1);
479 isl_seq_cpy(T->row[1], facet, dim);
480 isl_seq_cpy(T->row[2], ridge, dim);
481 T = isl_mat_right_inverse(ctx, T);
482 set = isl_set_preimage(set, T);
486 lp = wrap_constraints(ctx, set);
487 obj = isl_vec_alloc(ctx, dim*set->n);
490 for (i = 0; i < set->n; ++i) {
491 isl_seq_clr(obj->block.data+dim*i, 2);
492 isl_int_set_si(obj->block.data[dim*i+2], 1);
493 isl_seq_clr(obj->block.data+dim*i+3, dim-3);
497 res = isl_solve_lp((struct isl_basic_map *)lp, 0,
498 obj->block.data, ctx->one, &num, &den);
499 if (res == isl_lp_ok) {
500 isl_int_neg(num, num);
501 isl_seq_combine(facet, num, facet, den, ridge, dim);
505 isl_vec_free(ctx, obj);
506 isl_basic_set_free(lp);
508 isl_assert(ctx, res == isl_lp_ok || res == isl_lp_unbounded,
512 isl_basic_set_free(lp);
513 isl_mat_free(ctx, T);
518 /* Given a set of d linearly independent bounding constraints of the
519 * convex hull of "set", compute the constraint of a facet of "set".
521 * We first compute the intersection with the first bounding hyperplane
522 * and remove the component corresponding to this hyperplane from
523 * other bounds (in homogeneous space).
524 * We then wrap around one of the remaining bounding constraints
525 * and continue the process until all bounding constraints have been
526 * taken into account.
527 * The resulting linear combination of the bounding constraints will
528 * correspond to a facet of the convex hull.
530 static struct isl_mat *initial_facet_constraint(struct isl_ctx *ctx,
531 struct isl_set *set, struct isl_mat *bounds)
533 struct isl_set *slice = NULL;
534 struct isl_basic_set *face = NULL;
535 struct isl_mat *m, *U, *Q;
537 unsigned dim = isl_set_n_dim(set);
539 isl_assert(ctx, set->n > 0, goto error);
540 isl_assert(ctx, bounds->n_row == dim, goto error);
542 while (bounds->n_row > 1) {
543 slice = isl_set_copy(set);
544 slice = isl_set_add_equality(ctx, slice, bounds->row[0]);
545 face = isl_set_affine_hull(slice);
548 if (face->n_eq == 1) {
549 isl_basic_set_free(face);
552 m = isl_mat_alloc(ctx, 1 + face->n_eq, 1 + dim);
555 isl_int_set_si(m->row[0][0], 1);
556 isl_seq_clr(m->row[0]+1, dim);
557 for (i = 0; i < face->n_eq; ++i)
558 isl_seq_cpy(m->row[1 + i], face->eq[i], 1 + dim);
559 U = isl_mat_right_inverse(ctx, m);
560 Q = isl_mat_right_inverse(ctx, isl_mat_copy(ctx, U));
561 U = isl_mat_drop_cols(ctx, U, 1 + face->n_eq,
563 Q = isl_mat_drop_rows(ctx, Q, 1 + face->n_eq,
565 U = isl_mat_drop_cols(ctx, U, 0, 1);
566 Q = isl_mat_drop_rows(ctx, Q, 0, 1);
567 bounds = isl_mat_product(ctx, bounds, U);
568 bounds = isl_mat_product(ctx, bounds, Q);
569 while (isl_seq_first_non_zero(bounds->row[bounds->n_row-1],
570 bounds->n_col) == -1) {
572 isl_assert(ctx, bounds->n_row > 1, goto error);
574 if (!wrap_facet(ctx, set, bounds->row[0],
575 bounds->row[bounds->n_row-1]))
577 isl_basic_set_free(face);
582 isl_basic_set_free(face);
583 isl_mat_free(ctx, bounds);
587 /* Given the bounding constraint "c" of a facet of the convex hull of "set",
588 * compute a hyperplane description of the facet, i.e., compute the facets
591 * We compute an affine transformation that transforms the constraint
600 * by computing the right inverse U of a matrix that starts with the rows
613 * Since z_1 is zero, we can drop this variable as well as the corresponding
614 * column of U to obtain
622 * with Q' equal to Q, but without the corresponding row.
623 * After computing the facets of the facet in the z' space,
624 * we convert them back to the x space through Q.
626 static struct isl_basic_set *compute_facet(struct isl_ctx *ctx,
627 struct isl_set *set, isl_int *c)
629 struct isl_mat *m, *U, *Q;
630 struct isl_basic_set *facet;
633 set = isl_set_copy(set);
634 dim = isl_set_n_dim(set);
635 m = isl_mat_alloc(ctx, 2, 1 + dim);
638 isl_int_set_si(m->row[0][0], 1);
639 isl_seq_clr(m->row[0]+1, dim);
640 isl_seq_cpy(m->row[1], c, 1+dim);
641 U = isl_mat_right_inverse(ctx, m);
642 Q = isl_mat_right_inverse(ctx, isl_mat_copy(ctx, U));
643 U = isl_mat_drop_cols(ctx, U, 1, 1);
644 Q = isl_mat_drop_rows(ctx, Q, 1, 1);
645 set = isl_set_preimage(set, U);
646 facet = uset_convex_hull_wrap(set);
647 facet = isl_basic_set_preimage(facet, Q);
654 /* Given an initial facet constraint, compute the remaining facets.
655 * We do this by running through all facets found so far and computing
656 * the adjacent facets through wrapping, adding those facets that we
657 * hadn't already found before.
659 * This function can still be significantly optimized by checking which of
660 * the facets of the basic sets are also facets of the convex hull and
661 * using all the facets so far to help in constructing the facets of the
664 * using the technique in section "3.1 Ridge Generation" of
665 * "Extended Convex Hull" by Fukuda et al.
667 static struct isl_basic_set *extend(struct isl_ctx *ctx, struct isl_set *set,
668 struct isl_mat *initial)
672 struct isl_basic_set *hull = NULL;
673 struct isl_basic_set *facet = NULL;
678 isl_assert(ctx, set->n > 0, goto error);
681 for (i = 0; i < set->n; ++i) {
682 n_ineq += set->p[i]->n_eq;
683 n_ineq += set->p[i]->n_ineq;
685 dim = isl_set_n_dim(set);
686 isl_assert(ctx, 1 + dim == initial->n_col, goto error);
687 hull = isl_basic_set_alloc(ctx, 0, dim, 0, 0, n_ineq);
688 hull = isl_basic_set_set_rational(hull);
691 k = isl_basic_set_alloc_inequality(hull);
694 isl_seq_cpy(hull->ineq[k], initial->row[0], initial->n_col);
695 for (i = 0; i < hull->n_ineq; ++i) {
696 facet = compute_facet(ctx, set, hull->ineq[i]);
699 if (facet->n_ineq + hull->n_ineq > n_ineq) {
700 hull = isl_basic_set_extend(hull,
701 0, dim, 0, 0, facet->n_ineq);
702 n_ineq = hull->n_ineq + facet->n_ineq;
704 for (j = 0; j < facet->n_ineq; ++j) {
705 k = isl_basic_set_alloc_inequality(hull);
708 isl_seq_cpy(hull->ineq[k], hull->ineq[i], 1+dim);
709 if (!wrap_facet(ctx, set, hull->ineq[k], facet->ineq[j]))
711 for (f = 0; f < k; ++f)
712 if (isl_seq_eq(hull->ineq[f], hull->ineq[k],
716 isl_basic_set_free_inequality(hull, 1);
718 isl_basic_set_free(facet);
720 hull = isl_basic_set_simplify(hull);
721 hull = isl_basic_set_finalize(hull);
724 isl_basic_set_free(facet);
725 isl_basic_set_free(hull);
729 /* Special case for computing the convex hull of a one dimensional set.
730 * We simply collect the lower and upper bounds of each basic set
731 * and the biggest of those.
733 static struct isl_basic_set *convex_hull_1d(struct isl_ctx *ctx,
736 struct isl_mat *c = NULL;
737 isl_int *lower = NULL;
738 isl_int *upper = NULL;
741 struct isl_basic_set *hull;
743 for (i = 0; i < set->n; ++i) {
744 set->p[i] = isl_basic_set_simplify(set->p[i]);
748 set = isl_set_remove_empty_parts(set);
751 isl_assert(ctx, set->n > 0, goto error);
752 c = isl_mat_alloc(ctx, 2, 2);
756 if (set->p[0]->n_eq > 0) {
757 isl_assert(ctx, set->p[0]->n_eq == 1, goto error);
760 if (isl_int_is_pos(set->p[0]->eq[0][1])) {
761 isl_seq_cpy(lower, set->p[0]->eq[0], 2);
762 isl_seq_neg(upper, set->p[0]->eq[0], 2);
764 isl_seq_neg(lower, set->p[0]->eq[0], 2);
765 isl_seq_cpy(upper, set->p[0]->eq[0], 2);
768 for (j = 0; j < set->p[0]->n_ineq; ++j) {
769 if (isl_int_is_pos(set->p[0]->ineq[j][1])) {
771 isl_seq_cpy(lower, set->p[0]->ineq[j], 2);
774 isl_seq_cpy(upper, set->p[0]->ineq[j], 2);
781 for (i = 0; i < set->n; ++i) {
782 struct isl_basic_set *bset = set->p[i];
786 for (j = 0; j < bset->n_eq; ++j) {
790 isl_int_mul(a, lower[0], bset->eq[j][1]);
791 isl_int_mul(b, lower[1], bset->eq[j][0]);
792 if (isl_int_lt(a, b) && isl_int_is_pos(bset->eq[j][1]))
793 isl_seq_cpy(lower, bset->eq[j], 2);
794 if (isl_int_gt(a, b) && isl_int_is_neg(bset->eq[j][1]))
795 isl_seq_neg(lower, bset->eq[j], 2);
798 isl_int_mul(a, upper[0], bset->eq[j][1]);
799 isl_int_mul(b, upper[1], bset->eq[j][0]);
800 if (isl_int_lt(a, b) && isl_int_is_pos(bset->eq[j][1]))
801 isl_seq_neg(upper, bset->eq[j], 2);
802 if (isl_int_gt(a, b) && isl_int_is_neg(bset->eq[j][1]))
803 isl_seq_cpy(upper, bset->eq[j], 2);
806 for (j = 0; j < bset->n_ineq; ++j) {
807 if (isl_int_is_pos(bset->ineq[j][1]))
809 if (isl_int_is_neg(bset->ineq[j][1]))
811 if (lower && isl_int_is_pos(bset->ineq[j][1])) {
812 isl_int_mul(a, lower[0], bset->ineq[j][1]);
813 isl_int_mul(b, lower[1], bset->ineq[j][0]);
814 if (isl_int_lt(a, b))
815 isl_seq_cpy(lower, bset->ineq[j], 2);
817 if (upper && isl_int_is_neg(bset->ineq[j][1])) {
818 isl_int_mul(a, upper[0], bset->ineq[j][1]);
819 isl_int_mul(b, upper[1], bset->ineq[j][0]);
820 if (isl_int_gt(a, b))
821 isl_seq_cpy(upper, bset->ineq[j], 2);
832 hull = isl_basic_set_alloc(ctx, 0, 1, 0, 0, 2);
833 hull = isl_basic_set_set_rational(hull);
837 k = isl_basic_set_alloc_inequality(hull);
838 isl_seq_cpy(hull->ineq[k], lower, 2);
841 k = isl_basic_set_alloc_inequality(hull);
842 isl_seq_cpy(hull->ineq[k], upper, 2);
844 hull = isl_basic_set_finalize(hull);
846 isl_mat_free(ctx, c);
850 isl_mat_free(ctx, c);
854 /* Project out final n dimensions using Fourier-Motzkin */
855 static struct isl_set *set_project_out(struct isl_ctx *ctx,
856 struct isl_set *set, unsigned n)
858 return isl_set_remove_dims(set, isl_set_n_dim(set) - n, n);
861 static struct isl_basic_set *convex_hull_0d(struct isl_set *set)
863 struct isl_basic_set *convex_hull;
868 if (isl_set_is_empty(set))
869 convex_hull = isl_basic_set_empty(isl_dim_copy(set->dim));
871 convex_hull = isl_basic_set_universe(isl_dim_copy(set->dim));
876 /* Compute the convex hull of a pair of basic sets without any parameters or
877 * integer divisions using Fourier-Motzkin elimination.
878 * The convex hull is the set of all points that can be written as
879 * the sum of points from both basic sets (in homogeneous coordinates).
880 * We set up the constraints in a space with dimensions for each of
881 * the three sets and then project out the dimensions corresponding
882 * to the two original basic sets, retaining only those corresponding
883 * to the convex hull.
885 static struct isl_basic_set *convex_hull_pair(struct isl_basic_set *bset1,
886 struct isl_basic_set *bset2)
889 struct isl_basic_set *bset[2];
890 struct isl_basic_set *hull = NULL;
893 if (!bset1 || !bset2)
896 dim = isl_basic_set_n_dim(bset1);
897 hull = isl_basic_set_alloc(bset1->ctx, 0, 2 + 3 * dim, 0,
898 1 + dim + bset1->n_eq + bset2->n_eq,
899 2 + bset1->n_ineq + bset2->n_ineq);
902 for (i = 0; i < 2; ++i) {
903 for (j = 0; j < bset[i]->n_eq; ++j) {
904 k = isl_basic_set_alloc_equality(hull);
907 isl_seq_clr(hull->eq[k], (i+1) * (1+dim));
908 isl_seq_clr(hull->eq[k]+(i+2)*(1+dim), (1-i)*(1+dim));
909 isl_seq_cpy(hull->eq[k]+(i+1)*(1+dim), bset[i]->eq[j],
912 for (j = 0; j < bset[i]->n_ineq; ++j) {
913 k = isl_basic_set_alloc_inequality(hull);
916 isl_seq_clr(hull->ineq[k], (i+1) * (1+dim));
917 isl_seq_clr(hull->ineq[k]+(i+2)*(1+dim), (1-i)*(1+dim));
918 isl_seq_cpy(hull->ineq[k]+(i+1)*(1+dim),
919 bset[i]->ineq[j], 1+dim);
921 k = isl_basic_set_alloc_inequality(hull);
924 isl_seq_clr(hull->ineq[k], 1+2+3*dim);
925 isl_int_set_si(hull->ineq[k][(i+1)*(1+dim)], 1);
927 for (j = 0; j < 1+dim; ++j) {
928 k = isl_basic_set_alloc_equality(hull);
931 isl_seq_clr(hull->eq[k], 1+2+3*dim);
932 isl_int_set_si(hull->eq[k][j], -1);
933 isl_int_set_si(hull->eq[k][1+dim+j], 1);
934 isl_int_set_si(hull->eq[k][2*(1+dim)+j], 1);
936 hull = isl_basic_set_set_rational(hull);
937 hull = isl_basic_set_remove_dims(hull, dim, 2*(1+dim));
938 hull = isl_basic_set_convex_hull(hull);
939 isl_basic_set_free(bset1);
940 isl_basic_set_free(bset2);
943 isl_basic_set_free(bset1);
944 isl_basic_set_free(bset2);
945 isl_basic_set_free(hull);
949 /* Compute the convex hull of a set without any parameters or
950 * integer divisions using Fourier-Motzkin elimination.
951 * In each step, we combined two basic sets until only one
954 static struct isl_basic_set *uset_convex_hull_elim(struct isl_set *set)
956 struct isl_basic_set *convex_hull = NULL;
958 convex_hull = isl_set_copy_basic_set(set);
959 set = isl_set_drop_basic_set(set, convex_hull);
963 struct isl_basic_set *t;
964 t = isl_set_copy_basic_set(set);
967 set = isl_set_drop_basic_set(set, t);
970 convex_hull = convex_hull_pair(convex_hull, t);
976 isl_basic_set_free(convex_hull);
980 static struct isl_basic_set *uset_convex_hull_wrap_with_bounds(
981 struct isl_set *set, struct isl_mat *bounds)
983 struct isl_basic_set *convex_hull = NULL;
985 isl_assert(set->ctx, bounds->n_row == isl_set_n_dim(set), goto error);
986 bounds = initial_facet_constraint(set->ctx, set, bounds);
989 convex_hull = extend(set->ctx, set, bounds);
990 isl_mat_free(set->ctx, bounds);
995 isl_mat_free(set->ctx, bounds);
1000 static int isl_basic_set_is_bounded(struct isl_basic_set *bset)
1002 struct isl_tab *tab;
1005 tab = isl_tab_from_recession_cone((struct isl_basic_map *)bset);
1006 bounded = isl_tab_cone_is_bounded(bset->ctx, tab);
1007 isl_tab_free(bset->ctx, tab);
1011 static int isl_set_is_bounded(struct isl_set *set)
1015 for (i = 0; i < set->n; ++i) {
1016 int bounded = isl_basic_set_is_bounded(set->p[i]);
1017 if (!bounded || bounded < 0)
1023 /* Compute the convex hull of a set without any parameters or
1024 * integer divisions. Depending on whether the set is bounded,
1025 * we pass control to the wrapping based convex hull or
1026 * the Fourier-Motzkin elimination based convex hull.
1027 * We also handle a few special cases before checking the boundedness.
1029 static struct isl_basic_set *uset_convex_hull(struct isl_set *set)
1032 struct isl_basic_set *convex_hull = NULL;
1033 struct isl_mat *bounds = NULL;
1035 if (isl_set_n_dim(set) == 0)
1036 return convex_hull_0d(set);
1038 set = isl_set_set_rational(set);
1042 set = isl_set_normalize(set);
1046 convex_hull = isl_basic_set_copy(set->p[0]);
1050 if (isl_set_n_dim(set) == 1)
1051 return convex_hull_1d(set->ctx, set);
1053 if (!isl_set_is_bounded(set))
1054 return uset_convex_hull_elim(set);
1056 bounds = independent_bounds(set->ctx, set);
1059 return uset_convex_hull_wrap_with_bounds(set, bounds);
1062 isl_basic_set_free(convex_hull);
1066 /* This is the core procedure, where "set" is a "pure" set, i.e.,
1067 * without parameters or divs and where the convex hull of set is
1068 * known to be full-dimensional.
1070 static struct isl_basic_set *uset_convex_hull_wrap(struct isl_set *set)
1073 struct isl_basic_set *convex_hull = NULL;
1074 struct isl_mat *bounds;
1076 if (isl_set_n_dim(set) == 0) {
1077 convex_hull = isl_basic_set_universe(isl_dim_copy(set->dim));
1079 convex_hull = isl_basic_set_set_rational(convex_hull);
1083 set = isl_set_set_rational(set);
1087 set = isl_set_normalize(set);
1091 convex_hull = isl_basic_set_copy(set->p[0]);
1095 if (isl_set_n_dim(set) == 1)
1096 return convex_hull_1d(set->ctx, set);
1098 bounds = independent_bounds(set->ctx, set);
1101 return uset_convex_hull_wrap_with_bounds(set, bounds);
1107 /* Compute the convex hull of set "set" with affine hull "affine_hull",
1108 * We first remove the equalities (transforming the set), compute the
1109 * convex hull of the transformed set and then add the equalities back
1110 * (after performing the inverse transformation.
1112 static struct isl_basic_set *modulo_affine_hull(struct isl_ctx *ctx,
1113 struct isl_set *set, struct isl_basic_set *affine_hull)
1117 struct isl_basic_set *dummy;
1118 struct isl_basic_set *convex_hull;
1120 dummy = isl_basic_set_remove_equalities(
1121 isl_basic_set_copy(affine_hull), &T, &T2);
1124 isl_basic_set_free(dummy);
1125 set = isl_set_preimage(set, T);
1126 convex_hull = uset_convex_hull(set);
1127 convex_hull = isl_basic_set_preimage(convex_hull, T2);
1128 convex_hull = isl_basic_set_intersect(convex_hull, affine_hull);
1131 isl_basic_set_free(affine_hull);
1136 /* Compute the convex hull of a map.
1138 * The implementation was inspired by "Extended Convex Hull" by Fukuda et al.,
1139 * specifically, the wrapping of facets to obtain new facets.
1141 struct isl_basic_map *isl_map_convex_hull(struct isl_map *map)
1143 struct isl_basic_set *bset;
1144 struct isl_basic_map *model = NULL;
1145 struct isl_basic_set *affine_hull = NULL;
1146 struct isl_basic_map *convex_hull = NULL;
1147 struct isl_set *set = NULL;
1148 struct isl_ctx *ctx;
1155 convex_hull = isl_basic_map_empty_like_map(map);
1160 map = isl_map_align_divs(map);
1161 model = isl_basic_map_copy(map->p[0]);
1162 set = isl_map_underlying_set(map);
1166 affine_hull = isl_set_affine_hull(isl_set_copy(set));
1169 if (affine_hull->n_eq != 0)
1170 bset = modulo_affine_hull(ctx, set, affine_hull);
1172 isl_basic_set_free(affine_hull);
1173 bset = uset_convex_hull(set);
1176 convex_hull = isl_basic_map_overlying_set(bset, model);
1178 ISL_F_CLR(convex_hull, ISL_BASIC_MAP_RATIONAL);
1182 isl_basic_map_free(model);
1186 struct isl_basic_set *isl_set_convex_hull(struct isl_set *set)
1188 return (struct isl_basic_set *)
1189 isl_map_convex_hull((struct isl_map *)set);
1192 /* Compute a superset of the convex hull of map that is described
1193 * by only translates of the constraints in the constituents of map.
1195 * The implementation is not very efficient. In particular, if
1196 * constraints with the same normal appear in more than one
1197 * basic map, they will be (re)examined each time.
1199 struct isl_basic_map *isl_map_simple_hull(struct isl_map *map)
1201 struct isl_set *set = NULL;
1202 struct isl_basic_map *model = NULL;
1203 struct isl_basic_map *hull;
1204 struct isl_basic_set *bset = NULL;
1212 hull = isl_basic_map_empty_like_map(map);
1217 hull = isl_basic_map_copy(map->p[0]);
1222 map = isl_map_align_divs(map);
1223 model = isl_basic_map_copy(map->p[0]);
1226 for (i = 0; i < map->n; ++i) {
1229 n_ineq += map->p[i]->n_ineq;
1232 set = isl_map_underlying_set(map);
1236 bset = isl_set_affine_hull(isl_set_copy(set));
1239 dim = isl_basic_set_n_dim(bset);
1240 bset = isl_basic_set_extend(bset, 0, dim, 0, 0, n_ineq);
1244 for (i = 0; i < set->n; ++i) {
1245 for (j = 0; j < set->p[i]->n_ineq; ++j) {
1249 k = isl_basic_set_alloc_inequality(bset);
1252 isl_seq_cpy(bset->ineq[k], set->p[i]->ineq[j], 1 + dim);
1253 is_bound = uset_is_bound(set->ctx, set, bset->ineq[k],
1258 isl_basic_set_free_inequality(bset, 1);
1262 bset = isl_basic_set_simplify(bset);
1263 bset = isl_basic_set_finalize(bset);
1264 bset = isl_basic_set_convex_hull(bset);
1266 hull = isl_basic_map_overlying_set(bset, model);
1271 isl_basic_set_free(bset);
1273 isl_basic_map_free(model);
1277 struct isl_basic_set *isl_set_simple_hull(struct isl_set *set)
1279 return (struct isl_basic_set *)
1280 isl_map_simple_hull((struct isl_map *)set);