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Diffstat (limited to 'polly/lib/External/isl/isl_convex_hull.c')
| -rw-r--r-- | polly/lib/External/isl/isl_convex_hull.c | 2829 |
1 files changed, 2829 insertions, 0 deletions
diff --git a/polly/lib/External/isl/isl_convex_hull.c b/polly/lib/External/isl/isl_convex_hull.c new file mode 100644 index 00000000000..b1254be1783 --- /dev/null +++ b/polly/lib/External/isl/isl_convex_hull.c @@ -0,0 +1,2829 @@ +/* + * 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 <isl_ctx_private.h> +#include <isl_map_private.h> +#include <isl_lp_private.h> +#include <isl/map.h> +#include <isl_mat_private.h> +#include <isl_vec_private.h> +#include <isl/set.h> +#include <isl_seq.h> +#include <isl_options_private.h> +#include "isl_equalities.h" +#include "isl_tab.h" +#include <isl_sort.h> + +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) + isl_assert(ctx, facet->n_eq == 0, goto error); + 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 <s, r> > 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 <s, r> > 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 = bset1->ctx; + dir = valid_direction(isl_basic_set_copy(bset1), + isl_basic_set_copy(bset2)); + if (!dir) + goto error; + T = isl_mat_alloc(bset1->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; +} + +/* 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; + struct isl_ctx *ctx; + + map = isl_map_detect_equalities(map); + map = isl_map_align_divs(map); + if (!map) + goto error; + + ctx = map->ctx; + if (map->n == 0) { + convex_hull = isl_basic_map_empty_like_map(map); + isl_map_free(map); + return convex_hull; + } + + 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. + * 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) + return NULL; + if (map->n == 0) { + hull = isl_basic_map_empty_like_map(map); + isl_map_free(map); + return hull; + } + if (map->n == 1) { + hull = isl_basic_map_copy(map->p[0]); + isl_map_free(map); + return hull; + } + + map = isl_map_detect_equalities(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); +} + +/* 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 "set" 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 set 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_set *set_unshifted_simple_hull_from_basic_set_list( + __isl_take isl_set *set, __isl_take isl_basic_set_list *list) +{ + isl_basic_set *model; + isl_basic_set *hull; + + if (!set || !list) + goto error; + + if (isl_basic_set_list_n_basic_set(list) == 0) { + isl_space *space; + + space = isl_set_get_space(set); + isl_set_free(set); + isl_basic_set_list_free(list); + return isl_basic_set_universe(space); + } + if (isl_set_plain_is_empty(set)) { + isl_set_free(set); + return isl_basic_set_list_intersect(list); + } + + set = isl_set_align_divs_to_basic_set_list(set, list); + if (!set) + goto error; + list = isl_basic_set_list_align_divs_to_basic_set(list, set->p[0]); + + model = isl_basic_set_list_get_basic_set(list, 0); + + set = isl_set_to_underlying_set(set); + list = isl_basic_set_list_underlying_set(list); + + hull = uset_unshifted_simple_hull_from_basic_set_list(set, list); + hull = isl_basic_map_overlying_set(hull, model); + + return hull; +error: + isl_set_free(set); + isl_basic_set_list_free(list); + return NULL; +} + +/* Return a sequence of the basic sets that make up the sets in "list". + */ +static __isl_give isl_basic_set_list *collect_basic_sets( + __isl_take isl_set_list *list) +{ + int i, n; + isl_ctx *ctx; + isl_basic_set_list *bset_list; + + if (!list) + return NULL; + n = isl_set_list_n_set(list); + ctx = isl_set_list_get_ctx(list); + bset_list = isl_basic_set_list_alloc(ctx, 0); + + for (i = 0; i < n; ++i) { + isl_set *set; + isl_basic_set_list *list_i; + + set = isl_set_list_get_set(list, i); + set = isl_set_compute_divs(set); + list_i = isl_set_get_basic_set_list(set); + isl_set_free(set); + bset_list = isl_basic_set_list_concat(bset_list, list_i); + } + + isl_set_list_free(list); + return bset_list; +} + +/* Compute a superset of the convex hull of "set" that is described + * by only constraints in the elements of "list". + * + * If "set" 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 sets in the set list and + * continue with set_unshifted_simple_hull_from_basic_set_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) +{ + isl_basic_set_list *bset_list; + int is_universe; + + is_universe = isl_set_plain_is_universe(set); + if (is_universe < 0) + set = isl_set_free(set); + if (is_universe < 0 || is_universe) { + isl_set_list_free(list); + return isl_set_unshifted_simple_hull(set); + } + + bset_list = collect_basic_sets(list); + return set_unshifted_simple_hull_from_basic_set_list(set, bset_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; +} |
