add dependence analysis
authorSven Verdoolaege <skimo@kotnet.org>
Sun, 31 Jan 2010 20:15:48 +0000 (21:15 +0100)
committerSven Verdoolaege <skimo@kotnet.org>
Sun, 31 Jan 2010 22:28:25 +0000 (23:28 +0100)
Makefile.am
doc/user.pod
include/isl_flow.h [new file with mode: 0644]
isl_flow.c [new file with mode: 0644]

index 4c0cf46..0f3ca74 100644 (file)
@@ -43,6 +43,7 @@ libisl_la_SOURCES = \
        isl_div.c \
        isl_equalities.c \
        isl_equalities.h \
+       isl_flow.c \
        isl_gmp.c \
        isl_hash.c \
        isl_ilp.c \
@@ -136,6 +137,7 @@ pkginclude_HEADERS = \
        include/isl_ctx.h \
        include/isl_dim.h \
        include/isl_div.h \
+       include/isl_flow.h \
        include/isl_ilp.h \
        include/isl_int.h \
        include/isl_hash.h \
index fbeceb3..1d31d10 100644 (file)
@@ -934,6 +934,86 @@ of all elements associated to that element.
        __isl_give isl_map *isl_map_lexmax(
                __isl_take isl_map *map);
 
+=head2 Dependence Analysis
+
+C<isl> contains specialized functionality for performing
+array dataflow analysis.  That is, given a I<sink> access relation
+and a collection of possible I<source> access relations,
+C<isl> can compute relations that describe
+for each iteration of the sink access, which iteration
+of which of the source access relations was the last
+to access the same data element before the given iteration
+of the sink access.
+To compute standard flow dependences, the sink should be
+a read, while the sources should be writes.
+
+       #include <isl_flow.h>
+
+       __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_give isl_access_info *isl_access_info_add_source(
+               __isl_take isl_access_info *acc,
+               __isl_take isl_map *source, void *source_user);
+
+       __isl_give isl_flow *isl_access_info_compute_flow(
+               __isl_take isl_access_info *acc);
+
+       int isl_flow_foreach(__isl_keep isl_flow *deps,
+               int (*fn)(__isl_take isl_map *dep, void *dep_user,
+                         void *user),
+               void *user);
+       __isl_give isl_set *isl_flow_get_no_source(
+               __isl_keep isl_flow *deps);
+       void isl_flow_free(__isl_take isl_flow *deps);
+
+The function C<isl_access_info_compute_flow> performs the actual
+dependence analysis.  The other functions are used to construct
+the input for this function or to read off the output.
+
+The input is collected in an C<isl_access_info>, which can
+be created through a call to C<isl_access_info_alloc>.
+The arguments to this functions are the sink access relation
+C<sink>, a token C<sink_user> used to identify the sink
+access to the user, a callback function for specifying the
+relative order of source and sink accesses, and the number
+of source access relations that will be added.
+The callback function has type C<int (*)(void *first, void *second)>.
+The function is called with two user supplied tokens identifying
+either a source or the sink and it should return the shared nesting
+level and the relative order of the two accesses.
+In particular, let I<n> be the number of loops shared by
+the two accesses.  If C<first> precedes C<second> textually,
+then the function should return I<2 * n + 1>; otherwise,
+it should return I<2 * n>.
+The sources can be added to the C<isl_access_info> by performing
+(at most) C<max_source> calls to C<isl_access_info_add_source>.
+The C<source_user> token is again used to identify
+the source access.  The range of the source access relation
+C<source> should have the same dimension as the range
+of the sink access relation.
+
+The result of the dependence analysis is collected in an
+C<isl_flow>.  There may be elements in the domain of
+the sink access for which no preceding source access could be
+find.  The set of these elements can be obtained through
+a call to C<isl_flow_get_no_source>.
+In the case of standard flow dependence analysis,
+this set corresponds to the reads from uninitialized
+array elements.
+The actual flow dependences can be extracted using
+C<isl_flow_foreach>.  This function will call the user-specified
+callback function C<fn> for each B<non-empty> dependence between
+a source and the sink.  The callback function is called
+with three arguments, the actual flow dependence relation
+mapping source iterations to sink iterations, a token
+identifying the source and an additional C<void *> with value
+equal to the third argument of the C<isl_flow_foreach> call.
+
+After finishing with an C<isl_flow>, the user should call
+C<isl_flow_free> to free all associated memory.
+
 =head1 Applications
 
 Although C<isl> is mainly meant to be used as a library,
diff --git a/include/isl_flow.h b/include/isl_flow.h
new file mode 100644 (file)
index 0000000..67f8480
--- /dev/null
@@ -0,0 +1,38 @@
+#ifndef ISL_FLOW_H
+#define ISL_FLOW_H
+
+#include <isl_set.h>
+#include <isl_map.h>
+
+#if defined(__cplusplus)
+extern "C" {
+#endif
+
+/* Let n (>= 0) be the number of iterators shared by first and second.
+ * If first precedes second textually return 2 * n + 1,
+ * otherwise return 2 * n.
+ */
+typedef int (*isl_access_level_before)(void *first, void *second);
+
+struct isl_access_info;
+typedef struct isl_access_info isl_access_info;
+struct isl_flow;
+typedef struct isl_flow isl_flow;
+
+__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_give isl_access_info *isl_access_info_add_source(
+       __isl_take isl_access_info *acc, __isl_take isl_map *source,
+       void *source_user);
+__isl_give isl_flow *isl_access_info_compute_flow(__isl_take isl_access_info *acc);
+int isl_flow_foreach(__isl_keep isl_flow *deps,
+       int (*fn)(__isl_take isl_map *dep, void *dep_user, void *user),
+       void *user);
+__isl_give isl_set *isl_flow_get_no_source(__isl_keep isl_flow *deps);
+void isl_flow_free(__isl_take isl_flow *deps);
+
+#if defined(__cplusplus)
+}
+#endif
+
+#endif
diff --git a/isl_flow.c b/isl_flow.c
new file mode 100644 (file)
index 0000000..111aa30
--- /dev/null
@@ -0,0 +1,550 @@
+/*
+ * 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_flow.h>
+
+/* A private structure to keep track of a mapping together with
+ * a user-specified identifier.
+ */
+struct isl_labeled_map {
+       struct isl_map  *map;
+       void            *data;
+};
+
+/* A structure containing the input for dependence analysis:
+ * - a sink
+ * - n_source (<= max_source) sources
+ * - a function for determining the relative order of sources and sink
+ */
+struct isl_access_info {
+       struct isl_labeled_map  sink;
+       isl_access_level_before level_before;
+       int                     max_source;
+       int                     n_source;
+       struct isl_labeled_map  source[1];
+};
+
+/* A structure containing the output of dependence analysis:
+ * - n_source flow dependences
+ * - a subset of the sink for which no source could be found
+ */
+struct isl_flow {
+       struct isl_set          *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)
+{
+       struct isl_access_info *acc;
+
+       if (!sink)
+               return NULL;
+
+       isl_assert(sink->ctx, max_source >= 0, goto error);
+
+       acc = isl_alloc(sink->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_source = 0;
+
+       return acc;
+error:
+       isl_map_free(sink);
+       return NULL;
+}
+
+/* Free the given isl_access_info structure.
+ * This function is static because the user is expected to call
+ * isl_access_info_compute_flow on any isl_access_info structure
+ * he creates.
+ */
+static 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_source; ++i)
+               isl_map_free(acc->source[i].map);
+       free(acc);
+}
+
+/* Add another source to an isl_access_info structure.
+ * 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,
+       void *source_user)
+{
+       if (!acc)
+               return NULL;
+       isl_assert(acc->sink.map->ctx, acc->n_source < acc->max_source, goto error);
+
+       acc->source[acc->n_source].map = source;
+       acc->source[acc->n_source].data = source_user;
+       acc->n_source++;
+
+       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.
+ */
+static int access_sort_cmp(const void *p1, const void *p2)
+{
+       const struct isl_access_sort_info *i1, *i2;
+       int level1, level2;
+       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);
+
+       return (level1 % 2) ? -1 : 1;
+}
+
+/* Sort the 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;
+       struct isl_access_sort_info *array;
+
+       if (!acc)
+               return NULL;
+       if (acc->n_source <= 1)
+               return acc;
+
+       array = isl_alloc_array(acc->sink.map->ctx,
+                               struct isl_access_sort_info, acc->n_source);
+       if (!array)
+               goto error;
+
+       for (i = 0; i < acc->n_source; ++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_source, sizeof(struct isl_access_sort_info),
+               access_sort_cmp);
+
+       for (i = 0; i < acc->n_source; ++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.
+ * 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 = acc->sink.map->ctx;
+       dep = isl_calloc_type(ctx, struct isl_flow);
+       if (!dep)
+               return NULL;
+
+       dep->dep = isl_alloc_array(ctx, struct isl_labeled_map, acc->n_source);
+       if (!dep->dep)
+               goto error;
+
+       dep->n_source = acc->n_source;
+       for (i = 0; i < acc->n_source; ++i) {
+               struct isl_dim *dim;
+               dim = isl_dim_join(isl_dim_copy(acc->source[i].map->dim),
+                           isl_dim_reverse(isl_dim_copy(acc->sink.map->dim)));
+               dep->dep[i].map = isl_map_empty(dim);
+               dep->dep[i].data = acc->source[i].data;
+       }
+
+       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, void *dep_user, void *user),
+       void *user)
+{
+       int i;
+
+       if (!deps)
+               return -1;
+
+       for (i = 0; i < deps->n_source; ++i) {
+               if (isl_map_fast_is_empty(deps->dep[i].map))
+                       continue;
+               if (fn(isl_map_copy(deps->dep[i].map), 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_set *isl_flow_get_no_source(__isl_keep isl_flow *deps)
+{
+       if (!deps)
+               return NULL;
+       
+       return isl_set_copy(deps->no_source);
+}
+
+void isl_flow_free(__isl_take isl_flow *deps)
+{
+       int i;
+
+       if (!deps)
+               return;
+       isl_set_free(deps->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);
+}
+
+/* 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 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_dim_copy(dep_map->dim), 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 source j and iterations
+ * of the sink, compute the last iteration of source k preceding
+ * the sink at level before_level for any of the sink iterations,
+ * but following the corresponding iteration of 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_dim_copy(acc->source[k].map->dim),
+                   isl_dim_reverse(isl_dim_copy(acc->source[j].map->dim)));
+       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_dim_copy(dep_map->dim), 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_fast_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_fast_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;
+}
+
+/* Given a "sink" access, a list of n "source" accesses,
+ * compute for each iteration of the sink access,
+ * the source access in the list that last accessed the
+ * same 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.
+ *
+ * The 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
+ */
+__isl_give isl_flow *isl_access_info_compute_flow(__isl_take isl_access_info *acc)
+{
+       struct isl_ctx *ctx;
+       struct isl_set *todo;
+       int level, j;
+       int depth;
+       struct isl_map **temp_rel;
+       struct isl_flow *res;
+
+       acc = isl_access_info_sort_sources(acc);
+
+       res = isl_flow_alloc(acc);
+       if (!res)
+               goto error;
+       ctx = acc->sink.map->ctx;
+
+       depth = 2 * isl_map_dim(acc->sink.map, isl_dim_in) + 1;
+       todo = isl_map_domain(isl_map_copy(acc->sink.map));
+       if (isl_set_fast_is_empty(todo))
+               goto done;
+
+       temp_rel = isl_alloc_array(ctx, struct isl_map *, acc->n_source);
+
+       for (level = depth; level >= 1; --level) {
+               for (j = acc->n_source-1; j >=0; --j)
+                       temp_rel[j] = isl_map_empty_like(res->dep[j].map);
+
+               for (j = acc->n_source - 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, todo, j, level, &rest);
+                       temp_rel[j] = isl_map_union_disjoint(temp_rel[j], T);
+                       todo = rest;
+
+                       intermediate_sources(acc, temp_rel, j, level);
+
+                       if (isl_set_fast_is_empty(todo))
+                               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, temp_rel, j, level);
+               }
+
+               for (j = acc->n_source - 1; j >= 0; --j)
+                       res->dep[j].map = isl_map_union_disjoint(res->dep[j].map,
+                                                            temp_rel[j]);
+               if (isl_set_fast_is_empty(todo))
+                       break;
+       }
+
+       free(temp_rel);
+done:
+       res->no_source = todo;
+       isl_access_info_free(acc);
+       return res;
+error:
+       isl_access_info_free(acc);
+       return NULL;
+}