/* * Copyright 2008-2009 Katholieke Universiteit Leuven * Copyright 2014 INRIA Rocquencourt * * Use of this software is governed by the MIT license * * Written by Sven Verdoolaege, K.U.Leuven, Departement * Computerwetenschappen, Celestijnenlaan 200A, B-3001 Leuven, Belgium * and Inria Paris - Rocquencourt, Domaine de Voluceau - Rocquencourt, * B.P. 105 - 78153 Le Chesnay, France */ #include #include #include #include #include #include #include #include #include #include "isl_equalities.h" #include "isl_tab.h" #include static struct isl_basic_set *uset_convex_hull_wrap_bounded(struct isl_set *set); /* Return 1 if constraint c is redundant with respect to the constraints * in bmap. If c is a lower [upper] bound in some variable and bmap * does not have a lower [upper] bound in that variable, then c cannot * be redundant and we do not need solve any lp. */ int isl_basic_map_constraint_is_redundant(struct isl_basic_map **bmap, isl_int *c, isl_int *opt_n, isl_int *opt_d) { enum isl_lp_result res; unsigned total; int i, j; if (!bmap) return -1; total = isl_basic_map_total_dim(*bmap); for (i = 0; i < total; ++i) { int sign; if (isl_int_is_zero(c[1+i])) continue; sign = isl_int_sgn(c[1+i]); for (j = 0; j < (*bmap)->n_ineq; ++j) if (sign == isl_int_sgn((*bmap)->ineq[j][1+i])) break; if (j == (*bmap)->n_ineq) break; } if (i < total) return 0; res = isl_basic_map_solve_lp(*bmap, 0, c, (*bmap)->ctx->one, opt_n, opt_d, NULL); if (res == isl_lp_unbounded) return 0; if (res == isl_lp_error) return -1; if (res == isl_lp_empty) { *bmap = isl_basic_map_set_to_empty(*bmap); return 0; } return !isl_int_is_neg(*opt_n); } int isl_basic_set_constraint_is_redundant(struct isl_basic_set **bset, isl_int *c, isl_int *opt_n, isl_int *opt_d) { return isl_basic_map_constraint_is_redundant( (struct isl_basic_map **)bset, c, opt_n, opt_d); } /* Remove redundant * constraints. If the minimal value along the normal of a constraint * is the same if the constraint is removed, then the constraint is redundant. * * Alternatively, we could have intersected the basic map with the * corresponding equality and the checked if the dimension was that * of a facet. */ __isl_give isl_basic_map *isl_basic_map_remove_redundancies( __isl_take isl_basic_map *bmap) { struct isl_tab *tab; if (!bmap) return NULL; bmap = isl_basic_map_gauss(bmap, NULL); if (ISL_F_ISSET(bmap, ISL_BASIC_MAP_EMPTY)) return bmap; if (ISL_F_ISSET(bmap, ISL_BASIC_MAP_NO_REDUNDANT)) return bmap; if (bmap->n_ineq <= 1) return bmap; tab = isl_tab_from_basic_map(bmap, 0); if (isl_tab_detect_implicit_equalities(tab) < 0) goto error; if (isl_tab_detect_redundant(tab) < 0) goto error; bmap = isl_basic_map_update_from_tab(bmap, tab); isl_tab_free(tab); ISL_F_SET(bmap, ISL_BASIC_MAP_NO_IMPLICIT); ISL_F_SET(bmap, ISL_BASIC_MAP_NO_REDUNDANT); return bmap; error: isl_tab_free(tab); isl_basic_map_free(bmap); return NULL; } __isl_give isl_basic_set *isl_basic_set_remove_redundancies( __isl_take isl_basic_set *bset) { return (struct isl_basic_set *) isl_basic_map_remove_redundancies((struct isl_basic_map *)bset); } /* Remove redundant constraints in each of the basic maps. */ __isl_give isl_map *isl_map_remove_redundancies(__isl_take isl_map *map) { return isl_map_inline_foreach_basic_map(map, &isl_basic_map_remove_redundancies); } __isl_give isl_set *isl_set_remove_redundancies(__isl_take isl_set *set) { return isl_map_remove_redundancies(set); } /* Check if the set set is bound in the direction of the affine * constraint c and if so, set the constant term such that the * resulting constraint is a bounding constraint for the set. */ static int uset_is_bound(struct isl_set *set, isl_int *c, unsigned len) { int first; int j; isl_int opt; isl_int opt_denom; isl_int_init(opt); isl_int_init(opt_denom); first = 1; for (j = 0; j < set->n; ++j) { enum isl_lp_result res; if (ISL_F_ISSET(set->p[j], ISL_BASIC_SET_EMPTY)) continue; res = isl_basic_set_solve_lp(set->p[j], 0, c, set->ctx->one, &opt, &opt_denom, NULL); if (res == isl_lp_unbounded) break; if (res == isl_lp_error) goto error; if (res == isl_lp_empty) { set->p[j] = isl_basic_set_set_to_empty(set->p[j]); if (!set->p[j]) goto error; continue; } if (first || isl_int_is_neg(opt)) { if (!isl_int_is_one(opt_denom)) isl_seq_scale(c, c, opt_denom, len); isl_int_sub(c[0], c[0], opt); } first = 0; } isl_int_clear(opt); isl_int_clear(opt_denom); return j >= set->n; error: isl_int_clear(opt); isl_int_clear(opt_denom); return -1; } __isl_give isl_basic_map *isl_basic_map_set_rational( __isl_take isl_basic_set *bmap) { if (!bmap) return NULL; if (ISL_F_ISSET(bmap, ISL_BASIC_MAP_RATIONAL)) return bmap; bmap = isl_basic_map_cow(bmap); if (!bmap) return NULL; ISL_F_SET(bmap, ISL_BASIC_MAP_RATIONAL); return isl_basic_map_finalize(bmap); } __isl_give isl_basic_set *isl_basic_set_set_rational( __isl_take isl_basic_set *bset) { return isl_basic_map_set_rational(bset); } __isl_give isl_map *isl_map_set_rational(__isl_take isl_map *map) { int i; map = isl_map_cow(map); if (!map) return NULL; for (i = 0; i < map->n; ++i) { map->p[i] = isl_basic_map_set_rational(map->p[i]); if (!map->p[i]) goto error; } return map; error: isl_map_free(map); return NULL; } __isl_give isl_set *isl_set_set_rational(__isl_take isl_set *set) { return isl_map_set_rational(set); } static struct isl_basic_set *isl_basic_set_add_equality( struct isl_basic_set *bset, isl_int *c) { int i; unsigned dim; if (!bset) return NULL; if (ISL_F_ISSET(bset, ISL_BASIC_SET_EMPTY)) return bset; isl_assert(bset->ctx, isl_basic_set_n_param(bset) == 0, goto error); isl_assert(bset->ctx, bset->n_div == 0, goto error); dim = isl_basic_set_n_dim(bset); bset = isl_basic_set_cow(bset); bset = isl_basic_set_extend(bset, 0, dim, 0, 1, 0); i = isl_basic_set_alloc_equality(bset); if (i < 0) goto error; isl_seq_cpy(bset->eq[i], c, 1 + dim); return bset; error: isl_basic_set_free(bset); return NULL; } static struct isl_set *isl_set_add_basic_set_equality(struct isl_set *set, isl_int *c) { int i; set = isl_set_cow(set); if (!set) return NULL; for (i = 0; i < set->n; ++i) { set->p[i] = isl_basic_set_add_equality(set->p[i], c); if (!set->p[i]) goto error; } return set; error: isl_set_free(set); return NULL; } /* Given a union of basic sets, construct the constraints for wrapping * a facet around one of its ridges. * In particular, if each of n the d-dimensional basic sets i in "set" * contains the origin, satisfies the constraints x_1 >= 0 and x_2 >= 0 * and is defined by the constraints * [ 1 ] * A_i [ x ] >= 0 * * then the resulting set is of dimension n*(1+d) and has as constraints * * [ a_i ] * A_i [ x_i ] >= 0 * * a_i >= 0 * * \sum_i x_{i,1} = 1 */ static struct isl_basic_set *wrap_constraints(struct isl_set *set) { struct isl_basic_set *lp; unsigned n_eq; unsigned n_ineq; int i, j, k; unsigned dim, lp_dim; if (!set) return NULL; dim = 1 + isl_set_n_dim(set); n_eq = 1; n_ineq = set->n; for (i = 0; i < set->n; ++i) { n_eq += set->p[i]->n_eq; n_ineq += set->p[i]->n_ineq; } lp = isl_basic_set_alloc(set->ctx, 0, dim * set->n, 0, n_eq, n_ineq); lp = isl_basic_set_set_rational(lp); if (!lp) return NULL; lp_dim = isl_basic_set_n_dim(lp); k = isl_basic_set_alloc_equality(lp); isl_int_set_si(lp->eq[k][0], -1); for (i = 0; i < set->n; ++i) { isl_int_set_si(lp->eq[k][1+dim*i], 0); isl_int_set_si(lp->eq[k][1+dim*i+1], 1); isl_seq_clr(lp->eq[k]+1+dim*i+2, dim-2); } for (i = 0; i < set->n; ++i) { k = isl_basic_set_alloc_inequality(lp); isl_seq_clr(lp->ineq[k], 1+lp_dim); isl_int_set_si(lp->ineq[k][1+dim*i], 1); for (j = 0; j < set->p[i]->n_eq; ++j) { k = isl_basic_set_alloc_equality(lp); isl_seq_clr(lp->eq[k], 1+dim*i); isl_seq_cpy(lp->eq[k]+1+dim*i, set->p[i]->eq[j], dim); isl_seq_clr(lp->eq[k]+1+dim*(i+1), dim*(set->n-i-1)); } for (j = 0; j < set->p[i]->n_ineq; ++j) { k = isl_basic_set_alloc_inequality(lp); isl_seq_clr(lp->ineq[k], 1+dim*i); isl_seq_cpy(lp->ineq[k]+1+dim*i, set->p[i]->ineq[j], dim); isl_seq_clr(lp->ineq[k]+1+dim*(i+1), dim*(set->n-i-1)); } } return lp; } /* Given a facet "facet" of the convex hull of "set" and a facet "ridge" * of that facet, compute the other facet of the convex hull that contains * the ridge. * * We first transform the set such that the facet constraint becomes * * x_1 >= 0 * * I.e., the facet lies in * * x_1 = 0 * * and on that facet, the constraint that defines the ridge is * * x_2 >= 0 * * (This transformation is not strictly needed, all that is needed is * that the ridge contains the origin.) * * Since the ridge contains the origin, the cone of the convex hull * will be of the form * * x_1 >= 0 * x_2 >= a x_1 * * with this second constraint defining the new facet. * The constant a is obtained by settting x_1 in the cone of the * convex hull to 1 and minimizing x_2. * Now, each element in the cone of the convex hull is the sum * of elements in the cones of the basic sets. * If a_i is the dilation factor of basic set i, then the problem * we need to solve is * * min \sum_i x_{i,2} * st * \sum_i x_{i,1} = 1 * a_i >= 0 * [ a_i ] * A [ x_i ] >= 0 * * with * [ 1 ] * A_i [ x_i ] >= 0 * * the constraints of each (transformed) basic set. * If a = n/d, then the constraint defining the new facet (in the transformed * space) is * * -n x_1 + d x_2 >= 0 * * In the original space, we need to take the same combination of the * corresponding constraints "facet" and "ridge". * * If a = -infty = "-1/0", then we just return the original facet constraint. * This means that the facet is unbounded, but has a bounded intersection * with the union of sets. */ isl_int *isl_set_wrap_facet(__isl_keep isl_set *set, isl_int *facet, isl_int *ridge) { int i; isl_ctx *ctx; struct isl_mat *T = NULL; struct isl_basic_set *lp = NULL; struct isl_vec *obj; enum isl_lp_result res; isl_int num, den; unsigned dim; if (!set) return NULL; ctx = set->ctx; set = isl_set_copy(set); set = isl_set_set_rational(set); dim = 1 + isl_set_n_dim(set); T = isl_mat_alloc(ctx, 3, dim); if (!T) goto error; isl_int_set_si(T->row[0][0], 1); isl_seq_clr(T->row[0]+1, dim - 1); isl_seq_cpy(T->row[1], facet, dim); isl_seq_cpy(T->row[2], ridge, dim); T = isl_mat_right_inverse(T); set = isl_set_preimage(set, T); T = NULL; if (!set) goto error; lp = wrap_constraints(set); obj = isl_vec_alloc(ctx, 1 + dim*set->n); if (!obj) goto error; isl_int_set_si(obj->block.data[0], 0); for (i = 0; i < set->n; ++i) { isl_seq_clr(obj->block.data + 1 + dim*i, 2); isl_int_set_si(obj->block.data[1 + dim*i+2], 1); isl_seq_clr(obj->block.data + 1 + dim*i+3, dim-3); } isl_int_init(num); isl_int_init(den); res = isl_basic_set_solve_lp(lp, 0, obj->block.data, ctx->one, &num, &den, NULL); if (res == isl_lp_ok) { isl_int_neg(num, num); isl_seq_combine(facet, num, facet, den, ridge, dim); isl_seq_normalize(ctx, facet, dim); } isl_int_clear(num); isl_int_clear(den); isl_vec_free(obj); isl_basic_set_free(lp); isl_set_free(set); if (res == isl_lp_error) return NULL; isl_assert(ctx, res == isl_lp_ok || res == isl_lp_unbounded, return NULL); return facet; error: isl_basic_set_free(lp); isl_mat_free(T); isl_set_free(set); return NULL; } /* Compute the constraint of a facet of "set". * * We first compute the intersection with a bounding constraint * that is orthogonal to one of the coordinate axes. * If the affine hull of this intersection has only one equality, * we have found a facet. * Otherwise, we wrap the current bounding constraint around * one of the equalities of the face (one that is not equal to * the current bounding constraint). * This process continues until we have found a facet. * The dimension of the intersection increases by at least * one on each iteration, so termination is guaranteed. */ static __isl_give isl_mat *initial_facet_constraint(__isl_keep isl_set *set) { struct isl_set *slice = NULL; struct isl_basic_set *face = NULL; int i; unsigned dim = isl_set_n_dim(set); int is_bound; isl_mat *bounds = NULL; isl_assert(set->ctx, set->n > 0, goto error); bounds = isl_mat_alloc(set->ctx, 1, 1 + dim); if (!bounds) return NULL; isl_seq_clr(bounds->row[0], dim); isl_int_set_si(bounds->row[0][1 + dim - 1], 1); is_bound = uset_is_bound(set, bounds->row[0], 1 + dim); if (is_bound < 0) goto error; isl_assert(set->ctx, is_bound, goto error); isl_seq_normalize(set->ctx, bounds->row[0], 1 + dim); bounds->n_row = 1; for (;;) { slice = isl_set_copy(set); slice = isl_set_add_basic_set_equality(slice, bounds->row[0]); face = isl_set_affine_hull(slice); if (!face) goto error; if (face->n_eq == 1) { isl_basic_set_free(face); break; } for (i = 0; i < face->n_eq; ++i) if (!isl_seq_eq(bounds->row[0], face->eq[i], 1 + dim) && !isl_seq_is_neg(bounds->row[0], face->eq[i], 1 + dim)) break; isl_assert(set->ctx, i < face->n_eq, goto error); if (!isl_set_wrap_facet(set, bounds->row[0], face->eq[i])) goto error; isl_seq_normalize(set->ctx, bounds->row[0], bounds->n_col); isl_basic_set_free(face); } return bounds; error: isl_basic_set_free(face); isl_mat_free(bounds); return NULL; } /* Given the bounding constraint "c" of a facet of the convex hull of "set", * compute a hyperplane description of the facet, i.e., compute the facets * of the facet. * * We compute an affine transformation that transforms the constraint * * [ 1 ] * c [ x ] = 0 * * to the constraint * * z_1 = 0 * * by computing the right inverse U of a matrix that starts with the rows * * [ 1 0 ] * [ c ] * * Then * [ 1 ] [ 1 ] * [ x ] = U [ z ] * and * [ 1 ] [ 1 ] * [ z ] = Q [ x ] * * with Q = U^{-1} * Since z_1 is zero, we can drop this variable as well as the corresponding * column of U to obtain * * [ 1 ] [ 1 ] * [ x ] = U' [ z' ] * and * [ 1 ] [ 1 ] * [ z' ] = Q' [ x ] * * with Q' equal to Q, but without the corresponding row. * After computing the facets of the facet in the z' space, * we convert them back to the x space through Q. */ static struct isl_basic_set *compute_facet(struct isl_set *set, isl_int *c) { struct isl_mat *m, *U, *Q; struct isl_basic_set *facet = NULL; struct isl_ctx *ctx; unsigned dim; ctx = set->ctx; set = isl_set_copy(set); dim = isl_set_n_dim(set); m = isl_mat_alloc(set->ctx, 2, 1 + dim); if (!m) goto error; isl_int_set_si(m->row[0][0], 1); isl_seq_clr(m->row[0]+1, dim); isl_seq_cpy(m->row[1], c, 1+dim); U = isl_mat_right_inverse(m); Q = isl_mat_right_inverse(isl_mat_copy(U)); U = isl_mat_drop_cols(U, 1, 1); Q = isl_mat_drop_rows(Q, 1, 1); set = isl_set_preimage(set, U); facet = uset_convex_hull_wrap_bounded(set); facet = isl_basic_set_preimage(facet, Q); if (facet && facet->n_eq != 0) isl_die(ctx, isl_error_internal, "unexpected equality", return isl_basic_set_free(facet)); return facet; error: isl_basic_set_free(facet); isl_set_free(set); return NULL; } /* Given an initial facet constraint, compute the remaining facets. * We do this by running through all facets found so far and computing * the adjacent facets through wrapping, adding those facets that we * hadn't already found before. * * For each facet we have found so far, we first compute its facets * in the resulting convex hull. That is, we compute the ridges * of the resulting convex hull contained in the facet. * We also compute the corresponding facet in the current approximation * of the convex hull. There is no need to wrap around the ridges * in this facet since that would result in a facet that is already * present in the current approximation. * * This function can still be significantly optimized by checking which of * the facets of the basic sets are also facets of the convex hull and * using all the facets so far to help in constructing the facets of the * facets * and/or * using the technique in section "3.1 Ridge Generation" of * "Extended Convex Hull" by Fukuda et al. */ static struct isl_basic_set *extend(struct isl_basic_set *hull, struct isl_set *set) { int i, j, f; int k; struct isl_basic_set *facet = NULL; struct isl_basic_set *hull_facet = NULL; unsigned dim; if (!hull) return NULL; isl_assert(set->ctx, set->n > 0, goto error); dim = isl_set_n_dim(set); for (i = 0; i < hull->n_ineq; ++i) { facet = compute_facet(set, hull->ineq[i]); facet = isl_basic_set_add_equality(facet, hull->ineq[i]); facet = isl_basic_set_gauss(facet, NULL); facet = isl_basic_set_normalize_constraints(facet); hull_facet = isl_basic_set_copy(hull); hull_facet = isl_basic_set_add_equality(hull_facet, hull->ineq[i]); hull_facet = isl_basic_set_gauss(hull_facet, NULL); hull_facet = isl_basic_set_normalize_constraints(hull_facet); if (!facet || !hull_facet) goto error; hull = isl_basic_set_cow(hull); hull = isl_basic_set_extend_space(hull, isl_space_copy(hull->dim), 0, 0, facet->n_ineq); if (!hull) goto error; for (j = 0; j < facet->n_ineq; ++j) { for (f = 0; f < hull_facet->n_ineq; ++f) if (isl_seq_eq(facet->ineq[j], hull_facet->ineq[f], 1 + dim)) break; if (f < hull_facet->n_ineq) continue; k = isl_basic_set_alloc_inequality(hull); if (k < 0) goto error; isl_seq_cpy(hull->ineq[k], hull->ineq[i], 1+dim); if (!isl_set_wrap_facet(set, hull->ineq[k], facet->ineq[j])) goto error; } isl_basic_set_free(hull_facet); isl_basic_set_free(facet); } hull = isl_basic_set_simplify(hull); hull = isl_basic_set_finalize(hull); return hull; error: isl_basic_set_free(hull_facet); isl_basic_set_free(facet); isl_basic_set_free(hull); return NULL; } /* Special case for computing the convex hull of a one dimensional set. * We simply collect the lower and upper bounds of each basic set * and the biggest of those. */ static struct isl_basic_set *convex_hull_1d(struct isl_set *set) { struct isl_mat *c = NULL; isl_int *lower = NULL; isl_int *upper = NULL; int i, j, k; isl_int a, b; struct isl_basic_set *hull; for (i = 0; i < set->n; ++i) { set->p[i] = isl_basic_set_simplify(set->p[i]); if (!set->p[i]) goto error; } set = isl_set_remove_empty_parts(set); if (!set) goto error; isl_assert(set->ctx, set->n > 0, goto error); c = isl_mat_alloc(set->ctx, 2, 2); if (!c) goto error; if (set->p[0]->n_eq > 0) { isl_assert(set->ctx, set->p[0]->n_eq == 1, goto error); lower = c->row[0]; upper = c->row[1]; if (isl_int_is_pos(set->p[0]->eq[0][1])) { isl_seq_cpy(lower, set->p[0]->eq[0], 2); isl_seq_neg(upper, set->p[0]->eq[0], 2); } else { isl_seq_neg(lower, set->p[0]->eq[0], 2); isl_seq_cpy(upper, set->p[0]->eq[0], 2); } } else { for (j = 0; j < set->p[0]->n_ineq; ++j) { if (isl_int_is_pos(set->p[0]->ineq[j][1])) { lower = c->row[0]; isl_seq_cpy(lower, set->p[0]->ineq[j], 2); } else { upper = c->row[1]; isl_seq_cpy(upper, set->p[0]->ineq[j], 2); } } } isl_int_init(a); isl_int_init(b); for (i = 0; i < set->n; ++i) { struct isl_basic_set *bset = set->p[i]; int has_lower = 0; int has_upper = 0; for (j = 0; j < bset->n_eq; ++j) { has_lower = 1; has_upper = 1; if (lower) { isl_int_mul(a, lower[0], bset->eq[j][1]); isl_int_mul(b, lower[1], bset->eq[j][0]); if (isl_int_lt(a, b) && isl_int_is_pos(bset->eq[j][1])) isl_seq_cpy(lower, bset->eq[j], 2); if (isl_int_gt(a, b) && isl_int_is_neg(bset->eq[j][1])) isl_seq_neg(lower, bset->eq[j], 2); } if (upper) { isl_int_mul(a, upper[0], bset->eq[j][1]); isl_int_mul(b, upper[1], bset->eq[j][0]); if (isl_int_lt(a, b) && isl_int_is_pos(bset->eq[j][1])) isl_seq_neg(upper, bset->eq[j], 2); if (isl_int_gt(a, b) && isl_int_is_neg(bset->eq[j][1])) isl_seq_cpy(upper, bset->eq[j], 2); } } for (j = 0; j < bset->n_ineq; ++j) { if (isl_int_is_pos(bset->ineq[j][1])) has_lower = 1; if (isl_int_is_neg(bset->ineq[j][1])) has_upper = 1; if (lower && isl_int_is_pos(bset->ineq[j][1])) { isl_int_mul(a, lower[0], bset->ineq[j][1]); isl_int_mul(b, lower[1], bset->ineq[j][0]); if (isl_int_lt(a, b)) isl_seq_cpy(lower, bset->ineq[j], 2); } if (upper && isl_int_is_neg(bset->ineq[j][1])) { isl_int_mul(a, upper[0], bset->ineq[j][1]); isl_int_mul(b, upper[1], bset->ineq[j][0]); if (isl_int_gt(a, b)) isl_seq_cpy(upper, bset->ineq[j], 2); } } if (!has_lower) lower = NULL; if (!has_upper) upper = NULL; } isl_int_clear(a); isl_int_clear(b); hull = isl_basic_set_alloc(set->ctx, 0, 1, 0, 0, 2); hull = isl_basic_set_set_rational(hull); if (!hull) goto error; if (lower) { k = isl_basic_set_alloc_inequality(hull); isl_seq_cpy(hull->ineq[k], lower, 2); } if (upper) { k = isl_basic_set_alloc_inequality(hull); isl_seq_cpy(hull->ineq[k], upper, 2); } hull = isl_basic_set_finalize(hull); isl_set_free(set); isl_mat_free(c); return hull; error: isl_set_free(set); isl_mat_free(c); return NULL; } static struct isl_basic_set *convex_hull_0d(struct isl_set *set) { struct isl_basic_set *convex_hull; if (!set) return NULL; if (isl_set_is_empty(set)) convex_hull = isl_basic_set_empty(isl_space_copy(set->dim)); else convex_hull = isl_basic_set_universe(isl_space_copy(set->dim)); isl_set_free(set); return convex_hull; } /* Compute the convex hull of a pair of basic sets without any parameters or * integer divisions using Fourier-Motzkin elimination. * The convex hull is the set of all points that can be written as * the sum of points from both basic sets (in homogeneous coordinates). * We set up the constraints in a space with dimensions for each of * the three sets and then project out the dimensions corresponding * to the two original basic sets, retaining only those corresponding * to the convex hull. */ static struct isl_basic_set *convex_hull_pair_elim(struct isl_basic_set *bset1, struct isl_basic_set *bset2) { int i, j, k; struct isl_basic_set *bset[2]; struct isl_basic_set *hull = NULL; unsigned dim; if (!bset1 || !bset2) goto error; dim = isl_basic_set_n_dim(bset1); hull = isl_basic_set_alloc(bset1->ctx, 0, 2 + 3 * dim, 0, 1 + dim + bset1->n_eq + bset2->n_eq, 2 + bset1->n_ineq + bset2->n_ineq); bset[0] = bset1; bset[1] = bset2; for (i = 0; i < 2; ++i) { for (j = 0; j < bset[i]->n_eq; ++j) { k = isl_basic_set_alloc_equality(hull); if (k < 0) goto error; isl_seq_clr(hull->eq[k], (i+1) * (1+dim)); isl_seq_clr(hull->eq[k]+(i+2)*(1+dim), (1-i)*(1+dim)); isl_seq_cpy(hull->eq[k]+(i+1)*(1+dim), bset[i]->eq[j], 1+dim); } for (j = 0; j < bset[i]->n_ineq; ++j) { k = isl_basic_set_alloc_inequality(hull); if (k < 0) goto error; isl_seq_clr(hull->ineq[k], (i+1) * (1+dim)); isl_seq_clr(hull->ineq[k]+(i+2)*(1+dim), (1-i)*(1+dim)); isl_seq_cpy(hull->ineq[k]+(i+1)*(1+dim), bset[i]->ineq[j], 1+dim); } k = isl_basic_set_alloc_inequality(hull); if (k < 0) goto error; isl_seq_clr(hull->ineq[k], 1+2+3*dim); isl_int_set_si(hull->ineq[k][(i+1)*(1+dim)], 1); } for (j = 0; j < 1+dim; ++j) { k = isl_basic_set_alloc_equality(hull); if (k < 0) goto error; isl_seq_clr(hull->eq[k], 1+2+3*dim); isl_int_set_si(hull->eq[k][j], -1); isl_int_set_si(hull->eq[k][1+dim+j], 1); isl_int_set_si(hull->eq[k][2*(1+dim)+j], 1); } hull = isl_basic_set_set_rational(hull); hull = isl_basic_set_remove_dims(hull, isl_dim_set, dim, 2*(1+dim)); hull = isl_basic_set_remove_redundancies(hull); isl_basic_set_free(bset1); isl_basic_set_free(bset2); return hull; error: isl_basic_set_free(bset1); isl_basic_set_free(bset2); isl_basic_set_free(hull); return NULL; } /* Is the set bounded for each value of the parameters? */ int isl_basic_set_is_bounded(__isl_keep isl_basic_set *bset) { struct isl_tab *tab; int bounded; if (!bset) return -1; if (isl_basic_set_plain_is_empty(bset)) return 1; tab = isl_tab_from_recession_cone(bset, 1); bounded = isl_tab_cone_is_bounded(tab); isl_tab_free(tab); return bounded; } /* Is the image bounded for each value of the parameters and * the domain variables? */ int isl_basic_map_image_is_bounded(__isl_keep isl_basic_map *bmap) { unsigned nparam = isl_basic_map_dim(bmap, isl_dim_param); unsigned n_in = isl_basic_map_dim(bmap, isl_dim_in); int bounded; bmap = isl_basic_map_copy(bmap); bmap = isl_basic_map_cow(bmap); bmap = isl_basic_map_move_dims(bmap, isl_dim_param, nparam, isl_dim_in, 0, n_in); bounded = isl_basic_set_is_bounded((isl_basic_set *)bmap); isl_basic_map_free(bmap); return bounded; } /* Is the set bounded for each value of the parameters? */ int isl_set_is_bounded(__isl_keep isl_set *set) { int i; if (!set) return -1; for (i = 0; i < set->n; ++i) { int bounded = isl_basic_set_is_bounded(set->p[i]); if (!bounded || bounded < 0) return bounded; } return 1; } /* Compute the lineality space of the convex hull of bset1 and bset2. * * We first compute the intersection of the recession cone of bset1 * with the negative of the recession cone of bset2 and then compute * the linear hull of the resulting cone. */ static struct isl_basic_set *induced_lineality_space( struct isl_basic_set *bset1, struct isl_basic_set *bset2) { int i, k; struct isl_basic_set *lin = NULL; unsigned dim; if (!bset1 || !bset2) goto error; dim = isl_basic_set_total_dim(bset1); lin = isl_basic_set_alloc_space(isl_basic_set_get_space(bset1), 0, bset1->n_eq + bset2->n_eq, bset1->n_ineq + bset2->n_ineq); lin = isl_basic_set_set_rational(lin); if (!lin) goto error; for (i = 0; i < bset1->n_eq; ++i) { k = isl_basic_set_alloc_equality(lin); if (k < 0) goto error; isl_int_set_si(lin->eq[k][0], 0); isl_seq_cpy(lin->eq[k] + 1, bset1->eq[i] + 1, dim); } for (i = 0; i < bset1->n_ineq; ++i) { k = isl_basic_set_alloc_inequality(lin); if (k < 0) goto error; isl_int_set_si(lin->ineq[k][0], 0); isl_seq_cpy(lin->ineq[k] + 1, bset1->ineq[i] + 1, dim); } for (i = 0; i < bset2->n_eq; ++i) { k = isl_basic_set_alloc_equality(lin); if (k < 0) goto error; isl_int_set_si(lin->eq[k][0], 0); isl_seq_neg(lin->eq[k] + 1, bset2->eq[i] + 1, dim); } for (i = 0; i < bset2->n_ineq; ++i) { k = isl_basic_set_alloc_inequality(lin); if (k < 0) goto error; isl_int_set_si(lin->ineq[k][0], 0); isl_seq_neg(lin->ineq[k] + 1, bset2->ineq[i] + 1, dim); } isl_basic_set_free(bset1); isl_basic_set_free(bset2); return isl_basic_set_affine_hull(lin); error: isl_basic_set_free(lin); isl_basic_set_free(bset1); isl_basic_set_free(bset2); return NULL; } static struct isl_basic_set *uset_convex_hull(struct isl_set *set); /* Given a set and a linear space "lin" of dimension n > 0, * project the linear space from the set, compute the convex hull * and then map the set back to the original space. * * Let * * M x = 0 * * describe the linear space. We first compute the Hermite normal * form H = M U of M = H Q, to obtain * * H Q x = 0 * * The last n rows of H will be zero, so the last n variables of x' = Q x * are the one we want to project out. We do this by transforming each * basic set A x >= b to A U x' >= b and then removing the last n dimensions. * After computing the convex hull in x'_1, i.e., A' x'_1 >= b', * we transform the hull back to the original space as A' Q_1 x >= b', * with Q_1 all but the last n rows of Q. */ static struct isl_basic_set *modulo_lineality(struct isl_set *set, struct isl_basic_set *lin) { unsigned total = isl_basic_set_total_dim(lin); unsigned lin_dim; struct isl_basic_set *hull; struct isl_mat *M, *U, *Q; if (!set || !lin) goto error; lin_dim = total - lin->n_eq; M = isl_mat_sub_alloc6(set->ctx, lin->eq, 0, lin->n_eq, 1, total); M = isl_mat_left_hermite(M, 0, &U, &Q); if (!M) goto error; isl_mat_free(M); isl_basic_set_free(lin); Q = isl_mat_drop_rows(Q, Q->n_row - lin_dim, lin_dim); U = isl_mat_lin_to_aff(U); Q = isl_mat_lin_to_aff(Q); set = isl_set_preimage(set, U); set = isl_set_remove_dims(set, isl_dim_set, total - lin_dim, lin_dim); hull = uset_convex_hull(set); hull = isl_basic_set_preimage(hull, Q); return hull; error: isl_basic_set_free(lin); isl_set_free(set); return NULL; } /* Given two polyhedra with as constraints h_{ij} x >= 0 in homegeneous space, * set up an LP for solving * * \sum_j \alpha_{1j} h_{1j} = \sum_j \alpha_{2j} h_{2j} * * \alpha{i0} corresponds to the (implicit) positivity constraint 1 >= 0 * The next \alpha{ij} correspond to the equalities and come in pairs. * The final \alpha{ij} correspond to the inequalities. */ static struct isl_basic_set *valid_direction_lp( struct isl_basic_set *bset1, struct isl_basic_set *bset2) { isl_space *dim; struct isl_basic_set *lp; unsigned d; int n; int i, j, k; if (!bset1 || !bset2) goto error; d = 1 + isl_basic_set_total_dim(bset1); n = 2 + 2 * bset1->n_eq + bset1->n_ineq + 2 * bset2->n_eq + bset2->n_ineq; dim = isl_space_set_alloc(bset1->ctx, 0, n); lp = isl_basic_set_alloc_space(dim, 0, d, n); if (!lp) goto error; for (i = 0; i < n; ++i) { k = isl_basic_set_alloc_inequality(lp); if (k < 0) goto error; isl_seq_clr(lp->ineq[k] + 1, n); isl_int_set_si(lp->ineq[k][0], -1); isl_int_set_si(lp->ineq[k][1 + i], 1); } for (i = 0; i < d; ++i) { k = isl_basic_set_alloc_equality(lp); if (k < 0) goto error; n = 0; isl_int_set_si(lp->eq[k][n], 0); n++; /* positivity constraint 1 >= 0 */ isl_int_set_si(lp->eq[k][n], i == 0); n++; for (j = 0; j < bset1->n_eq; ++j) { isl_int_set(lp->eq[k][n], bset1->eq[j][i]); n++; isl_int_neg(lp->eq[k][n], bset1->eq[j][i]); n++; } for (j = 0; j < bset1->n_ineq; ++j) { isl_int_set(lp->eq[k][n], bset1->ineq[j][i]); n++; } /* positivity constraint 1 >= 0 */ isl_int_set_si(lp->eq[k][n], -(i == 0)); n++; for (j = 0; j < bset2->n_eq; ++j) { isl_int_neg(lp->eq[k][n], bset2->eq[j][i]); n++; isl_int_set(lp->eq[k][n], bset2->eq[j][i]); n++; } for (j = 0; j < bset2->n_ineq; ++j) { isl_int_neg(lp->eq[k][n], bset2->ineq[j][i]); n++; } } lp = isl_basic_set_gauss(lp, NULL); isl_basic_set_free(bset1); isl_basic_set_free(bset2); return lp; error: isl_basic_set_free(bset1); isl_basic_set_free(bset2); return NULL; } /* Compute a vector s in the homogeneous space such that > 0 * for all rays in the homogeneous space of the two cones that correspond * to the input polyhedra bset1 and bset2. * * We compute s as a vector that satisfies * * s = \sum_j \alpha_{ij} h_{ij} for i = 1,2 (*) * * with h_{ij} the normals of the facets of polyhedron i * (including the "positivity constraint" 1 >= 0) and \alpha_{ij} * strictly positive numbers. For simplicity we impose \alpha_{ij} >= 1. * We first set up an LP with as variables the \alpha{ij}. * In this formulation, for each polyhedron i, * the first constraint is the positivity constraint, followed by pairs * of variables for the equalities, followed by variables for the inequalities. * We then simply pick a feasible solution and compute s using (*). * * Note that we simply pick any valid direction and make no attempt * to pick a "good" or even the "best" valid direction. */ static struct isl_vec *valid_direction( struct isl_basic_set *bset1, struct isl_basic_set *bset2) { struct isl_basic_set *lp; struct isl_tab *tab; struct isl_vec *sample = NULL; struct isl_vec *dir; unsigned d; int i; int n; if (!bset1 || !bset2) goto error; lp = valid_direction_lp(isl_basic_set_copy(bset1), isl_basic_set_copy(bset2)); tab = isl_tab_from_basic_set(lp, 0); sample = isl_tab_get_sample_value(tab); isl_tab_free(tab); isl_basic_set_free(lp); if (!sample) goto error; d = isl_basic_set_total_dim(bset1); dir = isl_vec_alloc(bset1->ctx, 1 + d); if (!dir) goto error; isl_seq_clr(dir->block.data + 1, dir->size - 1); n = 1; /* positivity constraint 1 >= 0 */ isl_int_set(dir->block.data[0], sample->block.data[n]); n++; for (i = 0; i < bset1->n_eq; ++i) { isl_int_sub(sample->block.data[n], sample->block.data[n], sample->block.data[n+1]); isl_seq_combine(dir->block.data, bset1->ctx->one, dir->block.data, sample->block.data[n], bset1->eq[i], 1 + d); n += 2; } for (i = 0; i < bset1->n_ineq; ++i) isl_seq_combine(dir->block.data, bset1->ctx->one, dir->block.data, sample->block.data[n++], bset1->ineq[i], 1 + d); isl_vec_free(sample); isl_seq_normalize(bset1->ctx, dir->el, dir->size); isl_basic_set_free(bset1); isl_basic_set_free(bset2); return dir; error: isl_vec_free(sample); isl_basic_set_free(bset1); isl_basic_set_free(bset2); return NULL; } /* Given a polyhedron b_i + A_i x >= 0 and a map T = S^{-1}, * compute b_i' + A_i' x' >= 0, with * * [ b_i A_i ] [ y' ] [ y' ] * [ 1 0 ] S^{-1} [ x' ] >= 0 or [ b_i' A_i' ] [ x' ] >= 0 * * In particular, add the "positivity constraint" and then perform * the mapping. */ static struct isl_basic_set *homogeneous_map(struct isl_basic_set *bset, struct isl_mat *T) { int k; if (!bset) goto error; bset = isl_basic_set_extend_constraints(bset, 0, 1); k = isl_basic_set_alloc_inequality(bset); if (k < 0) goto error; isl_seq_clr(bset->ineq[k] + 1, isl_basic_set_total_dim(bset)); isl_int_set_si(bset->ineq[k][0], 1); bset = isl_basic_set_preimage(bset, T); return bset; error: isl_mat_free(T); isl_basic_set_free(bset); return NULL; } /* Compute the convex hull of a pair of basic sets without any parameters or * integer divisions, where the convex hull is known to be pointed, * but the basic sets may be unbounded. * * We turn this problem into the computation of a convex hull of a pair * _bounded_ polyhedra by "changing the direction of the homogeneous * dimension". This idea is due to Matthias Koeppe. * * Consider the cones in homogeneous space that correspond to the * input polyhedra. The rays of these cones are also rays of the * polyhedra if the coordinate that corresponds to the homogeneous * dimension is zero. That is, if the inner product of the rays * with the homogeneous direction is zero. * The cones in the homogeneous space can also be considered to * correspond to other pairs of polyhedra by chosing a different * homogeneous direction. To ensure that both of these polyhedra * are bounded, we need to make sure that all rays of the cones * correspond to vertices and not to rays. * Let s be a direction such that > 0 for all rays r of both cones. * Then using s as a homogeneous direction, we obtain a pair of polytopes. * The vector s is computed in valid_direction. * * Note that we need to consider _all_ rays of the cones and not just * the rays that correspond to rays in the polyhedra. If we were to * only consider those rays and turn them into vertices, then we * may inadvertently turn some vertices into rays. * * The standard homogeneous direction is the unit vector in the 0th coordinate. * We therefore transform the two polyhedra such that the selected * direction is mapped onto this standard direction and then proceed * with the normal computation. * Let S be a non-singular square matrix with s as its first row, * then we want to map the polyhedra to the space * * [ y' ] [ y ] [ y ] [ y' ] * [ x' ] = S [ x ] i.e., [ x ] = S^{-1} [ x' ] * * We take S to be the unimodular completion of s to limit the growth * of the coefficients in the following computations. * * Let b_i + A_i x >= 0 be the constraints of polyhedron i. * We first move to the homogeneous dimension * * b_i y + A_i x >= 0 [ b_i A_i ] [ y ] [ 0 ] * y >= 0 or [ 1 0 ] [ x ] >= [ 0 ] * * Then we change directoin * * [ b_i A_i ] [ y' ] [ y' ] * [ 1 0 ] S^{-1} [ x' ] >= 0 or [ b_i' A_i' ] [ x' ] >= 0 * * Then we compute the convex hull of the polytopes b_i' + A_i' x' >= 0 * resulting in b' + A' x' >= 0, which we then convert back * * [ y ] [ y ] * [ b' A' ] S [ x ] >= 0 or [ b A ] [ x ] >= 0 * * The polyhedron b + A x >= 0 is then the convex hull of the input polyhedra. */ static struct isl_basic_set *convex_hull_pair_pointed( struct isl_basic_set *bset1, struct isl_basic_set *bset2) { struct isl_ctx *ctx = NULL; struct isl_vec *dir = NULL; struct isl_mat *T = NULL; struct isl_mat *T2 = NULL; struct isl_basic_set *hull; struct isl_set *set; if (!bset1 || !bset2) goto error; ctx = isl_basic_set_get_ctx(bset1); dir = valid_direction(isl_basic_set_copy(bset1), isl_basic_set_copy(bset2)); if (!dir) goto error; T = isl_mat_alloc(ctx, dir->size, dir->size); if (!T) goto error; isl_seq_cpy(T->row[0], dir->block.data, dir->size); T = isl_mat_unimodular_complete(T, 1); T2 = isl_mat_right_inverse(isl_mat_copy(T)); bset1 = homogeneous_map(bset1, isl_mat_copy(T2)); bset2 = homogeneous_map(bset2, T2); set = isl_set_alloc_space(isl_basic_set_get_space(bset1), 2, 0); set = isl_set_add_basic_set(set, bset1); set = isl_set_add_basic_set(set, bset2); hull = uset_convex_hull(set); hull = isl_basic_set_preimage(hull, T); isl_vec_free(dir); return hull; error: isl_vec_free(dir); isl_basic_set_free(bset1); isl_basic_set_free(bset2); return NULL; } static struct isl_basic_set *uset_convex_hull_wrap(struct isl_set *set); static struct isl_basic_set *modulo_affine_hull( struct isl_set *set, struct isl_basic_set *affine_hull); /* Compute the convex hull of a pair of basic sets without any parameters or * integer divisions. * * This function is called from uset_convex_hull_unbounded, which * means that the complete convex hull is unbounded. Some pairs * of basic sets may still be bounded, though. * They may even lie inside a lower dimensional space, in which * case they need to be handled inside their affine hull since * the main algorithm assumes that the result is full-dimensional. * * If the convex hull of the two basic sets would have a non-trivial * lineality space, we first project out this lineality space. */ static struct isl_basic_set *convex_hull_pair(struct isl_basic_set *bset1, struct isl_basic_set *bset2) { isl_basic_set *lin, *aff; int bounded1, bounded2; if (bset1->ctx->opt->convex == ISL_CONVEX_HULL_FM) return convex_hull_pair_elim(bset1, bset2); aff = isl_set_affine_hull(isl_basic_set_union(isl_basic_set_copy(bset1), isl_basic_set_copy(bset2))); if (!aff) goto error; if (aff->n_eq != 0) return modulo_affine_hull(isl_basic_set_union(bset1, bset2), aff); isl_basic_set_free(aff); bounded1 = isl_basic_set_is_bounded(bset1); bounded2 = isl_basic_set_is_bounded(bset2); if (bounded1 < 0 || bounded2 < 0) goto error; if (bounded1 && bounded2) return uset_convex_hull_wrap(isl_basic_set_union(bset1, bset2)); if (bounded1 || bounded2) return convex_hull_pair_pointed(bset1, bset2); lin = induced_lineality_space(isl_basic_set_copy(bset1), isl_basic_set_copy(bset2)); if (!lin) goto error; if (isl_basic_set_is_universe(lin)) { isl_basic_set_free(bset1); isl_basic_set_free(bset2); return lin; } if (lin->n_eq < isl_basic_set_total_dim(lin)) { struct isl_set *set; set = isl_set_alloc_space(isl_basic_set_get_space(bset1), 2, 0); set = isl_set_add_basic_set(set, bset1); set = isl_set_add_basic_set(set, bset2); return modulo_lineality(set, lin); } isl_basic_set_free(lin); return convex_hull_pair_pointed(bset1, bset2); error: isl_basic_set_free(bset1); isl_basic_set_free(bset2); return NULL; } /* Compute the lineality space of a basic set. * We currently do not allow the basic set to have any divs. * We basically just drop the constants and turn every inequality * into an equality. */ struct isl_basic_set *isl_basic_set_lineality_space(struct isl_basic_set *bset) { int i, k; struct isl_basic_set *lin = NULL; unsigned dim; if (!bset) goto error; isl_assert(bset->ctx, bset->n_div == 0, goto error); dim = isl_basic_set_total_dim(bset); lin = isl_basic_set_alloc_space(isl_basic_set_get_space(bset), 0, dim, 0); if (!lin) goto error; for (i = 0; i < bset->n_eq; ++i) { k = isl_basic_set_alloc_equality(lin); if (k < 0) goto error; isl_int_set_si(lin->eq[k][0], 0); isl_seq_cpy(lin->eq[k] + 1, bset->eq[i] + 1, dim); } lin = isl_basic_set_gauss(lin, NULL); if (!lin) goto error; for (i = 0; i < bset->n_ineq && lin->n_eq < dim; ++i) { k = isl_basic_set_alloc_equality(lin); if (k < 0) goto error; isl_int_set_si(lin->eq[k][0], 0); isl_seq_cpy(lin->eq[k] + 1, bset->ineq[i] + 1, dim); lin = isl_basic_set_gauss(lin, NULL); if (!lin) goto error; } isl_basic_set_free(bset); return lin; error: isl_basic_set_free(lin); isl_basic_set_free(bset); return NULL; } /* Compute the (linear) hull of the lineality spaces of the basic sets in the * "underlying" set "set". */ static struct isl_basic_set *uset_combined_lineality_space(struct isl_set *set) { int i; struct isl_set *lin = NULL; if (!set) return NULL; if (set->n == 0) { isl_space *dim = isl_set_get_space(set); isl_set_free(set); return isl_basic_set_empty(dim); } lin = isl_set_alloc_space(isl_set_get_space(set), set->n, 0); for (i = 0; i < set->n; ++i) lin = isl_set_add_basic_set(lin, isl_basic_set_lineality_space(isl_basic_set_copy(set->p[i]))); isl_set_free(set); return isl_set_affine_hull(lin); } /* Compute the convex hull of a set without any parameters or * integer divisions. * In each step, we combined two basic sets until only one * basic set is left. * The input basic sets are assumed not to have a non-trivial * lineality space. If any of the intermediate results has * a non-trivial lineality space, it is projected out. */ static struct isl_basic_set *uset_convex_hull_unbounded(struct isl_set *set) { struct isl_basic_set *convex_hull = NULL; convex_hull = isl_set_copy_basic_set(set); set = isl_set_drop_basic_set(set, convex_hull); if (!set) goto error; while (set->n > 0) { struct isl_basic_set *t; t = isl_set_copy_basic_set(set); if (!t) goto error; set = isl_set_drop_basic_set(set, t); if (!set) goto error; convex_hull = convex_hull_pair(convex_hull, t); if (set->n == 0) break; t = isl_basic_set_lineality_space(isl_basic_set_copy(convex_hull)); if (!t) goto error; if (isl_basic_set_is_universe(t)) { isl_basic_set_free(convex_hull); convex_hull = t; break; } if (t->n_eq < isl_basic_set_total_dim(t)) { set = isl_set_add_basic_set(set, convex_hull); return modulo_lineality(set, t); } isl_basic_set_free(t); } isl_set_free(set); return convex_hull; error: isl_set_free(set); isl_basic_set_free(convex_hull); return NULL; } /* Compute an initial hull for wrapping containing a single initial * facet. * This function assumes that the given set is bounded. */ static struct isl_basic_set *initial_hull(struct isl_basic_set *hull, struct isl_set *set) { struct isl_mat *bounds = NULL; unsigned dim; int k; if (!hull) goto error; bounds = initial_facet_constraint(set); if (!bounds) goto error; k = isl_basic_set_alloc_inequality(hull); if (k < 0) goto error; dim = isl_set_n_dim(set); isl_assert(set->ctx, 1 + dim == bounds->n_col, goto error); isl_seq_cpy(hull->ineq[k], bounds->row[0], bounds->n_col); isl_mat_free(bounds); return hull; error: isl_basic_set_free(hull); isl_mat_free(bounds); return NULL; } struct max_constraint { struct isl_mat *c; int count; int ineq; }; static int max_constraint_equal(const void *entry, const void *val) { struct max_constraint *a = (struct max_constraint *)entry; isl_int *b = (isl_int *)val; return isl_seq_eq(a->c->row[0] + 1, b, a->c->n_col - 1); } static void update_constraint(struct isl_ctx *ctx, struct isl_hash_table *table, isl_int *con, unsigned len, int n, int ineq) { struct isl_hash_table_entry *entry; struct max_constraint *c; uint32_t c_hash; c_hash = isl_seq_get_hash(con + 1, len); entry = isl_hash_table_find(ctx, table, c_hash, max_constraint_equal, con + 1, 0); if (!entry) return; c = entry->data; if (c->count < n) { isl_hash_table_remove(ctx, table, entry); return; } c->count++; if (isl_int_gt(c->c->row[0][0], con[0])) return; if (isl_int_eq(c->c->row[0][0], con[0])) { if (ineq) c->ineq = ineq; return; } c->c = isl_mat_cow(c->c); isl_int_set(c->c->row[0][0], con[0]); c->ineq = ineq; } /* Check whether the constraint hash table "table" constains the constraint * "con". */ static int has_constraint(struct isl_ctx *ctx, struct isl_hash_table *table, isl_int *con, unsigned len, int n) { struct isl_hash_table_entry *entry; struct max_constraint *c; uint32_t c_hash; c_hash = isl_seq_get_hash(con + 1, len); entry = isl_hash_table_find(ctx, table, c_hash, max_constraint_equal, con + 1, 0); if (!entry) return 0; c = entry->data; if (c->count < n) return 0; return isl_int_eq(c->c->row[0][0], con[0]); } /* Check for inequality constraints of a basic set without equalities * such that the same or more stringent copies of the constraint appear * in all of the basic sets. Such constraints are necessarily facet * constraints of the convex hull. * * If the resulting basic set is by chance identical to one of * the basic sets in "set", then we know that this basic set contains * all other basic sets and is therefore the convex hull of set. * In this case we set *is_hull to 1. */ static struct isl_basic_set *common_constraints(struct isl_basic_set *hull, struct isl_set *set, int *is_hull) { int i, j, s, n; int min_constraints; int best; struct max_constraint *constraints = NULL; struct isl_hash_table *table = NULL; unsigned total; *is_hull = 0; for (i = 0; i < set->n; ++i) if (set->p[i]->n_eq == 0) break; if (i >= set->n) return hull; min_constraints = set->p[i]->n_ineq; best = i; for (i = best + 1; i < set->n; ++i) { if (set->p[i]->n_eq != 0) continue; if (set->p[i]->n_ineq >= min_constraints) continue; min_constraints = set->p[i]->n_ineq; best = i; } constraints = isl_calloc_array(hull->ctx, struct max_constraint, min_constraints); if (!constraints) return hull; table = isl_alloc_type(hull->ctx, struct isl_hash_table); if (isl_hash_table_init(hull->ctx, table, min_constraints)) goto error; total = isl_space_dim(set->dim, isl_dim_all); for (i = 0; i < set->p[best]->n_ineq; ++i) { constraints[i].c = isl_mat_sub_alloc6(hull->ctx, set->p[best]->ineq + i, 0, 1, 0, 1 + total); if (!constraints[i].c) goto error; constraints[i].ineq = 1; } for (i = 0; i < min_constraints; ++i) { struct isl_hash_table_entry *entry; uint32_t c_hash; c_hash = isl_seq_get_hash(constraints[i].c->row[0] + 1, total); entry = isl_hash_table_find(hull->ctx, table, c_hash, max_constraint_equal, constraints[i].c->row[0] + 1, 1); if (!entry) goto error; isl_assert(hull->ctx, !entry->data, goto error); entry->data = &constraints[i]; } n = 0; for (s = 0; s < set->n; ++s) { if (s == best) continue; for (i = 0; i < set->p[s]->n_eq; ++i) { isl_int *eq = set->p[s]->eq[i]; for (j = 0; j < 2; ++j) { isl_seq_neg(eq, eq, 1 + total); update_constraint(hull->ctx, table, eq, total, n, 0); } } for (i = 0; i < set->p[s]->n_ineq; ++i) { isl_int *ineq = set->p[s]->ineq[i]; update_constraint(hull->ctx, table, ineq, total, n, set->p[s]->n_eq == 0); } ++n; } for (i = 0; i < min_constraints; ++i) { if (constraints[i].count < n) continue; if (!constraints[i].ineq) continue; j = isl_basic_set_alloc_inequality(hull); if (j < 0) goto error; isl_seq_cpy(hull->ineq[j], constraints[i].c->row[0], 1 + total); } for (s = 0; s < set->n; ++s) { if (set->p[s]->n_eq) continue; if (set->p[s]->n_ineq != hull->n_ineq) continue; for (i = 0; i < set->p[s]->n_ineq; ++i) { isl_int *ineq = set->p[s]->ineq[i]; if (!has_constraint(hull->ctx, table, ineq, total, n)) break; } if (i == set->p[s]->n_ineq) *is_hull = 1; } isl_hash_table_clear(table); for (i = 0; i < min_constraints; ++i) isl_mat_free(constraints[i].c); free(constraints); free(table); return hull; error: isl_hash_table_clear(table); free(table); if (constraints) for (i = 0; i < min_constraints; ++i) isl_mat_free(constraints[i].c); free(constraints); return hull; } /* Create a template for the convex hull of "set" and fill it up * obvious facet constraints, if any. If the result happens to * be the convex hull of "set" then *is_hull is set to 1. */ static struct isl_basic_set *proto_hull(struct isl_set *set, int *is_hull) { struct isl_basic_set *hull; unsigned n_ineq; int i; n_ineq = 1; for (i = 0; i < set->n; ++i) { n_ineq += set->p[i]->n_eq; n_ineq += set->p[i]->n_ineq; } hull = isl_basic_set_alloc_space(isl_space_copy(set->dim), 0, 0, n_ineq); hull = isl_basic_set_set_rational(hull); if (!hull) return NULL; return common_constraints(hull, set, is_hull); } static struct isl_basic_set *uset_convex_hull_wrap(struct isl_set *set) { struct isl_basic_set *hull; int is_hull; hull = proto_hull(set, &is_hull); if (hull && !is_hull) { if (hull->n_ineq == 0) hull = initial_hull(hull, set); hull = extend(hull, set); } isl_set_free(set); return hull; } /* Compute the convex hull of a set without any parameters or * integer divisions. Depending on whether the set is bounded, * we pass control to the wrapping based convex hull or * the Fourier-Motzkin elimination based convex hull. * We also handle a few special cases before checking the boundedness. */ static struct isl_basic_set *uset_convex_hull(struct isl_set *set) { struct isl_basic_set *convex_hull = NULL; struct isl_basic_set *lin; if (isl_set_n_dim(set) == 0) return convex_hull_0d(set); set = isl_set_coalesce(set); set = isl_set_set_rational(set); if (!set) goto error; if (!set) return NULL; if (set->n == 1) { convex_hull = isl_basic_set_copy(set->p[0]); isl_set_free(set); return convex_hull; } if (isl_set_n_dim(set) == 1) return convex_hull_1d(set); if (isl_set_is_bounded(set) && set->ctx->opt->convex == ISL_CONVEX_HULL_WRAP) return uset_convex_hull_wrap(set); lin = uset_combined_lineality_space(isl_set_copy(set)); if (!lin) goto error; if (isl_basic_set_is_universe(lin)) { isl_set_free(set); return lin; } if (lin->n_eq < isl_basic_set_total_dim(lin)) return modulo_lineality(set, lin); isl_basic_set_free(lin); return uset_convex_hull_unbounded(set); error: isl_set_free(set); isl_basic_set_free(convex_hull); return NULL; } /* This is the core procedure, where "set" is a "pure" set, i.e., * without parameters or divs and where the convex hull of set is * known to be full-dimensional. */ static struct isl_basic_set *uset_convex_hull_wrap_bounded(struct isl_set *set) { struct isl_basic_set *convex_hull = NULL; if (!set) goto error; if (isl_set_n_dim(set) == 0) { convex_hull = isl_basic_set_universe(isl_space_copy(set->dim)); isl_set_free(set); convex_hull = isl_basic_set_set_rational(convex_hull); return convex_hull; } set = isl_set_set_rational(set); set = isl_set_coalesce(set); if (!set) goto error; if (set->n == 1) { convex_hull = isl_basic_set_copy(set->p[0]); isl_set_free(set); convex_hull = isl_basic_map_remove_redundancies(convex_hull); return convex_hull; } if (isl_set_n_dim(set) == 1) return convex_hull_1d(set); return uset_convex_hull_wrap(set); error: isl_set_free(set); return NULL; } /* Compute the convex hull of set "set" with affine hull "affine_hull", * We first remove the equalities (transforming the set), compute the * convex hull of the transformed set and then add the equalities back * (after performing the inverse transformation. */ static struct isl_basic_set *modulo_affine_hull( struct isl_set *set, struct isl_basic_set *affine_hull) { struct isl_mat *T; struct isl_mat *T2; struct isl_basic_set *dummy; struct isl_basic_set *convex_hull; dummy = isl_basic_set_remove_equalities( isl_basic_set_copy(affine_hull), &T, &T2); if (!dummy) goto error; isl_basic_set_free(dummy); set = isl_set_preimage(set, T); convex_hull = uset_convex_hull(set); convex_hull = isl_basic_set_preimage(convex_hull, T2); convex_hull = isl_basic_set_intersect(convex_hull, affine_hull); return convex_hull; error: isl_basic_set_free(affine_hull); isl_set_free(set); return NULL; } /* Return an empty basic map living in the same space as "map". */ static __isl_give isl_basic_map *replace_map_by_empty_basic_map( __isl_take isl_map *map) { isl_space *space; space = isl_map_get_space(map); isl_map_free(map); return isl_basic_map_empty(space); } /* Compute the convex hull of a map. * * The implementation was inspired by "Extended Convex Hull" by Fukuda et al., * specifically, the wrapping of facets to obtain new facets. */ struct isl_basic_map *isl_map_convex_hull(struct isl_map *map) { struct isl_basic_set *bset; struct isl_basic_map *model = NULL; struct isl_basic_set *affine_hull = NULL; struct isl_basic_map *convex_hull = NULL; struct isl_set *set = NULL; map = isl_map_detect_equalities(map); map = isl_map_align_divs(map); if (!map) goto error; if (map->n == 0) return replace_map_by_empty_basic_map(map); model = isl_basic_map_copy(map->p[0]); set = isl_map_underlying_set(map); if (!set) goto error; affine_hull = isl_set_affine_hull(isl_set_copy(set)); if (!affine_hull) goto error; if (affine_hull->n_eq != 0) bset = modulo_affine_hull(set, affine_hull); else { isl_basic_set_free(affine_hull); bset = uset_convex_hull(set); } convex_hull = isl_basic_map_overlying_set(bset, model); if (!convex_hull) return NULL; ISL_F_SET(convex_hull, ISL_BASIC_MAP_NO_IMPLICIT); ISL_F_SET(convex_hull, ISL_BASIC_MAP_ALL_EQUALITIES); ISL_F_CLR(convex_hull, ISL_BASIC_MAP_RATIONAL); return convex_hull; error: isl_set_free(set); isl_basic_map_free(model); return NULL; } struct isl_basic_set *isl_set_convex_hull(struct isl_set *set) { return (struct isl_basic_set *) isl_map_convex_hull((struct isl_map *)set); } __isl_give isl_basic_map *isl_map_polyhedral_hull(__isl_take isl_map *map) { isl_basic_map *hull; hull = isl_map_convex_hull(map); return isl_basic_map_remove_divs(hull); } __isl_give isl_basic_set *isl_set_polyhedral_hull(__isl_take isl_set *set) { return (isl_basic_set *)isl_map_polyhedral_hull((isl_map *)set); } struct sh_data_entry { struct isl_hash_table *table; struct isl_tab *tab; }; /* Holds the data needed during the simple hull computation. * In particular, * n the number of basic sets in the original set * hull_table a hash table of already computed constraints * in the simple hull * p for each basic set, * table a hash table of the constraints * tab the tableau corresponding to the basic set */ struct sh_data { struct isl_ctx *ctx; unsigned n; struct isl_hash_table *hull_table; struct sh_data_entry p[1]; }; static void sh_data_free(struct sh_data *data) { int i; if (!data) return; isl_hash_table_free(data->ctx, data->hull_table); for (i = 0; i < data->n; ++i) { isl_hash_table_free(data->ctx, data->p[i].table); isl_tab_free(data->p[i].tab); } free(data); } struct ineq_cmp_data { unsigned len; isl_int *p; }; static int has_ineq(const void *entry, const void *val) { isl_int *row = (isl_int *)entry; struct ineq_cmp_data *v = (struct ineq_cmp_data *)val; return isl_seq_eq(row + 1, v->p + 1, v->len) || isl_seq_is_neg(row + 1, v->p + 1, v->len); } static int hash_ineq(struct isl_ctx *ctx, struct isl_hash_table *table, isl_int *ineq, unsigned len) { uint32_t c_hash; struct ineq_cmp_data v; struct isl_hash_table_entry *entry; v.len = len; v.p = ineq; c_hash = isl_seq_get_hash(ineq + 1, len); entry = isl_hash_table_find(ctx, table, c_hash, has_ineq, &v, 1); if (!entry) return - 1; entry->data = ineq; return 0; } /* Fill hash table "table" with the constraints of "bset". * Equalities are added as two inequalities. * The value in the hash table is a pointer to the (in)equality of "bset". */ static int hash_basic_set(struct isl_hash_table *table, struct isl_basic_set *bset) { int i, j; unsigned dim = isl_basic_set_total_dim(bset); for (i = 0; i < bset->n_eq; ++i) { for (j = 0; j < 2; ++j) { isl_seq_neg(bset->eq[i], bset->eq[i], 1 + dim); if (hash_ineq(bset->ctx, table, bset->eq[i], dim) < 0) return -1; } } for (i = 0; i < bset->n_ineq; ++i) { if (hash_ineq(bset->ctx, table, bset->ineq[i], dim) < 0) return -1; } return 0; } static struct sh_data *sh_data_alloc(struct isl_set *set, unsigned n_ineq) { struct sh_data *data; int i; data = isl_calloc(set->ctx, struct sh_data, sizeof(struct sh_data) + (set->n - 1) * sizeof(struct sh_data_entry)); if (!data) return NULL; data->ctx = set->ctx; data->n = set->n; data->hull_table = isl_hash_table_alloc(set->ctx, n_ineq); if (!data->hull_table) goto error; for (i = 0; i < set->n; ++i) { data->p[i].table = isl_hash_table_alloc(set->ctx, 2 * set->p[i]->n_eq + set->p[i]->n_ineq); if (!data->p[i].table) goto error; if (hash_basic_set(data->p[i].table, set->p[i]) < 0) goto error; } return data; error: sh_data_free(data); return NULL; } /* Check if inequality "ineq" is a bound for basic set "j" or if * it can be relaxed (by increasing the constant term) to become * a bound for that basic set. In the latter case, the constant * term is updated. * Relaxation of the constant term is only allowed if "shift" is set. * * Return 1 if "ineq" is a bound * 0 if "ineq" may attain arbitrarily small values on basic set "j" * -1 if some error occurred */ static int is_bound(struct sh_data *data, struct isl_set *set, int j, isl_int *ineq, int shift) { enum isl_lp_result res; isl_int opt; if (!data->p[j].tab) { data->p[j].tab = isl_tab_from_basic_set(set->p[j], 0); if (!data->p[j].tab) return -1; } isl_int_init(opt); res = isl_tab_min(data->p[j].tab, ineq, data->ctx->one, &opt, NULL, 0); if (res == isl_lp_ok && isl_int_is_neg(opt)) { if (shift) isl_int_sub(ineq[0], ineq[0], opt); else res = isl_lp_unbounded; } isl_int_clear(opt); return (res == isl_lp_ok || res == isl_lp_empty) ? 1 : res == isl_lp_unbounded ? 0 : -1; } /* Check if inequality "ineq" from basic set "i" is or can be relaxed to * become a bound on the whole set. If so, add the (relaxed) inequality * to "hull". Relaxation is only allowed if "shift" is set. * * We first check if "hull" already contains a translate of the inequality. * If so, we are done. * Then, we check if any of the previous basic sets contains a translate * of the inequality. If so, then we have already considered this * inequality and we are done. * Otherwise, for each basic set other than "i", we check if the inequality * is a bound on the basic set. * For previous basic sets, we know that they do not contain a translate * of the inequality, so we directly call is_bound. * For following basic sets, we first check if a translate of the * inequality appears in its description and if so directly update * the inequality accordingly. */ static struct isl_basic_set *add_bound(struct isl_basic_set *hull, struct sh_data *data, struct isl_set *set, int i, isl_int *ineq, int shift) { uint32_t c_hash; struct ineq_cmp_data v; struct isl_hash_table_entry *entry; int j, k; if (!hull) return NULL; v.len = isl_basic_set_total_dim(hull); v.p = ineq; c_hash = isl_seq_get_hash(ineq + 1, v.len); entry = isl_hash_table_find(hull->ctx, data->hull_table, c_hash, has_ineq, &v, 0); if (entry) return hull; for (j = 0; j < i; ++j) { entry = isl_hash_table_find(hull->ctx, data->p[j].table, c_hash, has_ineq, &v, 0); if (entry) break; } if (j < i) return hull; k = isl_basic_set_alloc_inequality(hull); if (k < 0) goto error; isl_seq_cpy(hull->ineq[k], ineq, 1 + v.len); for (j = 0; j < i; ++j) { int bound; bound = is_bound(data, set, j, hull->ineq[k], shift); if (bound < 0) goto error; if (!bound) break; } if (j < i) { isl_basic_set_free_inequality(hull, 1); return hull; } for (j = i + 1; j < set->n; ++j) { int bound, neg; isl_int *ineq_j; entry = isl_hash_table_find(hull->ctx, data->p[j].table, c_hash, has_ineq, &v, 0); if (entry) { ineq_j = entry->data; neg = isl_seq_is_neg(ineq_j + 1, hull->ineq[k] + 1, v.len); if (neg) isl_int_neg(ineq_j[0], ineq_j[0]); if (isl_int_gt(ineq_j[0], hull->ineq[k][0])) isl_int_set(hull->ineq[k][0], ineq_j[0]); if (neg) isl_int_neg(ineq_j[0], ineq_j[0]); continue; } bound = is_bound(data, set, j, hull->ineq[k], shift); if (bound < 0) goto error; if (!bound) break; } if (j < set->n) { isl_basic_set_free_inequality(hull, 1); return hull; } entry = isl_hash_table_find(hull->ctx, data->hull_table, c_hash, has_ineq, &v, 1); if (!entry) goto error; entry->data = hull->ineq[k]; return hull; error: isl_basic_set_free(hull); return NULL; } /* Check if any inequality from basic set "i" is or can be relaxed to * become a bound on the whole set. If so, add the (relaxed) inequality * to "hull". Relaxation is only allowed if "shift" is set. */ static struct isl_basic_set *add_bounds(struct isl_basic_set *bset, struct sh_data *data, struct isl_set *set, int i, int shift) { int j, k; unsigned dim = isl_basic_set_total_dim(bset); for (j = 0; j < set->p[i]->n_eq; ++j) { for (k = 0; k < 2; ++k) { isl_seq_neg(set->p[i]->eq[j], set->p[i]->eq[j], 1+dim); bset = add_bound(bset, data, set, i, set->p[i]->eq[j], shift); } } for (j = 0; j < set->p[i]->n_ineq; ++j) bset = add_bound(bset, data, set, i, set->p[i]->ineq[j], shift); return bset; } /* Compute a superset of the convex hull of set that is described * by only (translates of) the constraints in the constituents of set. * Translation is only allowed if "shift" is set. */ static __isl_give isl_basic_set *uset_simple_hull(__isl_take isl_set *set, int shift) { struct sh_data *data = NULL; struct isl_basic_set *hull = NULL; unsigned n_ineq; int i; if (!set) return NULL; n_ineq = 0; for (i = 0; i < set->n; ++i) { if (!set->p[i]) goto error; n_ineq += 2 * set->p[i]->n_eq + set->p[i]->n_ineq; } hull = isl_basic_set_alloc_space(isl_space_copy(set->dim), 0, 0, n_ineq); if (!hull) goto error; data = sh_data_alloc(set, n_ineq); if (!data) goto error; for (i = 0; i < set->n; ++i) hull = add_bounds(hull, data, set, i, shift); sh_data_free(data); isl_set_free(set); return hull; error: sh_data_free(data); isl_basic_set_free(hull); isl_set_free(set); return NULL; } /* Compute a superset of the convex hull of map that is described * by only (translates of) the constraints in the constituents of map. * Handle trivial cases where map is NULL or contains at most one disjunct. */ static __isl_give isl_basic_map *map_simple_hull_trivial( __isl_take isl_map *map) { isl_basic_map *hull; if (!map) return NULL; if (map->n == 0) return replace_map_by_empty_basic_map(map); hull = isl_basic_map_copy(map->p[0]); isl_map_free(map); return hull; } /* Compute a superset of the convex hull of map that is described * by only (translates of) the constraints in the constituents of map. * Translation is only allowed if "shift" is set. */ static __isl_give isl_basic_map *map_simple_hull(__isl_take isl_map *map, int shift) { struct isl_set *set = NULL; struct isl_basic_map *model = NULL; struct isl_basic_map *hull; struct isl_basic_map *affine_hull; struct isl_basic_set *bset = NULL; if (!map || map->n <= 1) return map_simple_hull_trivial(map); map = isl_map_detect_equalities(map); if (!map || map->n <= 1) return map_simple_hull_trivial(map); affine_hull = isl_map_affine_hull(isl_map_copy(map)); map = isl_map_align_divs(map); model = map ? isl_basic_map_copy(map->p[0]) : NULL; set = isl_map_underlying_set(map); bset = uset_simple_hull(set, shift); hull = isl_basic_map_overlying_set(bset, model); hull = isl_basic_map_intersect(hull, affine_hull); hull = isl_basic_map_remove_redundancies(hull); if (!hull) return NULL; ISL_F_SET(hull, ISL_BASIC_MAP_NO_IMPLICIT); ISL_F_SET(hull, ISL_BASIC_MAP_ALL_EQUALITIES); hull = isl_basic_map_finalize(hull); return hull; } /* Compute a superset of the convex hull of map that is described * by only translates of the constraints in the constituents of map. */ __isl_give isl_basic_map *isl_map_simple_hull(__isl_take isl_map *map) { return map_simple_hull(map, 1); } struct isl_basic_set *isl_set_simple_hull(struct isl_set *set) { return (struct isl_basic_set *) isl_map_simple_hull((struct isl_map *)set); } /* Compute a superset of the convex hull of map that is described * by only the constraints in the constituents of map. */ __isl_give isl_basic_map *isl_map_unshifted_simple_hull( __isl_take isl_map *map) { return map_simple_hull(map, 0); } __isl_give isl_basic_set *isl_set_unshifted_simple_hull( __isl_take isl_set *set) { return isl_map_unshifted_simple_hull(set); } /* Drop all inequalities from "bmap1" that do not also appear in "bmap2". * A constraint that appears with different constant terms * in "bmap1" and "bmap2" is also kept, with the least restrictive * (i.e., greatest) constant term. * "bmap1" and "bmap2" are assumed to have the same (known) * integer divisions. * The constraints of both "bmap1" and "bmap2" are assumed * to have been sorted using isl_basic_map_sort_constraints. * * Run through the inequality constraints of "bmap1" and "bmap2" * in sorted order. * Each constraint of "bmap1" without a matching constraint in "bmap2" * is removed. * If a match is found, the constraint is kept. If needed, the constant * term of the constraint is adjusted. */ static __isl_give isl_basic_map *select_shared_inequalities( __isl_take isl_basic_map *bmap1, __isl_keep isl_basic_map *bmap2) { int i1, i2; bmap1 = isl_basic_map_cow(bmap1); if (!bmap1 || !bmap2) return isl_basic_map_free(bmap1); i1 = bmap1->n_ineq - 1; i2 = bmap2->n_ineq - 1; while (bmap1 && i1 >= 0 && i2 >= 0) { int cmp; cmp = isl_basic_map_constraint_cmp(bmap1, bmap1->ineq[i1], bmap2->ineq[i2]); if (cmp < 0) { --i2; continue; } if (cmp > 0) { if (isl_basic_map_drop_inequality(bmap1, i1) < 0) bmap1 = isl_basic_map_free(bmap1); --i1; continue; } if (isl_int_lt(bmap1->ineq[i1][0], bmap2->ineq[i2][0])) isl_int_set(bmap1->ineq[i1][0], bmap2->ineq[i2][0]); --i1; --i2; } for (; i1 >= 0; --i1) if (isl_basic_map_drop_inequality(bmap1, i1) < 0) bmap1 = isl_basic_map_free(bmap1); return bmap1; } /* Drop all equalities from "bmap1" that do not also appear in "bmap2". * "bmap1" and "bmap2" are assumed to have the same (known) * integer divisions. * * Run through the equality constraints of "bmap1" and "bmap2". * Each constraint of "bmap1" without a matching constraint in "bmap2" * is removed. */ static __isl_give isl_basic_map *select_shared_equalities( __isl_take isl_basic_map *bmap1, __isl_keep isl_basic_map *bmap2) { int i1, i2; unsigned total; bmap1 = isl_basic_map_cow(bmap1); if (!bmap1 || !bmap2) return isl_basic_map_free(bmap1); total = isl_basic_map_total_dim(bmap1); i1 = bmap1->n_eq - 1; i2 = bmap2->n_eq - 1; while (bmap1 && i1 >= 0 && i2 >= 0) { int last1, last2; last1 = isl_seq_last_non_zero(bmap1->eq[i1] + 1, total); last2 = isl_seq_last_non_zero(bmap2->eq[i2] + 1, total); if (last1 > last2) { --i2; continue; } if (last1 < last2) { if (isl_basic_map_drop_equality(bmap1, i1) < 0) bmap1 = isl_basic_map_free(bmap1); --i1; continue; } if (!isl_seq_eq(bmap1->eq[i1], bmap2->eq[i2], 1 + total)) { if (isl_basic_map_drop_equality(bmap1, i1) < 0) bmap1 = isl_basic_map_free(bmap1); } --i1; --i2; } for (; i1 >= 0; --i1) if (isl_basic_map_drop_equality(bmap1, i1) < 0) bmap1 = isl_basic_map_free(bmap1); return bmap1; } /* Compute a superset of "bmap1" and "bmap2" that is described * by only the constraints that appear in both "bmap1" and "bmap2". * * First drop constraints that involve unknown integer divisions * since it is not trivial to check whether two such integer divisions * in different basic maps are the same. * Then align the remaining (known) divs and sort the constraints. * Finally drop all inequalities and equalities from "bmap1" that * do not also appear in "bmap2". */ __isl_give isl_basic_map *isl_basic_map_plain_unshifted_simple_hull( __isl_take isl_basic_map *bmap1, __isl_take isl_basic_map *bmap2) { bmap1 = isl_basic_map_drop_constraint_involving_unknown_divs(bmap1); bmap2 = isl_basic_map_drop_constraint_involving_unknown_divs(bmap2); bmap2 = isl_basic_map_align_divs(bmap2, bmap1); bmap1 = isl_basic_map_align_divs(bmap1, bmap2); bmap1 = isl_basic_map_gauss(bmap1, NULL); bmap2 = isl_basic_map_gauss(bmap2, NULL); bmap1 = isl_basic_map_sort_constraints(bmap1); bmap2 = isl_basic_map_sort_constraints(bmap2); bmap1 = select_shared_inequalities(bmap1, bmap2); bmap1 = select_shared_equalities(bmap1, bmap2); isl_basic_map_free(bmap2); bmap1 = isl_basic_map_finalize(bmap1); return bmap1; } /* Compute a superset of the convex hull of "map" that is described * by only the constraints in the constituents of "map". * In particular, the result is composed of constraints that appear * in each of the basic maps of "map" * * Constraints that involve unknown integer divisions are dropped * since it is not trivial to check whether two such integer divisions * in different basic maps are the same. * * The hull is initialized from the first basic map and then * updated with respect to the other basic maps in turn. */ __isl_give isl_basic_map *isl_map_plain_unshifted_simple_hull( __isl_take isl_map *map) { int i; isl_basic_map *hull; if (!map) return NULL; if (map->n <= 1) return map_simple_hull_trivial(map); map = isl_map_drop_constraint_involving_unknown_divs(map); hull = isl_basic_map_copy(map->p[0]); for (i = 1; i < map->n; ++i) { isl_basic_map *bmap_i; bmap_i = isl_basic_map_copy(map->p[i]); hull = isl_basic_map_plain_unshifted_simple_hull(hull, bmap_i); } isl_map_free(map); return hull; } /* Check if "ineq" is a bound on "set" and, if so, add it to "hull". * * For each basic set in "set", we first check if the basic set * contains a translate of "ineq". If this translate is more relaxed, * then we assume that "ineq" is not a bound on this basic set. * Otherwise, we know that it is a bound. * If the basic set does not contain a translate of "ineq", then * we call is_bound to perform the test. */ static __isl_give isl_basic_set *add_bound_from_constraint( __isl_take isl_basic_set *hull, struct sh_data *data, __isl_keep isl_set *set, isl_int *ineq) { int i, k; isl_ctx *ctx; uint32_t c_hash; struct ineq_cmp_data v; if (!hull || !set) return isl_basic_set_free(hull); v.len = isl_basic_set_total_dim(hull); v.p = ineq; c_hash = isl_seq_get_hash(ineq + 1, v.len); ctx = isl_basic_set_get_ctx(hull); for (i = 0; i < set->n; ++i) { int bound; struct isl_hash_table_entry *entry; entry = isl_hash_table_find(ctx, data->p[i].table, c_hash, &has_ineq, &v, 0); if (entry) { isl_int *ineq_i = entry->data; int neg, more_relaxed; neg = isl_seq_is_neg(ineq_i + 1, ineq + 1, v.len); if (neg) isl_int_neg(ineq_i[0], ineq_i[0]); more_relaxed = isl_int_gt(ineq_i[0], ineq[0]); if (neg) isl_int_neg(ineq_i[0], ineq_i[0]); if (more_relaxed) break; else continue; } bound = is_bound(data, set, i, ineq, 0); if (bound < 0) return isl_basic_set_free(hull); if (!bound) break; } if (i < set->n) return hull; k = isl_basic_set_alloc_inequality(hull); if (k < 0) return isl_basic_set_free(hull); isl_seq_cpy(hull->ineq[k], ineq, 1 + v.len); return hull; } /* Compute a superset of the convex hull of "set" that is described * by only some of the "n_ineq" constraints in the list "ineq", where "set" * has no parameters or integer divisions. * * The inequalities in "ineq" are assumed to have been sorted such * that constraints with the same linear part appear together and * that among constraints with the same linear part, those with * smaller constant term appear first. * * We reuse the same data structure that is used by uset_simple_hull, * but we do not need the hull table since we will not consider the * same constraint more than once. We therefore allocate it with zero size. * * We run through the constraints and try to add them one by one, * skipping identical constraints. If we have added a constraint and * the next constraint is a more relaxed translate, then we skip this * next constraint as well. */ static __isl_give isl_basic_set *uset_unshifted_simple_hull_from_constraints( __isl_take isl_set *set, int n_ineq, isl_int **ineq) { int i; int last_added = 0; struct sh_data *data = NULL; isl_basic_set *hull = NULL; unsigned dim; hull = isl_basic_set_alloc_space(isl_set_get_space(set), 0, 0, n_ineq); if (!hull) goto error; data = sh_data_alloc(set, 0); if (!data) goto error; dim = isl_set_dim(set, isl_dim_set); for (i = 0; i < n_ineq; ++i) { int hull_n_ineq = hull->n_ineq; int parallel; parallel = i > 0 && isl_seq_eq(ineq[i - 1] + 1, ineq[i] + 1, dim); if (parallel && (last_added || isl_int_eq(ineq[i - 1][0], ineq[i][0]))) continue; hull = add_bound_from_constraint(hull, data, set, ineq[i]); if (!hull) goto error; last_added = hull->n_ineq > hull_n_ineq; } sh_data_free(data); isl_set_free(set); return hull; error: sh_data_free(data); isl_set_free(set); isl_basic_set_free(hull); return NULL; } /* Collect pointers to all the inequalities in the elements of "list" * in "ineq". For equalities, store both a pointer to the equality and * a pointer to its opposite, which is first copied to "mat". * "ineq" and "mat" are assumed to have been preallocated to the right size * (the number of inequalities + 2 times the number of equalites and * the number of equalities, respectively). */ static __isl_give isl_mat *collect_inequalities(__isl_take isl_mat *mat, __isl_keep isl_basic_set_list *list, isl_int **ineq) { int i, j, n, n_eq, n_ineq; if (!mat) return NULL; n_eq = 0; n_ineq = 0; n = isl_basic_set_list_n_basic_set(list); for (i = 0; i < n; ++i) { isl_basic_set *bset; bset = isl_basic_set_list_get_basic_set(list, i); if (!bset) return isl_mat_free(mat); for (j = 0; j < bset->n_eq; ++j) { ineq[n_ineq++] = mat->row[n_eq]; ineq[n_ineq++] = bset->eq[j]; isl_seq_neg(mat->row[n_eq++], bset->eq[j], mat->n_col); } for (j = 0; j < bset->n_ineq; ++j) ineq[n_ineq++] = bset->ineq[j]; isl_basic_set_free(bset); } return mat; } /* Comparison routine for use as an isl_sort callback. * * Constraints with the same linear part are sorted together and * among constraints with the same linear part, those with smaller * constant term are sorted first. */ static int cmp_ineq(const void *a, const void *b, void *arg) { unsigned dim = *(unsigned *) arg; isl_int * const *ineq1 = a; isl_int * const *ineq2 = b; int cmp; cmp = isl_seq_cmp((*ineq1) + 1, (*ineq2) + 1, dim); if (cmp != 0) return cmp; return isl_int_cmp((*ineq1)[0], (*ineq2)[0]); } /* Compute a superset of the convex hull of "set" that is described * by only constraints in the elements of "list", where "set" has * no parameters or integer divisions. * * We collect all the constraints in those elements and then * sort the constraints such that constraints with the same linear part * are sorted together and that those with smaller constant term are * sorted first. */ static __isl_give isl_basic_set *uset_unshifted_simple_hull_from_basic_set_list( __isl_take isl_set *set, __isl_take isl_basic_set_list *list) { int i, n, n_eq, n_ineq; unsigned dim; isl_ctx *ctx; isl_mat *mat = NULL; isl_int **ineq = NULL; isl_basic_set *hull; if (!set) goto error; ctx = isl_set_get_ctx(set); n_eq = 0; n_ineq = 0; n = isl_basic_set_list_n_basic_set(list); for (i = 0; i < n; ++i) { isl_basic_set *bset; bset = isl_basic_set_list_get_basic_set(list, i); if (!bset) goto error; n_eq += bset->n_eq; n_ineq += 2 * bset->n_eq + bset->n_ineq; isl_basic_set_free(bset); } ineq = isl_alloc_array(ctx, isl_int *, n_ineq); if (n_ineq > 0 && !ineq) goto error; dim = isl_set_dim(set, isl_dim_set); mat = isl_mat_alloc(ctx, n_eq, 1 + dim); mat = collect_inequalities(mat, list, ineq); if (!mat) goto error; if (isl_sort(ineq, n_ineq, sizeof(ineq[0]), &cmp_ineq, &dim) < 0) goto error; hull = uset_unshifted_simple_hull_from_constraints(set, n_ineq, ineq); isl_mat_free(mat); free(ineq); isl_basic_set_list_free(list); return hull; error: isl_mat_free(mat); free(ineq); isl_set_free(set); isl_basic_set_list_free(list); return NULL; } /* Compute a superset of the convex hull of "map" that is described * by only constraints in the elements of "list". * * If the list is empty, then we can only describe the universe set. * If the input map is empty, then all constraints are valid, so * we return the intersection of the elements in "list". * * Otherwise, we align all divs and temporarily treat them * as regular variables, computing the unshifted simple hull in * uset_unshifted_simple_hull_from_basic_set_list. */ static __isl_give isl_basic_map *map_unshifted_simple_hull_from_basic_map_list( __isl_take isl_map *map, __isl_take isl_basic_map_list *list) { isl_basic_map *model; isl_basic_map *hull; isl_set *set; isl_basic_set_list *bset_list; if (!map || !list) goto error; if (isl_basic_map_list_n_basic_map(list) == 0) { isl_space *space; space = isl_map_get_space(map); isl_map_free(map); isl_basic_map_list_free(list); return isl_basic_map_universe(space); } if (isl_map_plain_is_empty(map)) { isl_map_free(map); return isl_basic_map_list_intersect(list); } map = isl_map_align_divs_to_basic_map_list(map, list); if (!map) goto error; list = isl_basic_map_list_align_divs_to_basic_map(list, map->p[0]); model = isl_basic_map_list_get_basic_map(list, 0); set = isl_map_underlying_set(map); bset_list = isl_basic_map_list_underlying_set(list); hull = uset_unshifted_simple_hull_from_basic_set_list(set, bset_list); hull = isl_basic_map_overlying_set(hull, model); return hull; error: isl_map_free(map); isl_basic_map_list_free(list); return NULL; } /* Return a sequence of the basic maps that make up the maps in "list". */ static __isl_give isl_basic_set_list *collect_basic_maps( __isl_take isl_map_list *list) { int i, n; isl_ctx *ctx; isl_basic_map_list *bmap_list; if (!list) return NULL; n = isl_map_list_n_map(list); ctx = isl_map_list_get_ctx(list); bmap_list = isl_basic_map_list_alloc(ctx, 0); for (i = 0; i < n; ++i) { isl_map *map; isl_basic_map_list *list_i; map = isl_map_list_get_map(list, i); map = isl_map_compute_divs(map); list_i = isl_map_get_basic_map_list(map); isl_map_free(map); bmap_list = isl_basic_map_list_concat(bmap_list, list_i); } isl_map_list_free(list); return bmap_list; } /* Compute a superset of the convex hull of "map" that is described * by only constraints in the elements of "list". * * If "map" is the universe, then the convex hull (and therefore * any superset of the convexhull) is the universe as well. * * Otherwise, we collect all the basic maps in the map list and * continue with map_unshifted_simple_hull_from_basic_map_list. */ __isl_give isl_basic_map *isl_map_unshifted_simple_hull_from_map_list( __isl_take isl_map *map, __isl_take isl_map_list *list) { isl_basic_map_list *bmap_list; int is_universe; is_universe = isl_map_plain_is_universe(map); if (is_universe < 0) map = isl_map_free(map); if (is_universe < 0 || is_universe) { isl_map_list_free(list); return isl_map_unshifted_simple_hull(map); } bmap_list = collect_basic_maps(list); return map_unshifted_simple_hull_from_basic_map_list(map, bmap_list); } /* Compute a superset of the convex hull of "set" that is described * by only constraints in the elements of "list". */ __isl_give isl_basic_set *isl_set_unshifted_simple_hull_from_set_list( __isl_take isl_set *set, __isl_take isl_set_list *list) { return isl_map_unshifted_simple_hull_from_map_list(set, list); } /* Given a set "set", return parametric bounds on the dimension "dim". */ static struct isl_basic_set *set_bounds(struct isl_set *set, int dim) { unsigned set_dim = isl_set_dim(set, isl_dim_set); set = isl_set_copy(set); set = isl_set_eliminate_dims(set, dim + 1, set_dim - (dim + 1)); set = isl_set_eliminate_dims(set, 0, dim); return isl_set_convex_hull(set); } /* Computes a "simple hull" and then check if each dimension in the * resulting hull is bounded by a symbolic constant. If not, the * hull is intersected with the corresponding bounds on the whole set. */ struct isl_basic_set *isl_set_bounded_simple_hull(struct isl_set *set) { int i, j; struct isl_basic_set *hull; unsigned nparam, left; int removed_divs = 0; hull = isl_set_simple_hull(isl_set_copy(set)); if (!hull) goto error; nparam = isl_basic_set_dim(hull, isl_dim_param); for (i = 0; i < isl_basic_set_dim(hull, isl_dim_set); ++i) { int lower = 0, upper = 0; struct isl_basic_set *bounds; left = isl_basic_set_total_dim(hull) - nparam - i - 1; for (j = 0; j < hull->n_eq; ++j) { if (isl_int_is_zero(hull->eq[j][1 + nparam + i])) continue; if (isl_seq_first_non_zero(hull->eq[j]+1+nparam+i+1, left) == -1) break; } if (j < hull->n_eq) continue; for (j = 0; j < hull->n_ineq; ++j) { if (isl_int_is_zero(hull->ineq[j][1 + nparam + i])) continue; if (isl_seq_first_non_zero(hull->ineq[j]+1+nparam+i+1, left) != -1 || isl_seq_first_non_zero(hull->ineq[j]+1+nparam, i) != -1) continue; if (isl_int_is_pos(hull->ineq[j][1 + nparam + i])) lower = 1; else upper = 1; if (lower && upper) break; } if (lower && upper) continue; if (!removed_divs) { set = isl_set_remove_divs(set); if (!set) goto error; removed_divs = 1; } bounds = set_bounds(set, i); hull = isl_basic_set_intersect(hull, bounds); if (!hull) goto error; } isl_set_free(set); return hull; error: isl_set_free(set); return NULL; }