isl_map_transitive_closure: reformulate exactness test in terms of extended paths

[isl.git] / isl_transitive_closure.c

/*
 * Copyright 2010      INRIA Saclay
 *
 * Use of this software is governed by the GNU LGPLv2.1 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_map.h"
#include "isl_map_private.h"
#include "isl_seq.h"
 
/* 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)
{
        struct isl_dim *dim;
        struct isl_basic_map *bmap;
        unsigned d;
        unsigned nparam;
        int k;
        isl_int *c;

        if (!map)
                return NULL;

        dim = isl_map_get_dim(map);
        d = isl_dim_size(dim, isl_dim_in);
        nparam = isl_dim_size(dim, isl_dim_param);
        bmap = isl_basic_map_alloc_dim(dim, 0, 1, 1);
        if (exactly) {
                k = isl_basic_map_alloc_equality(bmap);
                c = bmap->eq[k];
        } else {
                k = isl_basic_map_alloc_inequality(bmap);
                c = bmap->ineq[k];
        }
        if (k < 0)
                goto error;
        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(map, isl_dim_in, 1);
        map = isl_map_add(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);

        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;
error:
        isl_map_free(app);
        isl_map_free(map);
        return -1;
}

/*
 * 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_dim *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_dim_size(dim, isl_dim_in);
        n = steps->n_row;
        nparam = isl_dim_size(dim, isl_dim_param);

        path = isl_basic_map_alloc_dim(isl_dim_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_dim_free(dim);

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

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

/* 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 IMPURE otherwise.
 */
static int purity(__isl_keep isl_basic_set *bset, isl_int *c)
{
        unsigned d;
        unsigned n_div;
        unsigned nparam;

        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);

        if (isl_seq_first_non_zero(c + 1 + nparam + d, n_div) != -1)
                return IMPURE;
        if (isl_seq_first_non_zero(c + 1, nparam) == -1)
                return PURE_VAR;
        if (isl_seq_first_non_zero(c + 1 + nparam, d) == -1)
                return PURE_PARAM;
        return IMPURE;
}

/* 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 >= and B p + b >= 0 and
 *                              C x + C'p + c >= 0 }
 *
 * 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 k >= 1 }
 *      union { [x] -> [x] }
 *
 * Existentially quantified variables in \delta are currently ignored.
 * This is safe, but leads to an additional overapproximation.
 */
static __isl_give isl_map *path_along_delta(__isl_take isl_dim *dim,
        __isl_take isl_basic_set *delta)
{
        isl_basic_map *path = NULL;
        unsigned d;
        unsigned n_div;
        unsigned nparam;
        unsigned off;
        int i, k;

        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_dim(isl_dim_copy(dim), n_div + d + 1,
                        d + 1 + 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);
        }

        for (i = 0; i < delta->n_eq; ++i) {
                int p = purity(delta, delta->eq[i]);
                if (p == IMPURE)
                        continue;
                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));
                if (p == PURE_VAR) {
                        isl_seq_cpy(path->eq[k] + off,
                                    delta->eq[i] + 1 + nparam, d);
                        isl_int_set(path->eq[k][off + d], delta->eq[i][0]);
                } else
                        isl_seq_cpy(path->eq[k], delta->eq[i], 1 + nparam);
        }

        for (i = 0; i < delta->n_ineq; ++i) {
                int p = purity(delta, delta->ineq[i]);
                if (p == IMPURE)
                        continue;
                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 (p == PURE_VAR) {
                        isl_seq_cpy(path->ineq[k] + off,
                                    delta->ineq[i] + 1 + nparam, d);
                        isl_int_set(path->ineq[k][off + d], delta->ineq[i][0]);
                } else
                        isl_seq_cpy(path->ineq[k], delta->ineq[i], 1 + nparam);
        }

        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][0], -1);
        isl_int_set_si(path->ineq[k][off + d], 1);
                        
        isl_basic_set_free(delta);
        path = isl_basic_map_finalize(path);
        return isl_basic_map_union(path,
                                isl_basic_map_identity(isl_dim_domain(dim)));
error:
        isl_dim_free(dim);
        isl_basic_set_free(delta);
        isl_basic_map_free(path);
        return NULL;
}

/* Given a dimenion 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_dim *dim,
        unsigned param)
{
        struct isl_basic_map *bmap;
        unsigned d;
        unsigned nparam;
        int k;

        d = isl_dim_size(dim, isl_dim_in);
        nparam = isl_dim_size(dim, isl_dim_param);
        bmap = isl_basic_map_alloc_dim(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_dim *dim,
        __isl_keep isl_map *map, int *project)
{
        struct isl_mat *steps = NULL;
        struct isl_map *path = NULL;
        unsigned d;
        int i, j, n;

        d = isl_map_dim(map, isl_dim_in);

        path = isl_map_identity(isl_dim_domain(isl_dim_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) {
                        int fixed;

                        fixed = isl_basic_set_fast_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_dim_copy(dim), delta));
                } else {
                        isl_basic_set_free(delta);
                        ++n;
                }
        }

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

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

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

/* 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 the union of the identity map and
 * 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 } union
 *      { (x) -> (x) }
 */
static __isl_give isl_map *construct_component(__isl_take isl_dim *dim,
        __isl_keep isl_map *map, int *project)
{
        struct isl_set *domain = NULL;
        struct isl_set *range = NULL;
        struct isl_map *app = NULL;
        struct isl_map *path = NULL;

        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);
        app = isl_map_from_domain_and_range(domain, range);
        app = isl_map_add(app, isl_dim_in, 1);
        app = isl_map_add(app, isl_dim_out, 1);

        path = construct_extended_path(isl_dim_copy(dim), map, project);
        app = isl_map_intersect(app, path);

        return isl_map_union(app, isl_map_identity(isl_dim_domain(dim)));
}

/* Structure for representing the nodes in the graph being traversed
 * using Tarjan's algorithm.
 * index represents the order in which nodes are visited.
 * min_index is the index of the root of a (sub)component.
 * on_stack indicates whether the node is currently on the stack.
 */
struct basic_map_sort_node {
        int index;
        int min_index;
        int on_stack;
};
/* Structure for representing the graph being traversed
 * using Tarjan's algorithm.
 * len is the number of nodes
 * node is an array of nodes
 * stack contains the nodes on the path from the root to the current node
 * sp is the stack pointer
 * index is the index of the last node visited
 * order contains the elements of the components separated by -1
 * op represents the current position in order
 */
struct basic_map_sort {
        int len;
        struct basic_map_sort_node *node;
        int *stack;
        int sp;
        int index;
        int *order;
        int op;
};

static void basic_map_sort_free(struct basic_map_sort *s)
{
        if (!s)
                return;
        free(s->node);
        free(s->stack);
        free(s->order);
        free(s);
}

static struct basic_map_sort *basic_map_sort_alloc(struct isl_ctx *ctx, int len)
{
        struct basic_map_sort *s;
        int i;

        s = isl_calloc_type(ctx, struct basic_map_sort);
        if (!s)
                return NULL;
        s->len = len;
        s->node = isl_alloc_array(ctx, struct basic_map_sort_node, len);
        if (!s->node)
                goto error;
        for (i = 0; i < len; ++i)
                s->node[i].index = -1;
        s->stack = isl_alloc_array(ctx, int, len);
        if (!s->stack)
                goto error;
        s->order = isl_alloc_array(ctx, int, 2 * len);
        if (!s->order)
                goto error;

        s->sp = 0;
        s->index = 0;
        s->op = 0;

        return s;
error:
        basic_map_sort_free(s);
        return NULL;
}

/* Check whether in the computation of the transitive closure
 * "bmap1" (R_1) should follow (or be part of the same component as)
 * "bmap2" (R_2).
 *
 * That is check whether
 *
 *      R_1 \circ R_2
 *
 * is non-empty and that moreover, it is non-empty on the set
 * of elements that do not get mapped to the same set of elements
 * by both "R_1 \circ R_2" and "R_2 \circ R_1".
 * For elements that do get mapped to the same elements by these
 * two compositions, R_1 and R_2 are commutative, so if these
 * elements are the only ones for which R_1 \circ R_2 is non-empty,
 * then you may just as well apply R_1 first.
 */
static int basic_map_follows(__isl_keep isl_basic_map *bmap1,
        __isl_keep isl_basic_map *bmap2)
{
        struct isl_map *map12 = NULL;
        struct isl_map *map21 = NULL;
        struct isl_map *d = NULL;
        struct isl_set *dom = NULL;
        int empty;

        map21 = isl_map_from_basic_map(
                        isl_basic_map_apply_range(
                                isl_basic_map_copy(bmap2),
                                isl_basic_map_copy(bmap1)));
        empty = isl_map_is_empty(map21);
        if (empty < 0)
                goto error;
        if (empty) {
                isl_map_free(map21);
                return 0;
        }

        map12 = isl_map_from_basic_map(
                        isl_basic_map_apply_range(
                                isl_basic_map_copy(bmap1),
                                isl_basic_map_copy(bmap2)));
        d = isl_map_subtract(isl_map_copy(map12), isl_map_copy(map21));
        d = isl_map_union(d,
                isl_map_subtract(isl_map_copy(map21), isl_map_copy(map12)));
        dom = isl_map_domain(d);

        map21 = isl_map_intersect_domain(map21, dom);
        empty = isl_map_is_empty(map21);

        isl_map_free(map12);
        isl_map_free(map21);

        return empty < 0 ? -1 : !empty;
error:
        isl_map_free(map21);
        return -1;
}

/* Perform Tarjan's algorithm for computing the strongly connected components
 * in the graph with the disjuncts of "map" as vertices and with an
 * edge between any pair of disjuncts such that the first has
 * to be applied after the second.
 */
static int power_components_tarjan(struct basic_map_sort *s,
        __isl_keep isl_map *map, int i)
{
        int j;

        s->node[i].index = s->index;
        s->node[i].min_index = s->index;
        s->node[i].on_stack = 1;
        s->index++;
        s->stack[s->sp++] = i;

        for (j = s->len - 1; j >= 0; --j) {
                int f;

                if (j == i)
                        continue;
                if (s->node[j].index >= 0 &&
                        (!s->node[j].on_stack ||
                         s->node[j].index > s->node[i].min_index))
                        continue;

                f = basic_map_follows(map->p[i], map->p[j]);
                if (f < 0)
                        return -1;
                if (!f)
                        continue;

                if (s->node[j].index < 0) {
                        power_components_tarjan(s, map, j);
                        if (s->node[j].min_index < s->node[i].min_index)
                                s->node[i].min_index = s->node[j].min_index;
                } else if (s->node[j].index < s->node[i].min_index)
                        s->node[i].min_index = s->node[j].index;
        }

        if (s->node[i].index != s->node[i].min_index)
                return 0;

        do {
                j = s->stack[--s->sp];
                s->node[j].on_stack = 0;
                s->order[s->op++] = j;
        } while (j != i);
        s->order[s->op++] = -1;

        return 0;
}

/* 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 the union of the identity map and
 * 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 } union
 *      { (x) -> (x) }
 *
 * We first split the map into strongly connected components, perform
 * the above on each component and the join the results in the correct
 * order.  The power of each of the components needs to be extended
 * with the identity map because a path in the global result need
 * not go through every component.
 * The final result will then also contain the identity map, but
 * this part will be removed when the length of the path is forced
 * to be strictly positive.
 */
static __isl_give isl_map *construct_power_components(__isl_take isl_dim *dim,
        __isl_keep isl_map *map, int *project)
{
        int i, n;
        struct isl_map *path = NULL;
        struct basic_map_sort *s = NULL;

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

        s = basic_map_sort_alloc(map->ctx, map->n);
        if (!s)
                goto error;
        for (i = map->n - 1; i >= 0; --i) {
                if (s->node[i].index >= 0)
                        continue;
                if (power_components_tarjan(s, map, i) < 0)
                        goto error;
        }

        i = 0;
        n = map->n;
        path = isl_map_identity(isl_dim_domain(isl_dim_copy(dim)));
        while (n) {
                struct isl_map *comp;
                comp = isl_map_alloc_dim(isl_map_get_dim(map), n, 0);
                while (s->order[i] != -1) {
                        comp = isl_map_add_basic_map(comp,
                                    isl_basic_map_copy(map->p[s->order[i]]));
                        --n;
                        ++i;
                }
                path = isl_map_apply_range(path,
                            construct_component(isl_dim_copy(dim), comp, project));
                isl_map_free(comp);
                ++i;
        }

        basic_map_sort_free(s);
        isl_dim_free(dim);

        return path;
error:
        basic_map_sort_free(s);
        isl_dim_free(dim);
        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 }
 *
 * We first 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).
 * In the final step, this difference is equated to the parameter "param"
 * and made positive.  The extra coordinates are subsequently projected out.
 */
static __isl_give isl_map *construct_power(__isl_keep isl_map *map,
        unsigned param, int *exact, int project)
{
        struct isl_map *app = NULL;
        struct isl_map *diff;
        struct isl_dim *dim = NULL;
        unsigned d;

        if (!map)
                return NULL;

        dim = isl_map_get_dim(map);

        d = isl_dim_size(dim, isl_dim_in);
        dim = isl_dim_add(dim, isl_dim_in, 1);
        dim = isl_dim_add(dim, isl_dim_out, 1);

        app = construct_power_components(isl_dim_copy(dim), map,
                                        exact ? &project : NULL);

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

        diff = equate_parameter_to_length(dim, param);
        app = isl_map_intersect(app, diff);
        app = isl_map_project_out(app, isl_dim_in, d, 1);
        app = isl_map_project_out(app, isl_dim_out, d, 1);

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

/* Compute the positive powers of "map", or an overapproximation.
 * The power is given by parameter "param".  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.
 */
static __isl_give isl_map *map_power(__isl_take isl_map *map, unsigned param,
        int *exact, int project)
{
        struct isl_map *app = NULL;

        if (exact)
                *exact = 1;

        map = isl_map_remove_empty_parts(map);
        if (!map)
                return NULL;

        if (isl_map_fast_is_empty(map))
                return map;

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

        app = construct_power(map, param, 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 power is given by parameter "param".  If the result is exact,
 * then *exact is set to 1.
 */
__isl_give isl_map *isl_map_power(__isl_take isl_map *map, unsigned param,
        int *exact)
{
        return map_power(map, param, exact, 0);
}

/* Compute the transitive closure  of "map", or an overapproximation.
 * If the result is exact, then *exact is set to 1.
 * Simply compute the powers of map and then project out the parameter
 * describing the power.
 */
__isl_give isl_map *isl_map_transitive_closure(__isl_take isl_map *map,
        int *exact)
{
        unsigned param;

        if (!map)
                goto error;

        param = isl_map_dim(map, isl_dim_param);
        map = isl_map_add(map, isl_dim_param, 1);
        map = map_power(map, param, exact, 1);
        map = isl_map_project_out(map, isl_dim_param, param, 1);

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