isl_affine_hull.c: uset_affine_hull: avoid use of isl_basic_set_universe_like

[isl.git] / isl_flow.c

/*
 * Copyright 2005-2007 Universiteit Leiden
 * Copyright 2008-2009 Katholieke Universiteit Leuven
 * Copyright 2010      INRIA Saclay
 * Copyright 2012      Universiteit Leiden
 * Copyright 2014      Ecole Normale Superieure
 *
 * Use of this software is governed by the MIT license
 *
 * Written by Sven Verdoolaege, Leiden Institute of Advanced Computer Science,
 * Universiteit Leiden, Niels Bohrweg 1, 2333 CA Leiden, The Netherlands
 * and K.U.Leuven, Departement Computerwetenschappen, Celestijnenlaan 200A,
 * B-3001 Leuven, Belgium
 * and INRIA Saclay - Ile-de-France, Parc Club Orsay Universite,
 * ZAC des vignes, 4 rue Jacques Monod, 91893 Orsay, France 
 * and Ecole Normale Superieure, 45 rue d'Ulm, 75230 Paris, France
 */

#include <isl/set.h>
#include <isl/map.h>
#include <isl/union_set.h>
#include <isl/union_map.h>
#include <isl/flow.h>
#include <isl/schedule_node.h>
#include <isl_sort.h>

enum isl_restriction_type {
        isl_restriction_type_empty,
        isl_restriction_type_none,
        isl_restriction_type_input,
        isl_restriction_type_output
};

struct isl_restriction {
        enum isl_restriction_type type;

        isl_set *source;
        isl_set *sink;
};

/* Create a restriction of the given type.
 */
static __isl_give isl_restriction *isl_restriction_alloc(
        __isl_take isl_map *source_map, enum isl_restriction_type type)
{
        isl_ctx *ctx;
        isl_restriction *restr;

        if (!source_map)
                return NULL;

        ctx = isl_map_get_ctx(source_map);
        restr = isl_calloc_type(ctx, struct isl_restriction);
        if (!restr)
                goto error;

        restr->type = type;

        isl_map_free(source_map);
        return restr;
error:
        isl_map_free(source_map);
        return NULL;
}

/* Create a restriction that doesn't restrict anything.
 */
__isl_give isl_restriction *isl_restriction_none(__isl_take isl_map *source_map)
{
        return isl_restriction_alloc(source_map, isl_restriction_type_none);
}

/* Create a restriction that removes everything.
 */
__isl_give isl_restriction *isl_restriction_empty(
        __isl_take isl_map *source_map)
{
        return isl_restriction_alloc(source_map, isl_restriction_type_empty);
}

/* Create a restriction on the input of the maximization problem
 * based on the given source and sink restrictions.
 */
__isl_give isl_restriction *isl_restriction_input(
        __isl_take isl_set *source_restr, __isl_take isl_set *sink_restr)
{
        isl_ctx *ctx;
        isl_restriction *restr;

        if (!source_restr || !sink_restr)
                goto error;

        ctx = isl_set_get_ctx(source_restr);
        restr = isl_calloc_type(ctx, struct isl_restriction);
        if (!restr)
                goto error;

        restr->type = isl_restriction_type_input;
        restr->source = source_restr;
        restr->sink = sink_restr;

        return restr;
error:
        isl_set_free(source_restr);
        isl_set_free(sink_restr);
        return NULL;
}

/* Create a restriction on the output of the maximization problem
 * based on the given source restriction.
 */
__isl_give isl_restriction *isl_restriction_output(
        __isl_take isl_set *source_restr)
{
        isl_ctx *ctx;
        isl_restriction *restr;

        if (!source_restr)
                return NULL;

        ctx = isl_set_get_ctx(source_restr);
        restr = isl_calloc_type(ctx, struct isl_restriction);
        if (!restr)
                goto error;

        restr->type = isl_restriction_type_output;
        restr->source = source_restr;

        return restr;
error:
        isl_set_free(source_restr);
        return NULL;
}

__isl_null isl_restriction *isl_restriction_free(
        __isl_take isl_restriction *restr)
{
        if (!restr)
                return NULL;

        isl_set_free(restr->source);
        isl_set_free(restr->sink);
        free(restr);
        return NULL;
}

isl_ctx *isl_restriction_get_ctx(__isl_keep isl_restriction *restr)
{
        return restr ? isl_set_get_ctx(restr->source) : NULL;
}

/* A private structure to keep track of a mapping together with
 * a user-specified identifier and a boolean indicating whether
 * the map represents a must or may access/dependence.
 */
struct isl_labeled_map {
        struct isl_map  *map;
        void            *data;
        int             must;
};

/* A structure containing the input for dependence analysis:
 * - a sink
 * - n_must + n_may (<= max_source) sources
 * - a function for determining the relative order of sources and sink
 * The must sources are placed before the may sources.
 *
 * domain_map is an auxiliary map that maps the sink access relation
 * to the domain of this access relation.
 *
 * restrict_fn is a callback that (if not NULL) will be called
 * right before any lexicographical maximization.
 */
struct isl_access_info {
        isl_map                         *domain_map;
        struct isl_labeled_map          sink;
        isl_access_level_before         level_before;

        isl_access_restrict             restrict_fn;
        void                            *restrict_user;

        int                             max_source;
        int                             n_must;
        int                             n_may;
        struct isl_labeled_map          source[1];
};

/* A structure containing the output of dependence analysis:
 * - n_source dependences
 * - a wrapped subset of the sink for which definitely no source could be found
 * - a wrapped subset of the sink for which possibly no source could be found
 */
struct isl_flow {
        isl_set                 *must_no_source;
        isl_set                 *may_no_source;
        int                     n_source;
        struct isl_labeled_map  *dep;
};

/* Construct an isl_access_info structure and fill it up with
 * the given data.  The number of sources is set to 0.
 */
__isl_give isl_access_info *isl_access_info_alloc(__isl_take isl_map *sink,
        void *sink_user, isl_access_level_before fn, int max_source)
{
        isl_ctx *ctx;
        struct isl_access_info *acc;

        if (!sink)
                return NULL;

        ctx = isl_map_get_ctx(sink);
        isl_assert(ctx, max_source >= 0, goto error);

        acc = isl_calloc(ctx, struct isl_access_info,
                        sizeof(struct isl_access_info) +
                        (max_source - 1) * sizeof(struct isl_labeled_map));
        if (!acc)
                goto error;

        acc->sink.map = sink;
        acc->sink.data = sink_user;
        acc->level_before = fn;
        acc->max_source = max_source;
        acc->n_must = 0;
        acc->n_may = 0;

        return acc;
error:
        isl_map_free(sink);
        return NULL;
}

/* Free the given isl_access_info structure.
 */
__isl_null isl_access_info *isl_access_info_free(
        __isl_take isl_access_info *acc)
{
        int i;

        if (!acc)
                return NULL;
        isl_map_free(acc->domain_map);
        isl_map_free(acc->sink.map);
        for (i = 0; i < acc->n_must + acc->n_may; ++i)
                isl_map_free(acc->source[i].map);
        free(acc);
        return NULL;
}

isl_ctx *isl_access_info_get_ctx(__isl_keep isl_access_info *acc)
{
        return acc ? isl_map_get_ctx(acc->sink.map) : NULL;
}

__isl_give isl_access_info *isl_access_info_set_restrict(
        __isl_take isl_access_info *acc, isl_access_restrict fn, void *user)
{
        if (!acc)
                return NULL;
        acc->restrict_fn = fn;
        acc->restrict_user = user;
        return acc;
}

/* Add another source to an isl_access_info structure, making
 * sure the "must" sources are placed before the "may" sources.
 * This function may be called at most max_source times on a
 * given isl_access_info structure, with max_source as specified
 * in the call to isl_access_info_alloc that constructed the structure.
 */
__isl_give isl_access_info *isl_access_info_add_source(
        __isl_take isl_access_info *acc, __isl_take isl_map *source,
        int must, void *source_user)
{
        isl_ctx *ctx;

        if (!acc)
                goto error;
        ctx = isl_map_get_ctx(acc->sink.map);
        isl_assert(ctx, acc->n_must + acc->n_may < acc->max_source, goto error);
        
        if (must) {
                if (acc->n_may)
                        acc->source[acc->n_must + acc->n_may] =
                                acc->source[acc->n_must];
                acc->source[acc->n_must].map = source;
                acc->source[acc->n_must].data = source_user;
                acc->source[acc->n_must].must = 1;
                acc->n_must++;
        } else {
                acc->source[acc->n_must + acc->n_may].map = source;
                acc->source[acc->n_must + acc->n_may].data = source_user;
                acc->source[acc->n_must + acc->n_may].must = 0;
                acc->n_may++;
        }

        return acc;
error:
        isl_map_free(source);
        isl_access_info_free(acc);
        return NULL;
}

/* Return -n, 0 or n (with n a positive value), depending on whether
 * the source access identified by p1 should be sorted before, together
 * or after that identified by p2.
 *
 * If p1 appears before p2, then it should be sorted first.
 * For more generic initial schedules, it is possible that neither
 * p1 nor p2 appears before the other, or at least not in any obvious way.
 * We therefore also check if p2 appears before p1, in which case p2
 * should be sorted first.
 * If not, we try to order the two statements based on the description
 * of the iteration domains.  This results in an arbitrary, but fairly
 * stable ordering.
 */
static int access_sort_cmp(const void *p1, const void *p2, void *user)
{
        isl_access_info *acc = user;
        const struct isl_labeled_map *i1, *i2;
        int level1, level2;
        uint32_t h1, h2;
        i1 = (const struct isl_labeled_map *) p1;
        i2 = (const struct isl_labeled_map *) p2;

        level1 = acc->level_before(i1->data, i2->data);
        if (level1 % 2)
                return -1;

        level2 = acc->level_before(i2->data, i1->data);
        if (level2 % 2)
                return 1;

        h1 = isl_map_get_hash(i1->map);
        h2 = isl_map_get_hash(i2->map);
        return h1 > h2 ? 1 : h1 < h2 ? -1 : 0;
}

/* Sort the must source accesses in their textual order.
 */
static __isl_give isl_access_info *isl_access_info_sort_sources(
        __isl_take isl_access_info *acc)
{
        if (!acc)
                return NULL;
        if (acc->n_must <= 1)
                return acc;

        if (isl_sort(acc->source, acc->n_must, sizeof(struct isl_labeled_map),
                    access_sort_cmp, acc) < 0)
                return isl_access_info_free(acc);

        return acc;
}

/* Align the parameters of the two spaces if needed and then call
 * isl_space_join.
 */
static __isl_give isl_space *space_align_and_join(__isl_take isl_space *left,
        __isl_take isl_space *right)
{
        if (isl_space_match(left, isl_dim_param, right, isl_dim_param))
                return isl_space_join(left, right);

        left = isl_space_align_params(left, isl_space_copy(right));
        right = isl_space_align_params(right, isl_space_copy(left));
        return isl_space_join(left, right);
}

/* Initialize an empty isl_flow structure corresponding to a given
 * isl_access_info structure.
 * For each must access, two dependences are created (initialized
 * to the empty relation), one for the resulting must dependences
 * and one for the resulting may dependences.  May accesses can
 * only lead to may dependences, so only one dependence is created
 * for each of them.
 * This function is private as isl_flow structures are only supposed
 * to be created by isl_access_info_compute_flow.
 */
static __isl_give isl_flow *isl_flow_alloc(__isl_keep isl_access_info *acc)
{
        int i, n;
        struct isl_ctx *ctx;
        struct isl_flow *dep;

        if (!acc)
                return NULL;

        ctx = isl_map_get_ctx(acc->sink.map);
        dep = isl_calloc_type(ctx, struct isl_flow);
        if (!dep)
                return NULL;

        n = 2 * acc->n_must + acc->n_may;
        dep->dep = isl_calloc_array(ctx, struct isl_labeled_map, n);
        if (n && !dep->dep)
                goto error;

        dep->n_source = n;
        for (i = 0; i < acc->n_must; ++i) {
                isl_space *dim;
                dim = space_align_and_join(
                        isl_map_get_space(acc->source[i].map),
                        isl_space_reverse(isl_map_get_space(acc->sink.map)));
                dep->dep[2 * i].map = isl_map_empty(dim);
                dep->dep[2 * i + 1].map = isl_map_copy(dep->dep[2 * i].map);
                dep->dep[2 * i].data = acc->source[i].data;
                dep->dep[2 * i + 1].data = acc->source[i].data;
                dep->dep[2 * i].must = 1;
                dep->dep[2 * i + 1].must = 0;
                if (!dep->dep[2 * i].map || !dep->dep[2 * i + 1].map)
                        goto error;
        }
        for (i = acc->n_must; i < acc->n_must + acc->n_may; ++i) {
                isl_space *dim;
                dim = space_align_and_join(
                        isl_map_get_space(acc->source[i].map),
                        isl_space_reverse(isl_map_get_space(acc->sink.map)));
                dep->dep[acc->n_must + i].map = isl_map_empty(dim);
                dep->dep[acc->n_must + i].data = acc->source[i].data;
                dep->dep[acc->n_must + i].must = 0;
                if (!dep->dep[acc->n_must + i].map)
                        goto error;
        }

        return dep;
error:
        isl_flow_free(dep);
        return NULL;
}

/* Iterate over all sources and for each resulting flow dependence
 * that is not empty, call the user specfied function.
 * The second argument in this function call identifies the source,
 * while the third argument correspond to the final argument of
 * the isl_flow_foreach call.
 */
int isl_flow_foreach(__isl_keep isl_flow *deps,
        int (*fn)(__isl_take isl_map *dep, int must, void *dep_user, void *user),
        void *user)
{
        int i;

        if (!deps)
                return -1;

        for (i = 0; i < deps->n_source; ++i) {
                if (isl_map_plain_is_empty(deps->dep[i].map))
                        continue;
                if (fn(isl_map_copy(deps->dep[i].map), deps->dep[i].must,
                                deps->dep[i].data, user) < 0)
                        return -1;
        }

        return 0;
}

/* Return a copy of the subset of the sink for which no source could be found.
 */
__isl_give isl_map *isl_flow_get_no_source(__isl_keep isl_flow *deps, int must)
{
        if (!deps)
                return NULL;
        
        if (must)
                return isl_set_unwrap(isl_set_copy(deps->must_no_source));
        else
                return isl_set_unwrap(isl_set_copy(deps->may_no_source));
}

void isl_flow_free(__isl_take isl_flow *deps)
{
        int i;

        if (!deps)
                return;
        isl_set_free(deps->must_no_source);
        isl_set_free(deps->may_no_source);
        if (deps->dep) {
                for (i = 0; i < deps->n_source; ++i)
                        isl_map_free(deps->dep[i].map);
                free(deps->dep);
        }
        free(deps);
}

isl_ctx *isl_flow_get_ctx(__isl_keep isl_flow *deps)
{
        return deps ? isl_set_get_ctx(deps->must_no_source) : NULL;
}

/* Return a map that enforces that the domain iteration occurs after
 * the range iteration at the given level.
 * If level is odd, then the domain iteration should occur after
 * the target iteration in their shared level/2 outermost loops.
 * In this case we simply need to enforce that these outermost
 * loop iterations are the same.
 * If level is even, then the loop iterator of the domain should
 * be greater than the loop iterator of the range at the last
 * of the level/2 shared loops, i.e., loop level/2 - 1.
 */
static __isl_give isl_map *after_at_level(__isl_take isl_space *dim, int level)
{
        struct isl_basic_map *bmap;

        if (level % 2)
                bmap = isl_basic_map_equal(dim, level/2);
        else
                bmap = isl_basic_map_more_at(dim, level/2 - 1);

        return isl_map_from_basic_map(bmap);
}

/* Compute the partial lexicographic maximum of "dep" on domain "sink",
 * but first check if the user has set acc->restrict_fn and if so
 * update either the input or the output of the maximization problem
 * with respect to the resulting restriction.
 *
 * Since the user expects a mapping from sink iterations to source iterations,
 * whereas the domain of "dep" is a wrapped map, mapping sink iterations
 * to accessed array elements, we first need to project out the accessed
 * sink array elements by applying acc->domain_map.
 * Similarly, the sink restriction specified by the user needs to be
 * converted back to the wrapped map.
 */
static __isl_give isl_map *restricted_partial_lexmax(
        __isl_keep isl_access_info *acc, __isl_take isl_map *dep,
        int source, __isl_take isl_set *sink, __isl_give isl_set **empty)
{
        isl_map *source_map;
        isl_restriction *restr;
        isl_set *sink_domain;
        isl_set *sink_restr;
        isl_map *res;

        if (!acc->restrict_fn)
                return isl_map_partial_lexmax(dep, sink, empty);

        source_map = isl_map_copy(dep);
        source_map = isl_map_apply_domain(source_map,
                                            isl_map_copy(acc->domain_map));
        sink_domain = isl_set_copy(sink);
        sink_domain = isl_set_apply(sink_domain, isl_map_copy(acc->domain_map));
        restr = acc->restrict_fn(source_map, sink_domain,
                                acc->source[source].data, acc->restrict_user);
        isl_set_free(sink_domain);
        isl_map_free(source_map);

        if (!restr)
                goto error;
        if (restr->type == isl_restriction_type_input) {
                dep = isl_map_intersect_range(dep, isl_set_copy(restr->source));
                sink_restr = isl_set_copy(restr->sink);
                sink_restr = isl_set_apply(sink_restr,
                                isl_map_reverse(isl_map_copy(acc->domain_map)));
                sink = isl_set_intersect(sink, sink_restr);
        } else if (restr->type == isl_restriction_type_empty) {
                isl_space *space = isl_map_get_space(dep);
                isl_map_free(dep);
                dep = isl_map_empty(space);
        }

        res = isl_map_partial_lexmax(dep, sink, empty);

        if (restr->type == isl_restriction_type_output)
                res = isl_map_intersect_range(res, isl_set_copy(restr->source));

        isl_restriction_free(restr);
        return res;
error:
        isl_map_free(dep);
        isl_set_free(sink);
        *empty = NULL;
        return NULL;
}

/* Compute the last iteration of must source j that precedes the sink
 * at the given level for sink iterations in set_C.
 * The subset of set_C for which no such iteration can be found is returned
 * in *empty.
 */
static struct isl_map *last_source(struct isl_access_info *acc, 
                                    struct isl_set *set_C,
                                    int j, int level, struct isl_set **empty)
{
        struct isl_map *read_map;
        struct isl_map *write_map;
        struct isl_map *dep_map;
        struct isl_map *after;
        struct isl_map *result;

        read_map = isl_map_copy(acc->sink.map);
        write_map = isl_map_copy(acc->source[j].map);
        write_map = isl_map_reverse(write_map);
        dep_map = isl_map_apply_range(read_map, write_map);
        after = after_at_level(isl_map_get_space(dep_map), level);
        dep_map = isl_map_intersect(dep_map, after);
        result = restricted_partial_lexmax(acc, dep_map, j, set_C, empty);
        result = isl_map_reverse(result);

        return result;
}

/* For a given mapping between iterations of must source j and iterations
 * of the sink, compute the last iteration of must source k preceding
 * the sink at level before_level for any of the sink iterations,
 * but following the corresponding iteration of must source j at level
 * after_level.
 */
static struct isl_map *last_later_source(struct isl_access_info *acc,
                                         struct isl_map *old_map,
                                         int j, int before_level,
                                         int k, int after_level,
                                         struct isl_set **empty)
{
        isl_space *dim;
        struct isl_set *set_C;
        struct isl_map *read_map;
        struct isl_map *write_map;
        struct isl_map *dep_map;
        struct isl_map *after_write;
        struct isl_map *before_read;
        struct isl_map *result;

        set_C = isl_map_range(isl_map_copy(old_map));
        read_map = isl_map_copy(acc->sink.map);
        write_map = isl_map_copy(acc->source[k].map);

        write_map = isl_map_reverse(write_map);
        dep_map = isl_map_apply_range(read_map, write_map);
        dim = space_align_and_join(isl_map_get_space(acc->source[k].map),
                    isl_space_reverse(isl_map_get_space(acc->source[j].map)));
        after_write = after_at_level(dim, after_level);
        after_write = isl_map_apply_range(after_write, old_map);
        after_write = isl_map_reverse(after_write);
        dep_map = isl_map_intersect(dep_map, after_write);
        before_read = after_at_level(isl_map_get_space(dep_map), before_level);
        dep_map = isl_map_intersect(dep_map, before_read);
        result = restricted_partial_lexmax(acc, dep_map, k, set_C, empty);
        result = isl_map_reverse(result);

        return result;
}

/* Given a shared_level between two accesses, return 1 if the
 * the first can precede the second at the requested target_level.
 * If the target level is odd, i.e., refers to a statement level
 * dimension, then first needs to precede second at the requested
 * level, i.e., shared_level must be equal to target_level.
 * If the target level is odd, then the two loops should share
 * at least the requested number of outer loops.
 */
static int can_precede_at_level(int shared_level, int target_level)
{
        if (shared_level < target_level)
                return 0;
        if ((target_level % 2) && shared_level > target_level)
                return 0;
        return 1;
}

/* Given a possible flow dependence temp_rel[j] between source j and the sink
 * at level sink_level, remove those elements for which
 * there is an iteration of another source k < j that is closer to the sink.
 * The flow dependences temp_rel[k] are updated with the improved sources.
 * Any improved source needs to precede the sink at the same level
 * and needs to follow source j at the same or a deeper level.
 * The lower this level, the later the execution date of source k.
 * We therefore consider lower levels first.
 *
 * If temp_rel[j] is empty, then there can be no improvement and
 * we return immediately.
 */
static int intermediate_sources(__isl_keep isl_access_info *acc,
        struct isl_map **temp_rel, int j, int sink_level)
{
        int k, level;
        int depth = 2 * isl_map_dim(acc->source[j].map, isl_dim_in) + 1;

        if (isl_map_plain_is_empty(temp_rel[j]))
                return 0;

        for (k = j - 1; k >= 0; --k) {
                int plevel, plevel2;
                plevel = acc->level_before(acc->source[k].data, acc->sink.data);
                if (!can_precede_at_level(plevel, sink_level))
                        continue;

                plevel2 = acc->level_before(acc->source[j].data,
                                                acc->source[k].data);

                for (level = sink_level; level <= depth; ++level) {
                        struct isl_map *T;
                        struct isl_set *trest;
                        struct isl_map *copy;

                        if (!can_precede_at_level(plevel2, level))
                                continue;

                        copy = isl_map_copy(temp_rel[j]);
                        T = last_later_source(acc, copy, j, sink_level, k,
                                              level, &trest);
                        if (isl_map_plain_is_empty(T)) {
                                isl_set_free(trest);
                                isl_map_free(T);
                                continue;
                        }
                        temp_rel[j] = isl_map_intersect_range(temp_rel[j], trest);
                        temp_rel[k] = isl_map_union_disjoint(temp_rel[k], T);
                }
        }

        return 0;
}

/* Compute all iterations of may source j that precedes the sink at the given
 * level for sink iterations in set_C.
 */
static __isl_give isl_map *all_sources(__isl_keep isl_access_info *acc,
                                    __isl_take isl_set *set_C, int j, int level)
{
        isl_map *read_map;
        isl_map *write_map;
        isl_map *dep_map;
        isl_map *after;

        read_map = isl_map_copy(acc->sink.map);
        read_map = isl_map_intersect_domain(read_map, set_C);
        write_map = isl_map_copy(acc->source[acc->n_must + j].map);
        write_map = isl_map_reverse(write_map);
        dep_map = isl_map_apply_range(read_map, write_map);
        after = after_at_level(isl_map_get_space(dep_map), level);
        dep_map = isl_map_intersect(dep_map, after);

        return isl_map_reverse(dep_map);
}

/* For a given mapping between iterations of must source k and iterations
 * of the sink, compute the all iteration of may source j preceding
 * the sink at level before_level for any of the sink iterations,
 * but following the corresponding iteration of must source k at level
 * after_level.
 */
static __isl_give isl_map *all_later_sources(__isl_keep isl_access_info *acc,
        __isl_take isl_map *old_map,
        int j, int before_level, int k, int after_level)
{
        isl_space *dim;
        isl_set *set_C;
        isl_map *read_map;
        isl_map *write_map;
        isl_map *dep_map;
        isl_map *after_write;
        isl_map *before_read;

        set_C = isl_map_range(isl_map_copy(old_map));
        read_map = isl_map_copy(acc->sink.map);
        read_map = isl_map_intersect_domain(read_map, set_C);
        write_map = isl_map_copy(acc->source[acc->n_must + j].map);

        write_map = isl_map_reverse(write_map);
        dep_map = isl_map_apply_range(read_map, write_map);
        dim = isl_space_join(isl_map_get_space(acc->source[acc->n_must + j].map),
                    isl_space_reverse(isl_map_get_space(acc->source[k].map)));
        after_write = after_at_level(dim, after_level);
        after_write = isl_map_apply_range(after_write, old_map);
        after_write = isl_map_reverse(after_write);
        dep_map = isl_map_intersect(dep_map, after_write);
        before_read = after_at_level(isl_map_get_space(dep_map), before_level);
        dep_map = isl_map_intersect(dep_map, before_read);
        return isl_map_reverse(dep_map);
}

/* Given the must and may dependence relations for the must accesses
 * for level sink_level, check if there are any accesses of may access j
 * that occur in between and return their union.
 * If some of these accesses are intermediate with respect to
 * (previously thought to be) must dependences, then these
 * must dependences are turned into may dependences.
 */
static __isl_give isl_map *all_intermediate_sources(
        __isl_keep isl_access_info *acc, __isl_take isl_map *map,
        struct isl_map **must_rel, struct isl_map **may_rel,
        int j, int sink_level)
{
        int k, level;
        int depth = 2 * isl_map_dim(acc->source[acc->n_must + j].map,
                                        isl_dim_in) + 1;

        for (k = 0; k < acc->n_must; ++k) {
                int plevel;

                if (isl_map_plain_is_empty(may_rel[k]) &&
                    isl_map_plain_is_empty(must_rel[k]))
                        continue;

                plevel = acc->level_before(acc->source[k].data,
                                        acc->source[acc->n_must + j].data);

                for (level = sink_level; level <= depth; ++level) {
                        isl_map *T;
                        isl_map *copy;
                        isl_set *ran;

                        if (!can_precede_at_level(plevel, level))
                                continue;

                        copy = isl_map_copy(may_rel[k]);
                        T = all_later_sources(acc, copy, j, sink_level, k, level);
                        map = isl_map_union(map, T);

                        copy = isl_map_copy(must_rel[k]);
                        T = all_later_sources(acc, copy, j, sink_level, k, level);
                        ran = isl_map_range(isl_map_copy(T));
                        map = isl_map_union(map, T);
                        may_rel[k] = isl_map_union_disjoint(may_rel[k],
                            isl_map_intersect_range(isl_map_copy(must_rel[k]),
                                                    isl_set_copy(ran)));
                        T = isl_map_from_domain_and_range(
                            isl_set_universe(
                                isl_space_domain(isl_map_get_space(must_rel[k]))),
                            ran);
                        must_rel[k] = isl_map_subtract(must_rel[k], T);
                }
        }

        return map;
}

/* Compute dependences for the case where all accesses are "may"
 * accesses, which boils down to computing memory based dependences.
 * The generic algorithm would also work in this case, but it would
 * be overkill to use it.
 */
static __isl_give isl_flow *compute_mem_based_dependences(
        __isl_keep isl_access_info *acc)
{
        int i;
        isl_set *mustdo;
        isl_set *maydo;
        isl_flow *res;

        res = isl_flow_alloc(acc);
        if (!res)
                return NULL;

        mustdo = isl_map_domain(isl_map_copy(acc->sink.map));
        maydo = isl_set_copy(mustdo);

        for (i = 0; i < acc->n_may; ++i) {
                int plevel;
                int is_before;
                isl_space *dim;
                isl_map *before;
                isl_map *dep;

                plevel = acc->level_before(acc->source[i].data, acc->sink.data);
                is_before = plevel & 1;
                plevel >>= 1;

                dim = isl_map_get_space(res->dep[i].map);
                if (is_before)
                        before = isl_map_lex_le_first(dim, plevel);
                else
                        before = isl_map_lex_lt_first(dim, plevel);
                dep = isl_map_apply_range(isl_map_copy(acc->source[i].map),
                        isl_map_reverse(isl_map_copy(acc->sink.map)));
                dep = isl_map_intersect(dep, before);
                mustdo = isl_set_subtract(mustdo,
                                            isl_map_range(isl_map_copy(dep)));
                res->dep[i].map = isl_map_union(res->dep[i].map, dep);
        }

        res->may_no_source = isl_set_subtract(maydo, isl_set_copy(mustdo));
        res->must_no_source = mustdo;

        return res;
}

/* Compute dependences for the case where there is at least one
 * "must" access.
 *
 * The core algorithm considers all levels in which a source may precede
 * the sink, where a level may either be a statement level or a loop level.
 * The outermost statement level is 1, the first loop level is 2, etc...
 * The algorithm basically does the following:
 * for all levels l of the read access from innermost to outermost
 *      for all sources w that may precede the sink access at that level
 *          compute the last iteration of the source that precedes the sink access
 *                                          at that level
 *          add result to possible last accesses at level l of source w
 *          for all sources w2 that we haven't considered yet at this level that may
 *                                          also precede the sink access
 *              for all levels l2 of w from l to innermost
 *                  for all possible last accesses dep of w at l
 *                      compute last iteration of w2 between the source and sink
 *                                                              of dep
 *                      add result to possible last accesses at level l of write w2
 *                      and replace possible last accesses dep by the remainder
 *
 *
 * The above algorithm is applied to the must access.  During the course
 * of the algorithm, we keep track of sink iterations that still
 * need to be considered.  These iterations are split into those that
 * haven't been matched to any source access (mustdo) and those that have only
 * been matched to may accesses (maydo).
 * At the end of each level, we also consider the may accesses.
 * In particular, we consider may accesses that precede the remaining
 * sink iterations, moving elements from mustdo to maydo when appropriate,
 * and may accesses that occur between a must source and a sink of any 
 * dependences found at the current level, turning must dependences into
 * may dependences when appropriate.
 * 
 */
static __isl_give isl_flow *compute_val_based_dependences(
        __isl_keep isl_access_info *acc)
{
        isl_ctx *ctx;
        isl_flow *res;
        isl_set *mustdo = NULL;
        isl_set *maydo = NULL;
        int level, j;
        int depth;
        isl_map **must_rel = NULL;
        isl_map **may_rel = NULL;

        if (!acc)
                return NULL;

        res = isl_flow_alloc(acc);
        if (!res)
                goto error;
        ctx = isl_map_get_ctx(acc->sink.map);

        depth = 2 * isl_map_dim(acc->sink.map, isl_dim_in) + 1;
        mustdo = isl_map_domain(isl_map_copy(acc->sink.map));
        maydo = isl_set_empty(isl_set_get_space(mustdo));
        if (!mustdo || !maydo)
                goto error;
        if (isl_set_plain_is_empty(mustdo))
                goto done;

        must_rel = isl_alloc_array(ctx, struct isl_map *, acc->n_must);
        may_rel = isl_alloc_array(ctx, struct isl_map *, acc->n_must);
        if (!must_rel || !may_rel)
                goto error;

        for (level = depth; level >= 1; --level) {
                for (j = acc->n_must-1; j >=0; --j) {
                        isl_space *space;
                        space = isl_map_get_space(res->dep[2 * j].map);
                        must_rel[j] = isl_map_empty(space);
                        may_rel[j] = isl_map_copy(must_rel[j]);
                }

                for (j = acc->n_must - 1; j >= 0; --j) {
                        struct isl_map *T;
                        struct isl_set *rest;
                        int plevel;

                        plevel = acc->level_before(acc->source[j].data,
                                                     acc->sink.data);
                        if (!can_precede_at_level(plevel, level))
                                continue;

                        T = last_source(acc, mustdo, j, level, &rest);
                        must_rel[j] = isl_map_union_disjoint(must_rel[j], T);
                        mustdo = rest;

                        intermediate_sources(acc, must_rel, j, level);

                        T = last_source(acc, maydo, j, level, &rest);
                        may_rel[j] = isl_map_union_disjoint(may_rel[j], T);
                        maydo = rest;

                        intermediate_sources(acc, may_rel, j, level);

                        if (isl_set_plain_is_empty(mustdo) &&
                            isl_set_plain_is_empty(maydo))
                                break;
                }
                for (j = j - 1; j >= 0; --j) {
                        int plevel;

                        plevel = acc->level_before(acc->source[j].data,
                                                     acc->sink.data);
                        if (!can_precede_at_level(plevel, level))
                                continue;

                        intermediate_sources(acc, must_rel, j, level);
                        intermediate_sources(acc, may_rel, j, level);
                }

                for (j = 0; j < acc->n_may; ++j) {
                        int plevel;
                        isl_map *T;
                        isl_set *ran;

                        plevel = acc->level_before(acc->source[acc->n_must + j].data,
                                                     acc->sink.data);
                        if (!can_precede_at_level(plevel, level))
                                continue;

                        T = all_sources(acc, isl_set_copy(maydo), j, level);
                        res->dep[2 * acc->n_must + j].map =
                            isl_map_union(res->dep[2 * acc->n_must + j].map, T);
                        T = all_sources(acc, isl_set_copy(mustdo), j, level);
                        ran = isl_map_range(isl_map_copy(T));
                        res->dep[2 * acc->n_must + j].map =
                            isl_map_union(res->dep[2 * acc->n_must + j].map, T);
                        mustdo = isl_set_subtract(mustdo, isl_set_copy(ran));
                        maydo = isl_set_union_disjoint(maydo, ran);

                        T = res->dep[2 * acc->n_must + j].map;
                        T = all_intermediate_sources(acc, T, must_rel, may_rel,
                                                        j, level);
                        res->dep[2 * acc->n_must + j].map = T;
                }

                for (j = acc->n_must - 1; j >= 0; --j) {
                        res->dep[2 * j].map =
                                isl_map_union_disjoint(res->dep[2 * j].map,
                                                             must_rel[j]);
                        res->dep[2 * j + 1].map =
                                isl_map_union_disjoint(res->dep[2 * j + 1].map,
                                                             may_rel[j]);
                }

                if (isl_set_plain_is_empty(mustdo) &&
                    isl_set_plain_is_empty(maydo))
                        break;
        }

        free(must_rel);
        free(may_rel);
done:
        res->must_no_source = mustdo;
        res->may_no_source = maydo;
        return res;
error:
        isl_flow_free(res);
        isl_set_free(mustdo);
        isl_set_free(maydo);
        free(must_rel);
        free(may_rel);
        return NULL;
}

/* Given a "sink" access, a list of n "source" accesses,
 * compute for each iteration of the sink access
 * and for each element accessed by that iteration,
 * the source access in the list that last accessed the
 * element accessed by the sink access before this sink access.
 * Each access is given as a map from the loop iterators
 * to the array indices.
 * The result is a list of n relations between source and sink
 * iterations and a subset of the domain of the sink access,
 * corresponding to those iterations that access an element
 * not previously accessed.
 *
 * To deal with multi-valued sink access relations, the sink iteration
 * domain is first extended with dimensions that correspond to the data
 * space.  After the computation is finished, these extra dimensions are
 * projected out again.
 */
__isl_give isl_flow *isl_access_info_compute_flow(__isl_take isl_access_info *acc)
{
        int j;
        struct isl_flow *res = NULL;

        if (!acc)
                return NULL;

        acc->domain_map = isl_map_domain_map(isl_map_copy(acc->sink.map));
        acc->sink.map = isl_map_range_map(acc->sink.map);
        if (!acc->sink.map)
                goto error;

        if (acc->n_must == 0)
                res = compute_mem_based_dependences(acc);
        else {
                acc = isl_access_info_sort_sources(acc);
                res = compute_val_based_dependences(acc);
        }
        if (!res)
                goto error;

        for (j = 0; j < res->n_source; ++j) {
                res->dep[j].map = isl_map_apply_range(res->dep[j].map,
                                        isl_map_copy(acc->domain_map));
                if (!res->dep[j].map)
                        goto error;
        }
        if (!res->must_no_source || !res->may_no_source)
                goto error;

        isl_access_info_free(acc);
        return res;
error:
        isl_access_info_free(acc);
        isl_flow_free(res);
        return NULL;
}


/* Keep track of some information about a schedule for a given
 * access.  In particular, keep track of which dimensions
 * have a constant value and of the actual constant values.
 */
struct isl_sched_info {
        int *is_cst;
        isl_vec *cst;
};

static void sched_info_free(__isl_take struct isl_sched_info *info)
{
        if (!info)
                return;
        isl_vec_free(info->cst);
        free(info->is_cst);
        free(info);
}

/* Extract information on the constant dimensions of the schedule
 * for a given access.  The "map" is of the form
 *
 *      [S -> D] -> A
 *
 * with S the schedule domain, D the iteration domain and A the data domain.
 */
static __isl_give struct isl_sched_info *sched_info_alloc(
        __isl_keep isl_map *map)
{
        isl_ctx *ctx;
        isl_space *dim;
        struct isl_sched_info *info;
        int i, n;

        if (!map)
                return NULL;

        dim = isl_space_unwrap(isl_space_domain(isl_map_get_space(map)));
        if (!dim)
                return NULL;
        n = isl_space_dim(dim, isl_dim_in);
        isl_space_free(dim);

        ctx = isl_map_get_ctx(map);
        info = isl_alloc_type(ctx, struct isl_sched_info);
        if (!info)
                return NULL;
        info->is_cst = isl_alloc_array(ctx, int, n);
        info->cst = isl_vec_alloc(ctx, n);
        if (n && (!info->is_cst || !info->cst))
                goto error;

        for (i = 0; i < n; ++i) {
                isl_val *v;

                v = isl_map_plain_get_val_if_fixed(map, isl_dim_in, i);
                if (!v)
                        goto error;
                info->is_cst[i] = !isl_val_is_nan(v);
                if (info->is_cst[i])
                        info->cst = isl_vec_set_element_val(info->cst, i, v);
                else
                        isl_val_free(v);
        }

        return info;
error:
        sched_info_free(info);
        return NULL;
}

/* This structure represents the input for a dependence analysis computation.
 *
 * "sink" represents the sink accesses.
 * "must_source" represents the definite source accesses.
 * "may_source" represents the possible source accesses.
 *
 * "schedule" or "schedule_map" represents the execution order.
 * Exactly one of these fields should be NULL.  The other field
 * determines the execution order.
 *
 * The domains of these four maps refer to the same iteration spaces(s).
 * The ranges of the first three maps also refer to the same data space(s).
 *
 * After a call to isl_union_access_info_introduce_schedule,
 * the "schedule_map" field no longer contains useful information.
 */
struct isl_union_access_info {
        isl_union_map *sink;
        isl_union_map *must_source;
        isl_union_map *may_source;

        isl_schedule *schedule;
        isl_union_map *schedule_map;
};

/* Free "access" and return NULL.
 */
__isl_null isl_union_access_info *isl_union_access_info_free(
        __isl_take isl_union_access_info *access)
{
        if (!access)
                return NULL;

        isl_union_map_free(access->sink);
        isl_union_map_free(access->must_source);
        isl_union_map_free(access->may_source);
        isl_schedule_free(access->schedule);
        isl_union_map_free(access->schedule_map);
        free(access);

        return NULL;
}

/* Return the isl_ctx to which "access" belongs.
 */
isl_ctx *isl_union_access_info_get_ctx(__isl_keep isl_union_access_info *access)
{
        return access ? isl_union_map_get_ctx(access->sink) : NULL;
}

/* Create a new isl_union_access_info with the given sink accesses and
 * and no source accesses or schedule information.
 *
 * By default, we use the schedule field of the isl_union_access_info,
 * but this may be overridden by a call
 * to isl_union_access_info_set_schedule_map.
 */
__isl_give isl_union_access_info *isl_union_access_info_from_sink(
        __isl_take isl_union_map *sink)
{
        isl_ctx *ctx;
        isl_space *space;
        isl_union_map *empty;
        isl_union_access_info *access;

        if (!sink)
                return NULL;
        ctx = isl_union_map_get_ctx(sink);
        access = isl_alloc_type(ctx, isl_union_access_info);
        if (!access)
                goto error;

        space = isl_union_map_get_space(sink);
        empty = isl_union_map_empty(isl_space_copy(space));
        access->sink = sink;
        access->must_source = isl_union_map_copy(empty);
        access->may_source = empty;
        access->schedule = isl_schedule_empty(space);
        access->schedule_map = NULL;

        if (!access->sink || !access->must_source ||
            !access->may_source || !access->schedule)
                return isl_union_access_info_free(access);

        return access;
error:
        isl_union_map_free(sink);
        return NULL;
}

/* Replace the definite source accesses of "access" by "must_source".
 */
__isl_give isl_union_access_info *isl_union_access_info_set_must_source(
        __isl_take isl_union_access_info *access,
        __isl_take isl_union_map *must_source)
{
        if (!access || !must_source)
                goto error;

        isl_union_map_free(access->must_source);
        access->must_source = must_source;

        return access;
error:
        isl_union_access_info_free(access);
        isl_union_map_free(must_source);
        return NULL;
}

/* Replace the possible source accesses of "access" by "may_source".
 */
__isl_give isl_union_access_info *isl_union_access_info_set_may_source(
        __isl_take isl_union_access_info *access,
        __isl_take isl_union_map *may_source)
{
        if (!access || !may_source)
                goto error;

        isl_union_map_free(access->may_source);
        access->may_source = may_source;

        return access;
error:
        isl_union_access_info_free(access);
        isl_union_map_free(may_source);
        return NULL;
}

/* Replace the schedule of "access" by "schedule".
 * Also free the schedule_map in case it was set last.
 */
__isl_give isl_union_access_info *isl_union_access_info_set_schedule(
        __isl_take isl_union_access_info *access,
        __isl_take isl_schedule *schedule)
{
        if (!access || !schedule)
                goto error;

        access->schedule_map = isl_union_map_free(access->schedule_map);
        isl_schedule_free(access->schedule);
        access->schedule = schedule;

        return access;
error:
        isl_union_access_info_free(access);
        isl_schedule_free(schedule);
        return NULL;
}

/* Replace the schedule map of "access" by "schedule_map".
 * Also free the schedule in case it was set last.
 */
__isl_give isl_union_access_info *isl_union_access_info_set_schedule_map(
        __isl_take isl_union_access_info *access,
        __isl_take isl_union_map *schedule_map)
{
        if (!access || !schedule_map)
                goto error;

        isl_union_map_free(access->schedule_map);
        access->schedule = isl_schedule_free(access->schedule);
        access->schedule_map = schedule_map;

        return access;
error:
        isl_union_access_info_free(access);
        isl_union_map_free(schedule_map);
        return NULL;
}

/* Update the fields of "access" such that they all have the same parameters,
 * keeping in mind that the schedule_map field may be NULL and ignoring
 * the schedule field.
 */
static __isl_give isl_union_access_info *isl_union_access_info_align_params(
        __isl_take isl_union_access_info *access)
{
        isl_space *space;

        if (!access)
                return NULL;

        space = isl_union_map_get_space(access->sink);
        space = isl_space_align_params(space,
                                isl_union_map_get_space(access->must_source));
        space = isl_space_align_params(space,
                                isl_union_map_get_space(access->may_source));
        if (access->schedule_map)
                space = isl_space_align_params(space,
                                isl_union_map_get_space(access->schedule_map));
        access->sink = isl_union_map_align_params(access->sink,
                                                        isl_space_copy(space));
        access->must_source = isl_union_map_align_params(access->must_source,
                                                        isl_space_copy(space));
        access->may_source = isl_union_map_align_params(access->may_source,
                                                        isl_space_copy(space));
        if (!access->schedule_map) {
                isl_space_free(space);
        } else {
                access->schedule_map =
                    isl_union_map_align_params(access->schedule_map, space);
                if (!access->schedule_map)
                        return isl_union_access_info_free(access);
        }

        if (!access->sink || !access->must_source || !access->may_source)
                return isl_union_access_info_free(access);

        return access;
}

/* Prepend the schedule dimensions to the iteration domains.
 *
 * That is, if the schedule is of the form
 *
 *      D -> S
 *
 * while the access relations are of the form
 *
 *      D -> A
 *
 * then the updated access relations are of the form
 *
 *      [S -> D] -> A
 *
 * The schedule map is also replaced by the map
 *
 *      [S -> D] -> D
 *
 * that is used during the internal computation.
 * Neither the original schedule map nor this updated schedule map
 * are used after the call to this function.
 */
static __isl_give isl_union_access_info *
isl_union_access_info_introduce_schedule(
        __isl_take isl_union_access_info *access)
{
        isl_union_map *sm;

        if (!access)
                return NULL;

        sm = isl_union_map_reverse(access->schedule_map);
        sm = isl_union_map_range_map(sm);
        access->sink = isl_union_map_apply_range(isl_union_map_copy(sm),
                                                access->sink);
        access->may_source = isl_union_map_apply_range(isl_union_map_copy(sm),
                                                access->may_source);
        access->must_source = isl_union_map_apply_range(isl_union_map_copy(sm),
                                                access->must_source);
        access->schedule_map = sm;

        if (!access->sink || !access->must_source ||
            !access->may_source || !access->schedule_map)
                return isl_union_access_info_free(access);

        return access;
}

/* This structure epresents the result of a dependence analysis computation.
 *
 * "must_dep" represents the definite dependences.
 * "may_dep" represents the non-definite dependences.
 * "must_no_source" represents the subset of the sink accesses for which
 * definitely no source was found.
 * "may_no_source" represents the subset of the sink accesses for which
 * possibly, but not definitely, no source was found.
 */
struct isl_union_flow {
        isl_union_map *must_dep;
        isl_union_map *may_dep;
        isl_union_map *must_no_source;
        isl_union_map *may_no_source;
};

/* Free "flow" and return NULL.
 */
__isl_null isl_union_flow *isl_union_flow_free(__isl_take isl_union_flow *flow)
{
        if (!flow)
                return NULL;
        isl_union_map_free(flow->must_dep);
        isl_union_map_free(flow->may_dep);
        isl_union_map_free(flow->must_no_source);
        isl_union_map_free(flow->may_no_source);
        free(flow);
        return NULL;
}

void isl_union_flow_dump(__isl_keep isl_union_flow *flow)
{
        if (!flow)
                return;

        fprintf(stderr, "must dependences: ");
        isl_union_map_dump(flow->must_dep);
        fprintf(stderr, "may dependences: ");
        isl_union_map_dump(flow->may_dep);
        fprintf(stderr, "must no source: ");
        isl_union_map_dump(flow->must_no_source);
        fprintf(stderr, "may no source: ");
        isl_union_map_dump(flow->may_no_source);
}

/* Return the definite dependences in "flow".
 */
__isl_give isl_union_map *isl_union_flow_get_must_dependence(
        __isl_keep isl_union_flow *flow)
{
        if (!flow)
                return NULL;
        return isl_union_map_copy(flow->must_dep);
}

/* Return the possible dependences in "flow", including the definite
 * dependences.
 */
__isl_give isl_union_map *isl_union_flow_get_may_dependence(
        __isl_keep isl_union_flow *flow)
{
        if (!flow)
                return NULL;
        return isl_union_map_union(isl_union_map_copy(flow->must_dep),
                                    isl_union_map_copy(flow->may_dep));
}

/* Return the non-definite dependences in "flow".
 */
static __isl_give isl_union_map *isl_union_flow_get_non_must_dependence(
        __isl_keep isl_union_flow *flow)
{
        if (!flow)
                return NULL;
        return isl_union_map_copy(flow->may_dep);
}

/* Return the subset of the sink accesses for which definitely
 * no source was found.
 */
__isl_give isl_union_map *isl_union_flow_get_must_no_source(
        __isl_keep isl_union_flow *flow)
{
        if (!flow)
                return NULL;
        return isl_union_map_copy(flow->must_no_source);
}

/* Return the subset of the sink accesses for which possibly
 * no source was found, including those for which definitely
 * no source was found.
 */
__isl_give isl_union_map *isl_union_flow_get_may_no_source(
        __isl_keep isl_union_flow *flow)
{
        if (!flow)
                return NULL;
        return isl_union_map_union(isl_union_map_copy(flow->must_no_source),
                                    isl_union_map_copy(flow->may_no_source));
}

/* Return the subset of the sink accesses for which possibly, but not
 * definitely, no source was found.
 */
static __isl_give isl_union_map *isl_union_flow_get_non_must_no_source(
        __isl_keep isl_union_flow *flow)
{
        if (!flow)
                return NULL;
        return isl_union_map_copy(flow->may_no_source);
}

/* Create a new isl_union_flow object, initialized with empty
 * dependence relations and sink subsets.
 */
static __isl_give isl_union_flow *isl_union_flow_alloc(
        __isl_take isl_space *space)
{
        isl_ctx *ctx;
        isl_union_map *empty;
        isl_union_flow *flow;

        if (!space)
                return NULL;
        ctx = isl_space_get_ctx(space);
        flow = isl_alloc_type(ctx, isl_union_flow);
        if (!flow)
                goto error;

        empty = isl_union_map_empty(space);
        flow->must_dep = isl_union_map_copy(empty);
        flow->may_dep = isl_union_map_copy(empty);
        flow->must_no_source = isl_union_map_copy(empty);
        flow->may_no_source = empty;

        if (!flow->must_dep || !flow->may_dep ||
            !flow->must_no_source || !flow->may_no_source)
                return isl_union_flow_free(flow);

        return flow;
error:
        isl_space_free(space);
        return NULL;
}

/* Drop the schedule dimensions from the iteration domains in "flow".
 * In particular, the schedule dimensions have been prepended
 * to the iteration domains prior to the dependence analysis by
 * replacing the iteration domain D, by the wrapped map [S -> D].
 * Replace these wrapped maps by the original D.
 */
static __isl_give isl_union_flow *isl_union_flow_drop_schedule(
        __isl_take isl_union_flow *flow)
{
        if (!flow)
                return NULL;

        flow->must_dep = isl_union_map_factor_range(flow->must_dep);
        flow->may_dep = isl_union_map_factor_range(flow->may_dep);
        flow->must_no_source =
                isl_union_map_domain_factor_range(flow->must_no_source);
        flow->may_no_source =
                isl_union_map_domain_factor_range(flow->may_no_source);

        if (!flow->must_dep || !flow->may_dep ||
            !flow->must_no_source || !flow->may_no_source)
                return isl_union_flow_free(flow);

        return flow;
}

struct isl_compute_flow_data {
        isl_union_map *must_source;
        isl_union_map *may_source;
        isl_union_flow *flow;

        int count;
        int must;
        isl_space *dim;
        struct isl_sched_info *sink_info;
        struct isl_sched_info **source_info;
        isl_access_info *accesses;
};

static int count_matching_array(__isl_take isl_map *map, void *user)
{
        int eq;
        isl_space *dim;
        struct isl_compute_flow_data *data;

        data = (struct isl_compute_flow_data *)user;

        dim = isl_space_range(isl_map_get_space(map));

        eq = isl_space_is_equal(dim, data->dim);

        isl_space_free(dim);
        isl_map_free(map);

        if (eq < 0)
                return -1;
        if (eq)
                data->count++;

        return 0;
}

static int collect_matching_array(__isl_take isl_map *map, void *user)
{
        int eq;
        isl_space *dim;
        struct isl_sched_info *info;
        struct isl_compute_flow_data *data;

        data = (struct isl_compute_flow_data *)user;

        dim = isl_space_range(isl_map_get_space(map));

        eq = isl_space_is_equal(dim, data->dim);

        isl_space_free(dim);

        if (eq < 0)
                goto error;
        if (!eq) {
                isl_map_free(map);
                return 0;
        }

        info = sched_info_alloc(map);
        data->source_info[data->count] = info;

        data->accesses = isl_access_info_add_source(data->accesses,
                                                    map, data->must, info);

        data->count++;

        return 0;
error:
        isl_map_free(map);
        return -1;
}

/* Determine the shared nesting level and the "textual order" of
 * the given accesses.
 *
 * We first determine the minimal schedule dimension for both accesses.
 *
 * If among those dimensions, we can find one where both have a fixed
 * value and if moreover those values are different, then the previous
 * dimension is the last shared nesting level and the textual order
 * is determined based on the order of the fixed values.
 * If no such fixed values can be found, then we set the shared
 * nesting level to the minimal schedule dimension, with no textual ordering.
 */
static int before(void *first, void *second)
{
        struct isl_sched_info *info1 = first;
        struct isl_sched_info *info2 = second;
        int n1, n2;
        int i;

        n1 = isl_vec_size(info1->cst);
        n2 = isl_vec_size(info2->cst);

        if (n2 < n1)
                n1 = n2;

        for (i = 0; i < n1; ++i) {
                int r;
                int cmp;

                if (!info1->is_cst[i])
                        continue;
                if (!info2->is_cst[i])
                        continue;
                cmp = isl_vec_cmp_element(info1->cst, info2->cst, i);
                if (cmp == 0)
                        continue;

                r = 2 * i + (cmp < 0);

                return r;
        }

        return 2 * n1;
}

/* Given a sink access, look for all the source accesses that access
 * the same array and perform dataflow analysis on them using
 * isl_access_info_compute_flow.
 */
static int compute_flow(__isl_take isl_map *map, void *user)
{
        int i;
        isl_ctx *ctx;
        struct isl_compute_flow_data *data;
        isl_flow *flow;
        isl_union_flow *df;

        data = (struct isl_compute_flow_data *)user;
        df = data->flow;

        ctx = isl_map_get_ctx(map);

        data->accesses = NULL;
        data->sink_info = NULL;
        data->source_info = NULL;
        data->count = 0;
        data->dim = isl_space_range(isl_map_get_space(map));

        if (isl_union_map_foreach_map(data->must_source,
                                        &count_matching_array, data) < 0)
                goto error;
        if (isl_union_map_foreach_map(data->may_source,
                                        &count_matching_array, data) < 0)
                goto error;

        data->sink_info = sched_info_alloc(map);
        data->source_info = isl_calloc_array(ctx, struct isl_sched_info *,
                                             data->count);

        data->accesses = isl_access_info_alloc(isl_map_copy(map),
                                data->sink_info, &before, data->count);
        if (!data->sink_info || (data->count && !data->source_info) ||
            !data->accesses)
                goto error;
        data->count = 0;
        data->must = 1;
        if (isl_union_map_foreach_map(data->must_source,
                                        &collect_matching_array, data) < 0)
                goto error;
        data->must = 0;
        if (isl_union_map_foreach_map(data->may_source,
                                        &collect_matching_array, data) < 0)
                goto error;

        flow = isl_access_info_compute_flow(data->accesses);
        data->accesses = NULL;

        if (!flow)
                goto error;

        df->must_no_source = isl_union_map_union(df->must_no_source,
                    isl_union_map_from_map(isl_flow_get_no_source(flow, 1)));
        df->may_no_source = isl_union_map_union(df->may_no_source,
                    isl_union_map_from_map(isl_flow_get_no_source(flow, 0)));

        for (i = 0; i < flow->n_source; ++i) {
                isl_union_map *dep;
                dep = isl_union_map_from_map(isl_map_copy(flow->dep[i].map));
                if (flow->dep[i].must)
                        df->must_dep = isl_union_map_union(df->must_dep, dep);
                else
                        df->may_dep = isl_union_map_union(df->may_dep, dep);
        }

        isl_flow_free(flow);

        sched_info_free(data->sink_info);
        if (data->source_info) {
                for (i = 0; i < data->count; ++i)
                        sched_info_free(data->source_info[i]);
                free(data->source_info);
        }
        isl_space_free(data->dim);
        isl_map_free(map);

        return 0;
error:
        isl_access_info_free(data->accesses);
        sched_info_free(data->sink_info);
        if (data->source_info) {
                for (i = 0; i < data->count; ++i)
                        sched_info_free(data->source_info[i]);
                free(data->source_info);
        }
        isl_space_free(data->dim);
        isl_map_free(map);

        return -1;
}

/* Remove the must accesses from the may accesses.
 *
 * A must access always trumps a may access, so there is no need
 * for a must access to also be considered as a may access.  Doing so
 * would only cost extra computations only to find out that
 * the duplicated may access does not make any difference.
 */
static __isl_give isl_union_access_info *isl_union_access_info_normalize(
        __isl_take isl_union_access_info *access)
{
        if (!access)
                return NULL;
        access->may_source = isl_union_map_subtract(access->may_source,
                                    isl_union_map_copy(access->must_source));
        if (!access->may_source)
                return isl_union_access_info_free(access);

        return access;
}

/* Given a description of the "sink" accesses, the "source" accesses and
 * a schedule, compute for each instance of a sink access
 * and for each element accessed by that instance,
 * the possible or definite source accesses that last accessed the
 * element accessed by the sink access before this sink access
 * in the sense that there is no intermediate definite source access.
 *
 * The must_no_source and may_no_source elements of the result
 * are subsets of access->sink.  The elements must_dep and may_dep
 * map domain elements of access->{may,must)_source to
 * domain elements of access->sink.
 *
 * This function is used when only the schedule map representation
 * is available.
 *
 * We first prepend the schedule dimensions to the domain
 * of the accesses so that we can easily compare their relative order.
 * Then we consider each sink access individually in compute_flow.
 */
static __isl_give isl_union_flow *compute_flow_union_map(
        __isl_take isl_union_access_info *access)
{
        struct isl_compute_flow_data data;

        access = isl_union_access_info_align_params(access);
        access = isl_union_access_info_introduce_schedule(access);
        if (!access)
                return NULL;

        data.must_source = access->must_source;
        data.may_source = access->may_source;

        data.flow = isl_union_flow_alloc(isl_union_map_get_space(access->sink));

        if (isl_union_map_foreach_map(access->sink, &compute_flow, &data) < 0)
                goto error;

        data.flow = isl_union_flow_drop_schedule(data.flow);

        isl_union_access_info_free(access);
        return data.flow;
error:
        isl_union_access_info_free(access);
        isl_union_flow_free(data.flow);
        return NULL;
}

/* A schedule access relation.
 *
 * The access relation "access" is of the form [S -> D] -> A,
 * where S corresponds to the prefix schedule at "node".
 * "must" is only relevant for source accesses and indicates
 * whether the access is a must source or a may source.
 */
struct isl_scheduled_access {
        isl_map *access;
        int must;
        isl_schedule_node *node;
};

/* Data structure for keeping track of individual scheduled sink and source
 * accesses when computing dependence analysis based on a schedule tree.
 *
 * "n_sink" is the number of used entries in "sink"
 * "n_source" is the number of used entries in "source"
 *
 * "set_sink", "must" and "node" are only used inside collect_sink_source,
 * to keep track of the current node and
 * of what extract_sink_source needs to do.
 */
struct isl_compute_flow_schedule_data {
        isl_union_access_info *access;

        int n_sink;
        int n_source;

        struct isl_scheduled_access *sink;
        struct isl_scheduled_access *source;

        int set_sink;
        int must;
        isl_schedule_node *node;
};

/* Align the parameters of all sinks with all sources.
 *
 * If there are no sinks or no sources, then no alignment is needed.
 */
static void isl_compute_flow_schedule_data_align_params(
        struct isl_compute_flow_schedule_data *data)
{
        int i;
        isl_space *space;

        if (data->n_sink == 0 || data->n_source == 0)
                return;

        space = isl_map_get_space(data->sink[0].access);

        for (i = 1; i < data->n_sink; ++i)
                space = isl_space_align_params(space,
                                isl_map_get_space(data->sink[i].access));
        for (i = 0; i < data->n_source; ++i)
                space = isl_space_align_params(space,
                                isl_map_get_space(data->source[i].access));

        for (i = 0; i < data->n_sink; ++i)
                data->sink[i].access =
                        isl_map_align_params(data->sink[i].access,
                                                        isl_space_copy(space));
        for (i = 0; i < data->n_source; ++i)
                data->source[i].access =
                        isl_map_align_params(data->source[i].access,
                                                        isl_space_copy(space));

        isl_space_free(space);
}

/* Free all the memory referenced from "data".
 * Do not free "data" itself as it may be allocated on the stack.
 */
static void isl_compute_flow_schedule_data_clear(
        struct isl_compute_flow_schedule_data *data)
{
        int i;

        if (!data->sink)
                return;

        for (i = 0; i < data->n_sink; ++i) {
                isl_map_free(data->sink[i].access);
                isl_schedule_node_free(data->sink[i].node);
        }

        for (i = 0; i < data->n_source; ++i) {
                isl_map_free(data->source[i].access);
                isl_schedule_node_free(data->source[i].node);
        }

        free(data->sink);
}

/* isl_schedule_foreach_schedule_node callback for counting
 * (an upper bound on) the number of sinks and sources.
 *
 * Sinks and sources are only extracted at leaves of the tree,
 * so we skip the node if it is not a leaf.
 * Otherwise we increment data->n_sink and data->n_source with
 * the number of spaces in the sink and source access domains
 * that reach this node.
 */
static int count_sink_source(__isl_keep isl_schedule_node *node, void *user)
{
        struct isl_compute_flow_schedule_data *data = user;
        isl_union_set *domain;
        isl_union_map *umap;
        int r = 0;

        if (isl_schedule_node_get_type(node) != isl_schedule_node_leaf)
                return 1;

        domain = isl_schedule_node_get_universe_domain(node);

        umap = isl_union_map_copy(data->access->sink);
        umap = isl_union_map_intersect_domain(umap, isl_union_set_copy(domain));
        data->n_sink += isl_union_map_n_map(umap);
        isl_union_map_free(umap);
        if (!umap)
                r = -1;

        umap = isl_union_map_copy(data->access->must_source);
        umap = isl_union_map_intersect_domain(umap, isl_union_set_copy(domain));
        data->n_source += isl_union_map_n_map(umap);
        isl_union_map_free(umap);
        if (!umap)
                r = -1;

        umap = isl_union_map_copy(data->access->may_source);
        umap = isl_union_map_intersect_domain(umap, isl_union_set_copy(domain));
        data->n_source += isl_union_map_n_map(umap);
        isl_union_map_free(umap);
        if (!umap)
                r = -1;

        isl_union_set_free(domain);

        return r;
}

/* Add a single scheduled sink or source (depending on data->set_sink)
 * with scheduled access relation "map", must property data->must and
 * schedule node data->node to the list of sinks or sources.
 */
static int extract_sink_source(__isl_take isl_map *map, void *user)
{
        struct isl_compute_flow_schedule_data *data = user;
        struct isl_scheduled_access *access;

        if (data->set_sink)
                access = data->sink + data->n_sink++;
        else
                access = data->source + data->n_source++;

        access->access = map;
        access->must = data->must;
        access->node = isl_schedule_node_copy(data->node);

        return 0;
}

/* isl_schedule_foreach_schedule_node callback for collecting
 * individual scheduled source and sink accesses.
 *
 * We only collect accesses at the leaves of the schedule tree.
 * We prepend the schedule dimensions at the leaf to the iteration
 * domains of the source and sink accesses and then extract
 * the individual accesses (per space).
 *
 * In particular, if the prefix schedule at the node is of the form
 *
 *      D -> S
 *
 * while the access relations are of the form
 *
 *      D -> A
 *
 * then the updated access relations are of the form
 *
 *      [S -> D] -> A
 *
 * Note that S consists of a single space such that introducing S
 * in the access relations does not increase the number of spaces.
 */
static int collect_sink_source(__isl_keep isl_schedule_node *node, void *user)
{
        struct isl_compute_flow_schedule_data *data = user;
        isl_union_map *prefix;
        isl_union_map *umap;
        int r = 0;

        if (isl_schedule_node_get_type(node) != isl_schedule_node_leaf)
                return 1;

        data->node = node;

        prefix = isl_schedule_node_get_prefix_schedule_union_map(node);
        prefix = isl_union_map_reverse(prefix);
        prefix = isl_union_map_range_map(prefix);

        data->set_sink = 1;
        umap = isl_union_map_copy(data->access->sink);
        umap = isl_union_map_apply_range(isl_union_map_copy(prefix), umap);
        if (isl_union_map_foreach_map(umap, &extract_sink_source, data) < 0)
                r = -1;
        isl_union_map_free(umap);

        data->set_sink = 0;
        data->must = 1;
        umap = isl_union_map_copy(data->access->must_source);
        umap = isl_union_map_apply_range(isl_union_map_copy(prefix), umap);
        if (isl_union_map_foreach_map(umap, &extract_sink_source, data) < 0)
                r = -1;
        isl_union_map_free(umap);

        data->set_sink = 0;
        data->must = 0;
        umap = isl_union_map_copy(data->access->may_source);
        umap = isl_union_map_apply_range(isl_union_map_copy(prefix), umap);
        if (isl_union_map_foreach_map(umap, &extract_sink_source, data) < 0)
                r = -1;
        isl_union_map_free(umap);

        isl_union_map_free(prefix);

        return r;
}

/* isl_access_info_compute_flow callback for determining whether
 * the shared nesting level and the ordering within that level
 * for two scheduled accesses for use in compute_single_flow.
 *
 * The tokens passed to this function refer to the leaves
 * in the schedule tree where the accesses take place.
 *
 * If n is the shared number of loops, then we need to return
 * "2 * n + 1" if "first" precedes "second" inside the innermost
 * shared loop and "2 * n" otherwise.
 *
 * The innermost shared ancestor may be the leaves themselves
 * if the accesses take place in the same leaf.  Otherwise,
 * it is either a set node or a sequence node.  Only in the case
 * of a sequence node do we consider one access to precede the other.
 */
static int before_node(void *first, void *second)
{
        isl_schedule_node *node1 = first;
        isl_schedule_node *node2 = second;
        isl_schedule_node *shared;
        int depth;
        int before = 0;

        shared = isl_schedule_node_get_shared_ancestor(node1, node2);
        if (!shared)
                return -1;

        depth = isl_schedule_node_get_schedule_depth(shared);
        if (isl_schedule_node_get_type(shared) == isl_schedule_node_sequence) {
                int pos1, pos2;

                pos1 = isl_schedule_node_get_ancestor_child_position(node1,
                                                                    shared);
                pos2 = isl_schedule_node_get_ancestor_child_position(node2,
                                                                    shared);
                before = pos1 < pos2;
        }

        isl_schedule_node_free(shared);

        return 2 * depth + before;
}

/* Add the scheduled sources from "data" that access
 * the same data space as "sink" to "access".
 */
static __isl_give isl_access_info *add_matching_sources(
        __isl_take isl_access_info *access, struct isl_scheduled_access *sink,
        struct isl_compute_flow_schedule_data *data)
{
        int i;
        isl_space *space;

        space = isl_space_range(isl_map_get_space(sink->access));
        for (i = 0; i < data->n_source; ++i) {
                struct isl_scheduled_access *source;
                isl_space *source_space;
                int eq;

                source = &data->source[i];
                source_space = isl_map_get_space(source->access);
                source_space = isl_space_range(source_space);
                eq = isl_space_is_equal(space, source_space);
                isl_space_free(source_space);

                if (!eq)
                        continue;
                if (eq < 0)
                        goto error;

                access = isl_access_info_add_source(access,
                    isl_map_copy(source->access), source->must, source->node);
        }

        isl_space_free(space);
        return access;
error:
        isl_space_free(space);
        isl_access_info_free(access);
        return NULL;
}

/* Given a scheduled sink access relation "sink", compute the corresponding
 * dependences on the sources in "data" and add the computed dependences
 * to "uf".
 */
static __isl_give isl_union_flow *compute_single_flow(
        __isl_take isl_union_flow *uf, struct isl_scheduled_access *sink,
        struct isl_compute_flow_schedule_data *data)
{
        int i;
        isl_access_info *access;
        isl_flow *flow;
        isl_map *map;

        if (!uf)
                return NULL;

        access = isl_access_info_alloc(isl_map_copy(sink->access), sink->node,
                                        &before_node, data->n_source);
        access = add_matching_sources(access, sink, data);

        flow = isl_access_info_compute_flow(access);
        if (!flow)
                return isl_union_flow_free(uf);

        map = isl_map_domain_factor_range(isl_flow_get_no_source(flow, 1));
        uf->must_no_source = isl_union_map_union(uf->must_no_source,
                                                isl_union_map_from_map(map));
        map = isl_map_domain_factor_range(isl_flow_get_no_source(flow, 0));
        uf->may_no_source = isl_union_map_union(uf->may_no_source,
                                                isl_union_map_from_map(map));

        for (i = 0; i < flow->n_source; ++i) {
                isl_union_map *dep;

                map = isl_map_factor_range(isl_map_copy(flow->dep[i].map));
                dep = isl_union_map_from_map(map);
                if (flow->dep[i].must)
                        uf->must_dep = isl_union_map_union(uf->must_dep, dep);
                else
                        uf->may_dep = isl_union_map_union(uf->may_dep, dep);
        }

        isl_flow_free(flow);

        return uf;
}

/* Given a description of the "sink" accesses, the "source" accesses and
 * a schedule, compute for each instance of a sink access
 * and for each element accessed by that instance,
 * the possible or definite source accesses that last accessed the
 * element accessed by the sink access before this sink access
 * in the sense that there is no intermediate definite source access.
 *
 * The must_no_source and may_no_source elements of the result
 * are subsets of access->sink.  The elements must_dep and may_dep
 * map domain elements of access->{may,must)_source to
 * domain elements of access->sink.
 *
 * This function is used when a schedule tree representation
 * is available.
 *
 * We extract the individual scheduled source and sink access relations and
 * then compute dependences for each scheduled sink individually.
 */
static __isl_give isl_union_flow *compute_flow_schedule(
        __isl_take isl_union_access_info *access)
{
        struct isl_compute_flow_schedule_data data = { access };
        int i, n;
        isl_ctx *ctx;
        isl_union_flow *flow;

        ctx = isl_union_access_info_get_ctx(access);

        data.n_sink = 0;
        data.n_source = 0;
        if (isl_schedule_foreach_schedule_node(access->schedule,
                                                &count_sink_source, &data) < 0)
                goto error;

        n = data.n_sink + data.n_source;
        data.sink = isl_calloc_array(ctx, struct isl_scheduled_access, n);
        if (n && !data.sink)
                goto error;
        data.source = data.sink + data.n_sink;

        data.n_sink = 0;
        data.n_source = 0;
        if (isl_schedule_foreach_schedule_node(access->schedule,
                                            &collect_sink_source, &data) < 0)
                goto error;

        flow = isl_union_flow_alloc(isl_union_map_get_space(access->sink));

        isl_compute_flow_schedule_data_align_params(&data);

        for (i = 0; i < data.n_sink; ++i)
                flow = compute_single_flow(flow, &data.sink[i], &data);

        isl_compute_flow_schedule_data_clear(&data);

        isl_union_access_info_free(access);
        return flow;
error:
        isl_union_access_info_free(access);
        isl_compute_flow_schedule_data_clear(&data);
        return NULL;
}

/* Given a description of the "sink" accesses, the "source" accesses and
 * a schedule, compute for each instance of a sink access
 * and for each element accessed by that instance,
 * the possible or definite source accesses that last accessed the
 * element accessed by the sink access before this sink access
 * in the sense that there is no intermediate definite source access.
 *
 * The must_no_source and may_no_source elements of the result
 * are subsets of access->sink.  The elements must_dep and may_dep
 * map domain elements of access->{may,must)_source to
 * domain elements of access->sink.
 *
 * We check whether the schedule is available as a schedule tree
 * or a schedule map and call the correpsonding function to perform
 * the analysis.
 */
__isl_give isl_union_flow *isl_union_access_info_compute_flow(
        __isl_take isl_union_access_info *access)
{
        access = isl_union_access_info_normalize(access);
        if (!access)
                return NULL;
        if (access->schedule)
                return compute_flow_schedule(access);
        else
                return compute_flow_union_map(access);
}

/* Given a collection of "sink" and "source" accesses,
 * compute for each iteration of a sink access
 * and for each element accessed by that iteration,
 * the source access in the list that last accessed the
 * element accessed by the sink access before this sink access.
 * Each access is given as a map from the loop iterators
 * to the array indices.
 * The result is a relations between source and sink
 * iterations and a subset of the domain of the sink accesses,
 * corresponding to those iterations that access an element
 * not previously accessed.
 *
 * We collect the inputs in an isl_union_access_info object,
 * call isl_union_access_info_compute_flow and extract
 * the outputs from the result.
 */
int isl_union_map_compute_flow(__isl_take isl_union_map *sink,
        __isl_take isl_union_map *must_source,
        __isl_take isl_union_map *may_source,
        __isl_take isl_union_map *schedule,
        __isl_give isl_union_map **must_dep, __isl_give isl_union_map **may_dep,
        __isl_give isl_union_map **must_no_source,
        __isl_give isl_union_map **may_no_source)
{
        isl_union_access_info *access;
        isl_union_flow *flow;

        access = isl_union_access_info_from_sink(sink);
        access = isl_union_access_info_set_must_source(access, must_source);
        access = isl_union_access_info_set_may_source(access, may_source);
        access = isl_union_access_info_set_schedule_map(access, schedule);
        flow = isl_union_access_info_compute_flow(access);

        if (must_dep)
                *must_dep = isl_union_flow_get_must_dependence(flow);
        if (may_dep)
                *may_dep = isl_union_flow_get_non_must_dependence(flow);
        if (must_no_source)
                *must_no_source = isl_union_flow_get_must_no_source(flow);
        if (may_no_source)
                *may_no_source = isl_union_flow_get_non_must_no_source(flow);

        isl_union_flow_free(flow);

        if ((must_dep && !*must_dep) || (may_dep && !*may_dep) ||
            (must_no_source && !*must_no_source) ||
            (may_no_source && !*may_no_source))
                goto error;

        return 0;
error:
        if (must_dep)
                *must_dep = isl_union_map_free(*must_dep);
        if (may_dep)
                *may_dep = isl_union_map_free(*may_dep);
        if (must_no_source)
                *must_no_source = isl_union_map_free(*must_no_source);
        if (may_no_source)
                *may_no_source = isl_union_map_free(*may_no_source);
        return -1;
}