add transitive closure example from Kelly's 1996 paper

[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"
 
/*
 * 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;
}

/* 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 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.
 *
 * 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_path(__isl_keep isl_map *map,
        unsigned param)
{
        struct isl_mat *steps = NULL;
        struct isl_map *path = NULL;
        struct isl_map *diff;
        struct isl_dim *dim = NULL;
        unsigned d;
        int i, j, n;

        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);

        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));
        }

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

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

/* Check whether "path" is acyclic.
 * That is, check whether
 *
 *      { d | d = y - x and (x,y) in path }
 *
 * does not contain the origin.
 */
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)
                delta = isl_set_fix_si(delta, isl_dim_set, i, 0);

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

        return acyclic;
}

/* 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_take isl_map *path, unsigned param, int project)
{
        isl_map *test;
        int exact;

        if (project) {
                project = is_acyclic(path);
                if (project < 0)
                        goto error;
        } else
                isl_map_free(path);

        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_set *domain = NULL;
        struct isl_set *range = NULL;
        struct isl_map *app = NULL;
        struct isl_map *path = 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);

        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));

        path = construct_path(map, param);
        app = isl_map_intersect(app, isl_map_copy(path));

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

        isl_set_free(domain);
        isl_set_free(range);
        isl_map_free(path);
        isl_map_free(map);
        return app;
error:
        isl_set_free(domain);
        isl_set_free(range);
        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;
}