2 * Copyright 2008-2009 Katholieke Universiteit Leuven
4 * Use of this software is governed by the GNU LGPLv2.1 license
6 * Written by Sven Verdoolaege, K.U.Leuven, Departement
7 * Computerwetenschappen, Celestijnenlaan 200A, B-3001 Leuven, Belgium
12 #include "isl_map_private.h"
16 #include "isl_equalities.h"
19 static struct isl_basic_set *uset_convex_hull_wrap_bounded(struct isl_set *set);
21 static void swap_ineq(struct isl_basic_map *bmap, unsigned i, unsigned j)
27 bmap->ineq[i] = bmap->ineq[j];
32 /* Return 1 if constraint c is redundant with respect to the constraints
33 * in bmap. If c is a lower [upper] bound in some variable and bmap
34 * does not have a lower [upper] bound in that variable, then c cannot
35 * be redundant and we do not need solve any lp.
37 int isl_basic_map_constraint_is_redundant(struct isl_basic_map **bmap,
38 isl_int *c, isl_int *opt_n, isl_int *opt_d)
40 enum isl_lp_result res;
47 total = isl_basic_map_total_dim(*bmap);
48 for (i = 0; i < total; ++i) {
50 if (isl_int_is_zero(c[1+i]))
52 sign = isl_int_sgn(c[1+i]);
53 for (j = 0; j < (*bmap)->n_ineq; ++j)
54 if (sign == isl_int_sgn((*bmap)->ineq[j][1+i]))
56 if (j == (*bmap)->n_ineq)
62 res = isl_basic_map_solve_lp(*bmap, 0, c, (*bmap)->ctx->one,
64 if (res == isl_lp_unbounded)
66 if (res == isl_lp_error)
68 if (res == isl_lp_empty) {
69 *bmap = isl_basic_map_set_to_empty(*bmap);
72 return !isl_int_is_neg(*opt_n);
75 int isl_basic_set_constraint_is_redundant(struct isl_basic_set **bset,
76 isl_int *c, isl_int *opt_n, isl_int *opt_d)
78 return isl_basic_map_constraint_is_redundant(
79 (struct isl_basic_map **)bset, c, opt_n, opt_d);
82 /* Compute the convex hull of a basic map, by removing the redundant
83 * constraints. If the minimal value along the normal of a constraint
84 * is the same if the constraint is removed, then the constraint is redundant.
86 * Alternatively, we could have intersected the basic map with the
87 * corresponding equality and the checked if the dimension was that
90 struct isl_basic_map *isl_basic_map_convex_hull(struct isl_basic_map *bmap)
97 bmap = isl_basic_map_gauss(bmap, NULL);
98 if (ISL_F_ISSET(bmap, ISL_BASIC_MAP_EMPTY))
100 if (ISL_F_ISSET(bmap, ISL_BASIC_MAP_NO_REDUNDANT))
102 if (bmap->n_ineq <= 1)
105 tab = isl_tab_from_basic_map(bmap);
106 tab = isl_tab_detect_implicit_equalities(tab);
107 if (isl_tab_detect_redundant(tab) < 0)
109 bmap = isl_basic_map_update_from_tab(bmap, tab);
111 ISL_F_SET(bmap, ISL_BASIC_MAP_NO_IMPLICIT);
112 ISL_F_SET(bmap, ISL_BASIC_MAP_NO_REDUNDANT);
116 isl_basic_map_free(bmap);
120 struct isl_basic_set *isl_basic_set_convex_hull(struct isl_basic_set *bset)
122 return (struct isl_basic_set *)
123 isl_basic_map_convex_hull((struct isl_basic_map *)bset);
126 /* Check if the set set is bound in the direction of the affine
127 * constraint c and if so, set the constant term such that the
128 * resulting constraint is a bounding constraint for the set.
130 static int uset_is_bound(struct isl_set *set, isl_int *c, unsigned len)
138 isl_int_init(opt_denom);
140 for (j = 0; j < set->n; ++j) {
141 enum isl_lp_result res;
143 if (ISL_F_ISSET(set->p[j], ISL_BASIC_SET_EMPTY))
146 res = isl_basic_set_solve_lp(set->p[j],
147 0, c, set->ctx->one, &opt, &opt_denom, NULL);
148 if (res == isl_lp_unbounded)
150 if (res == isl_lp_error)
152 if (res == isl_lp_empty) {
153 set->p[j] = isl_basic_set_set_to_empty(set->p[j]);
158 if (first || isl_int_is_neg(opt)) {
159 if (!isl_int_is_one(opt_denom))
160 isl_seq_scale(c, c, opt_denom, len);
161 isl_int_sub(c[0], c[0], opt);
166 isl_int_clear(opt_denom);
170 isl_int_clear(opt_denom);
174 /* Check if "c" is a direction that is independent of the previously found "n"
176 * If so, add it to the list, with the negative of the lower bound
177 * in the constant position, i.e., such that c corresponds to a bounding
178 * hyperplane (but not necessarily a facet).
179 * Assumes set "set" is bounded.
181 static int is_independent_bound(struct isl_set *set, isl_int *c,
182 struct isl_mat *dirs, int n)
187 isl_seq_cpy(dirs->row[n]+1, c+1, dirs->n_col-1);
189 int pos = isl_seq_first_non_zero(dirs->row[n]+1, dirs->n_col-1);
192 for (i = 0; i < n; ++i) {
194 pos_i = isl_seq_first_non_zero(dirs->row[i]+1, dirs->n_col-1);
199 isl_seq_elim(dirs->row[n]+1, dirs->row[i]+1, pos,
200 dirs->n_col-1, NULL);
201 pos = isl_seq_first_non_zero(dirs->row[n]+1, dirs->n_col-1);
207 is_bound = uset_is_bound(set, dirs->row[n], dirs->n_col);
210 isl_seq_normalize(set->ctx, dirs->row[n], dirs->n_col);
213 isl_int *t = dirs->row[n];
214 for (k = n; k > i; --k)
215 dirs->row[k] = dirs->row[k-1];
221 /* Compute and return a maximal set of linearly independent bounds
222 * on the set "set", based on the constraints of the basic sets
225 static struct isl_mat *independent_bounds(struct isl_set *set)
228 struct isl_mat *dirs = NULL;
229 unsigned dim = isl_set_n_dim(set);
231 dirs = isl_mat_alloc(set->ctx, dim, 1+dim);
236 for (i = 0; n < dim && i < set->n; ++i) {
238 struct isl_basic_set *bset = set->p[i];
240 for (j = 0; n < dim && j < bset->n_eq; ++j) {
241 f = is_independent_bound(set, bset->eq[j], dirs, n);
247 for (j = 0; n < dim && j < bset->n_ineq; ++j) {
248 f = is_independent_bound(set, bset->ineq[j], dirs, n);
262 struct isl_basic_set *isl_basic_set_set_rational(struct isl_basic_set *bset)
267 if (ISL_F_ISSET(bset, ISL_BASIC_MAP_RATIONAL))
270 bset = isl_basic_set_cow(bset);
274 ISL_F_SET(bset, ISL_BASIC_MAP_RATIONAL);
276 return isl_basic_set_finalize(bset);
279 static struct isl_set *isl_set_set_rational(struct isl_set *set)
283 set = isl_set_cow(set);
286 for (i = 0; i < set->n; ++i) {
287 set->p[i] = isl_basic_set_set_rational(set->p[i]);
297 static struct isl_basic_set *isl_basic_set_add_equality(
298 struct isl_basic_set *bset, isl_int *c)
303 if (ISL_F_ISSET(bset, ISL_BASIC_SET_EMPTY))
306 isl_assert(bset->ctx, isl_basic_set_n_param(bset) == 0, goto error);
307 isl_assert(bset->ctx, bset->n_div == 0, goto error);
308 dim = isl_basic_set_n_dim(bset);
309 bset = isl_basic_set_cow(bset);
310 bset = isl_basic_set_extend(bset, 0, dim, 0, 1, 0);
311 i = isl_basic_set_alloc_equality(bset);
314 isl_seq_cpy(bset->eq[i], c, 1 + dim);
317 isl_basic_set_free(bset);
321 static struct isl_set *isl_set_add_basic_set_equality(struct isl_set *set, isl_int *c)
325 set = isl_set_cow(set);
328 for (i = 0; i < set->n; ++i) {
329 set->p[i] = isl_basic_set_add_equality(set->p[i], c);
339 /* Given a union of basic sets, construct the constraints for wrapping
340 * a facet around one of its ridges.
341 * In particular, if each of n the d-dimensional basic sets i in "set"
342 * contains the origin, satisfies the constraints x_1 >= 0 and x_2 >= 0
343 * and is defined by the constraints
347 * then the resulting set is of dimension n*(1+d) and has as constraints
356 static struct isl_basic_set *wrap_constraints(struct isl_set *set)
358 struct isl_basic_set *lp;
362 unsigned dim, lp_dim;
367 dim = 1 + isl_set_n_dim(set);
370 for (i = 0; i < set->n; ++i) {
371 n_eq += set->p[i]->n_eq;
372 n_ineq += set->p[i]->n_ineq;
374 lp = isl_basic_set_alloc(set->ctx, 0, dim * set->n, 0, n_eq, n_ineq);
377 lp_dim = isl_basic_set_n_dim(lp);
378 k = isl_basic_set_alloc_equality(lp);
379 isl_int_set_si(lp->eq[k][0], -1);
380 for (i = 0; i < set->n; ++i) {
381 isl_int_set_si(lp->eq[k][1+dim*i], 0);
382 isl_int_set_si(lp->eq[k][1+dim*i+1], 1);
383 isl_seq_clr(lp->eq[k]+1+dim*i+2, dim-2);
385 for (i = 0; i < set->n; ++i) {
386 k = isl_basic_set_alloc_inequality(lp);
387 isl_seq_clr(lp->ineq[k], 1+lp_dim);
388 isl_int_set_si(lp->ineq[k][1+dim*i], 1);
390 for (j = 0; j < set->p[i]->n_eq; ++j) {
391 k = isl_basic_set_alloc_equality(lp);
392 isl_seq_clr(lp->eq[k], 1+dim*i);
393 isl_seq_cpy(lp->eq[k]+1+dim*i, set->p[i]->eq[j], dim);
394 isl_seq_clr(lp->eq[k]+1+dim*(i+1), dim*(set->n-i-1));
397 for (j = 0; j < set->p[i]->n_ineq; ++j) {
398 k = isl_basic_set_alloc_inequality(lp);
399 isl_seq_clr(lp->ineq[k], 1+dim*i);
400 isl_seq_cpy(lp->ineq[k]+1+dim*i, set->p[i]->ineq[j], dim);
401 isl_seq_clr(lp->ineq[k]+1+dim*(i+1), dim*(set->n-i-1));
407 /* Given a facet "facet" of the convex hull of "set" and a facet "ridge"
408 * of that facet, compute the other facet of the convex hull that contains
411 * We first transform the set such that the facet constraint becomes
415 * I.e., the facet lies in
419 * and on that facet, the constraint that defines the ridge is
423 * (This transformation is not strictly needed, all that is needed is
424 * that the ridge contains the origin.)
426 * Since the ridge contains the origin, the cone of the convex hull
427 * will be of the form
432 * with this second constraint defining the new facet.
433 * The constant a is obtained by settting x_1 in the cone of the
434 * convex hull to 1 and minimizing x_2.
435 * Now, each element in the cone of the convex hull is the sum
436 * of elements in the cones of the basic sets.
437 * If a_i is the dilation factor of basic set i, then the problem
438 * we need to solve is
451 * the constraints of each (transformed) basic set.
452 * If a = n/d, then the constraint defining the new facet (in the transformed
455 * -n x_1 + d x_2 >= 0
457 * In the original space, we need to take the same combination of the
458 * corresponding constraints "facet" and "ridge".
460 * Note that a is always finite, since we only apply the wrapping
461 * technique to a union of polytopes.
463 static isl_int *wrap_facet(struct isl_set *set, isl_int *facet, isl_int *ridge)
466 struct isl_mat *T = NULL;
467 struct isl_basic_set *lp = NULL;
469 enum isl_lp_result res;
473 set = isl_set_copy(set);
475 dim = 1 + isl_set_n_dim(set);
476 T = isl_mat_alloc(set->ctx, 3, dim);
479 isl_int_set_si(T->row[0][0], 1);
480 isl_seq_clr(T->row[0]+1, dim - 1);
481 isl_seq_cpy(T->row[1], facet, dim);
482 isl_seq_cpy(T->row[2], ridge, dim);
483 T = isl_mat_right_inverse(T);
484 set = isl_set_preimage(set, T);
488 lp = wrap_constraints(set);
489 obj = isl_vec_alloc(set->ctx, 1 + dim*set->n);
492 isl_int_set_si(obj->block.data[0], 0);
493 for (i = 0; i < set->n; ++i) {
494 isl_seq_clr(obj->block.data + 1 + dim*i, 2);
495 isl_int_set_si(obj->block.data[1 + dim*i+2], 1);
496 isl_seq_clr(obj->block.data + 1 + dim*i+3, dim-3);
500 res = isl_basic_set_solve_lp(lp, 0,
501 obj->block.data, set->ctx->one, &num, &den, NULL);
502 if (res == isl_lp_ok) {
503 isl_int_neg(num, num);
504 isl_seq_combine(facet, num, facet, den, ridge, dim);
509 isl_basic_set_free(lp);
511 isl_assert(set->ctx, res == isl_lp_ok, return NULL);
514 isl_basic_set_free(lp);
520 /* Drop rows in "rows" that are redundant with respect to earlier rows,
521 * assuming that "rows" is of full column rank.
523 * We compute the column echelon form. The non-redundant rows are
524 * those that are the first to contain a non-zero entry in a column.
525 * All the other rows can be removed.
527 static __isl_give isl_mat *drop_redundant_rows(__isl_take isl_mat *rows)
529 struct isl_mat *H = NULL;
537 isl_assert(rows->ctx, rows->n_row >= rows->n_col, goto error);
539 if (rows->n_row == rows->n_col)
542 H = isl_mat_left_hermite(isl_mat_copy(rows), 0, NULL, NULL);
546 last_row = rows->n_row;
547 for (col = rows->n_col - 1; col >= 0; --col) {
548 for (row = col; row < last_row; ++row)
549 if (!isl_int_is_zero(H->row[row][col]))
551 isl_assert(rows->ctx, row < last_row, goto error);
552 if (row + 1 < last_row) {
553 rows = isl_mat_drop_rows(rows, row + 1, last_row - (row + 1));
554 if (rows->n_row == rows->n_col)
569 /* Given a set of d linearly independent bounding constraints of the
570 * convex hull of "set", compute the constraint of a facet of "set".
572 * We first compute the intersection with the first bounding hyperplane
573 * and remove the component corresponding to this hyperplane from
574 * other bounds (in homogeneous space).
575 * We then wrap around one of the remaining bounding constraints
576 * and continue the process until all bounding constraints have been
577 * taken into account.
578 * The resulting linear combination of the bounding constraints will
579 * correspond to a facet of the convex hull.
581 static struct isl_mat *initial_facet_constraint(struct isl_set *set,
582 struct isl_mat *bounds)
584 struct isl_set *slice = NULL;
585 struct isl_basic_set *face = NULL;
586 struct isl_mat *m, *U, *Q;
588 unsigned dim = isl_set_n_dim(set);
590 isl_assert(set->ctx, set->n > 0, goto error);
591 isl_assert(set->ctx, bounds->n_row == dim, goto error);
593 while (bounds->n_row > 1) {
594 slice = isl_set_copy(set);
595 slice = isl_set_add_basic_set_equality(slice, bounds->row[0]);
596 face = isl_set_affine_hull(slice);
599 if (face->n_eq == 1) {
600 isl_basic_set_free(face);
603 m = isl_mat_alloc(set->ctx, 1 + face->n_eq, 1 + dim);
606 isl_int_set_si(m->row[0][0], 1);
607 isl_seq_clr(m->row[0]+1, dim);
608 for (i = 0; i < face->n_eq; ++i)
609 isl_seq_cpy(m->row[1 + i], face->eq[i], 1 + dim);
610 U = isl_mat_right_inverse(m);
611 Q = isl_mat_right_inverse(isl_mat_copy(U));
612 U = isl_mat_drop_cols(U, 1 + face->n_eq, dim - face->n_eq);
613 Q = isl_mat_drop_rows(Q, 1 + face->n_eq, dim - face->n_eq);
614 U = isl_mat_drop_cols(U, 0, 1);
615 Q = isl_mat_drop_rows(Q, 0, 1);
616 bounds = isl_mat_product(bounds, U);
617 bounds = drop_redundant_rows(bounds);
618 bounds = isl_mat_product(bounds, Q);
619 isl_assert(set->ctx, bounds->n_row > 1, goto error);
620 if (!wrap_facet(set, bounds->row[0],
621 bounds->row[bounds->n_row-1]))
623 isl_basic_set_free(face);
628 isl_basic_set_free(face);
629 isl_mat_free(bounds);
633 /* Given the bounding constraint "c" of a facet of the convex hull of "set",
634 * compute a hyperplane description of the facet, i.e., compute the facets
637 * We compute an affine transformation that transforms the constraint
646 * by computing the right inverse U of a matrix that starts with the rows
659 * Since z_1 is zero, we can drop this variable as well as the corresponding
660 * column of U to obtain
668 * with Q' equal to Q, but without the corresponding row.
669 * After computing the facets of the facet in the z' space,
670 * we convert them back to the x space through Q.
672 static struct isl_basic_set *compute_facet(struct isl_set *set, isl_int *c)
674 struct isl_mat *m, *U, *Q;
675 struct isl_basic_set *facet = NULL;
680 set = isl_set_copy(set);
681 dim = isl_set_n_dim(set);
682 m = isl_mat_alloc(set->ctx, 2, 1 + dim);
685 isl_int_set_si(m->row[0][0], 1);
686 isl_seq_clr(m->row[0]+1, dim);
687 isl_seq_cpy(m->row[1], c, 1+dim);
688 U = isl_mat_right_inverse(m);
689 Q = isl_mat_right_inverse(isl_mat_copy(U));
690 U = isl_mat_drop_cols(U, 1, 1);
691 Q = isl_mat_drop_rows(Q, 1, 1);
692 set = isl_set_preimage(set, U);
693 facet = uset_convex_hull_wrap_bounded(set);
694 facet = isl_basic_set_preimage(facet, Q);
695 isl_assert(ctx, facet->n_eq == 0, goto error);
698 isl_basic_set_free(facet);
703 /* Given an initial facet constraint, compute the remaining facets.
704 * We do this by running through all facets found so far and computing
705 * the adjacent facets through wrapping, adding those facets that we
706 * hadn't already found before.
708 * For each facet we have found so far, we first compute its facets
709 * in the resulting convex hull. That is, we compute the ridges
710 * of the resulting convex hull contained in the facet.
711 * We also compute the corresponding facet in the current approximation
712 * of the convex hull. There is no need to wrap around the ridges
713 * in this facet since that would result in a facet that is already
714 * present in the current approximation.
716 * This function can still be significantly optimized by checking which of
717 * the facets of the basic sets are also facets of the convex hull and
718 * using all the facets so far to help in constructing the facets of the
721 * using the technique in section "3.1 Ridge Generation" of
722 * "Extended Convex Hull" by Fukuda et al.
724 static struct isl_basic_set *extend(struct isl_basic_set *hull,
729 struct isl_basic_set *facet = NULL;
730 struct isl_basic_set *hull_facet = NULL;
736 isl_assert(set->ctx, set->n > 0, goto error);
738 dim = isl_set_n_dim(set);
740 for (i = 0; i < hull->n_ineq; ++i) {
741 facet = compute_facet(set, hull->ineq[i]);
742 facet = isl_basic_set_add_equality(facet, hull->ineq[i]);
743 facet = isl_basic_set_gauss(facet, NULL);
744 facet = isl_basic_set_normalize_constraints(facet);
745 hull_facet = isl_basic_set_copy(hull);
746 hull_facet = isl_basic_set_add_equality(hull_facet, hull->ineq[i]);
747 hull_facet = isl_basic_set_gauss(hull_facet, NULL);
748 hull_facet = isl_basic_set_normalize_constraints(hull_facet);
751 hull = isl_basic_set_cow(hull);
752 hull = isl_basic_set_extend_dim(hull,
753 isl_dim_copy(hull->dim), 0, 0, facet->n_ineq);
754 for (j = 0; j < facet->n_ineq; ++j) {
755 for (f = 0; f < hull_facet->n_ineq; ++f)
756 if (isl_seq_eq(facet->ineq[j],
757 hull_facet->ineq[f], 1 + dim))
759 if (f < hull_facet->n_ineq)
761 k = isl_basic_set_alloc_inequality(hull);
764 isl_seq_cpy(hull->ineq[k], hull->ineq[i], 1+dim);
765 if (!wrap_facet(set, hull->ineq[k], facet->ineq[j]))
768 isl_basic_set_free(hull_facet);
769 isl_basic_set_free(facet);
771 hull = isl_basic_set_simplify(hull);
772 hull = isl_basic_set_finalize(hull);
775 isl_basic_set_free(hull_facet);
776 isl_basic_set_free(facet);
777 isl_basic_set_free(hull);
781 /* Special case for computing the convex hull of a one dimensional set.
782 * We simply collect the lower and upper bounds of each basic set
783 * and the biggest of those.
785 static struct isl_basic_set *convex_hull_1d(struct isl_set *set)
787 struct isl_mat *c = NULL;
788 isl_int *lower = NULL;
789 isl_int *upper = NULL;
792 struct isl_basic_set *hull;
794 for (i = 0; i < set->n; ++i) {
795 set->p[i] = isl_basic_set_simplify(set->p[i]);
799 set = isl_set_remove_empty_parts(set);
802 isl_assert(set->ctx, set->n > 0, goto error);
803 c = isl_mat_alloc(set->ctx, 2, 2);
807 if (set->p[0]->n_eq > 0) {
808 isl_assert(set->ctx, set->p[0]->n_eq == 1, goto error);
811 if (isl_int_is_pos(set->p[0]->eq[0][1])) {
812 isl_seq_cpy(lower, set->p[0]->eq[0], 2);
813 isl_seq_neg(upper, set->p[0]->eq[0], 2);
815 isl_seq_neg(lower, set->p[0]->eq[0], 2);
816 isl_seq_cpy(upper, set->p[0]->eq[0], 2);
819 for (j = 0; j < set->p[0]->n_ineq; ++j) {
820 if (isl_int_is_pos(set->p[0]->ineq[j][1])) {
822 isl_seq_cpy(lower, set->p[0]->ineq[j], 2);
825 isl_seq_cpy(upper, set->p[0]->ineq[j], 2);
832 for (i = 0; i < set->n; ++i) {
833 struct isl_basic_set *bset = set->p[i];
837 for (j = 0; j < bset->n_eq; ++j) {
841 isl_int_mul(a, lower[0], bset->eq[j][1]);
842 isl_int_mul(b, lower[1], bset->eq[j][0]);
843 if (isl_int_lt(a, b) && isl_int_is_pos(bset->eq[j][1]))
844 isl_seq_cpy(lower, bset->eq[j], 2);
845 if (isl_int_gt(a, b) && isl_int_is_neg(bset->eq[j][1]))
846 isl_seq_neg(lower, bset->eq[j], 2);
849 isl_int_mul(a, upper[0], bset->eq[j][1]);
850 isl_int_mul(b, upper[1], bset->eq[j][0]);
851 if (isl_int_lt(a, b) && isl_int_is_pos(bset->eq[j][1]))
852 isl_seq_neg(upper, bset->eq[j], 2);
853 if (isl_int_gt(a, b) && isl_int_is_neg(bset->eq[j][1]))
854 isl_seq_cpy(upper, bset->eq[j], 2);
857 for (j = 0; j < bset->n_ineq; ++j) {
858 if (isl_int_is_pos(bset->ineq[j][1]))
860 if (isl_int_is_neg(bset->ineq[j][1]))
862 if (lower && isl_int_is_pos(bset->ineq[j][1])) {
863 isl_int_mul(a, lower[0], bset->ineq[j][1]);
864 isl_int_mul(b, lower[1], bset->ineq[j][0]);
865 if (isl_int_lt(a, b))
866 isl_seq_cpy(lower, bset->ineq[j], 2);
868 if (upper && isl_int_is_neg(bset->ineq[j][1])) {
869 isl_int_mul(a, upper[0], bset->ineq[j][1]);
870 isl_int_mul(b, upper[1], bset->ineq[j][0]);
871 if (isl_int_gt(a, b))
872 isl_seq_cpy(upper, bset->ineq[j], 2);
883 hull = isl_basic_set_alloc(set->ctx, 0, 1, 0, 0, 2);
884 hull = isl_basic_set_set_rational(hull);
888 k = isl_basic_set_alloc_inequality(hull);
889 isl_seq_cpy(hull->ineq[k], lower, 2);
892 k = isl_basic_set_alloc_inequality(hull);
893 isl_seq_cpy(hull->ineq[k], upper, 2);
895 hull = isl_basic_set_finalize(hull);
905 /* Project out final n dimensions using Fourier-Motzkin */
906 static struct isl_set *set_project_out(struct isl_ctx *ctx,
907 struct isl_set *set, unsigned n)
909 return isl_set_remove_dims(set, isl_set_n_dim(set) - n, n);
912 static struct isl_basic_set *convex_hull_0d(struct isl_set *set)
914 struct isl_basic_set *convex_hull;
919 if (isl_set_is_empty(set))
920 convex_hull = isl_basic_set_empty(isl_dim_copy(set->dim));
922 convex_hull = isl_basic_set_universe(isl_dim_copy(set->dim));
927 /* Compute the convex hull of a pair of basic sets without any parameters or
928 * integer divisions using Fourier-Motzkin elimination.
929 * The convex hull is the set of all points that can be written as
930 * the sum of points from both basic sets (in homogeneous coordinates).
931 * We set up the constraints in a space with dimensions for each of
932 * the three sets and then project out the dimensions corresponding
933 * to the two original basic sets, retaining only those corresponding
934 * to the convex hull.
936 static struct isl_basic_set *convex_hull_pair_elim(struct isl_basic_set *bset1,
937 struct isl_basic_set *bset2)
940 struct isl_basic_set *bset[2];
941 struct isl_basic_set *hull = NULL;
944 if (!bset1 || !bset2)
947 dim = isl_basic_set_n_dim(bset1);
948 hull = isl_basic_set_alloc(bset1->ctx, 0, 2 + 3 * dim, 0,
949 1 + dim + bset1->n_eq + bset2->n_eq,
950 2 + bset1->n_ineq + bset2->n_ineq);
953 for (i = 0; i < 2; ++i) {
954 for (j = 0; j < bset[i]->n_eq; ++j) {
955 k = isl_basic_set_alloc_equality(hull);
958 isl_seq_clr(hull->eq[k], (i+1) * (1+dim));
959 isl_seq_clr(hull->eq[k]+(i+2)*(1+dim), (1-i)*(1+dim));
960 isl_seq_cpy(hull->eq[k]+(i+1)*(1+dim), bset[i]->eq[j],
963 for (j = 0; j < bset[i]->n_ineq; ++j) {
964 k = isl_basic_set_alloc_inequality(hull);
967 isl_seq_clr(hull->ineq[k], (i+1) * (1+dim));
968 isl_seq_clr(hull->ineq[k]+(i+2)*(1+dim), (1-i)*(1+dim));
969 isl_seq_cpy(hull->ineq[k]+(i+1)*(1+dim),
970 bset[i]->ineq[j], 1+dim);
972 k = isl_basic_set_alloc_inequality(hull);
975 isl_seq_clr(hull->ineq[k], 1+2+3*dim);
976 isl_int_set_si(hull->ineq[k][(i+1)*(1+dim)], 1);
978 for (j = 0; j < 1+dim; ++j) {
979 k = isl_basic_set_alloc_equality(hull);
982 isl_seq_clr(hull->eq[k], 1+2+3*dim);
983 isl_int_set_si(hull->eq[k][j], -1);
984 isl_int_set_si(hull->eq[k][1+dim+j], 1);
985 isl_int_set_si(hull->eq[k][2*(1+dim)+j], 1);
987 hull = isl_basic_set_set_rational(hull);
988 hull = isl_basic_set_remove_dims(hull, dim, 2*(1+dim));
989 hull = isl_basic_set_convex_hull(hull);
990 isl_basic_set_free(bset1);
991 isl_basic_set_free(bset2);
994 isl_basic_set_free(bset1);
995 isl_basic_set_free(bset2);
996 isl_basic_set_free(hull);
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(bset);
1006 bounded = isl_tab_cone_is_bounded(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 lineality space of the convex hull of bset1 and bset2.
1025 * We first compute the intersection of the recession cone of bset1
1026 * with the negative of the recession cone of bset2 and then compute
1027 * the linear hull of the resulting cone.
1029 static struct isl_basic_set *induced_lineality_space(
1030 struct isl_basic_set *bset1, struct isl_basic_set *bset2)
1033 struct isl_basic_set *lin = NULL;
1036 if (!bset1 || !bset2)
1039 dim = isl_basic_set_total_dim(bset1);
1040 lin = isl_basic_set_alloc_dim(isl_basic_set_get_dim(bset1), 0,
1041 bset1->n_eq + bset2->n_eq,
1042 bset1->n_ineq + bset2->n_ineq);
1043 lin = isl_basic_set_set_rational(lin);
1046 for (i = 0; i < bset1->n_eq; ++i) {
1047 k = isl_basic_set_alloc_equality(lin);
1050 isl_int_set_si(lin->eq[k][0], 0);
1051 isl_seq_cpy(lin->eq[k] + 1, bset1->eq[i] + 1, dim);
1053 for (i = 0; i < bset1->n_ineq; ++i) {
1054 k = isl_basic_set_alloc_inequality(lin);
1057 isl_int_set_si(lin->ineq[k][0], 0);
1058 isl_seq_cpy(lin->ineq[k] + 1, bset1->ineq[i] + 1, dim);
1060 for (i = 0; i < bset2->n_eq; ++i) {
1061 k = isl_basic_set_alloc_equality(lin);
1064 isl_int_set_si(lin->eq[k][0], 0);
1065 isl_seq_neg(lin->eq[k] + 1, bset2->eq[i] + 1, dim);
1067 for (i = 0; i < bset2->n_ineq; ++i) {
1068 k = isl_basic_set_alloc_inequality(lin);
1071 isl_int_set_si(lin->ineq[k][0], 0);
1072 isl_seq_neg(lin->ineq[k] + 1, bset2->ineq[i] + 1, dim);
1075 isl_basic_set_free(bset1);
1076 isl_basic_set_free(bset2);
1077 return isl_basic_set_affine_hull(lin);
1079 isl_basic_set_free(lin);
1080 isl_basic_set_free(bset1);
1081 isl_basic_set_free(bset2);
1085 static struct isl_basic_set *uset_convex_hull(struct isl_set *set);
1087 /* Given a set and a linear space "lin" of dimension n > 0,
1088 * project the linear space from the set, compute the convex hull
1089 * and then map the set back to the original space.
1095 * describe the linear space. We first compute the Hermite normal
1096 * form H = M U of M = H Q, to obtain
1100 * The last n rows of H will be zero, so the last n variables of x' = Q x
1101 * are the one we want to project out. We do this by transforming each
1102 * basic set A x >= b to A U x' >= b and then removing the last n dimensions.
1103 * After computing the convex hull in x'_1, i.e., A' x'_1 >= b',
1104 * we transform the hull back to the original space as A' Q_1 x >= b',
1105 * with Q_1 all but the last n rows of Q.
1107 static struct isl_basic_set *modulo_lineality(struct isl_set *set,
1108 struct isl_basic_set *lin)
1110 unsigned total = isl_basic_set_total_dim(lin);
1112 struct isl_basic_set *hull;
1113 struct isl_mat *M, *U, *Q;
1117 lin_dim = total - lin->n_eq;
1118 M = isl_mat_sub_alloc(set->ctx, lin->eq, 0, lin->n_eq, 1, total);
1119 M = isl_mat_left_hermite(M, 0, &U, &Q);
1123 isl_basic_set_free(lin);
1125 Q = isl_mat_drop_rows(Q, Q->n_row - lin_dim, lin_dim);
1127 U = isl_mat_lin_to_aff(U);
1128 Q = isl_mat_lin_to_aff(Q);
1130 set = isl_set_preimage(set, U);
1131 set = isl_set_remove_dims(set, total - lin_dim, lin_dim);
1132 hull = uset_convex_hull(set);
1133 hull = isl_basic_set_preimage(hull, Q);
1137 isl_basic_set_free(lin);
1142 /* Given two polyhedra with as constraints h_{ij} x >= 0 in homegeneous space,
1143 * set up an LP for solving
1145 * \sum_j \alpha_{1j} h_{1j} = \sum_j \alpha_{2j} h_{2j}
1147 * \alpha{i0} corresponds to the (implicit) positivity constraint 1 >= 0
1148 * The next \alpha{ij} correspond to the equalities and come in pairs.
1149 * The final \alpha{ij} correspond to the inequalities.
1151 static struct isl_basic_set *valid_direction_lp(
1152 struct isl_basic_set *bset1, struct isl_basic_set *bset2)
1154 struct isl_dim *dim;
1155 struct isl_basic_set *lp;
1160 if (!bset1 || !bset2)
1162 d = 1 + isl_basic_set_total_dim(bset1);
1164 2 * bset1->n_eq + bset1->n_ineq + 2 * bset2->n_eq + bset2->n_ineq;
1165 dim = isl_dim_set_alloc(bset1->ctx, 0, n);
1166 lp = isl_basic_set_alloc_dim(dim, 0, d, n);
1169 for (i = 0; i < n; ++i) {
1170 k = isl_basic_set_alloc_inequality(lp);
1173 isl_seq_clr(lp->ineq[k] + 1, n);
1174 isl_int_set_si(lp->ineq[k][0], -1);
1175 isl_int_set_si(lp->ineq[k][1 + i], 1);
1177 for (i = 0; i < d; ++i) {
1178 k = isl_basic_set_alloc_equality(lp);
1182 isl_int_set_si(lp->eq[k][n++], 0);
1183 /* positivity constraint 1 >= 0 */
1184 isl_int_set_si(lp->eq[k][n++], i == 0);
1185 for (j = 0; j < bset1->n_eq; ++j) {
1186 isl_int_set(lp->eq[k][n++], bset1->eq[j][i]);
1187 isl_int_neg(lp->eq[k][n++], bset1->eq[j][i]);
1189 for (j = 0; j < bset1->n_ineq; ++j)
1190 isl_int_set(lp->eq[k][n++], bset1->ineq[j][i]);
1191 /* positivity constraint 1 >= 0 */
1192 isl_int_set_si(lp->eq[k][n++], -(i == 0));
1193 for (j = 0; j < bset2->n_eq; ++j) {
1194 isl_int_neg(lp->eq[k][n++], bset2->eq[j][i]);
1195 isl_int_set(lp->eq[k][n++], bset2->eq[j][i]);
1197 for (j = 0; j < bset2->n_ineq; ++j)
1198 isl_int_neg(lp->eq[k][n++], bset2->ineq[j][i]);
1200 lp = isl_basic_set_gauss(lp, NULL);
1201 isl_basic_set_free(bset1);
1202 isl_basic_set_free(bset2);
1205 isl_basic_set_free(bset1);
1206 isl_basic_set_free(bset2);
1210 /* Compute a vector s in the homogeneous space such that <s, r> > 0
1211 * for all rays in the homogeneous space of the two cones that correspond
1212 * to the input polyhedra bset1 and bset2.
1214 * We compute s as a vector that satisfies
1216 * s = \sum_j \alpha_{ij} h_{ij} for i = 1,2 (*)
1218 * with h_{ij} the normals of the facets of polyhedron i
1219 * (including the "positivity constraint" 1 >= 0) and \alpha_{ij}
1220 * strictly positive numbers. For simplicity we impose \alpha_{ij} >= 1.
1221 * We first set up an LP with as variables the \alpha{ij}.
1222 * In this formulateion, for each polyhedron i,
1223 * the first constraint is the positivity constraint, followed by pairs
1224 * of variables for the equalities, followed by variables for the inequalities.
1225 * We then simply pick a feasible solution and compute s using (*).
1227 * Note that we simply pick any valid direction and make no attempt
1228 * to pick a "good" or even the "best" valid direction.
1230 static struct isl_vec *valid_direction(
1231 struct isl_basic_set *bset1, struct isl_basic_set *bset2)
1233 struct isl_basic_set *lp;
1234 struct isl_tab *tab;
1235 struct isl_vec *sample = NULL;
1236 struct isl_vec *dir;
1241 if (!bset1 || !bset2)
1243 lp = valid_direction_lp(isl_basic_set_copy(bset1),
1244 isl_basic_set_copy(bset2));
1245 tab = isl_tab_from_basic_set(lp);
1246 sample = isl_tab_get_sample_value(tab);
1248 isl_basic_set_free(lp);
1251 d = isl_basic_set_total_dim(bset1);
1252 dir = isl_vec_alloc(bset1->ctx, 1 + d);
1255 isl_seq_clr(dir->block.data + 1, dir->size - 1);
1257 /* positivity constraint 1 >= 0 */
1258 isl_int_set(dir->block.data[0], sample->block.data[n++]);
1259 for (i = 0; i < bset1->n_eq; ++i) {
1260 isl_int_sub(sample->block.data[n],
1261 sample->block.data[n], sample->block.data[n+1]);
1262 isl_seq_combine(dir->block.data,
1263 bset1->ctx->one, dir->block.data,
1264 sample->block.data[n], bset1->eq[i], 1 + d);
1268 for (i = 0; i < bset1->n_ineq; ++i)
1269 isl_seq_combine(dir->block.data,
1270 bset1->ctx->one, dir->block.data,
1271 sample->block.data[n++], bset1->ineq[i], 1 + d);
1272 isl_vec_free(sample);
1273 isl_seq_normalize(bset1->ctx, dir->block.data + 1, dir->size - 1);
1274 isl_basic_set_free(bset1);
1275 isl_basic_set_free(bset2);
1278 isl_vec_free(sample);
1279 isl_basic_set_free(bset1);
1280 isl_basic_set_free(bset2);
1284 /* Given a polyhedron b_i + A_i x >= 0 and a map T = S^{-1},
1285 * compute b_i' + A_i' x' >= 0, with
1287 * [ b_i A_i ] [ y' ] [ y' ]
1288 * [ 1 0 ] S^{-1} [ x' ] >= 0 or [ b_i' A_i' ] [ x' ] >= 0
1290 * In particular, add the "positivity constraint" and then perform
1293 static struct isl_basic_set *homogeneous_map(struct isl_basic_set *bset,
1300 bset = isl_basic_set_extend_constraints(bset, 0, 1);
1301 k = isl_basic_set_alloc_inequality(bset);
1304 isl_seq_clr(bset->ineq[k] + 1, isl_basic_set_total_dim(bset));
1305 isl_int_set_si(bset->ineq[k][0], 1);
1306 bset = isl_basic_set_preimage(bset, T);
1310 isl_basic_set_free(bset);
1314 /* Compute the convex hull of a pair of basic sets without any parameters or
1315 * integer divisions, where the convex hull is known to be pointed,
1316 * but the basic sets may be unbounded.
1318 * We turn this problem into the computation of a convex hull of a pair
1319 * _bounded_ polyhedra by "changing the direction of the homogeneous
1320 * dimension". This idea is due to Matthias Koeppe.
1322 * Consider the cones in homogeneous space that correspond to the
1323 * input polyhedra. The rays of these cones are also rays of the
1324 * polyhedra if the coordinate that corresponds to the homogeneous
1325 * dimension is zero. That is, if the inner product of the rays
1326 * with the homogeneous direction is zero.
1327 * The cones in the homogeneous space can also be considered to
1328 * correspond to other pairs of polyhedra by chosing a different
1329 * homogeneous direction. To ensure that both of these polyhedra
1330 * are bounded, we need to make sure that all rays of the cones
1331 * correspond to vertices and not to rays.
1332 * Let s be a direction such that <s, r> > 0 for all rays r of both cones.
1333 * Then using s as a homogeneous direction, we obtain a pair of polytopes.
1334 * The vector s is computed in valid_direction.
1336 * Note that we need to consider _all_ rays of the cones and not just
1337 * the rays that correspond to rays in the polyhedra. If we were to
1338 * only consider those rays and turn them into vertices, then we
1339 * may inadvertently turn some vertices into rays.
1341 * The standard homogeneous direction is the unit vector in the 0th coordinate.
1342 * We therefore transform the two polyhedra such that the selected
1343 * direction is mapped onto this standard direction and then proceed
1344 * with the normal computation.
1345 * Let S be a non-singular square matrix with s as its first row,
1346 * then we want to map the polyhedra to the space
1348 * [ y' ] [ y ] [ y ] [ y' ]
1349 * [ x' ] = S [ x ] i.e., [ x ] = S^{-1} [ x' ]
1351 * We take S to be the unimodular completion of s to limit the growth
1352 * of the coefficients in the following computations.
1354 * Let b_i + A_i x >= 0 be the constraints of polyhedron i.
1355 * We first move to the homogeneous dimension
1357 * b_i y + A_i x >= 0 [ b_i A_i ] [ y ] [ 0 ]
1358 * y >= 0 or [ 1 0 ] [ x ] >= [ 0 ]
1360 * Then we change directoin
1362 * [ b_i A_i ] [ y' ] [ y' ]
1363 * [ 1 0 ] S^{-1} [ x' ] >= 0 or [ b_i' A_i' ] [ x' ] >= 0
1365 * Then we compute the convex hull of the polytopes b_i' + A_i' x' >= 0
1366 * resulting in b' + A' x' >= 0, which we then convert back
1369 * [ b' A' ] S [ x ] >= 0 or [ b A ] [ x ] >= 0
1371 * The polyhedron b + A x >= 0 is then the convex hull of the input polyhedra.
1373 static struct isl_basic_set *convex_hull_pair_pointed(
1374 struct isl_basic_set *bset1, struct isl_basic_set *bset2)
1376 struct isl_ctx *ctx = NULL;
1377 struct isl_vec *dir = NULL;
1378 struct isl_mat *T = NULL;
1379 struct isl_mat *T2 = NULL;
1380 struct isl_basic_set *hull;
1381 struct isl_set *set;
1383 if (!bset1 || !bset2)
1386 dir = valid_direction(isl_basic_set_copy(bset1),
1387 isl_basic_set_copy(bset2));
1390 T = isl_mat_alloc(bset1->ctx, dir->size, dir->size);
1393 isl_seq_cpy(T->row[0], dir->block.data, dir->size);
1394 T = isl_mat_unimodular_complete(T, 1);
1395 T2 = isl_mat_right_inverse(isl_mat_copy(T));
1397 bset1 = homogeneous_map(bset1, isl_mat_copy(T2));
1398 bset2 = homogeneous_map(bset2, T2);
1399 set = isl_set_alloc_dim(isl_basic_set_get_dim(bset1), 2, 0);
1400 set = isl_set_add_basic_set(set, bset1);
1401 set = isl_set_add_basic_set(set, bset2);
1402 hull = uset_convex_hull(set);
1403 hull = isl_basic_set_preimage(hull, T);
1410 isl_basic_set_free(bset1);
1411 isl_basic_set_free(bset2);
1415 /* Compute the convex hull of a pair of basic sets without any parameters or
1416 * integer divisions.
1418 * If the convex hull of the two basic sets would have a non-trivial
1419 * lineality space, we first project out this lineality space.
1421 static struct isl_basic_set *convex_hull_pair(struct isl_basic_set *bset1,
1422 struct isl_basic_set *bset2)
1424 struct isl_basic_set *lin;
1426 if (isl_basic_set_is_bounded(bset1) || isl_basic_set_is_bounded(bset2))
1427 return convex_hull_pair_pointed(bset1, bset2);
1429 lin = induced_lineality_space(isl_basic_set_copy(bset1),
1430 isl_basic_set_copy(bset2));
1433 if (isl_basic_set_is_universe(lin)) {
1434 isl_basic_set_free(bset1);
1435 isl_basic_set_free(bset2);
1438 if (lin->n_eq < isl_basic_set_total_dim(lin)) {
1439 struct isl_set *set;
1440 set = isl_set_alloc_dim(isl_basic_set_get_dim(bset1), 2, 0);
1441 set = isl_set_add_basic_set(set, bset1);
1442 set = isl_set_add_basic_set(set, bset2);
1443 return modulo_lineality(set, lin);
1445 isl_basic_set_free(lin);
1447 return convex_hull_pair_pointed(bset1, bset2);
1449 isl_basic_set_free(bset1);
1450 isl_basic_set_free(bset2);
1454 /* Compute the lineality space of a basic set.
1455 * We currently do not allow the basic set to have any divs.
1456 * We basically just drop the constants and turn every inequality
1459 struct isl_basic_set *isl_basic_set_lineality_space(struct isl_basic_set *bset)
1462 struct isl_basic_set *lin = NULL;
1467 isl_assert(bset->ctx, bset->n_div == 0, goto error);
1468 dim = isl_basic_set_total_dim(bset);
1470 lin = isl_basic_set_alloc_dim(isl_basic_set_get_dim(bset), 0, dim, 0);
1473 for (i = 0; i < bset->n_eq; ++i) {
1474 k = isl_basic_set_alloc_equality(lin);
1477 isl_int_set_si(lin->eq[k][0], 0);
1478 isl_seq_cpy(lin->eq[k] + 1, bset->eq[i] + 1, dim);
1480 lin = isl_basic_set_gauss(lin, NULL);
1483 for (i = 0; i < bset->n_ineq && lin->n_eq < dim; ++i) {
1484 k = isl_basic_set_alloc_equality(lin);
1487 isl_int_set_si(lin->eq[k][0], 0);
1488 isl_seq_cpy(lin->eq[k] + 1, bset->ineq[i] + 1, dim);
1489 lin = isl_basic_set_gauss(lin, NULL);
1493 isl_basic_set_free(bset);
1496 isl_basic_set_free(lin);
1497 isl_basic_set_free(bset);
1501 /* Compute the (linear) hull of the lineality spaces of the basic sets in the
1502 * "underlying" set "set".
1504 static struct isl_basic_set *uset_combined_lineality_space(struct isl_set *set)
1507 struct isl_set *lin = NULL;
1512 struct isl_dim *dim = isl_set_get_dim(set);
1514 return isl_basic_set_empty(dim);
1517 lin = isl_set_alloc_dim(isl_set_get_dim(set), set->n, 0);
1518 for (i = 0; i < set->n; ++i)
1519 lin = isl_set_add_basic_set(lin,
1520 isl_basic_set_lineality_space(isl_basic_set_copy(set->p[i])));
1522 return isl_set_affine_hull(lin);
1525 /* Compute the convex hull of a set without any parameters or
1526 * integer divisions.
1527 * In each step, we combined two basic sets until only one
1528 * basic set is left.
1529 * The input basic sets are assumed not to have a non-trivial
1530 * lineality space. If any of the intermediate results has
1531 * a non-trivial lineality space, it is projected out.
1533 static struct isl_basic_set *uset_convex_hull_unbounded(struct isl_set *set)
1535 struct isl_basic_set *convex_hull = NULL;
1537 convex_hull = isl_set_copy_basic_set(set);
1538 set = isl_set_drop_basic_set(set, convex_hull);
1541 while (set->n > 0) {
1542 struct isl_basic_set *t;
1543 t = isl_set_copy_basic_set(set);
1546 set = isl_set_drop_basic_set(set, t);
1549 convex_hull = convex_hull_pair(convex_hull, t);
1552 t = isl_basic_set_lineality_space(isl_basic_set_copy(convex_hull));
1555 if (isl_basic_set_is_universe(t)) {
1556 isl_basic_set_free(convex_hull);
1560 if (t->n_eq < isl_basic_set_total_dim(t)) {
1561 set = isl_set_add_basic_set(set, convex_hull);
1562 return modulo_lineality(set, t);
1564 isl_basic_set_free(t);
1570 isl_basic_set_free(convex_hull);
1574 /* Compute an initial hull for wrapping containing a single initial
1575 * facet by first computing bounds on the set and then using these
1576 * bounds to construct an initial facet.
1577 * This function is a remnant of an older implementation where the
1578 * bounds were also used to check whether the set was bounded.
1579 * Since this function will now only be called when we know the
1580 * set to be bounded, the initial facet should probably be constructed
1581 * by simply using the coordinate directions instead.
1583 static struct isl_basic_set *initial_hull(struct isl_basic_set *hull,
1584 struct isl_set *set)
1586 struct isl_mat *bounds = NULL;
1592 bounds = independent_bounds(set);
1595 isl_assert(set->ctx, bounds->n_row == isl_set_n_dim(set), goto error);
1596 bounds = initial_facet_constraint(set, bounds);
1599 k = isl_basic_set_alloc_inequality(hull);
1602 dim = isl_set_n_dim(set);
1603 isl_assert(set->ctx, 1 + dim == bounds->n_col, goto error);
1604 isl_seq_cpy(hull->ineq[k], bounds->row[0], bounds->n_col);
1605 isl_mat_free(bounds);
1609 isl_basic_set_free(hull);
1610 isl_mat_free(bounds);
1614 struct max_constraint {
1620 static int max_constraint_equal(const void *entry, const void *val)
1622 struct max_constraint *a = (struct max_constraint *)entry;
1623 isl_int *b = (isl_int *)val;
1625 return isl_seq_eq(a->c->row[0] + 1, b, a->c->n_col - 1);
1628 static void update_constraint(struct isl_ctx *ctx, struct isl_hash_table *table,
1629 isl_int *con, unsigned len, int n, int ineq)
1631 struct isl_hash_table_entry *entry;
1632 struct max_constraint *c;
1635 c_hash = isl_seq_get_hash(con + 1, len);
1636 entry = isl_hash_table_find(ctx, table, c_hash, max_constraint_equal,
1642 isl_hash_table_remove(ctx, table, entry);
1646 if (isl_int_gt(c->c->row[0][0], con[0]))
1648 if (isl_int_eq(c->c->row[0][0], con[0])) {
1653 c->c = isl_mat_cow(c->c);
1654 isl_int_set(c->c->row[0][0], con[0]);
1658 /* Check whether the constraint hash table "table" constains the constraint
1661 static int has_constraint(struct isl_ctx *ctx, struct isl_hash_table *table,
1662 isl_int *con, unsigned len, int n)
1664 struct isl_hash_table_entry *entry;
1665 struct max_constraint *c;
1668 c_hash = isl_seq_get_hash(con + 1, len);
1669 entry = isl_hash_table_find(ctx, table, c_hash, max_constraint_equal,
1676 return isl_int_eq(c->c->row[0][0], con[0]);
1679 /* Check for inequality constraints of a basic set without equalities
1680 * such that the same or more stringent copies of the constraint appear
1681 * in all of the basic sets. Such constraints are necessarily facet
1682 * constraints of the convex hull.
1684 * If the resulting basic set is by chance identical to one of
1685 * the basic sets in "set", then we know that this basic set contains
1686 * all other basic sets and is therefore the convex hull of set.
1687 * In this case we set *is_hull to 1.
1689 static struct isl_basic_set *common_constraints(struct isl_basic_set *hull,
1690 struct isl_set *set, int *is_hull)
1693 int min_constraints;
1695 struct max_constraint *constraints = NULL;
1696 struct isl_hash_table *table = NULL;
1701 for (i = 0; i < set->n; ++i)
1702 if (set->p[i]->n_eq == 0)
1706 min_constraints = set->p[i]->n_ineq;
1708 for (i = best + 1; i < set->n; ++i) {
1709 if (set->p[i]->n_eq != 0)
1711 if (set->p[i]->n_ineq >= min_constraints)
1713 min_constraints = set->p[i]->n_ineq;
1716 constraints = isl_calloc_array(hull->ctx, struct max_constraint,
1720 table = isl_alloc_type(hull->ctx, struct isl_hash_table);
1721 if (isl_hash_table_init(hull->ctx, table, min_constraints))
1724 total = isl_dim_total(set->dim);
1725 for (i = 0; i < set->p[best]->n_ineq; ++i) {
1726 constraints[i].c = isl_mat_sub_alloc(hull->ctx,
1727 set->p[best]->ineq + i, 0, 1, 0, 1 + total);
1728 if (!constraints[i].c)
1730 constraints[i].ineq = 1;
1732 for (i = 0; i < min_constraints; ++i) {
1733 struct isl_hash_table_entry *entry;
1735 c_hash = isl_seq_get_hash(constraints[i].c->row[0] + 1, total);
1736 entry = isl_hash_table_find(hull->ctx, table, c_hash,
1737 max_constraint_equal, constraints[i].c->row[0] + 1, 1);
1740 isl_assert(hull->ctx, !entry->data, goto error);
1741 entry->data = &constraints[i];
1745 for (s = 0; s < set->n; ++s) {
1749 for (i = 0; i < set->p[s]->n_eq; ++i) {
1750 isl_int *eq = set->p[s]->eq[i];
1751 for (j = 0; j < 2; ++j) {
1752 isl_seq_neg(eq, eq, 1 + total);
1753 update_constraint(hull->ctx, table,
1757 for (i = 0; i < set->p[s]->n_ineq; ++i) {
1758 isl_int *ineq = set->p[s]->ineq[i];
1759 update_constraint(hull->ctx, table, ineq, total, n,
1760 set->p[s]->n_eq == 0);
1765 for (i = 0; i < min_constraints; ++i) {
1766 if (constraints[i].count < n)
1768 if (!constraints[i].ineq)
1770 j = isl_basic_set_alloc_inequality(hull);
1773 isl_seq_cpy(hull->ineq[j], constraints[i].c->row[0], 1 + total);
1776 for (s = 0; s < set->n; ++s) {
1777 if (set->p[s]->n_eq)
1779 if (set->p[s]->n_ineq != hull->n_ineq)
1781 for (i = 0; i < set->p[s]->n_ineq; ++i) {
1782 isl_int *ineq = set->p[s]->ineq[i];
1783 if (!has_constraint(hull->ctx, table, ineq, total, n))
1786 if (i == set->p[s]->n_ineq)
1790 isl_hash_table_clear(table);
1791 for (i = 0; i < min_constraints; ++i)
1792 isl_mat_free(constraints[i].c);
1797 isl_hash_table_clear(table);
1800 for (i = 0; i < min_constraints; ++i)
1801 isl_mat_free(constraints[i].c);
1806 /* Create a template for the convex hull of "set" and fill it up
1807 * obvious facet constraints, if any. If the result happens to
1808 * be the convex hull of "set" then *is_hull is set to 1.
1810 static struct isl_basic_set *proto_hull(struct isl_set *set, int *is_hull)
1812 struct isl_basic_set *hull;
1817 for (i = 0; i < set->n; ++i) {
1818 n_ineq += set->p[i]->n_eq;
1819 n_ineq += set->p[i]->n_ineq;
1821 hull = isl_basic_set_alloc_dim(isl_dim_copy(set->dim), 0, 0, n_ineq);
1822 hull = isl_basic_set_set_rational(hull);
1825 return common_constraints(hull, set, is_hull);
1828 static struct isl_basic_set *uset_convex_hull_wrap(struct isl_set *set)
1830 struct isl_basic_set *hull;
1833 hull = proto_hull(set, &is_hull);
1834 if (hull && !is_hull) {
1835 if (hull->n_ineq == 0)
1836 hull = initial_hull(hull, set);
1837 hull = extend(hull, set);
1844 /* Compute the convex hull of a set without any parameters or
1845 * integer divisions. Depending on whether the set is bounded,
1846 * we pass control to the wrapping based convex hull or
1847 * the Fourier-Motzkin elimination based convex hull.
1848 * We also handle a few special cases before checking the boundedness.
1850 static struct isl_basic_set *uset_convex_hull(struct isl_set *set)
1852 struct isl_basic_set *convex_hull = NULL;
1853 struct isl_basic_set *lin;
1855 if (isl_set_n_dim(set) == 0)
1856 return convex_hull_0d(set);
1858 set = isl_set_coalesce(set);
1859 set = isl_set_set_rational(set);
1866 convex_hull = isl_basic_set_copy(set->p[0]);
1870 if (isl_set_n_dim(set) == 1)
1871 return convex_hull_1d(set);
1873 if (isl_set_is_bounded(set))
1874 return uset_convex_hull_wrap(set);
1876 lin = uset_combined_lineality_space(isl_set_copy(set));
1879 if (isl_basic_set_is_universe(lin)) {
1883 if (lin->n_eq < isl_basic_set_total_dim(lin))
1884 return modulo_lineality(set, lin);
1885 isl_basic_set_free(lin);
1887 return uset_convex_hull_unbounded(set);
1890 isl_basic_set_free(convex_hull);
1894 /* This is the core procedure, where "set" is a "pure" set, i.e.,
1895 * without parameters or divs and where the convex hull of set is
1896 * known to be full-dimensional.
1898 static struct isl_basic_set *uset_convex_hull_wrap_bounded(struct isl_set *set)
1900 struct isl_basic_set *convex_hull = NULL;
1902 if (isl_set_n_dim(set) == 0) {
1903 convex_hull = isl_basic_set_universe(isl_dim_copy(set->dim));
1905 convex_hull = isl_basic_set_set_rational(convex_hull);
1909 set = isl_set_set_rational(set);
1913 set = isl_set_coalesce(set);
1917 convex_hull = isl_basic_set_copy(set->p[0]);
1921 if (isl_set_n_dim(set) == 1)
1922 return convex_hull_1d(set);
1924 return uset_convex_hull_wrap(set);
1930 /* Compute the convex hull of set "set" with affine hull "affine_hull",
1931 * We first remove the equalities (transforming the set), compute the
1932 * convex hull of the transformed set and then add the equalities back
1933 * (after performing the inverse transformation.
1935 static struct isl_basic_set *modulo_affine_hull(struct isl_ctx *ctx,
1936 struct isl_set *set, struct isl_basic_set *affine_hull)
1940 struct isl_basic_set *dummy;
1941 struct isl_basic_set *convex_hull;
1943 dummy = isl_basic_set_remove_equalities(
1944 isl_basic_set_copy(affine_hull), &T, &T2);
1947 isl_basic_set_free(dummy);
1948 set = isl_set_preimage(set, T);
1949 convex_hull = uset_convex_hull(set);
1950 convex_hull = isl_basic_set_preimage(convex_hull, T2);
1951 convex_hull = isl_basic_set_intersect(convex_hull, affine_hull);
1954 isl_basic_set_free(affine_hull);
1959 /* Compute the convex hull of a map.
1961 * The implementation was inspired by "Extended Convex Hull" by Fukuda et al.,
1962 * specifically, the wrapping of facets to obtain new facets.
1964 struct isl_basic_map *isl_map_convex_hull(struct isl_map *map)
1966 struct isl_basic_set *bset;
1967 struct isl_basic_map *model = NULL;
1968 struct isl_basic_set *affine_hull = NULL;
1969 struct isl_basic_map *convex_hull = NULL;
1970 struct isl_set *set = NULL;
1971 struct isl_ctx *ctx;
1978 convex_hull = isl_basic_map_empty_like_map(map);
1983 map = isl_map_detect_equalities(map);
1984 map = isl_map_align_divs(map);
1985 model = isl_basic_map_copy(map->p[0]);
1986 set = isl_map_underlying_set(map);
1990 affine_hull = isl_set_affine_hull(isl_set_copy(set));
1993 if (affine_hull->n_eq != 0)
1994 bset = modulo_affine_hull(ctx, set, affine_hull);
1996 isl_basic_set_free(affine_hull);
1997 bset = uset_convex_hull(set);
2000 convex_hull = isl_basic_map_overlying_set(bset, model);
2002 ISL_F_SET(convex_hull, ISL_BASIC_MAP_NO_IMPLICIT);
2003 ISL_F_SET(convex_hull, ISL_BASIC_MAP_ALL_EQUALITIES);
2004 ISL_F_CLR(convex_hull, ISL_BASIC_MAP_RATIONAL);
2008 isl_basic_map_free(model);
2012 struct isl_basic_set *isl_set_convex_hull(struct isl_set *set)
2014 return (struct isl_basic_set *)
2015 isl_map_convex_hull((struct isl_map *)set);
2018 struct sh_data_entry {
2019 struct isl_hash_table *table;
2020 struct isl_tab *tab;
2023 /* Holds the data needed during the simple hull computation.
2025 * n the number of basic sets in the original set
2026 * hull_table a hash table of already computed constraints
2027 * in the simple hull
2028 * p for each basic set,
2029 * table a hash table of the constraints
2030 * tab the tableau corresponding to the basic set
2033 struct isl_ctx *ctx;
2035 struct isl_hash_table *hull_table;
2036 struct sh_data_entry p[1];
2039 static void sh_data_free(struct sh_data *data)
2045 isl_hash_table_free(data->ctx, data->hull_table);
2046 for (i = 0; i < data->n; ++i) {
2047 isl_hash_table_free(data->ctx, data->p[i].table);
2048 isl_tab_free(data->p[i].tab);
2053 struct ineq_cmp_data {
2058 static int has_ineq(const void *entry, const void *val)
2060 isl_int *row = (isl_int *)entry;
2061 struct ineq_cmp_data *v = (struct ineq_cmp_data *)val;
2063 return isl_seq_eq(row + 1, v->p + 1, v->len) ||
2064 isl_seq_is_neg(row + 1, v->p + 1, v->len);
2067 static int hash_ineq(struct isl_ctx *ctx, struct isl_hash_table *table,
2068 isl_int *ineq, unsigned len)
2071 struct ineq_cmp_data v;
2072 struct isl_hash_table_entry *entry;
2076 c_hash = isl_seq_get_hash(ineq + 1, len);
2077 entry = isl_hash_table_find(ctx, table, c_hash, has_ineq, &v, 1);
2084 /* Fill hash table "table" with the constraints of "bset".
2085 * Equalities are added as two inequalities.
2086 * The value in the hash table is a pointer to the (in)equality of "bset".
2088 static int hash_basic_set(struct isl_hash_table *table,
2089 struct isl_basic_set *bset)
2092 unsigned dim = isl_basic_set_total_dim(bset);
2094 for (i = 0; i < bset->n_eq; ++i) {
2095 for (j = 0; j < 2; ++j) {
2096 isl_seq_neg(bset->eq[i], bset->eq[i], 1 + dim);
2097 if (hash_ineq(bset->ctx, table, bset->eq[i], dim) < 0)
2101 for (i = 0; i < bset->n_ineq; ++i) {
2102 if (hash_ineq(bset->ctx, table, bset->ineq[i], dim) < 0)
2108 static struct sh_data *sh_data_alloc(struct isl_set *set, unsigned n_ineq)
2110 struct sh_data *data;
2113 data = isl_calloc(set->ctx, struct sh_data,
2114 sizeof(struct sh_data) +
2115 (set->n - 1) * sizeof(struct sh_data_entry));
2118 data->ctx = set->ctx;
2120 data->hull_table = isl_hash_table_alloc(set->ctx, n_ineq);
2121 if (!data->hull_table)
2123 for (i = 0; i < set->n; ++i) {
2124 data->p[i].table = isl_hash_table_alloc(set->ctx,
2125 2 * set->p[i]->n_eq + set->p[i]->n_ineq);
2126 if (!data->p[i].table)
2128 if (hash_basic_set(data->p[i].table, set->p[i]) < 0)
2137 /* Check if inequality "ineq" is a bound for basic set "j" or if
2138 * it can be relaxed (by increasing the constant term) to become
2139 * a bound for that basic set. In the latter case, the constant
2141 * Return 1 if "ineq" is a bound
2142 * 0 if "ineq" may attain arbitrarily small values on basic set "j"
2143 * -1 if some error occurred
2145 static int is_bound(struct sh_data *data, struct isl_set *set, int j,
2148 enum isl_lp_result res;
2151 if (!data->p[j].tab) {
2152 data->p[j].tab = isl_tab_from_basic_set(set->p[j]);
2153 if (!data->p[j].tab)
2159 res = isl_tab_min(data->p[j].tab, ineq, data->ctx->one,
2161 if (res == isl_lp_ok && isl_int_is_neg(opt))
2162 isl_int_sub(ineq[0], ineq[0], opt);
2166 return res == isl_lp_ok ? 1 :
2167 res == isl_lp_unbounded ? 0 : -1;
2170 /* Check if inequality "ineq" from basic set "i" can be relaxed to
2171 * become a bound on the whole set. If so, add the (relaxed) inequality
2174 * We first check if "hull" already contains a translate of the inequality.
2175 * If so, we are done.
2176 * Then, we check if any of the previous basic sets contains a translate
2177 * of the inequality. If so, then we have already considered this
2178 * inequality and we are done.
2179 * Otherwise, for each basic set other than "i", we check if the inequality
2180 * is a bound on the basic set.
2181 * For previous basic sets, we know that they do not contain a translate
2182 * of the inequality, so we directly call is_bound.
2183 * For following basic sets, we first check if a translate of the
2184 * inequality appears in its description and if so directly update
2185 * the inequality accordingly.
2187 static struct isl_basic_set *add_bound(struct isl_basic_set *hull,
2188 struct sh_data *data, struct isl_set *set, int i, isl_int *ineq)
2191 struct ineq_cmp_data v;
2192 struct isl_hash_table_entry *entry;
2198 v.len = isl_basic_set_total_dim(hull);
2200 c_hash = isl_seq_get_hash(ineq + 1, v.len);
2202 entry = isl_hash_table_find(hull->ctx, data->hull_table, c_hash,
2207 for (j = 0; j < i; ++j) {
2208 entry = isl_hash_table_find(hull->ctx, data->p[j].table,
2209 c_hash, has_ineq, &v, 0);
2216 k = isl_basic_set_alloc_inequality(hull);
2217 isl_seq_cpy(hull->ineq[k], ineq, 1 + v.len);
2221 for (j = 0; j < i; ++j) {
2223 bound = is_bound(data, set, j, hull->ineq[k]);
2230 isl_basic_set_free_inequality(hull, 1);
2234 for (j = i + 1; j < set->n; ++j) {
2237 entry = isl_hash_table_find(hull->ctx, data->p[j].table,
2238 c_hash, has_ineq, &v, 0);
2240 ineq_j = entry->data;
2241 neg = isl_seq_is_neg(ineq_j + 1,
2242 hull->ineq[k] + 1, v.len);
2244 isl_int_neg(ineq_j[0], ineq_j[0]);
2245 if (isl_int_gt(ineq_j[0], hull->ineq[k][0]))
2246 isl_int_set(hull->ineq[k][0], ineq_j[0]);
2248 isl_int_neg(ineq_j[0], ineq_j[0]);
2251 bound = is_bound(data, set, j, hull->ineq[k]);
2258 isl_basic_set_free_inequality(hull, 1);
2262 entry = isl_hash_table_find(hull->ctx, data->hull_table, c_hash,
2266 entry->data = hull->ineq[k];
2270 isl_basic_set_free(hull);
2274 /* Check if any inequality from basic set "i" can be relaxed to
2275 * become a bound on the whole set. If so, add the (relaxed) inequality
2278 static struct isl_basic_set *add_bounds(struct isl_basic_set *bset,
2279 struct sh_data *data, struct isl_set *set, int i)
2282 unsigned dim = isl_basic_set_total_dim(bset);
2284 for (j = 0; j < set->p[i]->n_eq; ++j) {
2285 for (k = 0; k < 2; ++k) {
2286 isl_seq_neg(set->p[i]->eq[j], set->p[i]->eq[j], 1+dim);
2287 add_bound(bset, data, set, i, set->p[i]->eq[j]);
2290 for (j = 0; j < set->p[i]->n_ineq; ++j)
2291 add_bound(bset, data, set, i, set->p[i]->ineq[j]);
2295 /* Compute a superset of the convex hull of set that is described
2296 * by only translates of the constraints in the constituents of set.
2298 static struct isl_basic_set *uset_simple_hull(struct isl_set *set)
2300 struct sh_data *data = NULL;
2301 struct isl_basic_set *hull = NULL;
2309 for (i = 0; i < set->n; ++i) {
2312 n_ineq += 2 * set->p[i]->n_eq + set->p[i]->n_ineq;
2315 hull = isl_basic_set_alloc_dim(isl_dim_copy(set->dim), 0, 0, n_ineq);
2319 data = sh_data_alloc(set, n_ineq);
2323 for (i = 0; i < set->n; ++i)
2324 hull = add_bounds(hull, data, set, i);
2332 isl_basic_set_free(hull);
2337 /* Compute a superset of the convex hull of map that is described
2338 * by only translates of the constraints in the constituents of map.
2340 struct isl_basic_map *isl_map_simple_hull(struct isl_map *map)
2342 struct isl_set *set = NULL;
2343 struct isl_basic_map *model = NULL;
2344 struct isl_basic_map *hull;
2345 struct isl_basic_map *affine_hull;
2346 struct isl_basic_set *bset = NULL;
2351 hull = isl_basic_map_empty_like_map(map);
2356 hull = isl_basic_map_copy(map->p[0]);
2361 map = isl_map_detect_equalities(map);
2362 affine_hull = isl_map_affine_hull(isl_map_copy(map));
2363 map = isl_map_align_divs(map);
2364 model = isl_basic_map_copy(map->p[0]);
2366 set = isl_map_underlying_set(map);
2368 bset = uset_simple_hull(set);
2370 hull = isl_basic_map_overlying_set(bset, model);
2372 hull = isl_basic_map_intersect(hull, affine_hull);
2373 hull = isl_basic_map_convex_hull(hull);
2374 ISL_F_SET(hull, ISL_BASIC_MAP_NO_IMPLICIT);
2375 ISL_F_SET(hull, ISL_BASIC_MAP_ALL_EQUALITIES);
2380 struct isl_basic_set *isl_set_simple_hull(struct isl_set *set)
2382 return (struct isl_basic_set *)
2383 isl_map_simple_hull((struct isl_map *)set);
2386 /* Given a set "set", return parametric bounds on the dimension "dim".
2388 static struct isl_basic_set *set_bounds(struct isl_set *set, int dim)
2390 unsigned set_dim = isl_set_dim(set, isl_dim_set);
2391 set = isl_set_copy(set);
2392 set = isl_set_eliminate_dims(set, dim + 1, set_dim - (dim + 1));
2393 set = isl_set_eliminate_dims(set, 0, dim);
2394 return isl_set_convex_hull(set);
2397 /* Computes a "simple hull" and then check if each dimension in the
2398 * resulting hull is bounded by a symbolic constant. If not, the
2399 * hull is intersected with the corresponding bounds on the whole set.
2401 struct isl_basic_set *isl_set_bounded_simple_hull(struct isl_set *set)
2404 struct isl_basic_set *hull;
2405 unsigned nparam, left;
2406 int removed_divs = 0;
2408 hull = isl_set_simple_hull(isl_set_copy(set));
2412 nparam = isl_basic_set_dim(hull, isl_dim_param);
2413 for (i = 0; i < isl_basic_set_dim(hull, isl_dim_set); ++i) {
2414 int lower = 0, upper = 0;
2415 struct isl_basic_set *bounds;
2417 left = isl_basic_set_total_dim(hull) - nparam - i - 1;
2418 for (j = 0; j < hull->n_eq; ++j) {
2419 if (isl_int_is_zero(hull->eq[j][1 + nparam + i]))
2421 if (isl_seq_first_non_zero(hull->eq[j]+1+nparam+i+1,
2428 for (j = 0; j < hull->n_ineq; ++j) {
2429 if (isl_int_is_zero(hull->ineq[j][1 + nparam + i]))
2431 if (isl_seq_first_non_zero(hull->ineq[j]+1+nparam+i+1,
2433 isl_seq_first_non_zero(hull->ineq[j]+1+nparam,
2436 if (isl_int_is_pos(hull->ineq[j][1 + nparam + i]))
2447 if (!removed_divs) {
2448 set = isl_set_remove_divs(set);
2453 bounds = set_bounds(set, i);
2454 hull = isl_basic_set_intersect(hull, bounds);