doc: document how to inspect sets and relations

[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"
 
/*
 * 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 union of translations (uniform dependences), return a matrix
 * with as many rows as there are disjuncts in the union and as many
 * columns as there are input dimensions (which should be equal to
 * the number of output dimensions).
 * Each row contains the translation performed by the corresponding disjunct.
 * If "map" turns out not to be a union of translations, then the contents
 * of the returned matrix are undefined and *ok is set to 0.
 */
static __isl_give isl_mat *extract_steps(__isl_keep isl_map *map, int *ok)
{
        int i, j;
        struct isl_mat *steps;
        unsigned dim = isl_map_dim(map, isl_dim_in);

        *ok = 1;

        steps = isl_mat_alloc(map->ctx, map->n, dim);
        if (!steps)
                return NULL;

        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 < dim; ++j) {
                        int fixed;

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

                isl_basic_set_free(delta);

                if (j < dim)
                        break;
        }

        if (i < map->n)
                *ok = 0;

        return steps;
error:
        isl_mat_free(steps);
        return NULL;
}

/* Given a set of n offsets v_i (the rows of "steps"), construct a relation
 * of the given dimension specification that maps a element x to any
 * element that can be reached by taking a positive number of steps
 * along any of the offsets, where the number of steps k is equal to
 * parameter "param".  That is, construct
 *
 * { [x] -> [y] : exists k_i >= 0, y = x + \sum_i k_i v_i, k = \sum_i k_i > 0 }
 *
 */
static __isl_give isl_map *path_along_steps(__isl_take isl_dim *dim,
        __isl_keep isl_mat *steps, unsigned param)
{
        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 + 1, n + 1);

        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);
                for (j = 0; j < n; ++j)
                        isl_int_set(path->eq[k][1 + nparam + 2 * d + j],
                                    steps->row[j][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 + param], -1);
        for (j = 0; j < n; ++j)
                isl_int_set_si(path->eq[k][1 + nparam + 2 * d + j], 1);

        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);
        }

        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 + param], 1);
        isl_int_set_si(path->ineq[k][0], -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;
}

/* Check whether any path of length at least one along "steps" is acyclic.
 * That is, check whether
 *
 *      \sum_i k_i \delta_i = 0
 *      \sum_i k_i >= 1
 *      k_i >= 0
 *
 * with \delta_i the rows of "steps", is infeasible.
 */
static int is_acyclic(__isl_keep isl_mat *steps)
{
        int acyclic;
        int i, j, k;
        struct isl_basic_set *test;

        if (!steps)
                return -1;

        test = isl_basic_set_alloc(steps->ctx, 0, steps->n_row, 0,
                                        steps->n_col, steps->n_row + 1);

        for (i = 0; i < steps->n_col; ++i) {
                k = isl_basic_set_alloc_equality(test);
                if (k < 0)
                        goto error;
                isl_int_set_si(test->eq[k][0], 0);
                for (j = 0; j < steps->n_row; ++j)
                        isl_int_set(test->eq[k][1 + j], steps->row[j][i]);
        }
        for (j = 0; j < steps->n_row; ++j) {
                k = isl_basic_set_alloc_inequality(test);
                if (k < 0)
                        goto error;
                isl_seq_clr(test->ineq[k], 1 + steps->n_row);
                isl_int_set_si(test->ineq[k][1 + j], 1);
        }

        k = isl_basic_set_alloc_inequality(test);
        if (k < 0)
                goto error;
        isl_int_set_si(test->ineq[k][0], -1);
        for (j = 0; j < steps->n_row; ++j)
                isl_int_set_si(test->ineq[k][1 + j], 1);

        acyclic = isl_basic_set_is_empty(test);

        isl_basic_set_free(test);
        return acyclic;
error:
        isl_basic_set_free(test);
        return -1;
}

/* Shift variable at position "pos" up by one.
 * That is, replace the corresponding variable v by v - 1.
 */
static __isl_give isl_basic_map *basic_map_shift_pos(
        __isl_take isl_basic_map *bmap, unsigned pos)
{
        int i;

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

        for (i = 0; i < bmap->n_eq; ++i)
                isl_int_sub(bmap->eq[i][0], bmap->eq[i][0], bmap->eq[i][pos]);

        for (i = 0; i < bmap->n_ineq; ++i)
                isl_int_sub(bmap->ineq[i][0],
                                bmap->ineq[i][0], bmap->ineq[i][pos]);

        for (i = 0; i < bmap->n_div; ++i) {
                if (isl_int_is_zero(bmap->div[i][0]))
                        continue;
                isl_int_sub(bmap->div[i][1],
                                bmap->div[i][1], bmap->div[i][1 + pos]);
        }

        return bmap;
}

/* Shift variable at position "pos" up by one.
 * That is, replace the corresponding variable v by v - 1.
 */
static __isl_give isl_map *map_shift_pos(__isl_take isl_map *map, unsigned pos)
{
        int i;

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

        for (i = 0; i < map->n; ++i) {
                map->p[i] = basic_map_shift_pos(map->p[i], pos);
                if (!map->p[i])
                        goto error;
        }
        ISL_F_CLR(map, ISL_MAP_NORMALIZED);
        return map;
error:
        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
 *
 */
static int check_power_exactness(__isl_take isl_map *map,
        __isl_take isl_map *app, unsigned param)
{
        int exact;
        isl_map *app_1;
        isl_map *app_2;

        app_1 = isl_map_fix_si(isl_map_copy(app), isl_dim_param, param, 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_2 = isl_map_lower_bound_si(isl_map_copy(app),
                                        isl_dim_param, param, 2);
        app_1 = map_shift_pos(app, 1 + param);
        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.
 */
static int check_exactness(__isl_take isl_map *map, __isl_take isl_map *app,
        __isl_keep isl_mat *steps, unsigned param, int project)
{
        isl_map *test;
        int exact;

        if (project) {
                project = is_acyclic(steps);
                if (project < 0)
                        goto error;
        }

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

        map = isl_map_project_out(map, isl_dim_param, param, 1);
        app = isl_map_project_out(app, isl_dim_param, param, 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;
}

/* 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_mat *steps = NULL;
        struct isl_set *domain = NULL;
        struct isl_set *range = NULL;
        struct isl_map *app = NULL;
        struct isl_map *path = NULL;
        int ok;

        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);

        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(isl_set_copy(domain),
                                            isl_set_copy(range));

        steps = extract_steps(map, &ok);
        if (!ok)
                goto not_exact;

        path = path_along_steps(isl_map_get_dim(map), steps, param);
        app = isl_map_intersect(app, isl_map_copy(path));

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

        if (0) {
not_exact:
                if (exact)
                        *exact = 0;
        }
        isl_set_free(domain);
        isl_set_free(range);
        isl_mat_free(steps);
        isl_map_free(path);
        isl_map_free(map);
        return app;
error:
        isl_set_free(domain);
        isl_set_free(range);
        isl_mat_free(steps);
        isl_map_free(path);
        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;
}