cloog-0.16.3/isl/isl_flow.c

/*
 * Copyright 2005-2007 Universiteit Leiden
 * Copyright 2008-2009 Katholieke Universiteit Leuven
 * Copyright 2010      INRIA Saclay
 *
 * Use of this software is governed by the GNU LGPLv2.1 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 
 */

#include <isl/set.h>
#include <isl/map.h>
#include <isl/flow.h>

/* 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.
 */
struct isl_access_info {
	struct isl_labeled_map	sink;
	isl_access_level_before	level_before;
	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_alloc(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.
 */
void isl_access_info_free(__isl_take isl_access_info *acc)
{
	int i;

	if (!acc)
		return;
	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);
}

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

/* 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)
		return NULL;
	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;
}

/* A temporary structure used while sorting the accesses in an isl_access_info.
 */
struct isl_access_sort_info {
	struct isl_map		*source_map;
	void			*source_data;
	struct isl_access_info	*acc;
};

/* 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 and p2 share a different number of levels with the sink,
 * then the one with the lowest number of shared levels should be
 * sorted first.
 * If they both share no levels, then the order is irrelevant.
 * Otherwise, 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)
{
	const struct isl_access_sort_info *i1, *i2;
	int level1, level2;
	uint32_t h1, h2;
	i1 = (const struct isl_access_sort_info *) p1;
	i2 = (const struct isl_access_sort_info *) p2;

	level1 = i1->acc->level_before(i1->source_data, i1->acc->sink.data);
	level2 = i2->acc->level_before(i2->source_data, i2->acc->sink.data);

	if (level1 != level2 || !level1)
		return level1 - level2;

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

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

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

/* Sort the must source accesses in order of increasing number of shared
 * levels with the sink access.
 * Source accesses with the same number of shared levels are sorted
 * in their textual order.
 */
static __isl_give isl_access_info *isl_access_info_sort_sources(
	__isl_take isl_access_info *acc)
{
	int i;
	isl_ctx *ctx;
	struct isl_access_sort_info *array;

	if (!acc)
		return NULL;
	if (acc->n_must <= 1)
		return acc;

	ctx = isl_map_get_ctx(acc->sink.map);
	array = isl_alloc_array(ctx, struct isl_access_sort_info, acc->n_must);
	if (!array)
		goto error;

	for (i = 0; i < acc->n_must; ++i) {
		array[i].source_map = acc->source[i].map;
		array[i].source_data = acc->source[i].data;
		array[i].acc = acc;
	}

	qsort(array, acc->n_must, sizeof(struct isl_access_sort_info),
		access_sort_cmp);

	for (i = 0; i < acc->n_must; ++i) {
		acc->source[i].map = array[i].source_map;
		acc->source[i].data = array[i].source_data;
	}

	free(array);

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

/* 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;
	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;

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

	dep->n_source = 2 * acc->n_must + acc->n_may;
	for (i = 0; i < acc->n_must; ++i) {
		struct isl_dim *dim;
		dim = isl_dim_join(isl_map_get_dim(acc->source[i].map),
			    isl_dim_reverse(isl_map_get_dim(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) {
		struct isl_dim *dim;
		dim = isl_dim_join(isl_map_get_dim(acc->source[i].map),
			    isl_dim_reverse(isl_map_get_dim(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(struct isl_dim *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 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_dim(dep_map), level);
	dep_map = isl_map_intersect(dep_map, after);
	result = isl_map_partial_lexmax(dep_map, 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)
{
	struct isl_dim *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 = isl_dim_join(isl_map_get_dim(acc->source[k].map),
		    isl_dim_reverse(isl_map_get_dim(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_dim(dep_map), before_level);
	dep_map = isl_map_intersect(dep_map, before_read);
	result = isl_map_partial_lexmax(dep_map, 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_dim(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_keep isl_map *old_map,
	int j, int before_level, int k, int after_level)
{
	isl_dim *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_dim_join(isl_map_get_dim(acc->source[acc->n_must + j].map),
		    isl_dim_reverse(isl_map_get_dim(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_dim(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_dim_domain(isl_map_get_dim(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_take isl_access_info *acc)
{
	int i;
	isl_set *mustdo;
	isl_set *maydo;
	isl_flow *res;

	res = isl_flow_alloc(acc);
	if (!res)
		goto error;

	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_dim *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_dim(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;

	isl_access_info_free(acc);

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

/* 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_take 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;

	acc = isl_access_info_sort_sources(acc);
	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_like(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) {
			must_rel[j] = isl_map_empty_like(res->dep[j].map);
			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;
	isl_access_info_free(acc);
	return res;
error:
	isl_access_info_free(acc);
	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;
	isl_map *domain_map = NULL;

	if (!acc)
		return NULL;

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

	for (j = 0; j < res->n_source; ++j) {
		res->dep[j].map = isl_map_apply_range(res->dep[j].map,
					isl_map_copy(domain_map));
		if (!res->dep[j].map)
			goto error2;
	}
	if (!res->must_no_source || !res->may_no_source)
		goto error2;

	isl_map_free(domain_map);
	return res;
error:
	isl_map_free(domain_map);
	isl_access_info_free(acc);
	return NULL;
error2:
	isl_map_free(domain_map);
	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_dim *dim;
	struct isl_sched_info *info;
	int i, n;

	if (!map)
		return NULL;

	dim = isl_dim_unwrap(isl_dim_domain(isl_map_get_dim(map)));
	if (!dim)
		return NULL;
	n = isl_dim_size(dim, isl_dim_in);
	isl_dim_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 (!info->is_cst || !info->cst)
		goto error;

	for (i = 0; i < n; ++i)
		info->is_cst[i] = isl_map_plain_is_fixed(map, isl_dim_in, i,
							&info->cst->el[i]);

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

struct isl_compute_flow_data {
	isl_union_map *must_source;
	isl_union_map *may_source;
	isl_union_map *must_dep;
	isl_union_map *may_dep;
	isl_union_map *must_no_source;
	isl_union_map *may_no_source;

	int count;
	int must;
	isl_dim *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_dim *dim;
	struct isl_compute_flow_data *data;

	data = (struct isl_compute_flow_data *)user;

	dim = isl_dim_range(isl_map_get_dim(map));

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

	isl_dim_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_dim *dim;
	struct isl_sched_info *info;
	struct isl_compute_flow_data *data;

	data = (struct isl_compute_flow_data *)user;

	dim = isl_dim_range(isl_map_get_dim(map));

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

	isl_dim_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 = info1->cst->size;
	n2 = info2->cst->size;

	if (n2 < n1)
		n1 = n2;

	for (i = 0; i < n1; ++i) {
		if (!info1->is_cst[i])
			continue;
		if (!info2->is_cst[i])
			continue;
		if (isl_int_eq(info1->cst->el[i], info2->cst->el[i]))
			continue;
		return 2 * i + isl_int_lt(info1->cst->el[i], info2->cst->el[i]);
	}

	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;

	data = (struct isl_compute_flow_data *)user;

	ctx = isl_map_get_ctx(map);

	data->accesses = NULL;
	data->sink_info = NULL;
	data->source_info = NULL;
	data->count = 0;
	data->dim = isl_dim_range(isl_map_get_dim(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->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;

	data->must_no_source = isl_union_map_union(data->must_no_source,
		    isl_union_map_from_map(isl_flow_get_no_source(flow, 1)));
	data->may_no_source = isl_union_map_union(data->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)
			data->must_dep = isl_union_map_union(data->must_dep, dep);
		else
			data->may_dep = isl_union_map_union(data->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_dim_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_dim_free(data->dim);
	isl_map_free(map);

	return -1;
}

/* 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 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.
 */
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_dim *dim;
	isl_union_map *range_map = NULL;
	struct isl_compute_flow_data data;

	sink = isl_union_map_align_params(sink,
					    isl_union_map_get_dim(must_source));
	sink = isl_union_map_align_params(sink,
					    isl_union_map_get_dim(may_source));
	sink = isl_union_map_align_params(sink,
					    isl_union_map_get_dim(schedule));
	dim = isl_union_map_get_dim(sink);
	must_source = isl_union_map_align_params(must_source, isl_dim_copy(dim));
	may_source = isl_union_map_align_params(may_source, isl_dim_copy(dim));
	schedule = isl_union_map_align_params(schedule, isl_dim_copy(dim));

	schedule = isl_union_map_reverse(schedule);
	range_map = isl_union_map_range_map(schedule);
	schedule = isl_union_map_reverse(isl_union_map_copy(range_map));
	sink = isl_union_map_apply_domain(sink, isl_union_map_copy(schedule));
	must_source = isl_union_map_apply_domain(must_source,
						isl_union_map_copy(schedule));
	may_source = isl_union_map_apply_domain(may_source, schedule);

	data.must_source = must_source;
	data.may_source = may_source;
	data.must_dep = must_dep ?
		isl_union_map_empty(isl_dim_copy(dim)) : NULL;
	data.may_dep = may_dep ? isl_union_map_empty(isl_dim_copy(dim)) : NULL;
	data.must_no_source = must_no_source ?
		isl_union_map_empty(isl_dim_copy(dim)) : NULL;
	data.may_no_source = may_no_source ?
		isl_union_map_empty(isl_dim_copy(dim)) : NULL;

	isl_dim_free(dim);

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

	isl_union_map_free(sink);
	isl_union_map_free(must_source);
	isl_union_map_free(may_source);

	if (must_dep) {
		data.must_dep = isl_union_map_apply_domain(data.must_dep,
					isl_union_map_copy(range_map));
		data.must_dep = isl_union_map_apply_range(data.must_dep,
					isl_union_map_copy(range_map));
		*must_dep = data.must_dep;
	}
	if (may_dep) {
		data.may_dep = isl_union_map_apply_domain(data.may_dep,
					isl_union_map_copy(range_map));
		data.may_dep = isl_union_map_apply_range(data.may_dep,
					isl_union_map_copy(range_map));
		*may_dep = data.may_dep;
	}
	if (must_no_source) {
		data.must_no_source = isl_union_map_apply_domain(
			data.must_no_source, isl_union_map_copy(range_map));
		*must_no_source = data.must_no_source;
	}
	if (may_no_source) {
		data.may_no_source = isl_union_map_apply_domain(
			data.may_no_source, isl_union_map_copy(range_map));
		*may_no_source = data.may_no_source;
	}

	isl_union_map_free(range_map);

	return 0;
error:
	isl_union_map_free(range_map);
	isl_union_map_free(sink);
	isl_union_map_free(must_source);
	isl_union_map_free(may_source);
	isl_union_map_free(data.must_dep);
	isl_union_map_free(data.may_dep);
	isl_union_map_free(data.must_no_source);
	isl_union_map_free(data.may_no_source);

	if (must_dep)
		*must_dep = NULL;
	if (may_dep)
		*may_dep = NULL;
	if (must_no_source)
		*must_no_source = NULL;
	if (may_no_source)
		*may_no_source = NULL;
	return -1;
}