include/isl/val.h: add missing include

[isl.git] / isl_transitive_closure.c

/*
 * Copyright 2010      INRIA Saclay
 *
 * Use of this software is governed by the MIT license
 *
 * Written by Sven Verdoolaege, INRIA Saclay - Ile-de-France,
 * Parc Club Orsay Universite, ZAC des vignes, 4 rue Jacques Monod,
 * 91893 Orsay, France 
 */

#include <isl_ctx_private.h>
#include <isl_map_private.h>
#include <isl/map.h>
#include <isl_seq.h>
#include <isl_space_private.h>
#include <isl_lp_private.h>
#include <isl/union_map.h>
#include <isl_mat_private.h>
#include <isl_vec_private.h>
#include <isl_options_private.h>
#include <isl_tarjan.h>

int isl_map_is_transitively_closed(__isl_keep isl_map *map)
{
        isl_map *map2;
        int closed;

        map2 = isl_map_apply_range(isl_map_copy(map), isl_map_copy(map));
        closed = isl_map_is_subset(map2, map);
        isl_map_free(map2);

        return closed;
}

int isl_union_map_is_transitively_closed(__isl_keep isl_union_map *umap)
{
        isl_union_map *umap2;
        int closed;

        umap2 = isl_union_map_apply_range(isl_union_map_copy(umap),
                                          isl_union_map_copy(umap));
        closed = isl_union_map_is_subset(umap2, umap);
        isl_union_map_free(umap2);

        return closed;
}
 
/* Given a map that represents a path with the length of the path
 * encoded as the difference between the last output coordindate
 * and the last input coordinate, set this length to either
 * exactly "length" (if "exactly" is set) or at least "length"
 * (if "exactly" is not set).
 */
static __isl_give isl_map *set_path_length(__isl_take isl_map *map,
        int exactly, int length)
{
        isl_space *dim;
        struct isl_basic_map *bmap;
        unsigned d;
        unsigned nparam;
        int k;
        isl_int *c;

        if (!map)
                return NULL;

        dim = isl_map_get_space(map);
        d = isl_space_dim(dim, isl_dim_in);
        nparam = isl_space_dim(dim, isl_dim_param);
        bmap = isl_basic_map_alloc_space(dim, 0, 1, 1);
        if (exactly) {
                k = isl_basic_map_alloc_equality(bmap);
                if (k < 0)
                        goto error;
                c = bmap->eq[k];
        } else {
                k = isl_basic_map_alloc_inequality(bmap);
                if (k < 0)
                        goto error;
                c = bmap->ineq[k];
        }
        isl_seq_clr(c, 1 + isl_basic_map_total_dim(bmap));
        isl_int_set_si(c[0], -length);
        isl_int_set_si(c[1 + nparam + d - 1], -1);
        isl_int_set_si(c[1 + nparam + d + d - 1], 1);

        bmap = isl_basic_map_finalize(bmap);
        map = isl_map_intersect(map, isl_map_from_basic_map(bmap));

        return map;
error:
        isl_basic_map_free(bmap);
        isl_map_free(map);
        return NULL;
}

/* Check whether the overapproximation of the power of "map" is exactly
 * the power of "map".  Let R be "map" and A_k the overapproximation.
 * The approximation is exact if
 *
 *      A_1 = R
 *      A_k = A_{k-1} \circ R                   k >= 2
 *
 * Since A_k is known to be an overapproximation, we only need to check
 *
 *      A_1 \subset R
 *      A_k \subset A_{k-1} \circ R             k >= 2
 *
 * In practice, "app" has an extra input and output coordinate
 * to encode the length of the path.  So, we first need to add
 * this coordinate to "map" and set the length of the path to
 * one.
 */
static int check_power_exactness(__isl_take isl_map *map,
        __isl_take isl_map *app)
{
        int exact;
        isl_map *app_1;
        isl_map *app_2;

        map = isl_map_add_dims(map, isl_dim_in, 1);
        map = isl_map_add_dims(map, isl_dim_out, 1);
        map = set_path_length(map, 1, 1);

        app_1 = set_path_length(isl_map_copy(app), 1, 1);

        exact = isl_map_is_subset(app_1, map);
        isl_map_free(app_1);

        if (!exact || exact < 0) {
                isl_map_free(app);
                isl_map_free(map);
                return exact;
        }

        app_1 = set_path_length(isl_map_copy(app), 0, 1);
        app_2 = set_path_length(app, 0, 2);
        app_1 = isl_map_apply_range(map, app_1);

        exact = isl_map_is_subset(app_2, app_1);

        isl_map_free(app_1);
        isl_map_free(app_2);

        return exact;
}

/* Check whether the overapproximation of the power of "map" is exactly
 * the power of "map", possibly after projecting out the power (if "project"
 * is set).
 *
 * If "project" is set and if "steps" can only result in acyclic paths,
 * then we check
 *
 *      A = R \cup (A \circ R)
 *
 * where A is the overapproximation with the power projected out, i.e.,
 * an overapproximation of the transitive closure.
 * More specifically, since A is known to be an overapproximation, we check
 *
 *      A \subset R \cup (A \circ R)
 *
 * Otherwise, we check if the power is exact.
 *
 * Note that "app" has an extra input and output coordinate to encode
 * the length of the part.  If we are only interested in the transitive
 * closure, then we can simply project out these coordinates first.
 */
static int check_exactness(__isl_take isl_map *map, __isl_take isl_map *app,
        int project)
{
        isl_map *test;
        int exact;
        unsigned d;

        if (!project)
                return check_power_exactness(map, app);

        d = isl_map_dim(map, isl_dim_in);
        app = set_path_length(app, 0, 1);
        app = isl_map_project_out(app, isl_dim_in, d, 1);
        app = isl_map_project_out(app, isl_dim_out, d, 1);

        app = isl_map_reset_space(app, isl_map_get_space(map));

        test = isl_map_apply_range(isl_map_copy(map), isl_map_copy(app));
        test = isl_map_union(test, isl_map_copy(map));

        exact = isl_map_is_subset(app, test);

        isl_map_free(app);
        isl_map_free(test);

        isl_map_free(map);

        return exact;
}

/*
 * The transitive closure implementation is based on the paper
 * "Computing the Transitive Closure of a Union of Affine Integer
 * Tuple Relations" by Anna Beletska, Denis Barthou, Wlodzimierz Bielecki and
 * Albert Cohen.
 */

/* Given a set of n offsets v_i (the rows of "steps"), construct a relation
 * of the given dimension specification (Z^{n+1} -> Z^{n+1})
 * that maps an element x to any element that can be reached
 * by taking a non-negative number of steps along any of
 * the extended offsets v'_i = [v_i 1].
 * That is, construct
 *
 * { [x] -> [y] : exists k_i >= 0, y = x + \sum_i k_i v'_i }
 *
 * For any element in this relation, the number of steps taken
 * is equal to the difference in the final coordinates.
 */
static __isl_give isl_map *path_along_steps(__isl_take isl_space *dim,
        __isl_keep isl_mat *steps)
{
        int i, j, k;
        struct isl_basic_map *path = NULL;
        unsigned d;
        unsigned n;
        unsigned nparam;

        if (!dim || !steps)
                goto error;

        d = isl_space_dim(dim, isl_dim_in);
        n = steps->n_row;
        nparam = isl_space_dim(dim, isl_dim_param);

        path = isl_basic_map_alloc_space(isl_space_copy(dim), n, d, n);

        for (i = 0; i < n; ++i) {
                k = isl_basic_map_alloc_div(path);
                if (k < 0)
                        goto error;
                isl_assert(steps->ctx, i == k, goto error);
                isl_int_set_si(path->div[k][0], 0);
        }

        for (i = 0; i < d; ++i) {
                k = isl_basic_map_alloc_equality(path);
                if (k < 0)
                        goto error;
                isl_seq_clr(path->eq[k], 1 + isl_basic_map_total_dim(path));
                isl_int_set_si(path->eq[k][1 + nparam + i], 1);
                isl_int_set_si(path->eq[k][1 + nparam + d + i], -1);
                if (i == d - 1)
                        for (j = 0; j < n; ++j)
                                isl_int_set_si(path->eq[k][1 + nparam + 2 * d + j], 1);
                else
                        for (j = 0; j < n; ++j)
                                isl_int_set(path->eq[k][1 + nparam + 2 * d + j],
                                            steps->row[j][i]);
        }

        for (i = 0; i < n; ++i) {
                k = isl_basic_map_alloc_inequality(path);
                if (k < 0)
                        goto error;
                isl_seq_clr(path->ineq[k], 1 + isl_basic_map_total_dim(path));
                isl_int_set_si(path->ineq[k][1 + nparam + 2 * d + i], 1);
        }

        isl_space_free(dim);

        path = isl_basic_map_simplify(path);
        path = isl_basic_map_finalize(path);
        return isl_map_from_basic_map(path);
error:
        isl_space_free(dim);
        isl_basic_map_free(path);
        return NULL;
}

#define IMPURE          0
#define PURE_PARAM      1
#define PURE_VAR        2
#define MIXED           3

/* Check whether the parametric constant term of constraint c is never
 * positive in "bset".
 */
static isl_bool parametric_constant_never_positive(
        __isl_keep isl_basic_set *bset, isl_int *c, int *div_purity)
{
        unsigned d;
        unsigned n_div;
        unsigned nparam;
        int i;
        int k;
        isl_bool empty;

        n_div = isl_basic_set_dim(bset, isl_dim_div);
        d = isl_basic_set_dim(bset, isl_dim_set);
        nparam = isl_basic_set_dim(bset, isl_dim_param);

        bset = isl_basic_set_copy(bset);
        bset = isl_basic_set_cow(bset);
        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_seq_cpy(bset->ineq[k], c, 1 + nparam);
        for (i = 0; i < n_div; ++i) {
                if (div_purity[i] != PURE_PARAM)
                        continue;
                isl_int_set(bset->ineq[k][1 + nparam + d + i],
                            c[1 + nparam + d + i]);
        }
        isl_int_sub_ui(bset->ineq[k][0], bset->ineq[k][0], 1);
        empty = isl_basic_set_is_empty(bset);
        isl_basic_set_free(bset);

        return empty;
error:
        isl_basic_set_free(bset);
        return isl_bool_error;
}

/* Return PURE_PARAM if only the coefficients of the parameters are non-zero.
 * Return PURE_VAR if only the coefficients of the set variables are non-zero.
 * Return MIXED if only the coefficients of the parameters and the set
 *      variables are non-zero and if moreover the parametric constant
 *      can never attain positive values.
 * Return IMPURE otherwise.
 */
static int purity(__isl_keep isl_basic_set *bset, isl_int *c, int *div_purity,
        int eq)
{
        unsigned d;
        unsigned n_div;
        unsigned nparam;
        isl_bool empty;
        int i;
        int p = 0, v = 0;

        n_div = isl_basic_set_dim(bset, isl_dim_div);
        d = isl_basic_set_dim(bset, isl_dim_set);
        nparam = isl_basic_set_dim(bset, isl_dim_param);

        for (i = 0; i < n_div; ++i) {
                if (isl_int_is_zero(c[1 + nparam + d + i]))
                        continue;
                switch (div_purity[i]) {
                case PURE_PARAM: p = 1; break;
                case PURE_VAR: v = 1; break;
                default: return IMPURE;
                }
        }
        if (!p && isl_seq_first_non_zero(c + 1, nparam) == -1)
                return PURE_VAR;
        if (!v && isl_seq_first_non_zero(c + 1 + nparam, d) == -1)
                return PURE_PARAM;

        empty = parametric_constant_never_positive(bset, c, div_purity);
        if (eq && empty >= 0 && !empty) {
                isl_seq_neg(c, c, 1 + nparam + d + n_div);
                empty = parametric_constant_never_positive(bset, c, div_purity);
        }

        return empty < 0 ? -1 : empty ? MIXED : IMPURE;
}

/* Return an array of integers indicating the type of each div in bset.
 * If the div is (recursively) defined in terms of only the parameters,
 * then the type is PURE_PARAM.
 * If the div is (recursively) defined in terms of only the set variables,
 * then the type is PURE_VAR.
 * Otherwise, the type is IMPURE.
 */
static __isl_give int *get_div_purity(__isl_keep isl_basic_set *bset)
{
        int i, j;
        int *div_purity;
        unsigned d;
        unsigned n_div;
        unsigned nparam;

        if (!bset)
                return NULL;

        n_div = isl_basic_set_dim(bset, isl_dim_div);
        d = isl_basic_set_dim(bset, isl_dim_set);
        nparam = isl_basic_set_dim(bset, isl_dim_param);

        div_purity = isl_alloc_array(bset->ctx, int, n_div);
        if (n_div && !div_purity)
                return NULL;

        for (i = 0; i < bset->n_div; ++i) {
                int p = 0, v = 0;
                if (isl_int_is_zero(bset->div[i][0])) {
                        div_purity[i] = IMPURE;
                        continue;
                }
                if (isl_seq_first_non_zero(bset->div[i] + 2, nparam) != -1)
                        p = 1;
                if (isl_seq_first_non_zero(bset->div[i] + 2 + nparam, d) != -1)
                        v = 1;
                for (j = 0; j < i; ++j) {
                        if (isl_int_is_zero(bset->div[i][2 + nparam + d + j]))
                                continue;
                        switch (div_purity[j]) {
                        case PURE_PARAM: p = 1; break;
                        case PURE_VAR: v = 1; break;
                        default: p = v = 1; break;
                        }
                }
                div_purity[i] = v ? p ? IMPURE : PURE_VAR : PURE_PARAM;
        }

        return div_purity;
}

/* Given a path with the as yet unconstrained length at position "pos",
 * check if setting the length to zero results in only the identity
 * mapping.
 */
static int empty_path_is_identity(__isl_keep isl_basic_map *path, unsigned pos)
{
        isl_basic_map *test = NULL;
        isl_basic_map *id = NULL;
        int k;
        int is_id;

        test = isl_basic_map_copy(path);
        test = isl_basic_map_extend_constraints(test, 1, 0);
        k = isl_basic_map_alloc_equality(test);
        if (k < 0)
                goto error;
        isl_seq_clr(test->eq[k], 1 + isl_basic_map_total_dim(test));
        isl_int_set_si(test->eq[k][pos], 1);
        test = isl_basic_map_gauss(test, NULL);
        id = isl_basic_map_identity(isl_basic_map_get_space(path));
        is_id = isl_basic_map_is_equal(test, id);
        isl_basic_map_free(test);
        isl_basic_map_free(id);
        return is_id;
error:
        isl_basic_map_free(test);
        return -1;
}

/* If any of the constraints is found to be impure then this function
 * sets *impurity to 1.
 *
 * If impurity is NULL then we are dealing with a non-parametric set
 * and so the constraints are obviously PURE_VAR.
 */
static __isl_give isl_basic_map *add_delta_constraints(
        __isl_take isl_basic_map *path,
        __isl_keep isl_basic_set *delta, unsigned off, unsigned nparam,
        unsigned d, int *div_purity, int eq, int *impurity)
{
        int i, k;
        int n = eq ? delta->n_eq : delta->n_ineq;
        isl_int **delta_c = eq ? delta->eq : delta->ineq;
        unsigned n_div;

        n_div = isl_basic_set_dim(delta, isl_dim_div);

        for (i = 0; i < n; ++i) {
                isl_int *path_c;
                int p = PURE_VAR;
                if (impurity)
                        p = purity(delta, delta_c[i], div_purity, eq);
                if (p < 0)
                        goto error;
                if (p != PURE_VAR && p != PURE_PARAM && !*impurity)
                        *impurity = 1;
                if (p == IMPURE)
                        continue;
                if (eq && p != MIXED) {
                        k = isl_basic_map_alloc_equality(path);
                        if (k < 0)
                                goto error;
                        path_c = path->eq[k];
                } else {
                        k = isl_basic_map_alloc_inequality(path);
                        if (k < 0)
                                goto error;
                        path_c = path->ineq[k];
                }
                isl_seq_clr(path_c, 1 + isl_basic_map_total_dim(path));
                if (p == PURE_VAR) {
                        isl_seq_cpy(path_c + off,
                                    delta_c[i] + 1 + nparam, d);
                        isl_int_set(path_c[off + d], delta_c[i][0]);
                } else if (p == PURE_PARAM) {
                        isl_seq_cpy(path_c, delta_c[i], 1 + nparam);
                } else {
                        isl_seq_cpy(path_c + off,
                                    delta_c[i] + 1 + nparam, d);
                        isl_seq_cpy(path_c, delta_c[i], 1 + nparam);
                }
                isl_seq_cpy(path_c + off - n_div,
                            delta_c[i] + 1 + nparam + d, n_div);
        }

        return path;
error:
        isl_basic_map_free(path);
        return NULL;
}

/* Given a set of offsets "delta", construct a relation of the
 * given dimension specification (Z^{n+1} -> Z^{n+1}) that
 * is an overapproximation of the relations that
 * maps an element x to any element that can be reached
 * by taking a non-negative number of steps along any of
 * the elements in "delta".
 * That is, construct an approximation of
 *
 *      { [x] -> [y] : exists f \in \delta, k \in Z :
 *                                      y = x + k [f, 1] and k >= 0 }
 *
 * For any element in this relation, the number of steps taken
 * is equal to the difference in the final coordinates.
 *
 * In particular, let delta be defined as
 *
 *      \delta = [p] -> { [x] : A x + a >= 0 and B p + b >= 0 and
 *                              C x + C'p + c >= 0 and
 *                              D x + D'p + d >= 0 }
 *
 * where the constraints C x + C'p + c >= 0 are such that the parametric
 * constant term of each constraint j, "C_j x + C'_j p + c_j",
 * can never attain positive values, then the relation is constructed as
 *
 *      { [x] -> [y] : exists [f, k] \in Z^{n+1} : y = x + f and
 *                      A f + k a >= 0 and B p + b >= 0 and
 *                      C f + C'p + c >= 0 and k >= 1 }
 *      union { [x] -> [x] }
 *
 * If the zero-length paths happen to correspond exactly to the identity
 * mapping, then we return
 *
 *      { [x] -> [y] : exists [f, k] \in Z^{n+1} : y = x + f and
 *                      A f + k a >= 0 and B p + b >= 0 and
 *                      C f + C'p + c >= 0 and k >= 0 }
 *
 * instead.
 *
 * Existentially quantified variables in \delta are handled by
 * classifying them as independent of the parameters, purely
 * parameter dependent and others.  Constraints containing
 * any of the other existentially quantified variables are removed.
 * This is safe, but leads to an additional overapproximation.
 *
 * If there are any impure constraints, then we also eliminate
 * the parameters from \delta, resulting in a set
 *
 *      \delta' = { [x] : E x + e >= 0 }
 *
 * and add the constraints
 *
 *                      E f + k e >= 0
 *
 * to the constructed relation.
 */
static __isl_give isl_map *path_along_delta(__isl_take isl_space *dim,
        __isl_take isl_basic_set *delta)
{
        isl_basic_map *path = NULL;
        unsigned d;
        unsigned n_div;
        unsigned nparam;
        unsigned off;
        int i, k;
        int is_id;
        int *div_purity = NULL;
        int impurity = 0;

        if (!delta)
                goto error;
        n_div = isl_basic_set_dim(delta, isl_dim_div);
        d = isl_basic_set_dim(delta, isl_dim_set);
        nparam = isl_basic_set_dim(delta, isl_dim_param);
        path = isl_basic_map_alloc_space(isl_space_copy(dim), n_div + d + 1,
                        d + 1 + delta->n_eq, delta->n_eq + delta->n_ineq + 1);
        off = 1 + nparam + 2 * (d + 1) + n_div;

        for (i = 0; i < n_div + d + 1; ++i) {
                k = isl_basic_map_alloc_div(path);
                if (k < 0)
                        goto error;
                isl_int_set_si(path->div[k][0], 0);
        }

        for (i = 0; i < d + 1; ++i) {
                k = isl_basic_map_alloc_equality(path);
                if (k < 0)
                        goto error;
                isl_seq_clr(path->eq[k], 1 + isl_basic_map_total_dim(path));
                isl_int_set_si(path->eq[k][1 + nparam + i], 1);
                isl_int_set_si(path->eq[k][1 + nparam + d + 1 + i], -1);
                isl_int_set_si(path->eq[k][off + i], 1);
        }

        div_purity = get_div_purity(delta);
        if (n_div && !div_purity)
                goto error;

        path = add_delta_constraints(path, delta, off, nparam, d,
                                     div_purity, 1, &impurity);
        path = add_delta_constraints(path, delta, off, nparam, d,
                                     div_purity, 0, &impurity);
        if (impurity) {
                isl_space *dim = isl_basic_set_get_space(delta);
                delta = isl_basic_set_project_out(delta,
                                                  isl_dim_param, 0, nparam);
                delta = isl_basic_set_add_dims(delta, isl_dim_param, nparam);
                delta = isl_basic_set_reset_space(delta, dim);
                if (!delta)
                        goto error;
                path = isl_basic_map_extend_constraints(path, delta->n_eq,
                                                        delta->n_ineq + 1);
                path = add_delta_constraints(path, delta, off, nparam, d,
                                             NULL, 1, NULL);
                path = add_delta_constraints(path, delta, off, nparam, d,
                                             NULL, 0, NULL);
                path = isl_basic_map_gauss(path, NULL);
        }

        is_id = empty_path_is_identity(path, off + d);
        if (is_id < 0)
                goto error;

        k = isl_basic_map_alloc_inequality(path);
        if (k < 0)
                goto error;
        isl_seq_clr(path->ineq[k], 1 + isl_basic_map_total_dim(path));
        if (!is_id)
                isl_int_set_si(path->ineq[k][0], -1);
        isl_int_set_si(path->ineq[k][off + d], 1);
                        
        free(div_purity);
        isl_basic_set_free(delta);
        path = isl_basic_map_finalize(path);
        if (is_id) {
                isl_space_free(dim);
                return isl_map_from_basic_map(path);
        }
        return isl_basic_map_union(path, isl_basic_map_identity(dim));
error:
        free(div_purity);
        isl_space_free(dim);
        isl_basic_set_free(delta);
        isl_basic_map_free(path);
        return NULL;
}

/* Given a dimension specification Z^{n+1} -> Z^{n+1} and a parameter "param",
 * construct a map that equates the parameter to the difference
 * in the final coordinates and imposes that this difference is positive.
 * That is, construct
 *
 *      { [x,x_s] -> [y,y_s] : k = y_s - x_s > 0 }
 */
static __isl_give isl_map *equate_parameter_to_length(__isl_take isl_space *dim,
        unsigned param)
{
        struct isl_basic_map *bmap;
        unsigned d;
        unsigned nparam;
        int k;

        d = isl_space_dim(dim, isl_dim_in);
        nparam = isl_space_dim(dim, isl_dim_param);
        bmap = isl_basic_map_alloc_space(dim, 0, 1, 1);
        k = isl_basic_map_alloc_equality(bmap);
        if (k < 0)
                goto error;
        isl_seq_clr(bmap->eq[k], 1 + isl_basic_map_total_dim(bmap));
        isl_int_set_si(bmap->eq[k][1 + param], -1);
        isl_int_set_si(bmap->eq[k][1 + nparam + d - 1], -1);
        isl_int_set_si(bmap->eq[k][1 + nparam + d + d - 1], 1);

        k = isl_basic_map_alloc_inequality(bmap);
        if (k < 0)
                goto error;
        isl_seq_clr(bmap->ineq[k], 1 + isl_basic_map_total_dim(bmap));
        isl_int_set_si(bmap->ineq[k][1 + param], 1);
        isl_int_set_si(bmap->ineq[k][0], -1);

        bmap = isl_basic_map_finalize(bmap);
        return isl_map_from_basic_map(bmap);
error:
        isl_basic_map_free(bmap);
        return NULL;
}

/* Check whether "path" is acyclic, where the last coordinates of domain
 * and range of path encode the number of steps taken.
 * That is, check whether
 *
 *      { d | d = y - x and (x,y) in path }
 *
 * does not contain any element with positive last coordinate (positive length)
 * and zero remaining coordinates (cycle).
 */
static int is_acyclic(__isl_take isl_map *path)
{
        int i;
        int acyclic;
        unsigned dim;
        struct isl_set *delta;

        delta = isl_map_deltas(path);
        dim = isl_set_dim(delta, isl_dim_set);
        for (i = 0; i < dim; ++i) {
                if (i == dim -1)
                        delta = isl_set_lower_bound_si(delta, isl_dim_set, i, 1);
                else
                        delta = isl_set_fix_si(delta, isl_dim_set, i, 0);
        }

        acyclic = isl_set_is_empty(delta);
        isl_set_free(delta);

        return acyclic;
}

/* Given a union of basic maps R = \cup_i R_i \subseteq D \times D
 * and a dimension specification (Z^{n+1} -> Z^{n+1}),
 * construct a map that is an overapproximation of the map
 * that takes an element from the space D \times Z to another
 * element from the same space, such that the first n coordinates of the
 * difference between them is a sum of differences between images
 * and pre-images in one of the R_i and such that the last coordinate
 * is equal to the number of steps taken.
 * That is, let
 *
 *      \Delta_i = { y - x | (x, y) in R_i }
 *
 * then the constructed map is an overapproximation of
 *
 *      { (x) -> (x + d) | \exists k_i >= 0, \delta_i \in \Delta_i :
 *                              d = (\sum_i k_i \delta_i, \sum_i k_i) }
 *
 * The elements of the singleton \Delta_i's are collected as the
 * rows of the steps matrix.  For all these \Delta_i's together,
 * a single path is constructed.
 * For each of the other \Delta_i's, we compute an overapproximation
 * of the paths along elements of \Delta_i.
 * Since each of these paths performs an addition, composition is
 * symmetric and we can simply compose all resulting paths in any order.
 */
static __isl_give isl_map *construct_extended_path(__isl_take isl_space *dim,
        __isl_keep isl_map *map, int *project)
{
        struct isl_mat *steps = NULL;
        struct isl_map *path = NULL;
        unsigned d;
        int i, j, n;

        if (!map)
                goto error;

        d = isl_map_dim(map, isl_dim_in);

        path = isl_map_identity(isl_space_copy(dim));

        steps = isl_mat_alloc(map->ctx, map->n, d);
        if (!steps)
                goto error;

        n = 0;
        for (i = 0; i < map->n; ++i) {
                struct isl_basic_set *delta;

                delta = isl_basic_map_deltas(isl_basic_map_copy(map->p[i]));

                for (j = 0; j < d; ++j) {
                        isl_bool fixed;

                        fixed = isl_basic_set_plain_dim_is_fixed(delta, j,
                                                            &steps->row[n][j]);
                        if (fixed < 0) {
                                isl_basic_set_free(delta);
                                goto error;
                        }
                        if (!fixed)
                                break;
                }


                if (j < d) {
                        path = isl_map_apply_range(path,
                                path_along_delta(isl_space_copy(dim), delta));
                        path = isl_map_coalesce(path);
                } else {
                        isl_basic_set_free(delta);
                        ++n;
                }
        }

        if (n > 0) {
                steps->n_row = n;
                path = isl_map_apply_range(path,
                                path_along_steps(isl_space_copy(dim), steps));
        }

        if (project && *project) {
                *project = is_acyclic(isl_map_copy(path));
                if (*project < 0)
                        goto error;
        }

        isl_space_free(dim);
        isl_mat_free(steps);
        return path;
error:
        isl_space_free(dim);
        isl_mat_free(steps);
        isl_map_free(path);
        return NULL;
}

static isl_bool isl_set_overlaps(__isl_keep isl_set *set1,
        __isl_keep isl_set *set2)
{
        isl_set *i;
        isl_bool no_overlap;

        if (!set1 || !set2)
                return isl_bool_error;

        if (!isl_space_tuple_is_equal(set1->dim, isl_dim_set,
                                        set2->dim, isl_dim_set))
                return isl_bool_false;

        i = isl_set_intersect(isl_set_copy(set1), isl_set_copy(set2));
        no_overlap = isl_set_is_empty(i);
        isl_set_free(i);

        return isl_bool_not(no_overlap);
}

/* Given a union of basic maps R = \cup_i R_i \subseteq D \times D
 * and a dimension specification (Z^{n+1} -> Z^{n+1}),
 * construct a map that is an overapproximation of the map
 * that takes an element from the dom R \times Z to an
 * element from ran R \times Z, such that the first n coordinates of the
 * difference between them is a sum of differences between images
 * and pre-images in one of the R_i and such that the last coordinate
 * is equal to the number of steps taken.
 * That is, let
 *
 *      \Delta_i = { y - x | (x, y) in R_i }
 *
 * then the constructed map is an overapproximation of
 *
 *      { (x) -> (x + d) | \exists k_i >= 0, \delta_i \in \Delta_i :
 *                              d = (\sum_i k_i \delta_i, \sum_i k_i) and
 *                              x in dom R and x + d in ran R and
 *                              \sum_i k_i >= 1 }
 */
static __isl_give isl_map *construct_component(__isl_take isl_space *dim,
        __isl_keep isl_map *map, int *exact, int project)
{
        struct isl_set *domain = NULL;
        struct isl_set *range = NULL;
        struct isl_map *app = NULL;
        struct isl_map *path = NULL;
        isl_bool overlaps;

        domain = isl_map_domain(isl_map_copy(map));
        domain = isl_set_coalesce(domain);
        range = isl_map_range(isl_map_copy(map));
        range = isl_set_coalesce(range);
        overlaps = isl_set_overlaps(domain, range);
        if (overlaps < 0 || !overlaps) {
                isl_set_free(domain);
                isl_set_free(range);
                isl_space_free(dim);

                if (overlaps < 0)
                        map = NULL;
                map = isl_map_copy(map);
                map = isl_map_add_dims(map, isl_dim_in, 1);
                map = isl_map_add_dims(map, isl_dim_out, 1);
                map = set_path_length(map, 1, 1);
                return map;
        }
        app = isl_map_from_domain_and_range(domain, range);
        app = isl_map_add_dims(app, isl_dim_in, 1);
        app = isl_map_add_dims(app, isl_dim_out, 1);

        path = construct_extended_path(isl_space_copy(dim), map,
                                        exact && *exact ? &project : NULL);
        app = isl_map_intersect(app, path);

        if (exact && *exact &&
            (*exact = check_exactness(isl_map_copy(map), isl_map_copy(app),
                                      project)) < 0)
                goto error;

        isl_space_free(dim);
        app = set_path_length(app, 0, 1);
        return app;
error:
        isl_space_free(dim);
        isl_map_free(app);
        return NULL;
}

/* Call construct_component and, if "project" is set, project out
 * the final coordinates.
 */
static __isl_give isl_map *construct_projected_component(
        __isl_take isl_space *dim,
        __isl_keep isl_map *map, int *exact, int project)
{
        isl_map *app;
        unsigned d;

        if (!dim)
                return NULL;
        d = isl_space_dim(dim, isl_dim_in);

        app = construct_component(dim, map, exact, project);
        if (project) {
                app = isl_map_project_out(app, isl_dim_in, d - 1, 1);
                app = isl_map_project_out(app, isl_dim_out, d - 1, 1);
        }
        return app;
}

/* Compute an extended version, i.e., with path lengths, of
 * an overapproximation of the transitive closure of "bmap"
 * with path lengths greater than or equal to zero and with
 * domain and range equal to "dom".
 */
static __isl_give isl_map *q_closure(__isl_take isl_space *dim,
        __isl_take isl_set *dom, __isl_keep isl_basic_map *bmap, int *exact)
{
        int project = 1;
        isl_map *path;
        isl_map *map;
        isl_map *app;

        dom = isl_set_add_dims(dom, isl_dim_set, 1);
        app = isl_map_from_domain_and_range(dom, isl_set_copy(dom));
        map = isl_map_from_basic_map(isl_basic_map_copy(bmap));
        path = construct_extended_path(dim, map, &project);
        app = isl_map_intersect(app, path);

        if ((*exact = check_exactness(map, isl_map_copy(app), project)) < 0)
                goto error;

        return app;
error:
        isl_map_free(app);
        return NULL;
}

/* Check whether qc has any elements of length at least one
 * with domain and/or range outside of dom and ran.
 */
static int has_spurious_elements(__isl_keep isl_map *qc,
        __isl_keep isl_set *dom, __isl_keep isl_set *ran)
{
        isl_set *s;
        int subset;
        unsigned d;

        if (!qc || !dom || !ran)
                return -1;

        d = isl_map_dim(qc, isl_dim_in);

        qc = isl_map_copy(qc);
        qc = set_path_length(qc, 0, 1);
        qc = isl_map_project_out(qc, isl_dim_in, d - 1, 1);
        qc = isl_map_project_out(qc, isl_dim_out, d - 1, 1);

        s = isl_map_domain(isl_map_copy(qc));
        subset = isl_set_is_subset(s, dom);
        isl_set_free(s);
        if (subset < 0)
                goto error;
        if (!subset) {
                isl_map_free(qc);
                return 1;
        }

        s = isl_map_range(qc);
        subset = isl_set_is_subset(s, ran);
        isl_set_free(s);

        return subset < 0 ? -1 : !subset;
error:
        isl_map_free(qc);
        return -1;
}

#define LEFT    2
#define RIGHT   1

/* For each basic map in "map", except i, check whether it combines
 * with the transitive closure that is reflexive on C combines
 * to the left and to the right.
 *
 * In particular, if
 *
 *      dom map_j \subseteq C
 *
 * then right[j] is set to 1.  Otherwise, if
 *
 *      ran map_i \cap dom map_j = \emptyset
 *
 * then right[j] is set to 0.  Otherwise, composing to the right
 * is impossible.
 *
 * Similar, for composing to the left, we have if
 *
 *      ran map_j \subseteq C
 *
 * then left[j] is set to 1.  Otherwise, if
 *
 *      dom map_i \cap ran map_j = \emptyset
 *
 * then left[j] is set to 0.  Otherwise, composing to the left
 * is impossible.
 *
 * The return value is or'd with LEFT if composing to the left
 * is possible and with RIGHT if composing to the right is possible.
 */
static int composability(__isl_keep isl_set *C, int i,
        isl_set **dom, isl_set **ran, int *left, int *right,
        __isl_keep isl_map *map)
{
        int j;
        int ok;

        ok = LEFT | RIGHT;
        for (j = 0; j < map->n && ok; ++j) {
                isl_bool overlaps, subset;
                if (j == i)
                        continue;

                if (ok & RIGHT) {
                        if (!dom[j])
                                dom[j] = isl_set_from_basic_set(
                                        isl_basic_map_domain(
                                                isl_basic_map_copy(map->p[j])));
                        if (!dom[j])
                                return -1;
                        overlaps = isl_set_overlaps(ran[i], dom[j]);
                        if (overlaps < 0)
                                return -1;
                        if (!overlaps)
                                right[j] = 0;
                        else {
                                subset = isl_set_is_subset(dom[j], C);
                                if (subset < 0)
                                        return -1;
                                if (subset)
                                        right[j] = 1;
                                else
                                        ok &= ~RIGHT;
                        }
                }

                if (ok & LEFT) {
                        if (!ran[j])
                                ran[j] = isl_set_from_basic_set(
                                        isl_basic_map_range(
                                                isl_basic_map_copy(map->p[j])));
                        if (!ran[j])
                                return -1;
                        overlaps = isl_set_overlaps(dom[i], ran[j]);
                        if (overlaps < 0)
                                return -1;
                        if (!overlaps)
                                left[j] = 0;
                        else {
                                subset = isl_set_is_subset(ran[j], C);
                                if (subset < 0)
                                        return -1;
                                if (subset)
                                        left[j] = 1;
                                else
                                        ok &= ~LEFT;
                        }
                }
        }

        return ok;
}

static __isl_give isl_map *anonymize(__isl_take isl_map *map)
{
        map = isl_map_reset(map, isl_dim_in);
        map = isl_map_reset(map, isl_dim_out);
        return map;
}

/* Return a map that is a union of the basic maps in "map", except i,
 * composed to left and right with qc based on the entries of "left"
 * and "right".
 */
static __isl_give isl_map *compose(__isl_keep isl_map *map, int i,
        __isl_take isl_map *qc, int *left, int *right)
{
        int j;
        isl_map *comp;

        comp = isl_map_empty(isl_map_get_space(map));
        for (j = 0; j < map->n; ++j) {
                isl_map *map_j;

                if (j == i)
                        continue;

                map_j = isl_map_from_basic_map(isl_basic_map_copy(map->p[j]));
                map_j = anonymize(map_j);
                if (left && left[j])
                        map_j = isl_map_apply_range(map_j, isl_map_copy(qc));
                if (right && right[j])
                        map_j = isl_map_apply_range(isl_map_copy(qc), map_j);
                comp = isl_map_union(comp, map_j);
        }

        comp = isl_map_compute_divs(comp);
        comp = isl_map_coalesce(comp);

        isl_map_free(qc);

        return comp;
}

/* Compute the transitive closure of "map" incrementally by
 * computing
 *
 *      map_i^+ \cup qc^+
 *
 * or
 *
 *      map_i^+ \cup ((id \cup map_i^) \circ qc^+)
 *
 * or
 *
 *      map_i^+ \cup (qc^+ \circ (id \cup map_i^))
 *
 * depending on whether left or right are NULL.
 */
static __isl_give isl_map *compute_incremental(
        __isl_take isl_space *dim, __isl_keep isl_map *map,
        int i, __isl_take isl_map *qc, int *left, int *right, int *exact)
{
        isl_map *map_i;
        isl_map *tc;
        isl_map *rtc = NULL;

        if (!map)
                goto error;
        isl_assert(map->ctx, left || right, goto error);

        map_i = isl_map_from_basic_map(isl_basic_map_copy(map->p[i]));
        tc = construct_projected_component(isl_space_copy(dim), map_i,
                                                exact, 1);
        isl_map_free(map_i);

        if (*exact)
                qc = isl_map_transitive_closure(qc, exact);

        if (!*exact) {
                isl_space_free(dim);
                isl_map_free(tc);
                isl_map_free(qc);
                return isl_map_universe(isl_map_get_space(map));
        }

        if (!left || !right)
                rtc = isl_map_union(isl_map_copy(tc),
                                    isl_map_identity(isl_map_get_space(tc)));
        if (!right)
                qc = isl_map_apply_range(rtc, qc);
        if (!left)
                qc = isl_map_apply_range(qc, rtc);
        qc = isl_map_union(tc, qc);

        isl_space_free(dim);

        return qc;
error:
        isl_space_free(dim);
        isl_map_free(qc);
        return NULL;
}

/* Given a map "map", try to find a basic map such that
 * map^+ can be computed as
 *
 * map^+ = map_i^+ \cup
 *    \bigcup_j ((map_i^+ \cup Id_C)^+ \circ map_j \circ (map_i^+ \cup Id_C))^+
 *
 * with C the simple hull of the domain and range of the input map.
 * map_i^ \cup Id_C is computed by allowing the path lengths to be zero
 * and by intersecting domain and range with C.
 * Of course, we need to check that this is actually equal to map_i^ \cup Id_C.
 * Also, we only use the incremental computation if all the transitive
 * closures are exact and if the number of basic maps in the union,
 * after computing the integer divisions, is smaller than the number
 * of basic maps in the input map.
 */
static int incemental_on_entire_domain(__isl_keep isl_space *dim,
        __isl_keep isl_map *map,
        isl_set **dom, isl_set **ran, int *left, int *right,
        __isl_give isl_map **res)
{
        int i;
        isl_set *C;
        unsigned d;

        *res = NULL;

        C = isl_set_union(isl_map_domain(isl_map_copy(map)),
                          isl_map_range(isl_map_copy(map)));
        C = isl_set_from_basic_set(isl_set_simple_hull(C));
        if (!C)
                return -1;
        if (C->n != 1) {
                isl_set_free(C);
                return 0;
        }

        d = isl_map_dim(map, isl_dim_in);

        for (i = 0; i < map->n; ++i) {
                isl_map *qc;
                int exact_i, spurious;
                int j;
                dom[i] = isl_set_from_basic_set(isl_basic_map_domain(
                                        isl_basic_map_copy(map->p[i])));
                ran[i] = isl_set_from_basic_set(isl_basic_map_range(
                                        isl_basic_map_copy(map->p[i])));
                qc = q_closure(isl_space_copy(dim), isl_set_copy(C),
                                map->p[i], &exact_i);
                if (!qc)
                        goto error;
                if (!exact_i) {
                        isl_map_free(qc);
                        continue;
                }
                spurious = has_spurious_elements(qc, dom[i], ran[i]);
                if (spurious) {
                        isl_map_free(qc);
                        if (spurious < 0)
                                goto error;
                        continue;
                }
                qc = isl_map_project_out(qc, isl_dim_in, d, 1);
                qc = isl_map_project_out(qc, isl_dim_out, d, 1);
                qc = isl_map_compute_divs(qc);
                for (j = 0; j < map->n; ++j)
                        left[j] = right[j] = 1;
                qc = compose(map, i, qc, left, right);
                if (!qc)
                        goto error;
                if (qc->n >= map->n) {
                        isl_map_free(qc);
                        continue;
                }
                *res = compute_incremental(isl_space_copy(dim), map, i, qc,
                                left, right, &exact_i);
                if (!*res)
                        goto error;
                if (exact_i)
                        break;
                isl_map_free(*res);
                *res = NULL;
        }

        isl_set_free(C);

        return *res != NULL;
error:
        isl_set_free(C);
        return -1;
}

/* Try and compute the transitive closure of "map" as
 *
 * map^+ = map_i^+ \cup
 *    \bigcup_j ((map_i^+ \cup Id_C)^+ \circ map_j \circ (map_i^+ \cup Id_C))^+
 *
 * with C either the simple hull of the domain and range of the entire
 * map or the simple hull of domain and range of map_i.
 */
static __isl_give isl_map *incremental_closure(__isl_take isl_space *dim,
        __isl_keep isl_map *map, int *exact, int project)
{
        int i;
        isl_set **dom = NULL;
        isl_set **ran = NULL;
        int *left = NULL;
        int *right = NULL;
        isl_set *C;
        unsigned d;
        isl_map *res = NULL;

        if (!project)
                return construct_projected_component(dim, map, exact, project);

        if (!map)
                goto error;
        if (map->n <= 1)
                return construct_projected_component(dim, map, exact, project);

        d = isl_map_dim(map, isl_dim_in);

        dom = isl_calloc_array(map->ctx, isl_set *, map->n);
        ran = isl_calloc_array(map->ctx, isl_set *, map->n);
        left = isl_calloc_array(map->ctx, int, map->n);
        right = isl_calloc_array(map->ctx, int, map->n);
        if (!ran || !dom || !left || !right)
                goto error;

        if (incemental_on_entire_domain(dim, map, dom, ran, left, right, &res) < 0)
                goto error;

        for (i = 0; !res && i < map->n; ++i) {
                isl_map *qc;
                int exact_i, spurious, comp;
                if (!dom[i])
                        dom[i] = isl_set_from_basic_set(
                                        isl_basic_map_domain(
                                                isl_basic_map_copy(map->p[i])));
                if (!dom[i])
                        goto error;
                if (!ran[i])
                        ran[i] = isl_set_from_basic_set(
                                        isl_basic_map_range(
                                                isl_basic_map_copy(map->p[i])));
                if (!ran[i])
                        goto error;
                C = isl_set_union(isl_set_copy(dom[i]),
                                      isl_set_copy(ran[i]));
                C = isl_set_from_basic_set(isl_set_simple_hull(C));
                if (!C)
                        goto error;
                if (C->n != 1) {
                        isl_set_free(C);
                        continue;
                }
                comp = composability(C, i, dom, ran, left, right, map);
                if (!comp || comp < 0) {
                        isl_set_free(C);
                        if (comp < 0)
                                goto error;
                        continue;
                }
                qc = q_closure(isl_space_copy(dim), C, map->p[i], &exact_i);
                if (!qc)
                        goto error;
                if (!exact_i) {
                        isl_map_free(qc);
                        continue;
                }
                spurious = has_spurious_elements(qc, dom[i], ran[i]);
                if (spurious) {
                        isl_map_free(qc);
                        if (spurious < 0)
                                goto error;
                        continue;
                }
                qc = isl_map_project_out(qc, isl_dim_in, d, 1);
                qc = isl_map_project_out(qc, isl_dim_out, d, 1);
                qc = isl_map_compute_divs(qc);
                qc = compose(map, i, qc, (comp & LEFT) ? left : NULL,
                                (comp & RIGHT) ? right : NULL);
                if (!qc)
                        goto error;
                if (qc->n >= map->n) {
                        isl_map_free(qc);
                        continue;
                }
                res = compute_incremental(isl_space_copy(dim), map, i, qc,
                                (comp & LEFT) ? left : NULL,
                                (comp & RIGHT) ? right : NULL, &exact_i);
                if (!res)
                        goto error;
                if (exact_i)
                        break;
                isl_map_free(res);
                res = NULL;
        }

        for (i = 0; i < map->n; ++i) {
                isl_set_free(dom[i]);
                isl_set_free(ran[i]);
        }
        free(dom);
        free(ran);
        free(left);
        free(right);

        if (res) {
                isl_space_free(dim);
                return res;
        }

        return construct_projected_component(dim, map, exact, project);
error:
        if (dom)
                for (i = 0; i < map->n; ++i)
                        isl_set_free(dom[i]);
        free(dom);
        if (ran)
                for (i = 0; i < map->n; ++i)
                        isl_set_free(ran[i]);
        free(ran);
        free(left);
        free(right);
        isl_space_free(dim);
        return NULL;
}

/* Given an array of sets "set", add "dom" at position "pos"
 * and search for elements at earlier positions that overlap with "dom".
 * If any can be found, then merge all of them, together with "dom", into
 * a single set and assign the union to the first in the array,
 * which becomes the new group leader for all groups involved in the merge.
 * During the search, we only consider group leaders, i.e., those with
 * group[i] = i, as the other sets have already been combined
 * with one of the group leaders.
 */
static int merge(isl_set **set, int *group, __isl_take isl_set *dom, int pos)
{
        int i;

        group[pos] = pos;
        set[pos] = isl_set_copy(dom);

        for (i = pos - 1; i >= 0; --i) {
                isl_bool o;

                if (group[i] != i)
                        continue;

                o = isl_set_overlaps(set[i], dom);
                if (o < 0)
                        goto error;
                if (!o)
                        continue;

                set[i] = isl_set_union(set[i], set[group[pos]]);
                set[group[pos]] = NULL;
                if (!set[i])
                        goto error;
                group[group[pos]] = i;
                group[pos] = i;
        }

        isl_set_free(dom);
        return 0;
error:
        isl_set_free(dom);
        return -1;
}

/* Replace each entry in the n by n grid of maps by the cross product
 * with the relation { [i] -> [i + 1] }.
 */
static int add_length(__isl_keep isl_map *map, isl_map ***grid, int n)
{
        int i, j, k;
        isl_space *dim;
        isl_basic_map *bstep;
        isl_map *step;
        unsigned nparam;

        if (!map)
                return -1;

        dim = isl_map_get_space(map);
        nparam = isl_space_dim(dim, isl_dim_param);
        dim = isl_space_drop_dims(dim, isl_dim_in, 0, isl_space_dim(dim, isl_dim_in));
        dim = isl_space_drop_dims(dim, isl_dim_out, 0, isl_space_dim(dim, isl_dim_out));
        dim = isl_space_add_dims(dim, isl_dim_in, 1);
        dim = isl_space_add_dims(dim, isl_dim_out, 1);
        bstep = isl_basic_map_alloc_space(dim, 0, 1, 0);
        k = isl_basic_map_alloc_equality(bstep);
        if (k < 0) {
                isl_basic_map_free(bstep);
                return -1;
        }
        isl_seq_clr(bstep->eq[k], 1 + isl_basic_map_total_dim(bstep));
        isl_int_set_si(bstep->eq[k][0], 1);
        isl_int_set_si(bstep->eq[k][1 + nparam], 1);
        isl_int_set_si(bstep->eq[k][1 + nparam + 1], -1);
        bstep = isl_basic_map_finalize(bstep);
        step = isl_map_from_basic_map(bstep);

        for (i = 0; i < n; ++i)
                for (j = 0; j < n; ++j)
                        grid[i][j] = isl_map_product(grid[i][j],
                                                     isl_map_copy(step));

        isl_map_free(step);

        return 0;
}

/* The core of the Floyd-Warshall algorithm.
 * Updates the given n x x matrix of relations in place.
 *
 * The algorithm iterates over all vertices.  In each step, the whole
 * matrix is updated to include all paths that go to the current vertex,
 * possibly stay there a while (including passing through earlier vertices)
 * and then come back.  At the start of each iteration, the diagonal
 * element corresponding to the current vertex is replaced by its
 * transitive closure to account for all indirect paths that stay
 * in the current vertex.
 */
static void floyd_warshall_iterate(isl_map ***grid, int n, int *exact)
{
        int r, p, q;

        for (r = 0; r < n; ++r) {
                int r_exact;
                grid[r][r] = isl_map_transitive_closure(grid[r][r],
                                (exact && *exact) ? &r_exact : NULL);
                if (exact && *exact && !r_exact)
                        *exact = 0;

                for (p = 0; p < n; ++p)
                        for (q = 0; q < n; ++q) {
                                isl_map *loop;
                                if (p == r && q == r)
                                        continue;
                                loop = isl_map_apply_range(
                                                isl_map_copy(grid[p][r]),
                                                isl_map_copy(grid[r][q]));
                                grid[p][q] = isl_map_union(grid[p][q], loop);
                                loop = isl_map_apply_range(
                                                isl_map_copy(grid[p][r]),
                                        isl_map_apply_range(
                                                isl_map_copy(grid[r][r]),
                                                isl_map_copy(grid[r][q])));
                                grid[p][q] = isl_map_union(grid[p][q], loop);
                                grid[p][q] = isl_map_coalesce(grid[p][q]);
                        }
        }
}

/* Given a partition of the domains and ranges of the basic maps in "map",
 * apply the Floyd-Warshall algorithm with the elements in the partition
 * as vertices.
 *
 * In particular, there are "n" elements in the partition and "group" is
 * an array of length 2 * map->n with entries in [0,n-1].
 *
 * We first construct a matrix of relations based on the partition information,
 * apply Floyd-Warshall on this matrix of relations and then take the
 * union of all entries in the matrix as the final result.
 *
 * If we are actually computing the power instead of the transitive closure,
 * i.e., when "project" is not set, then the result should have the
 * path lengths encoded as the difference between an extra pair of
 * coordinates.  We therefore apply the nested transitive closures
 * to relations that include these lengths.  In particular, we replace
 * the input relation by the cross product with the unit length relation
 * { [i] -> [i + 1] }.
 */
static __isl_give isl_map *floyd_warshall_with_groups(__isl_take isl_space *dim,
        __isl_keep isl_map *map, int *exact, int project, int *group, int n)
{
        int i, j, k;
        isl_map ***grid = NULL;
        isl_map *app;

        if (!map)
                goto error;

        if (n == 1) {
                free(group);
                return incremental_closure(dim, map, exact, project);
        }

        grid = isl_calloc_array(map->ctx, isl_map **, n);
        if (!grid)
                goto error;
        for (i = 0; i < n; ++i) {
                grid[i] = isl_calloc_array(map->ctx, isl_map *, n);
                if (!grid[i])
                        goto error;
                for (j = 0; j < n; ++j)
                        grid[i][j] = isl_map_empty(isl_map_get_space(map));
        }

        for (k = 0; k < map->n; ++k) {
                i = group[2 * k];
                j = group[2 * k + 1];
                grid[i][j] = isl_map_union(grid[i][j],
                                isl_map_from_basic_map(
                                        isl_basic_map_copy(map->p[k])));
        }

        if (!project && add_length(map, grid, n) < 0)
                goto error;

        floyd_warshall_iterate(grid, n, exact);

        app = isl_map_empty(isl_map_get_space(grid[0][0]));

        for (i = 0; i < n; ++i) {
                for (j = 0; j < n; ++j)
                        app = isl_map_union(app, grid[i][j]);
                free(grid[i]);
        }
        free(grid);

        free(group);
        isl_space_free(dim);

        return app;
error:
        if (grid)
                for (i = 0; i < n; ++i) {
                        if (!grid[i])
                                continue;
                        for (j = 0; j < n; ++j)
                                isl_map_free(grid[i][j]);
                        free(grid[i]);
                }
        free(grid);
        free(group);
        isl_space_free(dim);
        return NULL;
}

/* Partition the domains and ranges of the n basic relations in list
 * into disjoint cells.
 *
 * To find the partition, we simply consider all of the domains
 * and ranges in turn and combine those that overlap.
 * "set" contains the partition elements and "group" indicates
 * to which partition element a given domain or range belongs.
 * The domain of basic map i corresponds to element 2 * i in these arrays,
 * while the domain corresponds to element 2 * i + 1.
 * During the construction group[k] is either equal to k,
 * in which case set[k] contains the union of all the domains and
 * ranges in the corresponding group, or is equal to some l < k,
 * with l another domain or range in the same group.
 */
static int *setup_groups(isl_ctx *ctx, __isl_keep isl_basic_map **list, int n,
        isl_set ***set, int *n_group)
{
        int i;
        int *group = NULL;
        int g;

        *set = isl_calloc_array(ctx, isl_set *, 2 * n);
        group = isl_alloc_array(ctx, int, 2 * n);

        if (!*set || !group)
                goto error;

        for (i = 0; i < n; ++i) {
                isl_set *dom;
                dom = isl_set_from_basic_set(isl_basic_map_domain(
                                isl_basic_map_copy(list[i])));
                if (merge(*set, group, dom, 2 * i) < 0)
                        goto error;
                dom = isl_set_from_basic_set(isl_basic_map_range(
                                isl_basic_map_copy(list[i])));
                if (merge(*set, group, dom, 2 * i + 1) < 0)
                        goto error;
        }

        g = 0;
        for (i = 0; i < 2 * n; ++i)
                if (group[i] == i) {
                        if (g != i) {
                                (*set)[g] = (*set)[i];
                                (*set)[i] = NULL;
                        }
                        group[i] = g++;
                } else
                        group[i] = group[group[i]];

        *n_group = g;

        return group;
error:
        if (*set) {
                for (i = 0; i < 2 * n; ++i)
                        isl_set_free((*set)[i]);
                free(*set);
                *set = NULL;
        }
        free(group);
        return NULL;
}

/* Check if the domains and ranges of the basic maps in "map" can
 * be partitioned, and if so, apply Floyd-Warshall on the elements
 * of the partition.  Note that we also apply this algorithm
 * if we want to compute the power, i.e., when "project" is not set.
 * However, the results are unlikely to be exact since the recursive
 * calls inside the Floyd-Warshall algorithm typically result in
 * non-linear path lengths quite quickly.
 */
static __isl_give isl_map *floyd_warshall(__isl_take isl_space *dim,
        __isl_keep isl_map *map, int *exact, int project)
{
        int i;
        isl_set **set = NULL;
        int *group = NULL;
        int n;

        if (!map)
                goto error;
        if (map->n <= 1)
                return incremental_closure(dim, map, exact, project);

        group = setup_groups(map->ctx, map->p, map->n, &set, &n);
        if (!group)
                goto error;

        for (i = 0; i < 2 * map->n; ++i)
                isl_set_free(set[i]);

        free(set);

        return floyd_warshall_with_groups(dim, map, exact, project, group, n);
error:
        isl_space_free(dim);
        return NULL;
}

/* Structure for representing the nodes of the graph of which
 * strongly connected components are being computed.
 *
 * list contains the actual nodes
 * check_closed is set if we may have used the fact that
 * a pair of basic maps can be interchanged
 */
struct isl_tc_follows_data {
        isl_basic_map **list;
        int check_closed;
};

/* Check whether in the computation of the transitive closure
 * "list[i]" (R_1) should follow (or be part of the same component as)
 * "list[j]" (R_2).
 *
 * That is check whether
 *
 *      R_1 \circ R_2
 *
 * is a subset of
 *
 *      R_2 \circ R_1
 *
 * If so, then there is no reason for R_1 to immediately follow R_2
 * in any path.
 *
 * *check_closed is set if the subset relation holds while
 * R_1 \circ R_2 is not empty.
 */
static isl_bool basic_map_follows(int i, int j, void *user)
{
        struct isl_tc_follows_data *data = user;
        struct isl_map *map12 = NULL;
        struct isl_map *map21 = NULL;
        isl_bool subset;

        if (!isl_space_tuple_is_equal(data->list[i]->dim, isl_dim_in,
                                    data->list[j]->dim, isl_dim_out))
                return isl_bool_false;

        map21 = isl_map_from_basic_map(
                        isl_basic_map_apply_range(
                                isl_basic_map_copy(data->list[j]),
                                isl_basic_map_copy(data->list[i])));
        subset = isl_map_is_empty(map21);
        if (subset < 0)
                goto error;
        if (subset) {
                isl_map_free(map21);
                return isl_bool_false;
        }

        if (!isl_space_tuple_is_equal(data->list[i]->dim, isl_dim_in,
                                    data->list[i]->dim, isl_dim_out) ||
            !isl_space_tuple_is_equal(data->list[j]->dim, isl_dim_in,
                                    data->list[j]->dim, isl_dim_out)) {
                isl_map_free(map21);
                return isl_bool_true;
        }

        map12 = isl_map_from_basic_map(
                        isl_basic_map_apply_range(
                                isl_basic_map_copy(data->list[i]),
                                isl_basic_map_copy(data->list[j])));

        subset = isl_map_is_subset(map21, map12);

        isl_map_free(map12);
        isl_map_free(map21);

        if (subset)
                data->check_closed = 1;

        return subset < 0 ? isl_bool_error : !subset;
error:
        isl_map_free(map21);
        return isl_bool_error;
}

/* Given a union of basic maps R = \cup_i R_i \subseteq D \times D
 * and a dimension specification (Z^{n+1} -> Z^{n+1}),
 * construct a map that is an overapproximation of the map
 * that takes an element from the dom R \times Z to an
 * element from ran R \times Z, such that the first n coordinates of the
 * difference between them is a sum of differences between images
 * and pre-images in one of the R_i and such that the last coordinate
 * is equal to the number of steps taken.
 * If "project" is set, then these final coordinates are not included,
 * i.e., a relation of type Z^n -> Z^n is returned.
 * That is, let
 *
 *      \Delta_i = { y - x | (x, y) in R_i }
 *
 * then the constructed map is an overapproximation of
 *
 *      { (x) -> (x + d) | \exists k_i >= 0, \delta_i \in \Delta_i :
 *                              d = (\sum_i k_i \delta_i, \sum_i k_i) and
 *                              x in dom R and x + d in ran R }
 *
 * or
 *
 *      { (x) -> (x + d) | \exists k_i >= 0, \delta_i \in \Delta_i :
 *                              d = (\sum_i k_i \delta_i) and
 *                              x in dom R and x + d in ran R }
 *
 * if "project" is set.
 *
 * We first split the map into strongly connected components, perform
 * the above on each component and then join the results in the correct
 * order, at each join also taking in the union of both arguments
 * to allow for paths that do not go through one of the two arguments.
 */
static __isl_give isl_map *construct_power_components(__isl_take isl_space *dim,
        __isl_keep isl_map *map, int *exact, int project)
{
        int i, n, c;
        struct isl_map *path = NULL;
        struct isl_tc_follows_data data;
        struct isl_tarjan_graph *g = NULL;
        int *orig_exact;
        int local_exact;

        if (!map)
                goto error;
        if (map->n <= 1)
                return floyd_warshall(dim, map, exact, project);

        data.list = map->p;
        data.check_closed = 0;
        g = isl_tarjan_graph_init(map->ctx, map->n, &basic_map_follows, &data);
        if (!g)
                goto error;

        orig_exact = exact;
        if (data.check_closed && !exact)
                exact = &local_exact;

        c = 0;
        i = 0;
        n = map->n;
        if (project)
                path = isl_map_empty(isl_map_get_space(map));
        else
                path = isl_map_empty(isl_space_copy(dim));
        path = anonymize(path);
        while (n) {
                struct isl_map *comp;
                isl_map *path_comp, *path_comb;
                comp = isl_map_alloc_space(isl_map_get_space(map), n, 0);
                while (g->order[i] != -1) {
                        comp = isl_map_add_basic_map(comp,
                                    isl_basic_map_copy(map->p[g->order[i]]));
                        --n;
                        ++i;
                }
                path_comp = floyd_warshall(isl_space_copy(dim),
                                                comp, exact, project);
                path_comp = anonymize(path_comp);
                path_comb = isl_map_apply_range(isl_map_copy(path),
                                                isl_map_copy(path_comp));
                path = isl_map_union(path, path_comp);
                path = isl_map_union(path, path_comb);
                isl_map_free(comp);
                ++i;
                ++c;
        }

        if (c > 1 && data.check_closed && !*exact) {
                int closed;

                closed = isl_map_is_transitively_closed(path);
                if (closed < 0)
                        goto error;
                if (!closed) {
                        isl_tarjan_graph_free(g);
                        isl_map_free(path);
                        return floyd_warshall(dim, map, orig_exact, project);
                }
        }

        isl_tarjan_graph_free(g);
        isl_space_free(dim);

        return path;
error:
        isl_tarjan_graph_free(g);
        isl_space_free(dim);
        isl_map_free(path);
        return NULL;
}

/* Given a union of basic maps R = \cup_i R_i \subseteq D \times D,
 * construct a map that is an overapproximation of the map
 * that takes an element from the space D to another
 * element from the same space, such that the difference between
 * them is a strictly positive sum of differences between images
 * and pre-images in one of the R_i.
 * The number of differences in the sum is equated to parameter "param".
 * That is, let
 *
 *      \Delta_i = { y - x | (x, y) in R_i }
 *
 * then the constructed map is an overapproximation of
 *
 *      { (x) -> (x + d) | \exists k_i >= 0, \delta_i \in \Delta_i :
 *                              d = \sum_i k_i \delta_i and k = \sum_i k_i > 0 }
 * or
 *
 *      { (x) -> (x + d) | \exists k_i >= 0, \delta_i \in \Delta_i :
 *                              d = \sum_i k_i \delta_i and \sum_i k_i > 0 }
 *
 * if "project" is set.
 *
 * If "project" is not set, then
 * we construct an extended mapping with an extra coordinate
 * that indicates the number of steps taken.  In particular,
 * the difference in the last coordinate is equal to the number
 * of steps taken to move from a domain element to the corresponding
 * image element(s).
 */
static __isl_give isl_map *construct_power(__isl_keep isl_map *map,
        int *exact, int project)
{
        struct isl_map *app = NULL;
        isl_space *dim = NULL;

        if (!map)
                return NULL;

        dim = isl_map_get_space(map);

        dim = isl_space_add_dims(dim, isl_dim_in, 1);
        dim = isl_space_add_dims(dim, isl_dim_out, 1);

        app = construct_power_components(isl_space_copy(dim), map,
                                        exact, project);

        isl_space_free(dim);

        return app;
}

/* Compute the positive powers of "map", or an overapproximation.
 * If the result is exact, then *exact is set to 1.
 *
 * If project is set, then we are actually interested in the transitive
 * closure, so we can use a more relaxed exactness check.
 * The lengths of the paths are also projected out instead of being
 * encoded as the difference between an extra pair of final coordinates.
 */
static __isl_give isl_map *map_power(__isl_take isl_map *map,
        int *exact, int project)
{
        struct isl_map *app = NULL;

        if (exact)
                *exact = 1;

        if (!map)
                return NULL;

        isl_assert(map->ctx,
                isl_map_dim(map, isl_dim_in) == isl_map_dim(map, isl_dim_out),
                goto error);

        app = construct_power(map, exact, project);

        isl_map_free(map);
        return app;
error:
        isl_map_free(map);
        isl_map_free(app);
        return NULL;
}

/* Compute the positive powers of "map", or an overapproximation.
 * The result maps the exponent to a nested copy of the corresponding power.
 * If the result is exact, then *exact is set to 1.
 * map_power constructs an extended relation with the path lengths
 * encoded as the difference between the final coordinates.
 * In the final step, this difference is equated to an extra parameter
 * and made positive.  The extra coordinates are subsequently projected out
 * and the parameter is turned into the domain of the result.
 */
__isl_give isl_map *isl_map_power(__isl_take isl_map *map, int *exact)
{
        isl_space *target_dim;
        isl_space *dim;
        isl_map *diff;
        unsigned d;
        unsigned param;

        if (!map)
                return NULL;

        d = isl_map_dim(map, isl_dim_in);
        param = isl_map_dim(map, isl_dim_param);

        map = isl_map_compute_divs(map);
        map = isl_map_coalesce(map);

        if (isl_map_plain_is_empty(map)) {
                map = isl_map_from_range(isl_map_wrap(map));
                map = isl_map_add_dims(map, isl_dim_in, 1);
                map = isl_map_set_dim_name(map, isl_dim_in, 0, "k");
                return map;
        }

        target_dim = isl_map_get_space(map);
        target_dim = isl_space_from_range(isl_space_wrap(target_dim));
        target_dim = isl_space_add_dims(target_dim, isl_dim_in, 1);
        target_dim = isl_space_set_dim_name(target_dim, isl_dim_in, 0, "k");

        map = map_power(map, exact, 0);

        map = isl_map_add_dims(map, isl_dim_param, 1);
        dim = isl_map_get_space(map);
        diff = equate_parameter_to_length(dim, param);
        map = isl_map_intersect(map, diff);
        map = isl_map_project_out(map, isl_dim_in, d, 1);
        map = isl_map_project_out(map, isl_dim_out, d, 1);
        map = isl_map_from_range(isl_map_wrap(map));
        map = isl_map_move_dims(map, isl_dim_in, 0, isl_dim_param, param, 1);

        map = isl_map_reset_space(map, target_dim);

        return map;
}

/* Compute a relation that maps each element in the range of the input
 * relation to the lengths of all paths composed of edges in the input
 * relation that end up in the given range element.
 * The result may be an overapproximation, in which case *exact is set to 0.
 * The resulting relation is very similar to the power relation.
 * The difference are that the domain has been projected out, the
 * range has become the domain and the exponent is the range instead
 * of a parameter.
 */
__isl_give isl_map *isl_map_reaching_path_lengths(__isl_take isl_map *map,
        int *exact)
{
        isl_space *dim;
        isl_map *diff;
        unsigned d;
        unsigned param;

        if (!map)
                return NULL;

        d = isl_map_dim(map, isl_dim_in);
        param = isl_map_dim(map, isl_dim_param);

        map = isl_map_compute_divs(map);
        map = isl_map_coalesce(map);

        if (isl_map_plain_is_empty(map)) {
                if (exact)
                        *exact = 1;
                map = isl_map_project_out(map, isl_dim_out, 0, d);
                map = isl_map_add_dims(map, isl_dim_out, 1);
                return map;
        }

        map = map_power(map, exact, 0);

        map = isl_map_add_dims(map, isl_dim_param, 1);
        dim = isl_map_get_space(map);
        diff = equate_parameter_to_length(dim, param);
        map = isl_map_intersect(map, diff);
        map = isl_map_project_out(map, isl_dim_in, 0, d + 1);
        map = isl_map_project_out(map, isl_dim_out, d, 1);
        map = isl_map_reverse(map);
        map = isl_map_move_dims(map, isl_dim_out, 0, isl_dim_param, param, 1);

        return map;
}

/* Given a map, compute the smallest superset of this map that is of the form
 *
 *      { i -> j : L <= j - i <= U and exists a_p: j_p - i_p = M_p a_p }
 *
 * (where p ranges over the (non-parametric) dimensions),
 * compute the transitive closure of this map, i.e.,
 *
 *      { i -> j : exists k > 0:
 *              k L <= j - i <= k U and exists a: j_p - i_p = M_p a_p }
 *
 * and intersect domain and range of this transitive closure with
 * the given domain and range.
 *
 * If with_id is set, then try to include as much of the identity mapping
 * as possible, by computing
 *
 *      { i -> j : exists k >= 0:
 *              k L <= j - i <= k U and exists a: j_p - i_p = M_p a_p }
 *
 * instead (i.e., allow k = 0).
 *
 * In practice, we compute the difference set
 *
 *      delta  = { j - i | i -> j in map },
 *
 * look for stride constraint on the individual dimensions and compute
 * (constant) lower and upper bounds for each individual dimension,
 * adding a constraint for each bound not equal to infinity.
 */
static __isl_give isl_map *box_closure_on_domain(__isl_take isl_map *map,
        __isl_take isl_set *dom, __isl_take isl_set *ran, int with_id)
{
        int i;
        int k;
        unsigned d;
        unsigned nparam;
        unsigned total;
        isl_space *dim;
        isl_set *delta;
        isl_map *app = NULL;
        isl_basic_set *aff = NULL;
        isl_basic_map *bmap = NULL;
        isl_vec *obj = NULL;
        isl_int opt;

        isl_int_init(opt);

        delta = isl_map_deltas(isl_map_copy(map));

        aff = isl_set_affine_hull(isl_set_copy(delta));
        if (!aff)
                goto error;
        dim = isl_map_get_space(map);
        d = isl_space_dim(dim, isl_dim_in);
        nparam = isl_space_dim(dim, isl_dim_param);
        total = isl_space_dim(dim, isl_dim_all);
        bmap = isl_basic_map_alloc_space(dim,
                                        aff->n_div + 1, aff->n_div, 2 * d + 1);
        for (i = 0; i < aff->n_div + 1; ++i) {
                k = isl_basic_map_alloc_div(bmap);
                if (k < 0)
                        goto error;
                isl_int_set_si(bmap->div[k][0], 0);
        }
        for (i = 0; i < aff->n_eq; ++i) {
                if (!isl_basic_set_eq_is_stride(aff, i))
                        continue;
                k = isl_basic_map_alloc_equality(bmap);
                if (k < 0)
                        goto error;
                isl_seq_clr(bmap->eq[k], 1 + nparam);
                isl_seq_cpy(bmap->eq[k] + 1 + nparam + d,
                                aff->eq[i] + 1 + nparam, d);
                isl_seq_neg(bmap->eq[k] + 1 + nparam,
                                aff->eq[i] + 1 + nparam, d);
                isl_seq_cpy(bmap->eq[k] + 1 + nparam + 2 * d,
                                aff->eq[i] + 1 + nparam + d, aff->n_div);
                isl_int_set_si(bmap->eq[k][1 + total + aff->n_div], 0);
        }
        obj = isl_vec_alloc(map->ctx, 1 + nparam + d);
        if (!obj)
                goto error;
        isl_seq_clr(obj->el, 1 + nparam + d);
        for (i = 0; i < d; ++ i) {
                enum isl_lp_result res;

                isl_int_set_si(obj->el[1 + nparam + i], 1);

                res = isl_set_solve_lp(delta, 0, obj->el, map->ctx->one, &opt,
                                        NULL, NULL);
                if (res == isl_lp_error)
                        goto error;
                if (res == isl_lp_ok) {
                        k = isl_basic_map_alloc_inequality(bmap);
                        if (k < 0)
                                goto error;
                        isl_seq_clr(bmap->ineq[k],
                                        1 + nparam + 2 * d + bmap->n_div);
                        isl_int_set_si(bmap->ineq[k][1 + nparam + i], -1);
                        isl_int_set_si(bmap->ineq[k][1 + nparam + d + i], 1);
                        isl_int_neg(bmap->ineq[k][1 + nparam + 2 * d + aff->n_div], opt);
                }

                res = isl_set_solve_lp(delta, 1, obj->el, map->ctx->one, &opt,
                                        NULL, NULL);
                if (res == isl_lp_error)
                        goto error;
                if (res == isl_lp_ok) {
                        k = isl_basic_map_alloc_inequality(bmap);
                        if (k < 0)
                                goto error;
                        isl_seq_clr(bmap->ineq[k],
                                        1 + nparam + 2 * d + bmap->n_div);
                        isl_int_set_si(bmap->ineq[k][1 + nparam + i], 1);
                        isl_int_set_si(bmap->ineq[k][1 + nparam + d + i], -1);
                        isl_int_set(bmap->ineq[k][1 + nparam + 2 * d + aff->n_div], opt);
                }

                isl_int_set_si(obj->el[1 + nparam + i], 0);
        }
        k = isl_basic_map_alloc_inequality(bmap);
        if (k < 0)
                goto error;
        isl_seq_clr(bmap->ineq[k],
                        1 + nparam + 2 * d + bmap->n_div);
        if (!with_id)
                isl_int_set_si(bmap->ineq[k][0], -1);
        isl_int_set_si(bmap->ineq[k][1 + nparam + 2 * d + aff->n_div], 1);

        app = isl_map_from_domain_and_range(dom, ran);

        isl_vec_free(obj);
        isl_basic_set_free(aff);
        isl_map_free(map);
        bmap = isl_basic_map_finalize(bmap);
        isl_set_free(delta);
        isl_int_clear(opt);

        map = isl_map_from_basic_map(bmap);
        map = isl_map_intersect(map, app);

        return map;
error:
        isl_vec_free(obj);
        isl_basic_map_free(bmap);
        isl_basic_set_free(aff);
        isl_set_free(dom);
        isl_set_free(ran);
        isl_map_free(map);
        isl_set_free(delta);
        isl_int_clear(opt);
        return NULL;
}

/* Given a map, compute the smallest superset of this map that is of the form
 *
 *      { i -> j : L <= j - i <= U and exists a_p: j_p - i_p = M_p a_p }
 *
 * (where p ranges over the (non-parametric) dimensions),
 * compute the transitive closure of this map, i.e.,
 *
 *      { i -> j : exists k > 0:
 *              k L <= j - i <= k U and exists a: j_p - i_p = M_p a_p }
 *
 * and intersect domain and range of this transitive closure with
 * domain and range of the original map.
 */
static __isl_give isl_map *box_closure(__isl_take isl_map *map)
{
        isl_set *domain;
        isl_set *range;

        domain = isl_map_domain(isl_map_copy(map));
        domain = isl_set_coalesce(domain);
        range = isl_map_range(isl_map_copy(map));
        range = isl_set_coalesce(range);

        return box_closure_on_domain(map, domain, range, 0);
}

/* Given a map, compute the smallest superset of this map that is of the form
 *
 *      { i -> j : L <= j - i <= U and exists a_p: j_p - i_p = M_p a_p }
 *
 * (where p ranges over the (non-parametric) dimensions),
 * compute the transitive and partially reflexive closure of this map, i.e.,
 *
 *      { i -> j : exists k >= 0:
 *              k L <= j - i <= k U and exists a: j_p - i_p = M_p a_p }
 *
 * and intersect domain and range of this transitive closure with
 * the given domain.
 */
static __isl_give isl_map *box_closure_with_identity(__isl_take isl_map *map,
        __isl_take isl_set *dom)
{
        return box_closure_on_domain(map, dom, isl_set_copy(dom), 1);
}

/* Check whether app is the transitive closure of map.
 * In particular, check that app is acyclic and, if so,
 * check that
 *
 *      app \subset (map \cup (map \circ app))
 */
static int check_exactness_omega(__isl_keep isl_map *map,
        __isl_keep isl_map *app)
{
        isl_set *delta;
        int i;
        int is_empty, is_exact;
        unsigned d;
        isl_map *test;

        delta = isl_map_deltas(isl_map_copy(app));
        d = isl_set_dim(delta, isl_dim_set);
        for (i = 0; i < d; ++i)
                delta = isl_set_fix_si(delta, isl_dim_set, i, 0);
        is_empty = isl_set_is_empty(delta);
        isl_set_free(delta);
        if (is_empty < 0)
                return -1;
        if (!is_empty)
                return 0;

        test = isl_map_apply_range(isl_map_copy(app), isl_map_copy(map));
        test = isl_map_union(test, isl_map_copy(map));
        is_exact = isl_map_is_subset(app, test);
        isl_map_free(test);

        return is_exact;
}

/* Check if basic map M_i can be combined with all the other
 * basic maps such that
 *
 *      (\cup_j M_j)^+
 *
 * can be computed as
 *
 *      M_i \cup (\cup_{j \ne i} M_i^* \circ M_j \circ M_i^*)^+
 *
 * In particular, check if we can compute a compact representation
 * of
 *
 *              M_i^* \circ M_j \circ M_i^*
 *
 * for each j != i.
 * Let M_i^? be an extension of M_i^+ that allows paths
 * of length zero, i.e., the result of box_closure(., 1).
 * The criterion, as proposed by Kelly et al., is that
 * id = M_i^? - M_i^+ can be represented as a basic map
 * and that
 *
 *      id \circ M_j \circ id = M_j
 *
 * for each j != i.
 *
 * If this function returns 1, then tc and qc are set to
 * M_i^+ and M_i^?, respectively.
 */
static int can_be_split_off(__isl_keep isl_map *map, int i,
        __isl_give isl_map **tc, __isl_give isl_map **qc)
{
        isl_map *map_i, *id = NULL;
        int j = -1;
        isl_set *C;

        *tc = NULL;
        *qc = NULL;

        C = isl_set_union(isl_map_domain(isl_map_copy(map)),
                          isl_map_range(isl_map_copy(map)));
        C = isl_set_from_basic_set(isl_set_simple_hull(C));
        if (!C)
                goto error;

        map_i = isl_map_from_basic_map(isl_basic_map_copy(map->p[i]));
        *tc = box_closure(isl_map_copy(map_i));
        *qc = box_closure_with_identity(map_i, C);
        id = isl_map_subtract(isl_map_copy(*qc), isl_map_copy(*tc));

        if (!id || !*qc)
                goto error;
        if (id->n != 1 || (*qc)->n != 1)
                goto done;

        for (j = 0; j < map->n; ++j) {
                isl_map *map_j, *test;
                int is_ok;

                if (i == j)
                        continue;
                map_j = isl_map_from_basic_map(
                                        isl_basic_map_copy(map->p[j]));
                test = isl_map_apply_range(isl_map_copy(id),
                                                isl_map_copy(map_j));
                test = isl_map_apply_range(test, isl_map_copy(id));
                is_ok = isl_map_is_equal(test, map_j);
                isl_map_free(map_j);
                isl_map_free(test);
                if (is_ok < 0)
                        goto error;
                if (!is_ok)
                        break;
        }

done:
        isl_map_free(id);
        if (j == map->n)
                return 1;

        isl_map_free(*qc);
        isl_map_free(*tc);
        *qc = NULL;
        *tc = NULL;

        return 0;
error:
        isl_map_free(id);
        isl_map_free(*qc);
        isl_map_free(*tc);
        *qc = NULL;
        *tc = NULL;
        return -1;
}

static __isl_give isl_map *box_closure_with_check(__isl_take isl_map *map,
        int *exact)
{
        isl_map *app;

        app = box_closure(isl_map_copy(map));
        if (exact)
                *exact = check_exactness_omega(map, app);

        isl_map_free(map);
        return app;
}

/* Compute an overapproximation of the transitive closure of "map"
 * using a variation of the algorithm from
 * "Transitive Closure of Infinite Graphs and its Applications"
 * by Kelly et al.
 *
 * We first check whether we can can split of any basic map M_i and
 * compute
 *
 *      (\cup_j M_j)^+
 *
 * as
 *
 *      M_i \cup (\cup_{j \ne i} M_i^* \circ M_j \circ M_i^*)^+
 *
 * using a recursive call on the remaining map.
 *
 * If not, we simply call box_closure on the whole map.
 */
static __isl_give isl_map *transitive_closure_omega(__isl_take isl_map *map,
        int *exact)
{
        int i, j;
        int exact_i;
        isl_map *app;

        if (!map)
                return NULL;
        if (map->n == 1)
                return box_closure_with_check(map, exact);

        for (i = 0; i < map->n; ++i) {
                int ok;
                isl_map *qc, *tc;
                ok = can_be_split_off(map, i, &tc, &qc);
                if (ok < 0)
                        goto error;
                if (!ok)
                        continue;

                app = isl_map_alloc_space(isl_map_get_space(map), map->n - 1, 0);

                for (j = 0; j < map->n; ++j) {
                        if (j == i)
                                continue;
                        app = isl_map_add_basic_map(app,
                                                isl_basic_map_copy(map->p[j]));
                }

                app = isl_map_apply_range(isl_map_copy(qc), app);
                app = isl_map_apply_range(app, qc);

                app = isl_map_union(tc, transitive_closure_omega(app, NULL));
                exact_i = check_exactness_omega(map, app);
                if (exact_i == 1) {
                        if (exact)
                                *exact = exact_i;
                        isl_map_free(map);
                        return app;
                }
                isl_map_free(app);
                if (exact_i < 0)
                        goto error;
        }

        return box_closure_with_check(map, exact);
error:
        isl_map_free(map);
        return NULL;
}

/* Compute the transitive closure  of "map", or an overapproximation.
 * If the result is exact, then *exact is set to 1.
 * Simply use map_power to compute the powers of map, but tell
 * it to project out the lengths of the paths instead of equating
 * the length to a parameter.
 */
__isl_give isl_map *isl_map_transitive_closure(__isl_take isl_map *map,
        int *exact)
{
        isl_space *target_dim;
        int closed;

        if (!map)
                goto error;

        if (map->ctx->opt->closure == ISL_CLOSURE_BOX)
                return transitive_closure_omega(map, exact);

        map = isl_map_compute_divs(map);
        map = isl_map_coalesce(map);
        closed = isl_map_is_transitively_closed(map);
        if (closed < 0)
                goto error;
        if (closed) {
                if (exact)
                        *exact = 1;
                return map;
        }

        target_dim = isl_map_get_space(map);
        map = map_power(map, exact, 1);
        map = isl_map_reset_space(map, target_dim);

        return map;
error:
        isl_map_free(map);
        return NULL;
}

static isl_stat inc_count(__isl_take isl_map *map, void *user)
{
        int *n = user;

        *n += map->n;

        isl_map_free(map);

        return isl_stat_ok;
}

static isl_stat collect_basic_map(__isl_take isl_map *map, void *user)
{
        int i;
        isl_basic_map ***next = user;

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

        isl_map_free(map);
        return isl_stat_ok;
error:
        isl_map_free(map);
        return isl_stat_error;
}

/* Perform Floyd-Warshall on the given list of basic relations.
 * The basic relations may live in different dimensions,
 * but basic relations that get assigned to the diagonal of the
 * grid have domains and ranges of the same dimension and so
 * the standard algorithm can be used because the nested transitive
 * closures are only applied to diagonal elements and because all
 * compositions are peformed on relations with compatible domains and ranges.
 */
static __isl_give isl_union_map *union_floyd_warshall_on_list(isl_ctx *ctx,
        __isl_keep isl_basic_map **list, int n, int *exact)
{
        int i, j, k;
        int n_group;
        int *group = NULL;
        isl_set **set = NULL;
        isl_map ***grid = NULL;
        isl_union_map *app;

        group = setup_groups(ctx, list, n, &set, &n_group);
        if (!group)
                goto error;

        grid = isl_calloc_array(ctx, isl_map **, n_group);
        if (!grid)
                goto error;
        for (i = 0; i < n_group; ++i) {
                grid[i] = isl_calloc_array(ctx, isl_map *, n_group);
                if (!grid[i])
                        goto error;
                for (j = 0; j < n_group; ++j) {
                        isl_space *dim1, *dim2, *dim;
                        dim1 = isl_space_reverse(isl_set_get_space(set[i]));
                        dim2 = isl_set_get_space(set[j]);
                        dim = isl_space_join(dim1, dim2);
                        grid[i][j] = isl_map_empty(dim);
                }
        }

        for (k = 0; k < n; ++k) {
                i = group[2 * k];
                j = group[2 * k + 1];
                grid[i][j] = isl_map_union(grid[i][j],
                                isl_map_from_basic_map(
                                        isl_basic_map_copy(list[k])));
        }
        
        floyd_warshall_iterate(grid, n_group, exact);

        app = isl_union_map_empty(isl_map_get_space(grid[0][0]));

        for (i = 0; i < n_group; ++i) {
                for (j = 0; j < n_group; ++j)
                        app = isl_union_map_add_map(app, grid[i][j]);
                free(grid[i]);
        }
        free(grid);

        for (i = 0; i < 2 * n; ++i)
                isl_set_free(set[i]);
        free(set);

        free(group);
        return app;
error:
        if (grid)
                for (i = 0; i < n_group; ++i) {
                        if (!grid[i])
                                continue;
                        for (j = 0; j < n_group; ++j)
                                isl_map_free(grid[i][j]);
                        free(grid[i]);
                }
        free(grid);
        if (set) {
                for (i = 0; i < 2 * n; ++i)
                        isl_set_free(set[i]);
                free(set);
        }
        free(group);
        return NULL;
}

/* Perform Floyd-Warshall on the given union relation.
 * The implementation is very similar to that for non-unions.
 * The main difference is that it is applied unconditionally.
 * We first extract a list of basic maps from the union map
 * and then perform the algorithm on this list.
 */
static __isl_give isl_union_map *union_floyd_warshall(
        __isl_take isl_union_map *umap, int *exact)
{
        int i, n;
        isl_ctx *ctx;
        isl_basic_map **list = NULL;
        isl_basic_map **next;
        isl_union_map *res;

        n = 0;
        if (isl_union_map_foreach_map(umap, inc_count, &n) < 0)
                goto error;

        ctx = isl_union_map_get_ctx(umap);
        list = isl_calloc_array(ctx, isl_basic_map *, n);
        if (!list)
                goto error;

        next = list;
        if (isl_union_map_foreach_map(umap, collect_basic_map, &next) < 0)
                goto error;

        res = union_floyd_warshall_on_list(ctx, list, n, exact);

        if (list) {
                for (i = 0; i < n; ++i)
                        isl_basic_map_free(list[i]);
                free(list);
        }

        isl_union_map_free(umap);
        return res;
error:
        if (list) {
                for (i = 0; i < n; ++i)
                        isl_basic_map_free(list[i]);
                free(list);
        }
        isl_union_map_free(umap);
        return NULL;
}

/* Decompose the give union relation into strongly connected components.
 * The implementation is essentially the same as that of
 * construct_power_components with the major difference that all
 * operations are performed on union maps.
 */
static __isl_give isl_union_map *union_components(
        __isl_take isl_union_map *umap, int *exact)
{
        int i;
        int n;
        isl_ctx *ctx;
        isl_basic_map **list = NULL;
        isl_basic_map **next;
        isl_union_map *path = NULL;
        struct isl_tc_follows_data data;
        struct isl_tarjan_graph *g = NULL;
        int c, l;
        int recheck = 0;

        n = 0;
        if (isl_union_map_foreach_map(umap, inc_count, &n) < 0)
                goto error;

        if (n == 0)
                return umap;
        if (n <= 1)
                return union_floyd_warshall(umap, exact);

        ctx = isl_union_map_get_ctx(umap);
        list = isl_calloc_array(ctx, isl_basic_map *, n);
        if (!list)
                goto error;

        next = list;
        if (isl_union_map_foreach_map(umap, collect_basic_map, &next) < 0)
                goto error;

        data.list = list;
        data.check_closed = 0;
        g = isl_tarjan_graph_init(ctx, n, &basic_map_follows, &data);
        if (!g)
                goto error;

        c = 0;
        i = 0;
        l = n;
        path = isl_union_map_empty(isl_union_map_get_space(umap));
        while (l) {
                isl_union_map *comp;
                isl_union_map *path_comp, *path_comb;
                comp = isl_union_map_empty(isl_union_map_get_space(umap));
                while (g->order[i] != -1) {
                        comp = isl_union_map_add_map(comp,
                                    isl_map_from_basic_map(
                                        isl_basic_map_copy(list[g->order[i]])));
                        --l;
                        ++i;
                }
                path_comp = union_floyd_warshall(comp, exact);
                path_comb = isl_union_map_apply_range(isl_union_map_copy(path),
                                                isl_union_map_copy(path_comp));
                path = isl_union_map_union(path, path_comp);
                path = isl_union_map_union(path, path_comb);
                ++i;
                ++c;
        }

        if (c > 1 && data.check_closed && !*exact) {
                int closed;

                closed = isl_union_map_is_transitively_closed(path);
                if (closed < 0)
                        goto error;
                recheck = !closed;
        }

        isl_tarjan_graph_free(g);

        for (i = 0; i < n; ++i)
                isl_basic_map_free(list[i]);
        free(list);

        if (recheck) {
                isl_union_map_free(path);
                return union_floyd_warshall(umap, exact);
        }

        isl_union_map_free(umap);

        return path;
error:
        isl_tarjan_graph_free(g);
        if (list) {
                for (i = 0; i < n; ++i)
                        isl_basic_map_free(list[i]);
                free(list);
        }
        isl_union_map_free(umap);
        isl_union_map_free(path);
        return NULL;
}

/* Compute the transitive closure  of "umap", or an overapproximation.
 * If the result is exact, then *exact is set to 1.
 */
__isl_give isl_union_map *isl_union_map_transitive_closure(
        __isl_take isl_union_map *umap, int *exact)
{
        int closed;

        if (!umap)
                return NULL;

        if (exact)
                *exact = 1;

        umap = isl_union_map_compute_divs(umap);
        umap = isl_union_map_coalesce(umap);
        closed = isl_union_map_is_transitively_closed(umap);
        if (closed < 0)
                goto error;
        if (closed)
                return umap;
        umap = union_components(umap, exact);
        return umap;
error:
        isl_union_map_free(umap);
        return NULL;
}

struct isl_union_power {
        isl_union_map *pow;
        int *exact;
};

static isl_stat power(__isl_take isl_map *map, void *user)
{
        struct isl_union_power *up = user;

        map = isl_map_power(map, up->exact);
        up->pow = isl_union_map_from_map(map);

        return isl_stat_error;
}

/* Construct a map [x] -> [x+1], with parameters prescribed by "dim".
 */
static __isl_give isl_union_map *increment(__isl_take isl_space *dim)
{
        int k;
        isl_basic_map *bmap;

        dim = isl_space_add_dims(dim, isl_dim_in, 1);
        dim = isl_space_add_dims(dim, isl_dim_out, 1);
        bmap = isl_basic_map_alloc_space(dim, 0, 1, 0);
        k = isl_basic_map_alloc_equality(bmap);
        if (k < 0)
                goto error;
        isl_seq_clr(bmap->eq[k], isl_basic_map_total_dim(bmap));
        isl_int_set_si(bmap->eq[k][0], 1);
        isl_int_set_si(bmap->eq[k][isl_basic_map_offset(bmap, isl_dim_in)], 1);
        isl_int_set_si(bmap->eq[k][isl_basic_map_offset(bmap, isl_dim_out)], -1);
        return isl_union_map_from_map(isl_map_from_basic_map(bmap));
error:
        isl_basic_map_free(bmap);
        return NULL;
}

/* Construct a map [[x]->[y]] -> [y-x], with parameters prescribed by "dim".
 */
static __isl_give isl_union_map *deltas_map(__isl_take isl_space *dim)
{
        isl_basic_map *bmap;

        dim = isl_space_add_dims(dim, isl_dim_in, 1);
        dim = isl_space_add_dims(dim, isl_dim_out, 1);
        bmap = isl_basic_map_universe(dim);
        bmap = isl_basic_map_deltas_map(bmap);

        return isl_union_map_from_map(isl_map_from_basic_map(bmap));
}

/* Compute the positive powers of "map", or an overapproximation.
 * The result maps the exponent to a nested copy of the corresponding power.
 * If the result is exact, then *exact is set to 1.
 */
__isl_give isl_union_map *isl_union_map_power(__isl_take isl_union_map *umap,
        int *exact)
{
        int n;
        isl_union_map *inc;
        isl_union_map *dm;

        if (!umap)
                return NULL;
        n = isl_union_map_n_map(umap);
        if (n == 0)
                return umap;
        if (n == 1) {
                struct isl_union_power up = { NULL, exact };
                isl_union_map_foreach_map(umap, &power, &up);
                isl_union_map_free(umap);
                return up.pow;
        }
        inc = increment(isl_union_map_get_space(umap));
        umap = isl_union_map_product(inc, umap);
        umap = isl_union_map_transitive_closure(umap, exact);
        umap = isl_union_map_zip(umap);
        dm = deltas_map(isl_union_map_get_space(umap));
        umap = isl_union_map_apply_domain(umap, dm);
        
        return umap;
}

#undef TYPE
#define TYPE isl_map
#include "isl_power_templ.c"

#undef TYPE
#define TYPE isl_union_map
#include "isl_power_templ.c"