add isl_basic_map_add_constraint

[isl.git] / isl_convex_hull.c

#include "isl_lp.h"
#include "isl_map.h"
#include "isl_map_private.h"
#include "isl_mat.h"
#include "isl_set.h"
#include "isl_seq.h"
#include "isl_equalities.h"

static struct isl_basic_set *uset_convex_hull_wrap(struct isl_set *set);

static void swap_ineq(struct isl_basic_map *bmap, unsigned i, unsigned j)
{
        isl_int *t;

        if (i != j) {
                t = bmap->ineq[i];
                bmap->ineq[i] = bmap->ineq[j];
                bmap->ineq[j] = t;
        }
}

/* 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_solve_lp(*bmap, 0, c+1, (*bmap)->ctx->one, opt_n, opt_d);
        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;
        }
        if (opt_d)
                isl_int_addmul(*opt_n, *opt_d, c[0]);
        else
                isl_int_add(*opt_n, *opt_n, c[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);
}

/* Compute the convex hull of a basic map, by removing the 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.
 */
struct isl_basic_map *isl_basic_map_convex_hull(struct isl_basic_map *bmap)
{
        int i;
        isl_int opt_n;
        isl_int opt_d;
        struct isl_ctx *ctx;

        bmap = isl_basic_map_implicit_equalities(bmap);
        if (!bmap)
                return NULL;

        if (F_ISSET(bmap, ISL_BASIC_MAP_EMPTY))
                return bmap;
        if (F_ISSET(bmap, ISL_BASIC_MAP_NO_REDUNDANT))
                return bmap;

        ctx = bmap->ctx;
        isl_int_init(opt_n);
        isl_int_init(opt_d);
        for (i = bmap->n_ineq-1; i >= 0; --i) {
                int redundant;
                swap_ineq(bmap, i, bmap->n_ineq-1);
                bmap->n_ineq--;
                redundant = isl_basic_map_constraint_is_redundant(&bmap,
                                bmap->ineq[bmap->n_ineq], &opt_n, &opt_d);
                if (redundant == -1)
                        goto error;
                if (F_ISSET(bmap, ISL_BASIC_MAP_EMPTY))
                        break;
                bmap->n_ineq++;
                swap_ineq(bmap, i, bmap->n_ineq-1);
                if (redundant)
                        isl_basic_map_drop_inequality(bmap, i);
        }
        isl_int_clear(opt_n);
        isl_int_clear(opt_d);

        F_SET(bmap, ISL_BASIC_MAP_NO_REDUNDANT);
        return bmap;
error:
        isl_int_clear(opt_n);
        isl_int_clear(opt_d);
        isl_basic_map_free(bmap);
        return NULL;
}

struct isl_basic_set *isl_basic_set_convex_hull(struct isl_basic_set *bset)
{
        return (struct isl_basic_set *)
                isl_basic_map_convex_hull((struct isl_basic_map *)bset);
}

/* 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_ctx *ctx, 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 (F_ISSET(set->p[j], ISL_BASIC_SET_EMPTY))
                        continue;

                res = isl_solve_lp((struct isl_basic_map*)set->p[j],
                                0, c+1, ctx->one, &opt, &opt_denom);
                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 (!isl_int_is_one(opt_denom))
                        isl_seq_scale(c, c, opt_denom, len);
                if (first || isl_int_lt(opt, c[0]))
                        isl_int_set(c[0], opt);
                first = 0;
        }
        isl_int_clear(opt);
        isl_int_clear(opt_denom);
        isl_int_neg(c[0], c[0]);
        return j >= set->n;
error:
        isl_int_clear(opt);
        isl_int_clear(opt_denom);
        return -1;
}

/* Check if "c" is a direction with both a lower bound and an upper
 * bound in "set" that is independent of the previously found "n"
 * bounds in "dirs".
 * If so, add it to the list, with the negative of the lower bound
 * in the constant position, i.e., such that c corresponds to a bounding
 * hyperplane (but not necessarily a facet).
 */
static int is_independent_bound(struct isl_ctx *ctx,
        struct isl_set *set, isl_int *c,
        struct isl_mat *dirs, int n)
{
        int is_bound;
        int i = 0;

        isl_seq_cpy(dirs->row[n]+1, c+1, dirs->n_col-1);
        if (n != 0) {
                int pos = isl_seq_first_non_zero(dirs->row[n]+1, dirs->n_col-1);
                if (pos < 0)
                        return 0;
                for (i = 0; i < n; ++i) {
                        int pos_i;
                        pos_i = isl_seq_first_non_zero(dirs->row[i]+1, dirs->n_col-1);
                        if (pos_i < pos)
                                continue;
                        if (pos_i > pos)
                                break;
                        isl_seq_elim(dirs->row[n]+1, dirs->row[i]+1, pos,
                                        dirs->n_col-1, NULL);
                        pos = isl_seq_first_non_zero(dirs->row[n]+1, dirs->n_col-1);
                        if (pos < 0)
                                return 0;
                }
        }

        isl_seq_neg(dirs->row[n] + 1, dirs->row[n] + 1, dirs->n_col - 1);
        is_bound = uset_is_bound(ctx, set, dirs->row[n], dirs->n_col);
        isl_seq_neg(dirs->row[n] + 1, dirs->row[n] + 1, dirs->n_col - 1);
        if (is_bound != 1)
                return is_bound;
        is_bound = uset_is_bound(ctx, set, dirs->row[n], dirs->n_col);
        if (is_bound != 1)
                return is_bound;
        if (i < n) {
                int k;
                isl_int *t = dirs->row[n];
                for (k = n; k > i; --k)
                        dirs->row[k] = dirs->row[k-1];
                dirs->row[i] = t;
        }
        return 1;
}

/* Compute and return a maximal set of linearly independent bounds
 * on the set "set", based on the constraints of the basic sets
 * in "set".
 */
static struct isl_mat *independent_bounds(struct isl_ctx *ctx,
        struct isl_set *set)
{
        int i, j, n;
        struct isl_mat *dirs = NULL;
        unsigned dim = isl_set_n_dim(set);

        dirs = isl_mat_alloc(ctx, dim, 1+dim);
        if (!dirs)
                goto error;

        n = 0;
        for (i = 0; n < dim && i < set->n; ++i) {
                int f;
                struct isl_basic_set *bset = set->p[i];

                for (j = 0; n < dim && j < bset->n_eq; ++j) {
                        f = is_independent_bound(ctx, set, bset->eq[j],
                                                dirs, n);
                        if (f < 0)
                                goto error;
                        if (f)
                                ++n;
                }
                for (j = 0; n < dim && j < bset->n_ineq; ++j) {
                        f = is_independent_bound(ctx, set, bset->ineq[j],
                                                dirs, n);
                        if (f < 0)
                                goto error;
                        if (f)
                                ++n;
                }
        }
        dirs->n_row = n;
        return dirs;
error:
        isl_mat_free(ctx, dirs);
        return NULL;
}

static struct isl_basic_set *isl_basic_set_set_rational(
        struct isl_basic_set *bset)
{
        if (!bset)
                return NULL;

        if (F_ISSET(bset, ISL_BASIC_MAP_RATIONAL))
                return bset;

        bset = isl_basic_set_cow(bset);
        if (!bset)
                return NULL;

        F_SET(bset, ISL_BASIC_MAP_RATIONAL);

        return isl_basic_set_finalize(bset);
}

static struct isl_set *isl_set_set_rational(struct isl_set *set)
{
        int i;

        set = isl_set_cow(set);
        if (!set)
                return NULL;
        for (i = 0; i < set->n; ++i) {
                set->p[i] = isl_basic_set_set_rational(set->p[i]);
                if (!set->p[i])
                        goto error;
        }
        return set;
error:
        isl_set_free(set);
        return NULL;
}

static struct isl_basic_set *isl_basic_set_add_equality(struct isl_ctx *ctx,
        struct isl_basic_set *bset, isl_int *c)
{
        int i;
        unsigned total;
        unsigned dim;

        if (F_ISSET(bset, ISL_BASIC_SET_EMPTY))
                return bset;

        isl_assert(ctx, isl_basic_set_n_param(bset) == 0, goto error);
        isl_assert(ctx, bset->n_div == 0, goto error);
        dim = isl_basic_set_n_dim(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_equality(struct isl_ctx *ctx,
        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(ctx, 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 contraints
 *
 *                                  [ 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_ctx *ctx,
                                                        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(ctx, 0, dim * set->n, 0, n_eq, n_ineq);
        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.
 */
static isl_int *wrap_facet(struct isl_ctx *ctx, struct isl_set *set,
        isl_int *facet, isl_int *ridge)
{
        int i;
        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;

        set = isl_set_copy(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(ctx, T);
        set = isl_set_preimage(ctx, set, T);
        T = NULL;
        if (!set)
                goto error;
        lp = wrap_constraints(ctx, set);
        obj = isl_vec_alloc(ctx, dim*set->n);
        if (!obj)
                goto error;
        for (i = 0; i < set->n; ++i) {
                isl_seq_clr(obj->block.data+dim*i, 2);
                isl_int_set_si(obj->block.data[dim*i+2], 1);
                isl_seq_clr(obj->block.data+dim*i+3, dim-3);
        }
        isl_int_init(num);
        isl_int_init(den);
        res = isl_solve_lp((struct isl_basic_map *)lp, 0,
                                        obj->block.data, ctx->one, &num, &den);
        if (res == isl_lp_ok) {
                isl_int_neg(num, num);
                isl_seq_combine(facet, num, facet, den, ridge, dim);
        }
        isl_int_clear(num);
        isl_int_clear(den);
        isl_vec_free(ctx, obj);
        isl_basic_set_free(lp);
        isl_set_free(set);
        isl_assert(ctx, res == isl_lp_ok || res == isl_lp_unbounded, 
                   return NULL);
        return facet;
error:
        isl_basic_set_free(lp);
        isl_mat_free(ctx, T);
        isl_set_free(set);
        return NULL;
}

/* Given a set of d linearly independent bounding constraints of the
 * convex hull of "set", compute the constraint of a facet of "set".
 *
 * We first compute the intersection with the first bounding hyperplane
 * and remove the component corresponding to this hyperplane from
 * other bounds (in homogeneous space).
 * We then wrap around one of the remaining bounding constraints
 * and continue the process until all bounding constraints have been
 * taken into account.
 * The resulting linear combination of the bounding constraints will
 * correspond to a facet of the convex hull.
 */
static struct isl_mat *initial_facet_constraint(struct isl_ctx *ctx,
        struct isl_set *set, struct isl_mat *bounds)
{
        struct isl_set *slice = NULL;
        struct isl_basic_set *face = NULL;
        struct isl_mat *m, *U, *Q;
        int i;
        unsigned dim = isl_set_n_dim(set);

        isl_assert(ctx, set->n > 0, goto error);
        isl_assert(ctx, bounds->n_row == dim, goto error);

        while (bounds->n_row > 1) {
                slice = isl_set_copy(set);
                slice = isl_set_add_equality(ctx, 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;
                }
                m = isl_mat_alloc(ctx, 1 + face->n_eq, 1 + dim);
                if (!m)
                        goto error;
                isl_int_set_si(m->row[0][0], 1);
                isl_seq_clr(m->row[0]+1, dim);
                for (i = 0; i < face->n_eq; ++i)
                        isl_seq_cpy(m->row[1 + i], face->eq[i], 1 + dim);
                U = isl_mat_right_inverse(ctx, m);
                Q = isl_mat_right_inverse(ctx, isl_mat_copy(ctx, U));
                U = isl_mat_drop_cols(ctx, U, 1 + face->n_eq,
                                                dim - face->n_eq);
                Q = isl_mat_drop_rows(ctx, Q, 1 + face->n_eq,
                                                dim - face->n_eq);
                U = isl_mat_drop_cols(ctx, U, 0, 1);
                Q = isl_mat_drop_rows(ctx, Q, 0, 1);
                bounds = isl_mat_product(ctx, bounds, U);
                bounds = isl_mat_product(ctx, bounds, Q);
                while (isl_seq_first_non_zero(bounds->row[bounds->n_row-1],
                                              bounds->n_col) == -1) {
                        bounds->n_row--;
                        isl_assert(ctx, bounds->n_row > 1, goto error);
                }
                if (!wrap_facet(ctx, set, bounds->row[0],
                                          bounds->row[bounds->n_row-1]))
                        goto error;
                isl_basic_set_free(face);
                bounds->n_row--;
        }
        return bounds;
error:
        isl_basic_set_free(face);
        isl_mat_free(ctx, 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_ctx *ctx,
        struct isl_set *set, isl_int *c)
{
        struct isl_mat *m, *U, *Q;
        struct isl_basic_set *facet;
        unsigned dim;

        set = isl_set_copy(set);
        dim = isl_set_n_dim(set);
        m = isl_mat_alloc(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(ctx, m);
        Q = isl_mat_right_inverse(ctx, isl_mat_copy(ctx, U));
        U = isl_mat_drop_cols(ctx, U, 1, 1);
        Q = isl_mat_drop_rows(ctx, Q, 1, 1);
        set = isl_set_preimage(ctx, set, U);
        facet = uset_convex_hull_wrap(set);
        facet = isl_basic_set_preimage(ctx, facet, Q);
        return facet;
error:
        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.
 *
 * 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_ctx *ctx, struct isl_set *set,
        struct isl_mat *initial)
{
        int i, j, f;
        int k;
        struct isl_basic_set *hull = NULL;
        struct isl_basic_set *facet = NULL;
        unsigned n_ineq;
        unsigned total;
        unsigned dim;

        isl_assert(ctx, set->n > 0, goto error);

        n_ineq = 1;
        for (i = 0; i < set->n; ++i) {
                n_ineq += set->p[i]->n_eq;
                n_ineq += set->p[i]->n_ineq;
        }
        dim = isl_set_n_dim(set);
        isl_assert(ctx, 1 + dim == initial->n_col, goto error);
        hull = isl_basic_set_alloc(ctx, 0, dim, 0, 0, n_ineq);
        hull = isl_basic_set_set_rational(hull);
        if (!hull)
                goto error;
        k = isl_basic_set_alloc_inequality(hull);
        if (k < 0)
                goto error;
        isl_seq_cpy(hull->ineq[k], initial->row[0], initial->n_col);
        for (i = 0; i < hull->n_ineq; ++i) {
                facet = compute_facet(ctx, set, hull->ineq[i]);
                if (!facet)
                        goto error;
                if (facet->n_ineq + hull->n_ineq > n_ineq) {
                        hull = isl_basic_set_extend(hull,
                                0, dim, 0, 0, facet->n_ineq);
                        n_ineq = hull->n_ineq + facet->n_ineq;
                }
                for (j = 0; j < facet->n_ineq; ++j) {
                        k = isl_basic_set_alloc_inequality(hull);
                        if (k < 0)
                                goto error;
                        isl_seq_cpy(hull->ineq[k], hull->ineq[i], 1+dim);
                        if (!wrap_facet(ctx, set, hull->ineq[k], facet->ineq[j]))
                                goto error;
                        for (f = 0; f < k; ++f)
                                if (isl_seq_eq(hull->ineq[f], hull->ineq[k],
                                                1+dim))
                                        break;
                        if (f < k)
                                isl_basic_set_free_inequality(hull, 1);
                }
                isl_basic_set_free(facet);
        }
        hull = isl_basic_set_simplify(hull);
        hull = isl_basic_set_finalize(hull);
        return hull;
error:
        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_ctx *ctx,
        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(ctx, set->n > 0, goto error);
        c = isl_mat_alloc(ctx, 2, 2);
        if (!c)
                goto error;

        if (set->p[0]->n_eq > 0) {
                isl_assert(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(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(ctx, c);
        return hull;
error:
        isl_set_free(set);
        isl_mat_free(ctx, c);
        return NULL;
}

/* Project out final n dimensions using Fourier-Motzkin */
static struct isl_set *set_project_out(struct isl_ctx *ctx,
        struct isl_set *set, unsigned n)
{
        return isl_set_remove_dims(set, isl_set_n_dim(set) - n, n);
}

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_dim_copy(set->dim));
        else
                convex_hull = isl_basic_set_universe(isl_dim_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(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, dim, 2*(1+dim));
        hull = isl_basic_set_convex_hull(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;
}

/* Compute the convex hull of a set without any parameters or
 * integer divisions using Fourier-Motzkin elimination.
 * In each step, we combined two basic sets until only one
 * basic set is left.
 */
static struct isl_basic_set *uset_convex_hull_elim(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);
        }
        isl_set_free(set);
        return convex_hull;
error:
        isl_set_free(set);
        isl_basic_set_free(convex_hull);
        return NULL;
}

static struct isl_basic_set *uset_convex_hull_wrap_with_bounds(
        struct isl_set *set, struct isl_mat *bounds)
{
        struct isl_basic_set *convex_hull = NULL;

        isl_assert(set->ctx, bounds->n_row == isl_set_n_dim(set), goto error);
        bounds = initial_facet_constraint(set->ctx, set, bounds);
        if (!bounds)
                goto error;
        convex_hull = extend(set->ctx, set, bounds);
        isl_mat_free(set->ctx, bounds);
        isl_set_free(set);

        return convex_hull;
error:
        isl_set_free(set);
        return NULL;
}

/* 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)
{
        int i;
        struct isl_basic_set *convex_hull = NULL;
        struct isl_mat *bounds;

        if (isl_set_n_dim(set) == 0)
                return convex_hull_0d(set);

        set = isl_set_set_rational(set);

        if (!set)
                goto error;
        for (i = 0; i < set->n; ++i) {
                set->p[i] = isl_basic_set_convex_hull(set->p[i]);
                if (!set->p[i])
                        goto error;
        }
        set = isl_set_remove_empty_parts(set);
        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->ctx, set);

        bounds = independent_bounds(set->ctx, set);
        if (!bounds)
                goto error;
        if (bounds->n_row == isl_set_n_dim(set))
                return uset_convex_hull_wrap_with_bounds(set, bounds);
        isl_mat_free(set->ctx, bounds);

        return uset_convex_hull_elim(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(struct isl_set *set)
{
        int i;
        struct isl_basic_set *convex_hull = NULL;
        struct isl_mat *bounds;

        if (isl_set_n_dim(set) == 0) {
                convex_hull = isl_basic_set_universe(isl_dim_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);

        if (!set)
                goto error;
        for (i = 0; i < set->n; ++i) {
                set->p[i] = isl_basic_set_convex_hull(set->p[i]);
                if (!set->p[i])
                        goto error;
        }
        set = isl_set_remove_empty_parts(set);
        if (!set)
                goto error;
        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->ctx, set);

        bounds = independent_bounds(set->ctx, set);
        if (!bounds)
                goto error;
        return uset_convex_hull_wrap_with_bounds(set, bounds);
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_ctx *ctx,
        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(ctx, set, T);
        convex_hull = uset_convex_hull(set);
        convex_hull = isl_basic_set_preimage(ctx, 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;

        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;
        }

        map = isl_map_align_divs(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(ctx, set, affine_hull);
        else {
                isl_basic_set_free(affine_hull);
                bset = uset_convex_hull(set);
        }

        convex_hull = isl_basic_map_overlying_set(bset, model);

        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);
}

/* Compute a superset of the convex hull of map that is described
 * by only translates of the constraints in the constituents of map.
 *
 * The implementation is not very efficient.  In particular, if
 * constraints with the same normal appear in more than one
 * basic map, they will be (re)examined each time.
 */
struct isl_basic_map *isl_map_simple_hull(struct isl_map *map)
{
        struct isl_set *set = NULL;
        struct isl_basic_map *model = NULL;
        struct isl_basic_map *hull;
        struct isl_basic_set *bset = NULL;
        int i, j;
        unsigned n_ineq;
        unsigned dim;

        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_align_divs(map);
        model = isl_basic_map_copy(map->p[0]);

        n_ineq = 0;
        for (i = 0; i < map->n; ++i) {
                if (!map->p[i])
                        goto error;
                n_ineq += map->p[i]->n_ineq;
        }

        set = isl_map_underlying_set(map);
        if (!set)
                goto error;

        bset = isl_set_affine_hull(isl_set_copy(set));
        if (!bset)
                goto error;
        dim = isl_basic_set_n_dim(bset);
        bset = isl_basic_set_extend(bset, 0, dim, 0, 0, n_ineq);
        if (!bset)
                goto error;

        for (i = 0; i < set->n; ++i) {
                for (j = 0; j < set->p[i]->n_ineq; ++j) {
                        int k;
                        int is_bound;

                        k = isl_basic_set_alloc_inequality(bset);
                        if (k < 0)
                                goto error;
                        isl_seq_cpy(bset->ineq[k], set->p[i]->ineq[j], 1 + dim);
                        is_bound = uset_is_bound(set->ctx, set, bset->ineq[k],
                                                        1 + dim);
                        if (is_bound < 0)
                                goto error;
                        if (!is_bound)
                                isl_basic_set_free_inequality(bset, 1);
                }
        }

        bset = isl_basic_set_simplify(bset);
        bset = isl_basic_set_finalize(bset);
        bset = isl_basic_set_convex_hull(bset);

        hull = isl_basic_map_overlying_set(bset, model);

        isl_set_free(set);
        return hull;
error:
        isl_basic_set_free(bset);
        isl_set_free(set);
        isl_basic_map_free(model);
        return NULL;
}

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);
}