add isl_tab_lexmin_cut_to_integer

[isl.git] / isl_scheduler.c

/*
 * Copyright 2011      INRIA Saclay
 * Copyright 2012-2014 Ecole Normale Superieure
 * Copyright 2015-2016 Sven Verdoolaege
 * Copyright 2016      INRIA Paris
 * Copyright 2017      Sven Verdoolaege
 *
 * Use of this software is governed by the MIT license
 *
 * Written by Sven Verdoolaege, 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
 * and Centre de Recherche Inria de Paris, 2 rue Simone Iff - Voie DQ12,
 * CS 42112, 75589 Paris Cedex 12, France
 */

#include <isl_ctx_private.h>
#include <isl_map_private.h>
#include <isl_space_private.h>
#include <isl_aff_private.h>
#include <isl/hash.h>
#include <isl/constraint.h>
#include <isl/schedule.h>
#include <isl_schedule_constraints.h>
#include <isl/schedule_node.h>
#include <isl_mat_private.h>
#include <isl_vec_private.h>
#include <isl/set.h>
#include <isl/union_set.h>
#include <isl_seq.h>
#include <isl_tab.h>
#include <isl_dim_map.h>
#include <isl/map_to_basic_set.h>
#include <isl_sort.h>
#include <isl_options_private.h>
#include <isl_tarjan.h>
#include <isl_morph.h>
#include <isl/ilp.h>
#include <isl_val_private.h>

/*
 * The scheduling algorithm implemented in this file was inspired by
 * Bondhugula et al., "Automatic Transformations for Communication-Minimized
 * Parallelization and Locality Optimization in the Polyhedral Model".
 */


/* Internal information about a node that is used during the construction
 * of a schedule.
 * space represents the original space in which the domain lives;
 *      that is, the space is not affected by compression
 * sched is a matrix representation of the schedule being constructed
 *      for this node; if compressed is set, then this schedule is
 *      defined over the compressed domain space
 * sched_map is an isl_map representation of the same (partial) schedule
 *      sched_map may be NULL; if compressed is set, then this map
 *      is defined over the uncompressed domain space
 * rank is the number of linearly independent rows in the linear part
 *      of sched
 * the columns of cmap represent a change of basis for the schedule
 *      coefficients; the first rank columns span the linear part of
 *      the schedule rows
 * cinv is the inverse of cmap.
 * ctrans is the transpose of cmap.
 * start is the first variable in the LP problem in the sequences that
 *      represents the schedule coefficients of this node
 * nvar is the dimension of the domain
 * nparam is the number of parameters or 0 if we are not constructing
 *      a parametric schedule
 *
 * If compressed is set, then hull represents the constraints
 * that were used to derive the compression, while compress and
 * decompress map the original space to the compressed space and
 * vice versa.
 *
 * scc is the index of SCC (or WCC) this node belongs to
 *
 * "cluster" is only used inside extract_clusters and identifies
 * the cluster of SCCs that the node belongs to.
 *
 * coincident contains a boolean for each of the rows of the schedule,
 * indicating whether the corresponding scheduling dimension satisfies
 * the coincidence constraints in the sense that the corresponding
 * dependence distances are zero.
 *
 * If the schedule_treat_coalescing option is set, then
 * "sizes" contains the sizes of the (compressed) instance set
 * in each direction.  If there is no fixed size in a given direction,
 * then the corresponding size value is set to infinity.
 * If the schedule_treat_coalescing option or the schedule_max_coefficient
 * option is set, then "max" contains the maximal values for
 * schedule coefficients of the (compressed) variables.  If no bound
 * needs to be imposed on a particular variable, then the corresponding
 * value is negative.
 */
struct isl_sched_node {
        isl_space *space;
        int     compressed;
        isl_set *hull;
        isl_multi_aff *compress;
        isl_multi_aff *decompress;
        isl_mat *sched;
        isl_map *sched_map;
        int      rank;
        isl_mat *cmap;
        isl_mat *cinv;
        isl_mat *ctrans;
        int      start;
        int      nvar;
        int      nparam;

        int      scc;
        int      cluster;

        int     *coincident;

        isl_multi_val *sizes;
        isl_vec *max;
};

static int node_has_tuples(const void *entry, const void *val)
{
        struct isl_sched_node *node = (struct isl_sched_node *)entry;
        isl_space *space = (isl_space *) val;

        return isl_space_has_equal_tuples(node->space, space);
}

static int node_scc_exactly(struct isl_sched_node *node, int scc)
{
        return node->scc == scc;
}

static int node_scc_at_most(struct isl_sched_node *node, int scc)
{
        return node->scc <= scc;
}

static int node_scc_at_least(struct isl_sched_node *node, int scc)
{
        return node->scc >= scc;
}

/* An edge in the dependence graph.  An edge may be used to
 * ensure validity of the generated schedule, to minimize the dependence
 * distance or both
 *
 * map is the dependence relation, with i -> j in the map if j depends on i
 * tagged_condition and tagged_validity contain the union of all tagged
 *      condition or conditional validity dependence relations that
 *      specialize the dependence relation "map"; that is,
 *      if (i -> a) -> (j -> b) is an element of "tagged_condition"
 *      or "tagged_validity", then i -> j is an element of "map".
 *      If these fields are NULL, then they represent the empty relation.
 * src is the source node
 * dst is the sink node
 *
 * types is a bit vector containing the types of this edge.
 * validity is set if the edge is used to ensure correctness
 * coincidence is used to enforce zero dependence distances
 * proximity is set if the edge is used to minimize dependence distances
 * condition is set if the edge represents a condition
 *      for a conditional validity schedule constraint
 * local can only be set for condition edges and indicates that
 *      the dependence distance over the edge should be zero
 * conditional_validity is set if the edge is used to conditionally
 *      ensure correctness
 *
 * For validity edges, start and end mark the sequence of inequality
 * constraints in the LP problem that encode the validity constraint
 * corresponding to this edge.
 *
 * During clustering, an edge may be marked "no_merge" if it should
 * not be used to merge clusters.
 * The weight is also only used during clustering and it is
 * an indication of how many schedule dimensions on either side
 * of the schedule constraints can be aligned.
 * If the weight is negative, then this means that this edge was postponed
 * by has_bounded_distances or any_no_merge.  The original weight can
 * be retrieved by adding 1 + graph->max_weight, with "graph"
 * the graph containing this edge.
 */
struct isl_sched_edge {
        isl_map *map;
        isl_union_map *tagged_condition;
        isl_union_map *tagged_validity;

        struct isl_sched_node *src;
        struct isl_sched_node *dst;

        unsigned types;

        int start;
        int end;

        int no_merge;
        int weight;
};

/* Is "edge" marked as being of type "type"?
 */
static int is_type(struct isl_sched_edge *edge, enum isl_edge_type type)
{
        return ISL_FL_ISSET(edge->types, 1 << type);
}

/* Mark "edge" as being of type "type".
 */
static void set_type(struct isl_sched_edge *edge, enum isl_edge_type type)
{
        ISL_FL_SET(edge->types, 1 << type);
}

/* No longer mark "edge" as being of type "type"?
 */
static void clear_type(struct isl_sched_edge *edge, enum isl_edge_type type)
{
        ISL_FL_CLR(edge->types, 1 << type);
}

/* Is "edge" marked as a validity edge?
 */
static int is_validity(struct isl_sched_edge *edge)
{
        return is_type(edge, isl_edge_validity);
}

/* Mark "edge" as a validity edge.
 */
static void set_validity(struct isl_sched_edge *edge)
{
        set_type(edge, isl_edge_validity);
}

/* Is "edge" marked as a proximity edge?
 */
static int is_proximity(struct isl_sched_edge *edge)
{
        return is_type(edge, isl_edge_proximity);
}

/* Is "edge" marked as a local edge?
 */
static int is_local(struct isl_sched_edge *edge)
{
        return is_type(edge, isl_edge_local);
}

/* Mark "edge" as a local edge.
 */
static void set_local(struct isl_sched_edge *edge)
{
        set_type(edge, isl_edge_local);
}

/* No longer mark "edge" as a local edge.
 */
static void clear_local(struct isl_sched_edge *edge)
{
        clear_type(edge, isl_edge_local);
}

/* Is "edge" marked as a coincidence edge?
 */
static int is_coincidence(struct isl_sched_edge *edge)
{
        return is_type(edge, isl_edge_coincidence);
}

/* Is "edge" marked as a condition edge?
 */
static int is_condition(struct isl_sched_edge *edge)
{
        return is_type(edge, isl_edge_condition);
}

/* Is "edge" marked as a conditional validity edge?
 */
static int is_conditional_validity(struct isl_sched_edge *edge)
{
        return is_type(edge, isl_edge_conditional_validity);
}

/* Internal information about the dependence graph used during
 * the construction of the schedule.
 *
 * intra_hmap is a cache, mapping dependence relations to their dual,
 *      for dependences from a node to itself
 * inter_hmap is a cache, mapping dependence relations to their dual,
 *      for dependences between distinct nodes
 * if compression is involved then the key for these maps
 * is the original, uncompressed dependence relation, while
 * the value is the dual of the compressed dependence relation.
 *
 * n is the number of nodes
 * node is the list of nodes
 * maxvar is the maximal number of variables over all nodes
 * max_row is the allocated number of rows in the schedule
 * n_row is the current (maximal) number of linearly independent
 *      rows in the node schedules
 * n_total_row is the current number of rows in the node schedules
 * band_start is the starting row in the node schedules of the current band
 * root is set if this graph is the original dependence graph,
 *      without any splitting
 *
 * sorted contains a list of node indices sorted according to the
 *      SCC to which a node belongs
 *
 * n_edge is the number of edges
 * edge is the list of edges
 * max_edge contains the maximal number of edges of each type;
 *      in particular, it contains the number of edges in the inital graph.
 * edge_table contains pointers into the edge array, hashed on the source
 *      and sink spaces; there is one such table for each type;
 *      a given edge may be referenced from more than one table
 *      if the corresponding relation appears in more than one of the
 *      sets of dependences; however, for each type there is only
 *      a single edge between a given pair of source and sink space
 *      in the entire graph
 *
 * node_table contains pointers into the node array, hashed on the space tuples
 *
 * region contains a list of variable sequences that should be non-trivial
 *
 * lp contains the (I)LP problem used to obtain new schedule rows
 *
 * src_scc and dst_scc are the source and sink SCCs of an edge with
 *      conflicting constraints
 *
 * scc represents the number of components
 * weak is set if the components are weakly connected
 *
 * max_weight is used during clustering and represents the maximal
 * weight of the relevant proximity edges.
 */
struct isl_sched_graph {
        isl_map_to_basic_set *intra_hmap;
        isl_map_to_basic_set *inter_hmap;

        struct isl_sched_node *node;
        int n;
        int maxvar;
        int max_row;
        int n_row;

        int *sorted;

        int n_total_row;
        int band_start;

        int root;

        struct isl_sched_edge *edge;
        int n_edge;
        int max_edge[isl_edge_last + 1];
        struct isl_hash_table *edge_table[isl_edge_last + 1];

        struct isl_hash_table *node_table;
        struct isl_region *region;

        isl_basic_set *lp;

        int src_scc;
        int dst_scc;

        int scc;
        int weak;

        int max_weight;
};

/* Initialize node_table based on the list of nodes.
 */
static int graph_init_table(isl_ctx *ctx, struct isl_sched_graph *graph)
{
        int i;

        graph->node_table = isl_hash_table_alloc(ctx, graph->n);
        if (!graph->node_table)
                return -1;

        for (i = 0; i < graph->n; ++i) {
                struct isl_hash_table_entry *entry;
                uint32_t hash;

                hash = isl_space_get_tuple_hash(graph->node[i].space);
                entry = isl_hash_table_find(ctx, graph->node_table, hash,
                                            &node_has_tuples,
                                            graph->node[i].space, 1);
                if (!entry)
                        return -1;
                entry->data = &graph->node[i];
        }

        return 0;
}

/* Return a pointer to the node that lives within the given space,
 * or NULL if there is no such node.
 */
static struct isl_sched_node *graph_find_node(isl_ctx *ctx,
        struct isl_sched_graph *graph, __isl_keep isl_space *space)
{
        struct isl_hash_table_entry *entry;
        uint32_t hash;

        hash = isl_space_get_tuple_hash(space);
        entry = isl_hash_table_find(ctx, graph->node_table, hash,
                                    &node_has_tuples, space, 0);

        return entry ? entry->data : NULL;
}

static int edge_has_src_and_dst(const void *entry, const void *val)
{
        const struct isl_sched_edge *edge = entry;
        const struct isl_sched_edge *temp = val;

        return edge->src == temp->src && edge->dst == temp->dst;
}

/* Add the given edge to graph->edge_table[type].
 */
static isl_stat graph_edge_table_add(isl_ctx *ctx,
        struct isl_sched_graph *graph, enum isl_edge_type type,
        struct isl_sched_edge *edge)
{
        struct isl_hash_table_entry *entry;
        uint32_t hash;

        hash = isl_hash_init();
        hash = isl_hash_builtin(hash, edge->src);
        hash = isl_hash_builtin(hash, edge->dst);
        entry = isl_hash_table_find(ctx, graph->edge_table[type], hash,
                                    &edge_has_src_and_dst, edge, 1);
        if (!entry)
                return isl_stat_error;
        entry->data = edge;

        return isl_stat_ok;
}

/* Allocate the edge_tables based on the maximal number of edges of
 * each type.
 */
static int graph_init_edge_tables(isl_ctx *ctx, struct isl_sched_graph *graph)
{
        int i;

        for (i = 0; i <= isl_edge_last; ++i) {
                graph->edge_table[i] = isl_hash_table_alloc(ctx,
                                                            graph->max_edge[i]);
                if (!graph->edge_table[i])
                        return -1;
        }

        return 0;
}

/* If graph->edge_table[type] contains an edge from the given source
 * to the given destination, then return the hash table entry of this edge.
 * Otherwise, return NULL.
 */
static struct isl_hash_table_entry *graph_find_edge_entry(
        struct isl_sched_graph *graph,
        enum isl_edge_type type,
        struct isl_sched_node *src, struct isl_sched_node *dst)
{
        isl_ctx *ctx = isl_space_get_ctx(src->space);
        uint32_t hash;
        struct isl_sched_edge temp = { .src = src, .dst = dst };

        hash = isl_hash_init();
        hash = isl_hash_builtin(hash, temp.src);
        hash = isl_hash_builtin(hash, temp.dst);
        return isl_hash_table_find(ctx, graph->edge_table[type], hash,
                                    &edge_has_src_and_dst, &temp, 0);
}


/* If graph->edge_table[type] contains an edge from the given source
 * to the given destination, then return this edge.
 * Otherwise, return NULL.
 */
static struct isl_sched_edge *graph_find_edge(struct isl_sched_graph *graph,
        enum isl_edge_type type,
        struct isl_sched_node *src, struct isl_sched_node *dst)
{
        struct isl_hash_table_entry *entry;

        entry = graph_find_edge_entry(graph, type, src, dst);
        if (!entry)
                return NULL;

        return entry->data;
}

/* Check whether the dependence graph has an edge of the given type
 * between the given two nodes.
 */
static isl_bool graph_has_edge(struct isl_sched_graph *graph,
        enum isl_edge_type type,
        struct isl_sched_node *src, struct isl_sched_node *dst)
{
        struct isl_sched_edge *edge;
        isl_bool empty;

        edge = graph_find_edge(graph, type, src, dst);
        if (!edge)
                return 0;

        empty = isl_map_plain_is_empty(edge->map);
        if (empty < 0)
                return isl_bool_error;

        return !empty;
}

/* Look for any edge with the same src, dst and map fields as "model".
 *
 * Return the matching edge if one can be found.
 * Return "model" if no matching edge is found.
 * Return NULL on error.
 */
static struct isl_sched_edge *graph_find_matching_edge(
        struct isl_sched_graph *graph, struct isl_sched_edge *model)
{
        enum isl_edge_type i;
        struct isl_sched_edge *edge;

        for (i = isl_edge_first; i <= isl_edge_last; ++i) {
                int is_equal;

                edge = graph_find_edge(graph, i, model->src, model->dst);
                if (!edge)
                        continue;
                is_equal = isl_map_plain_is_equal(model->map, edge->map);
                if (is_equal < 0)
                        return NULL;
                if (is_equal)
                        return edge;
        }

        return model;
}

/* Remove the given edge from all the edge_tables that refer to it.
 */
static void graph_remove_edge(struct isl_sched_graph *graph,
        struct isl_sched_edge *edge)
{
        isl_ctx *ctx = isl_map_get_ctx(edge->map);
        enum isl_edge_type i;

        for (i = isl_edge_first; i <= isl_edge_last; ++i) {
                struct isl_hash_table_entry *entry;

                entry = graph_find_edge_entry(graph, i, edge->src, edge->dst);
                if (!entry)
                        continue;
                if (entry->data != edge)
                        continue;
                isl_hash_table_remove(ctx, graph->edge_table[i], entry);
        }
}

/* Check whether the dependence graph has any edge
 * between the given two nodes.
 */
static isl_bool graph_has_any_edge(struct isl_sched_graph *graph,
        struct isl_sched_node *src, struct isl_sched_node *dst)
{
        enum isl_edge_type i;
        isl_bool r;

        for (i = isl_edge_first; i <= isl_edge_last; ++i) {
                r = graph_has_edge(graph, i, src, dst);
                if (r < 0 || r)
                        return r;
        }

        return r;
}

/* Check whether the dependence graph has a validity edge
 * between the given two nodes.
 *
 * Conditional validity edges are essentially validity edges that
 * can be ignored if the corresponding condition edges are iteration private.
 * Here, we are only checking for the presence of validity
 * edges, so we need to consider the conditional validity edges too.
 * In particular, this function is used during the detection
 * of strongly connected components and we cannot ignore
 * conditional validity edges during this detection.
 */
static isl_bool graph_has_validity_edge(struct isl_sched_graph *graph,
        struct isl_sched_node *src, struct isl_sched_node *dst)
{
        isl_bool r;

        r = graph_has_edge(graph, isl_edge_validity, src, dst);
        if (r < 0 || r)
                return r;

        return graph_has_edge(graph, isl_edge_conditional_validity, src, dst);
}

static int graph_alloc(isl_ctx *ctx, struct isl_sched_graph *graph,
        int n_node, int n_edge)
{
        int i;

        graph->n = n_node;
        graph->n_edge = n_edge;
        graph->node = isl_calloc_array(ctx, struct isl_sched_node, graph->n);
        graph->sorted = isl_calloc_array(ctx, int, graph->n);
        graph->region = isl_alloc_array(ctx, struct isl_region, graph->n);
        graph->edge = isl_calloc_array(ctx,
                                        struct isl_sched_edge, graph->n_edge);

        graph->intra_hmap = isl_map_to_basic_set_alloc(ctx, 2 * n_edge);
        graph->inter_hmap = isl_map_to_basic_set_alloc(ctx, 2 * n_edge);

        if (!graph->node || !graph->region || (graph->n_edge && !graph->edge) ||
            !graph->sorted)
                return -1;

        for(i = 0; i < graph->n; ++i)
                graph->sorted[i] = i;

        return 0;
}

static void graph_free(isl_ctx *ctx, struct isl_sched_graph *graph)
{
        int i;

        isl_map_to_basic_set_free(graph->intra_hmap);
        isl_map_to_basic_set_free(graph->inter_hmap);

        if (graph->node)
                for (i = 0; i < graph->n; ++i) {
                        isl_space_free(graph->node[i].space);
                        isl_set_free(graph->node[i].hull);
                        isl_multi_aff_free(graph->node[i].compress);
                        isl_multi_aff_free(graph->node[i].decompress);
                        isl_mat_free(graph->node[i].sched);
                        isl_map_free(graph->node[i].sched_map);
                        isl_mat_free(graph->node[i].cmap);
                        isl_mat_free(graph->node[i].cinv);
                        isl_mat_free(graph->node[i].ctrans);
                        if (graph->root)
                                free(graph->node[i].coincident);
                        isl_multi_val_free(graph->node[i].sizes);
                        isl_vec_free(graph->node[i].max);
                }
        free(graph->node);
        free(graph->sorted);
        if (graph->edge)
                for (i = 0; i < graph->n_edge; ++i) {
                        isl_map_free(graph->edge[i].map);
                        isl_union_map_free(graph->edge[i].tagged_condition);
                        isl_union_map_free(graph->edge[i].tagged_validity);
                }
        free(graph->edge);
        free(graph->region);
        for (i = 0; i <= isl_edge_last; ++i)
                isl_hash_table_free(ctx, graph->edge_table[i]);
        isl_hash_table_free(ctx, graph->node_table);
        isl_basic_set_free(graph->lp);
}

/* For each "set" on which this function is called, increment
 * graph->n by one and update graph->maxvar.
 */
static isl_stat init_n_maxvar(__isl_take isl_set *set, void *user)
{
        struct isl_sched_graph *graph = user;
        int nvar = isl_set_dim(set, isl_dim_set);

        graph->n++;
        if (nvar > graph->maxvar)
                graph->maxvar = nvar;

        isl_set_free(set);

        return isl_stat_ok;
}

/* Compute the number of rows that should be allocated for the schedule.
 * In particular, we need one row for each variable or one row
 * for each basic map in the dependences.
 * Note that it is practically impossible to exhaust both
 * the number of dependences and the number of variables.
 */
static isl_stat compute_max_row(struct isl_sched_graph *graph,
        __isl_keep isl_schedule_constraints *sc)
{
        int n_edge;
        isl_stat r;
        isl_union_set *domain;

        graph->n = 0;
        graph->maxvar = 0;
        domain = isl_schedule_constraints_get_domain(sc);
        r = isl_union_set_foreach_set(domain, &init_n_maxvar, graph);
        isl_union_set_free(domain);
        if (r < 0)
                return isl_stat_error;
        n_edge = isl_schedule_constraints_n_basic_map(sc);
        if (n_edge < 0)
                return isl_stat_error;
        graph->max_row = n_edge + graph->maxvar;

        return isl_stat_ok;
}

/* Does "bset" have any defining equalities for its set variables?
 */
static isl_bool has_any_defining_equality(__isl_keep isl_basic_set *bset)
{
        int i, n;

        if (!bset)
                return isl_bool_error;

        n = isl_basic_set_dim(bset, isl_dim_set);
        for (i = 0; i < n; ++i) {
                isl_bool has;

                has = isl_basic_set_has_defining_equality(bset, isl_dim_set, i,
                                                        NULL);
                if (has < 0 || has)
                        return has;
        }

        return isl_bool_false;
}

/* Set the entries of node->max to the value of the schedule_max_coefficient
 * option, if set.
 */
static isl_stat set_max_coefficient(isl_ctx *ctx, struct isl_sched_node *node)
{
        int max;

        max = isl_options_get_schedule_max_coefficient(ctx);
        if (max == -1)
                return isl_stat_ok;

        node->max = isl_vec_alloc(ctx, node->nvar);
        node->max = isl_vec_set_si(node->max, max);
        if (!node->max)
                return isl_stat_error;

        return isl_stat_ok;
}

/* Set the entries of node->max to the minimum of the schedule_max_coefficient
 * option (if set) and half of the minimum of the sizes in the other
 * dimensions.  If the minimum of the sizes is one, half of the size
 * is zero and this value is reset to one.
 * If the global minimum is unbounded (i.e., if both
 * the schedule_max_coefficient is not set and the sizes in the other
 * dimensions are unbounded), then store a negative value.
 * If the schedule coefficient is close to the size of the instance set
 * in another dimension, then the schedule may represent a loop
 * coalescing transformation (especially if the coefficient
 * in that other dimension is one).  Forcing the coefficient to be
 * smaller than or equal to half the minimal size should avoid this
 * situation.
 */
static isl_stat compute_max_coefficient(isl_ctx *ctx,
        struct isl_sched_node *node)
{
        int max;
        int i, j;
        isl_vec *v;

        max = isl_options_get_schedule_max_coefficient(ctx);
        v = isl_vec_alloc(ctx, node->nvar);
        if (!v)
                return isl_stat_error;

        for (i = 0; i < node->nvar; ++i) {
                isl_int_set_si(v->el[i], max);
                isl_int_mul_si(v->el[i], v->el[i], 2);
        }

        for (i = 0; i < node->nvar; ++i) {
                isl_val *size;

                size = isl_multi_val_get_val(node->sizes, i);
                if (!size)
                        goto error;
                if (!isl_val_is_int(size)) {
                        isl_val_free(size);
                        continue;
                }
                for (j = 0; j < node->nvar; ++j) {
                        if (j == i)
                                continue;
                        if (isl_int_is_neg(v->el[j]) ||
                            isl_int_gt(v->el[j], size->n))
                                isl_int_set(v->el[j], size->n);
                }
                isl_val_free(size);
        }

        for (i = 0; i < node->nvar; ++i) {
                isl_int_fdiv_q_ui(v->el[i], v->el[i], 2);
                if (isl_int_is_zero(v->el[i]))
                        isl_int_set_si(v->el[i], 1);
        }

        node->max = v;
        return isl_stat_ok;
error:
        isl_vec_free(v);
        return isl_stat_error;
}

/* Compute and return the size of "set" in dimension "dim".
 * The size is taken to be the difference in values for that variable
 * for fixed values of the other variables.
 * In particular, the variable is first isolated from the other variables
 * in the range of a map
 *
 *      [i_0, ..., i_dim-1, i_dim+1, ...] -> [i_dim]
 *
 * and then duplicated
 *
 *      [i_0, ..., i_dim-1, i_dim+1, ...] -> [[i_dim] -> [i_dim']]
 *
 * The shared variables are then projected out and the maximal value
 * of i_dim' - i_dim is computed.
 */
static __isl_give isl_val *compute_size(__isl_take isl_set *set, int dim)
{
        isl_map *map;
        isl_local_space *ls;
        isl_aff *obj;
        isl_val *v;

        map = isl_set_project_onto_map(set, isl_dim_set, dim, 1);
        map = isl_map_project_out(map, isl_dim_in, dim, 1);
        map = isl_map_range_product(map, isl_map_copy(map));
        map = isl_set_unwrap(isl_map_range(map));
        set = isl_map_deltas(map);
        ls = isl_local_space_from_space(isl_set_get_space(set));
        obj = isl_aff_var_on_domain(ls, isl_dim_set, 0);
        v = isl_set_max_val(set, obj);
        isl_aff_free(obj);
        isl_set_free(set);

        return v;
}

/* Compute the size of the instance set "set" of "node", after compression,
 * as well as bounds on the corresponding coefficients, if needed.
 *
 * The sizes are needed when the schedule_treat_coalescing option is set.
 * The bounds are needed when the schedule_treat_coalescing option or
 * the schedule_max_coefficient option is set.
 *
 * If the schedule_treat_coalescing option is not set, then at most
 * the bounds need to be set and this is done in set_max_coefficient.
 * Otherwise, compress the domain if needed, compute the size
 * in each direction and store the results in node->size.
 * Finally, set the bounds on the coefficients based on the sizes
 * and the schedule_max_coefficient option in compute_max_coefficient.
 */
static isl_stat compute_sizes_and_max(isl_ctx *ctx, struct isl_sched_node *node,
        __isl_take isl_set *set)
{
        int j, n;
        isl_multi_val *mv;

        if (!isl_options_get_schedule_treat_coalescing(ctx)) {
                isl_set_free(set);
                return set_max_coefficient(ctx, node);
        }

        if (node->compressed)
                set = isl_set_preimage_multi_aff(set,
                                        isl_multi_aff_copy(node->decompress));
        mv = isl_multi_val_zero(isl_set_get_space(set));
        n = isl_set_dim(set, isl_dim_set);
        for (j = 0; j < n; ++j) {
                isl_val *v;

                v = compute_size(isl_set_copy(set), j);
                mv = isl_multi_val_set_val(mv, j, v);
        }
        node->sizes = mv;
        isl_set_free(set);
        if (!node->sizes)
                return isl_stat_error;
        return compute_max_coefficient(ctx, node);
}

/* Add a new node to the graph representing the given instance set.
 * "nvar" is the (possibly compressed) number of variables and
 * may be smaller than then number of set variables in "set"
 * if "compressed" is set.
 * If "compressed" is set, then "hull" represents the constraints
 * that were used to derive the compression, while "compress" and
 * "decompress" map the original space to the compressed space and
 * vice versa.
 * If "compressed" is not set, then "hull", "compress" and "decompress"
 * should be NULL.
 *
 * Compute the size of the instance set and bounds on the coefficients,
 * if needed.
 */
static isl_stat add_node(struct isl_sched_graph *graph,
        __isl_take isl_set *set, int nvar, int compressed,
        __isl_take isl_set *hull, __isl_take isl_multi_aff *compress,
        __isl_take isl_multi_aff *decompress)
{
        int nparam;
        isl_ctx *ctx;
        isl_mat *sched;
        isl_space *space;
        int *coincident;
        struct isl_sched_node *node;

        if (!set)
                return isl_stat_error;

        ctx = isl_set_get_ctx(set);
        nparam = isl_set_dim(set, isl_dim_param);
        if (!ctx->opt->schedule_parametric)
                nparam = 0;
        sched = isl_mat_alloc(ctx, 0, 1 + nparam + nvar);
        node = &graph->node[graph->n];
        graph->n++;
        space = isl_set_get_space(set);
        node->space = space;
        node->nvar = nvar;
        node->nparam = nparam;
        node->sched = sched;
        node->sched_map = NULL;
        coincident = isl_calloc_array(ctx, int, graph->max_row);
        node->coincident = coincident;
        node->compressed = compressed;
        node->hull = hull;
        node->compress = compress;
        node->decompress = decompress;
        if (compute_sizes_and_max(ctx, node, set) < 0)
                return isl_stat_error;

        if (!space || !sched || (graph->max_row && !coincident))
                return isl_stat_error;
        if (compressed && (!hull || !compress || !decompress))
                return isl_stat_error;

        return isl_stat_ok;
}

/* Construct an identifier for node "node", which will represent "set".
 * The name of the identifier is either "compressed" or
 * "compressed_<name>", with <name> the name of the space of "set".
 * The user pointer of the identifier points to "node".
 */
static __isl_give isl_id *construct_compressed_id(__isl_keep isl_set *set,
        struct isl_sched_node *node)
{
        isl_bool has_name;
        isl_ctx *ctx;
        isl_id *id;
        isl_printer *p;
        const char *name;
        char *id_name;

        has_name = isl_set_has_tuple_name(set);
        if (has_name < 0)
                return NULL;

        ctx = isl_set_get_ctx(set);
        if (!has_name)
                return isl_id_alloc(ctx, "compressed", node);

        p = isl_printer_to_str(ctx);
        name = isl_set_get_tuple_name(set);
        p = isl_printer_print_str(p, "compressed_");
        p = isl_printer_print_str(p, name);
        id_name = isl_printer_get_str(p);
        isl_printer_free(p);

        id = isl_id_alloc(ctx, id_name, node);
        free(id_name);

        return id;
}

/* Add a new node to the graph representing the given set.
 *
 * If any of the set variables is defined by an equality, then
 * we perform variable compression such that we can perform
 * the scheduling on the compressed domain.
 * In this case, an identifier is used that references the new node
 * such that each compressed space is unique and
 * such that the node can be recovered from the compressed space.
 */
static isl_stat extract_node(__isl_take isl_set *set, void *user)
{
        int nvar;
        isl_bool has_equality;
        isl_id *id;
        isl_basic_set *hull;
        isl_set *hull_set;
        isl_morph *morph;
        isl_multi_aff *compress, *decompress;
        struct isl_sched_graph *graph = user;

        hull = isl_set_affine_hull(isl_set_copy(set));
        hull = isl_basic_set_remove_divs(hull);
        nvar = isl_set_dim(set, isl_dim_set);
        has_equality = has_any_defining_equality(hull);

        if (has_equality < 0)
                goto error;
        if (!has_equality) {
                isl_basic_set_free(hull);
                return add_node(graph, set, nvar, 0, NULL, NULL, NULL);
        }

        id = construct_compressed_id(set, &graph->node[graph->n]);
        morph = isl_basic_set_variable_compression_with_id(hull,
                                                            isl_dim_set, id);
        isl_id_free(id);
        nvar = isl_morph_ran_dim(morph, isl_dim_set);
        compress = isl_morph_get_var_multi_aff(morph);
        morph = isl_morph_inverse(morph);
        decompress = isl_morph_get_var_multi_aff(morph);
        isl_morph_free(morph);

        hull_set = isl_set_from_basic_set(hull);
        return add_node(graph, set, nvar, 1, hull_set, compress, decompress);
error:
        isl_basic_set_free(hull);
        isl_set_free(set);
        return isl_stat_error;
}

struct isl_extract_edge_data {
        enum isl_edge_type type;
        struct isl_sched_graph *graph;
};

/* Merge edge2 into edge1, freeing the contents of edge2.
 * Return 0 on success and -1 on failure.
 *
 * edge1 and edge2 are assumed to have the same value for the map field.
 */
static int merge_edge(struct isl_sched_edge *edge1,
        struct isl_sched_edge *edge2)
{
        edge1->types |= edge2->types;
        isl_map_free(edge2->map);

        if (is_condition(edge2)) {
                if (!edge1->tagged_condition)
                        edge1->tagged_condition = edge2->tagged_condition;
                else
                        edge1->tagged_condition =
                                isl_union_map_union(edge1->tagged_condition,
                                                    edge2->tagged_condition);
        }

        if (is_conditional_validity(edge2)) {
                if (!edge1->tagged_validity)
                        edge1->tagged_validity = edge2->tagged_validity;
                else
                        edge1->tagged_validity =
                                isl_union_map_union(edge1->tagged_validity,
                                                    edge2->tagged_validity);
        }

        if (is_condition(edge2) && !edge1->tagged_condition)
                return -1;
        if (is_conditional_validity(edge2) && !edge1->tagged_validity)
                return -1;

        return 0;
}

/* Insert dummy tags in domain and range of "map".
 *
 * In particular, if "map" is of the form
 *
 *      A -> B
 *
 * then return
 *
 *      [A -> dummy_tag] -> [B -> dummy_tag]
 *
 * where the dummy_tags are identical and equal to any dummy tags
 * introduced by any other call to this function.
 */
static __isl_give isl_map *insert_dummy_tags(__isl_take isl_map *map)
{
        static char dummy;
        isl_ctx *ctx;
        isl_id *id;
        isl_space *space;
        isl_set *domain, *range;

        ctx = isl_map_get_ctx(map);

        id = isl_id_alloc(ctx, NULL, &dummy);
        space = isl_space_params(isl_map_get_space(map));
        space = isl_space_set_from_params(space);
        space = isl_space_set_tuple_id(space, isl_dim_set, id);
        space = isl_space_map_from_set(space);

        domain = isl_map_wrap(map);
        range = isl_map_wrap(isl_map_universe(space));
        map = isl_map_from_domain_and_range(domain, range);
        map = isl_map_zip(map);

        return map;
}

/* Given that at least one of "src" or "dst" is compressed, return
 * a map between the spaces of these nodes restricted to the affine
 * hull that was used in the compression.
 */
static __isl_give isl_map *extract_hull(struct isl_sched_node *src,
        struct isl_sched_node *dst)
{
        isl_set *dom, *ran;

        if (src->compressed)
                dom = isl_set_copy(src->hull);
        else
                dom = isl_set_universe(isl_space_copy(src->space));
        if (dst->compressed)
                ran = isl_set_copy(dst->hull);
        else
                ran = isl_set_universe(isl_space_copy(dst->space));

        return isl_map_from_domain_and_range(dom, ran);
}

/* Intersect the domains of the nested relations in domain and range
 * of "tagged" with "map".
 */
static __isl_give isl_map *map_intersect_domains(__isl_take isl_map *tagged,
        __isl_keep isl_map *map)
{
        isl_set *set;

        tagged = isl_map_zip(tagged);
        set = isl_map_wrap(isl_map_copy(map));
        tagged = isl_map_intersect_domain(tagged, set);
        tagged = isl_map_zip(tagged);
        return tagged;
}

/* Return a pointer to the node that lives in the domain space of "map"
 * or NULL if there is no such node.
 */
static struct isl_sched_node *find_domain_node(isl_ctx *ctx,
        struct isl_sched_graph *graph, __isl_keep isl_map *map)
{
        struct isl_sched_node *node;
        isl_space *space;

        space = isl_space_domain(isl_map_get_space(map));
        node = graph_find_node(ctx, graph, space);
        isl_space_free(space);

        return node;
}

/* Return a pointer to the node that lives in the range space of "map"
 * or NULL if there is no such node.
 */
static struct isl_sched_node *find_range_node(isl_ctx *ctx,
        struct isl_sched_graph *graph, __isl_keep isl_map *map)
{
        struct isl_sched_node *node;
        isl_space *space;

        space = isl_space_range(isl_map_get_space(map));
        node = graph_find_node(ctx, graph, space);
        isl_space_free(space);

        return node;
}

/* Add a new edge to the graph based on the given map
 * and add it to data->graph->edge_table[data->type].
 * If a dependence relation of a given type happens to be identical
 * to one of the dependence relations of a type that was added before,
 * then we don't create a new edge, but instead mark the original edge
 * as also representing a dependence of the current type.
 *
 * Edges of type isl_edge_condition or isl_edge_conditional_validity
 * may be specified as "tagged" dependence relations.  That is, "map"
 * may contain elements (i -> a) -> (j -> b), where i -> j denotes
 * the dependence on iterations and a and b are tags.
 * edge->map is set to the relation containing the elements i -> j,
 * while edge->tagged_condition and edge->tagged_validity contain
 * the union of all the "map" relations
 * for which extract_edge is called that result in the same edge->map.
 *
 * If the source or the destination node is compressed, then
 * intersect both "map" and "tagged" with the constraints that
 * were used to construct the compression.
 * This ensures that there are no schedule constraints defined
 * outside of these domains, while the scheduler no longer has
 * any control over those outside parts.
 */
static isl_stat extract_edge(__isl_take isl_map *map, void *user)
{
        isl_ctx *ctx = isl_map_get_ctx(map);
        struct isl_extract_edge_data *data = user;
        struct isl_sched_graph *graph = data->graph;
        struct isl_sched_node *src, *dst;
        struct isl_sched_edge *edge;
        isl_map *tagged = NULL;

        if (data->type == isl_edge_condition ||
            data->type == isl_edge_conditional_validity) {
                if (isl_map_can_zip(map)) {
                        tagged = isl_map_copy(map);
                        map = isl_set_unwrap(isl_map_domain(isl_map_zip(map)));
                } else {
                        tagged = insert_dummy_tags(isl_map_copy(map));
                }
        }

        src = find_domain_node(ctx, graph, map);
        dst = find_range_node(ctx, graph, map);

        if (!src || !dst) {
                isl_map_free(map);
                isl_map_free(tagged);
                return isl_stat_ok;
        }

        if (src->compressed || dst->compressed) {
                isl_map *hull;
                hull = extract_hull(src, dst);
                if (tagged)
                        tagged = map_intersect_domains(tagged, hull);
                map = isl_map_intersect(map, hull);
        }

        graph->edge[graph->n_edge].src = src;
        graph->edge[graph->n_edge].dst = dst;
        graph->edge[graph->n_edge].map = map;
        graph->edge[graph->n_edge].types = 0;
        graph->edge[graph->n_edge].tagged_condition = NULL;
        graph->edge[graph->n_edge].tagged_validity = NULL;
        set_type(&graph->edge[graph->n_edge], data->type);
        if (data->type == isl_edge_condition)
                graph->edge[graph->n_edge].tagged_condition =
                                        isl_union_map_from_map(tagged);
        if (data->type == isl_edge_conditional_validity)
                graph->edge[graph->n_edge].tagged_validity =
                                        isl_union_map_from_map(tagged);

        edge = graph_find_matching_edge(graph, &graph->edge[graph->n_edge]);
        if (!edge) {
                graph->n_edge++;
                return isl_stat_error;
        }
        if (edge == &graph->edge[graph->n_edge])
                return graph_edge_table_add(ctx, graph, data->type,
                                    &graph->edge[graph->n_edge++]);

        if (merge_edge(edge, &graph->edge[graph->n_edge]) < 0)
                return -1;

        return graph_edge_table_add(ctx, graph, data->type, edge);
}

/* Initialize the schedule graph "graph" from the schedule constraints "sc".
 *
 * The context is included in the domain before the nodes of
 * the graphs are extracted in order to be able to exploit
 * any possible additional equalities.
 * Note that this intersection is only performed locally here.
 */
static isl_stat graph_init(struct isl_sched_graph *graph,
        __isl_keep isl_schedule_constraints *sc)
{
        isl_ctx *ctx;
        isl_union_set *domain;
        isl_union_map *c;
        struct isl_extract_edge_data data;
        enum isl_edge_type i;
        isl_stat r;

        if (!sc)
                return isl_stat_error;

        ctx = isl_schedule_constraints_get_ctx(sc);

        domain = isl_schedule_constraints_get_domain(sc);
        graph->n = isl_union_set_n_set(domain);
        isl_union_set_free(domain);

        if (graph_alloc(ctx, graph, graph->n,
            isl_schedule_constraints_n_map(sc)) < 0)
                return isl_stat_error;

        if (compute_max_row(graph, sc) < 0)
                return isl_stat_error;
        graph->root = 1;
        graph->n = 0;
        domain = isl_schedule_constraints_get_domain(sc);
        domain = isl_union_set_intersect_params(domain,
                                    isl_schedule_constraints_get_context(sc));
        r = isl_union_set_foreach_set(domain, &extract_node, graph);
        isl_union_set_free(domain);
        if (r < 0)
                return isl_stat_error;
        if (graph_init_table(ctx, graph) < 0)
                return isl_stat_error;
        for (i = isl_edge_first; i <= isl_edge_last; ++i) {
                c = isl_schedule_constraints_get(sc, i);
                graph->max_edge[i] = isl_union_map_n_map(c);
                isl_union_map_free(c);
                if (!c)
                        return isl_stat_error;
        }
        if (graph_init_edge_tables(ctx, graph) < 0)
                return isl_stat_error;
        graph->n_edge = 0;
        data.graph = graph;
        for (i = isl_edge_first; i <= isl_edge_last; ++i) {
                isl_stat r;

                data.type = i;
                c = isl_schedule_constraints_get(sc, i);
                r = isl_union_map_foreach_map(c, &extract_edge, &data);
                isl_union_map_free(c);
                if (r < 0)
                        return isl_stat_error;
        }

        return isl_stat_ok;
}

/* Check whether there is any dependence from node[j] to node[i]
 * or from node[i] to node[j].
 */
static isl_bool node_follows_weak(int i, int j, void *user)
{
        isl_bool f;
        struct isl_sched_graph *graph = user;

        f = graph_has_any_edge(graph, &graph->node[j], &graph->node[i]);
        if (f < 0 || f)
                return f;
        return graph_has_any_edge(graph, &graph->node[i], &graph->node[j]);
}

/* Check whether there is a (conditional) validity dependence from node[j]
 * to node[i], forcing node[i] to follow node[j].
 */
static isl_bool node_follows_strong(int i, int j, void *user)
{
        struct isl_sched_graph *graph = user;

        return graph_has_validity_edge(graph, &graph->node[j], &graph->node[i]);
}

/* Use Tarjan's algorithm for computing the strongly connected components
 * in the dependence graph only considering those edges defined by "follows".
 */
static int detect_ccs(isl_ctx *ctx, struct isl_sched_graph *graph,
        isl_bool (*follows)(int i, int j, void *user))
{
        int i, n;
        struct isl_tarjan_graph *g = NULL;

        g = isl_tarjan_graph_init(ctx, graph->n, follows, graph);
        if (!g)
                return -1;

        graph->scc = 0;
        i = 0;
        n = graph->n;
        while (n) {
                while (g->order[i] != -1) {
                        graph->node[g->order[i]].scc = graph->scc;
                        --n;
                        ++i;
                }
                ++i;
                graph->scc++;
        }

        isl_tarjan_graph_free(g);

        return 0;
}

/* Apply Tarjan's algorithm to detect the strongly connected components
 * in the dependence graph.
 * Only consider the (conditional) validity dependences and clear "weak".
 */
static int detect_sccs(isl_ctx *ctx, struct isl_sched_graph *graph)
{
        graph->weak = 0;
        return detect_ccs(ctx, graph, &node_follows_strong);
}

/* Apply Tarjan's algorithm to detect the (weakly) connected components
 * in the dependence graph.
 * Consider all dependences and set "weak".
 */
static int detect_wccs(isl_ctx *ctx, struct isl_sched_graph *graph)
{
        graph->weak = 1;
        return detect_ccs(ctx, graph, &node_follows_weak);
}

static int cmp_scc(const void *a, const void *b, void *data)
{
        struct isl_sched_graph *graph = data;
        const int *i1 = a;
        const int *i2 = b;

        return graph->node[*i1].scc - graph->node[*i2].scc;
}

/* Sort the elements of graph->sorted according to the corresponding SCCs.
 */
static int sort_sccs(struct isl_sched_graph *graph)
{
        return isl_sort(graph->sorted, graph->n, sizeof(int), &cmp_scc, graph);
}

/* Given a dependence relation R from "node" to itself,
 * construct the set of coefficients of valid constraints for elements
 * in that dependence relation.
 * In particular, the result contains tuples of coefficients
 * c_0, c_n, c_x such that
 *
 *      c_0 + c_n n + c_x y - c_x x >= 0 for each (x,y) in R
 *
 * or, equivalently,
 *
 *      c_0 + c_n n + c_x d >= 0 for each d in delta R = { y - x | (x,y) in R }
 *
 * We choose here to compute the dual of delta R.
 * Alternatively, we could have computed the dual of R, resulting
 * in a set of tuples c_0, c_n, c_x, c_y, and then
 * plugged in (c_0, c_n, c_x, -c_x).
 *
 * If "node" has been compressed, then the dependence relation
 * is also compressed before the set of coefficients is computed.
 */
static __isl_give isl_basic_set *intra_coefficients(
        struct isl_sched_graph *graph, struct isl_sched_node *node,
        __isl_take isl_map *map)
{
        isl_set *delta;
        isl_map *key;
        isl_basic_set *coef;
        isl_maybe_isl_basic_set m;

        m = isl_map_to_basic_set_try_get(graph->intra_hmap, map);
        if (m.valid < 0 || m.valid) {
                isl_map_free(map);
                return m.value;
        }

        key = isl_map_copy(map);
        if (node->compressed) {
                map = isl_map_preimage_domain_multi_aff(map,
                                    isl_multi_aff_copy(node->decompress));
                map = isl_map_preimage_range_multi_aff(map,
                                    isl_multi_aff_copy(node->decompress));
        }
        delta = isl_set_remove_divs(isl_map_deltas(map));
        coef = isl_set_coefficients(delta);
        graph->intra_hmap = isl_map_to_basic_set_set(graph->intra_hmap, key,
                                        isl_basic_set_copy(coef));

        return coef;
}

/* Given a dependence relation R, construct the set of coefficients
 * of valid constraints for elements in that dependence relation.
 * In particular, the result contains tuples of coefficients
 * c_0, c_n, c_x, c_y such that
 *
 *      c_0 + c_n n + c_x x + c_y y >= 0 for each (x,y) in R
 *
 * If the source or destination nodes of "edge" have been compressed,
 * then the dependence relation is also compressed before
 * the set of coefficients is computed.
 */
static __isl_give isl_basic_set *inter_coefficients(
        struct isl_sched_graph *graph, struct isl_sched_edge *edge,
        __isl_take isl_map *map)
{
        isl_set *set;
        isl_map *key;
        isl_basic_set *coef;
        isl_maybe_isl_basic_set m;

        m = isl_map_to_basic_set_try_get(graph->inter_hmap, map);
        if (m.valid < 0 || m.valid) {
                isl_map_free(map);
                return m.value;
        }

        key = isl_map_copy(map);
        if (edge->src->compressed)
                map = isl_map_preimage_domain_multi_aff(map,
                                    isl_multi_aff_copy(edge->src->decompress));
        if (edge->dst->compressed)
                map = isl_map_preimage_range_multi_aff(map,
                                    isl_multi_aff_copy(edge->dst->decompress));
        set = isl_map_wrap(isl_map_remove_divs(map));
        coef = isl_set_coefficients(set);
        graph->inter_hmap = isl_map_to_basic_set_set(graph->inter_hmap, key,
                                        isl_basic_set_copy(coef));

        return coef;
}

/* Return the position of the coefficients of the variables in
 * the coefficients constraints "coef".
 *
 * The space of "coef" is of the form
 *
 *      { coefficients[[cst, params] -> S] }
 *
 * Return the position of S.
 */
static int coef_var_offset(__isl_keep isl_basic_set *coef)
{
        int offset;
        isl_space *space;

        space = isl_space_unwrap(isl_basic_set_get_space(coef));
        offset = isl_space_dim(space, isl_dim_in);
        isl_space_free(space);

        return offset;
}

/* Return the offset of the coefficients of the variables of "node"
 * within the (I)LP.
 *
 * Within each node, the coefficients have the following order:
 *      - c_i_0
 *      - c_i_n (if parametric)
 *      - positive and negative parts of c_i_x
 */
static int node_var_coef_offset(struct isl_sched_node *node)
{
        return node->start + 1 + node->nparam;
}

/* Construct an isl_dim_map for mapping constraints on coefficients
 * for "node" to the corresponding positions in graph->lp.
 * "offset" is the offset of the coefficients for the variables
 * in the input constraints.
 * "s" is the sign of the mapping.
 *
 * The input constraints are given in terms of the coefficients (c_0, c_n, c_x).
 * The mapping produced by this function essentially plugs in
 * (0, 0, c_i_x^+ - c_i_x^-) if s = 1 and
 * (0, 0, -c_i_x^+ + c_i_x^-) if s = -1.
 * In graph->lp, the c_i_x^- appear before their c_i_x^+ counterpart.
 *
 * The caller can extend the mapping to also map the other coefficients
 * (and therefore not plug in 0).
 */
static __isl_give isl_dim_map *intra_dim_map(isl_ctx *ctx,
        struct isl_sched_graph *graph, struct isl_sched_node *node,
        int offset, int s)
{
        int pos;
        unsigned total;
        isl_dim_map *dim_map;

        if (!node)
                return NULL;

        total = isl_basic_set_total_dim(graph->lp);
        pos = node_var_coef_offset(node);
        dim_map = isl_dim_map_alloc(ctx, total);
        isl_dim_map_range(dim_map, pos, 2, offset, 1, node->nvar, -s);
        isl_dim_map_range(dim_map, pos + 1, 2, offset, 1, node->nvar, s);

        return dim_map;
}

/* Construct an isl_dim_map for mapping constraints on coefficients
 * for "src" (node i) and "dst" (node j) to the corresponding positions
 * in graph->lp.
 * "offset" is the offset of the coefficients for the variables of "src"
 * in the input constraints.
 * "s" is the sign of the mapping.
 *
 * The input constraints are given in terms of the coefficients
 * (c_0, c_n, c_x, c_y).
 * The mapping produced by this function essentially plugs in
 * (c_j_0 - c_i_0, c_j_n - c_i_n,
 *  -(c_i_x^+ - c_i_x^-), c_j_x^+ - c_j_x^-) if s = 1 and
 * (-c_j_0 + c_i_0, -c_j_n + c_i_n,
 *  c_i_x^+ - c_i_x^-, -(c_j_x^+ - c_j_x^-)) if s = -1.
 * In graph->lp, the c_*^- appear before their c_*^+ counterpart.
 *
 * The caller can further extend the mapping.
 */
static __isl_give isl_dim_map *inter_dim_map(isl_ctx *ctx,
        struct isl_sched_graph *graph, struct isl_sched_node *src,
        struct isl_sched_node *dst, int offset, int s)
{
        int pos;
        unsigned total;
        isl_dim_map *dim_map;

        if (!src || !dst)
                return NULL;

        total = isl_basic_set_total_dim(graph->lp);
        dim_map = isl_dim_map_alloc(ctx, total);

        isl_dim_map_range(dim_map, dst->start, 0, 0, 0, 1, s);
        isl_dim_map_range(dim_map, dst->start + 1, 1, 1, 1, dst->nparam, s);
        pos = node_var_coef_offset(dst);
        isl_dim_map_range(dim_map, pos, 2, offset + src->nvar, 1,
                          dst->nvar, -s);
        isl_dim_map_range(dim_map, pos + 1, 2, offset + src->nvar, 1,
                          dst->nvar, s);

        isl_dim_map_range(dim_map, src->start, 0, 0, 0, 1, -s);
        isl_dim_map_range(dim_map, src->start + 1, 1, 1, 1, src->nparam, -s);
        pos = node_var_coef_offset(src);
        isl_dim_map_range(dim_map, pos, 2, offset, 1, src->nvar, s);
        isl_dim_map_range(dim_map, pos + 1, 2, offset, 1, src->nvar, -s);

        return dim_map;
}

/* Add the constraints from "src" to "dst" using "dim_map",
 * after making sure there is enough room in "dst" for the extra constraints.
 */
static __isl_give isl_basic_set *add_constraints_dim_map(
        __isl_take isl_basic_set *dst, __isl_take isl_basic_set *src,
        __isl_take isl_dim_map *dim_map)
{
        int n_eq, n_ineq;

        n_eq = isl_basic_set_n_equality(src);
        n_ineq = isl_basic_set_n_inequality(src);
        dst = isl_basic_set_extend_constraints(dst, n_eq, n_ineq);
        dst = isl_basic_set_add_constraints_dim_map(dst, src, dim_map);
        return dst;
}

/* Add constraints to graph->lp that force validity for the given
 * dependence from a node i to itself.
 * That is, add constraints that enforce
 *
 *      (c_i_0 + c_i_n n + c_i_x y) - (c_i_0 + c_i_n n + c_i_x x)
 *      = c_i_x (y - x) >= 0
 *
 * for each (x,y) in R.
 * We obtain general constraints on coefficients (c_0, c_n, c_x)
 * of valid constraints for (y - x) and then plug in (0, 0, c_i_x^+ - c_i_x^-),
 * where c_i_x = c_i_x^+ - c_i_x^-, with c_i_x^+ and c_i_x^- non-negative.
 * In graph->lp, the c_i_x^- appear before their c_i_x^+ counterpart.
 *
 * Actually, we do not construct constraints for the c_i_x themselves,
 * but for the coefficients of c_i_x written as a linear combination
 * of the columns in node->cmap.
 */
static isl_stat add_intra_validity_constraints(struct isl_sched_graph *graph,
        struct isl_sched_edge *edge)
{
        int offset;
        isl_map *map = isl_map_copy(edge->map);
        isl_ctx *ctx = isl_map_get_ctx(map);
        isl_dim_map *dim_map;
        isl_basic_set *coef;
        struct isl_sched_node *node = edge->src;

        coef = intra_coefficients(graph, node, map);

        offset = coef_var_offset(coef);

        coef = isl_basic_set_transform_dims(coef, isl_dim_set,
                                            offset, isl_mat_copy(node->cmap));
        if (!coef)
                return isl_stat_error;

        dim_map = intra_dim_map(ctx, graph, node, offset, 1);
        graph->lp = add_constraints_dim_map(graph->lp, coef, dim_map);

        return isl_stat_ok;
}

/* Add constraints to graph->lp that force validity for the given
 * dependence from node i to node j.
 * That is, add constraints that enforce
 *
 *      (c_j_0 + c_j_n n + c_j_x y) - (c_i_0 + c_i_n n + c_i_x x) >= 0
 *
 * for each (x,y) in R.
 * We obtain general constraints on coefficients (c_0, c_n, c_x, c_y)
 * of valid constraints for R and then plug in
 * (c_j_0 - c_i_0, c_j_n - c_i_n, -(c_i_x^+ - c_i_x^-), c_j_x^+ - c_j_x^-),
 * where c_* = c_*^+ - c_*^-, with c_*^+ and c_*^- non-negative.
 * In graph->lp, the c_*^- appear before their c_*^+ counterpart.
 *
 * Actually, we do not construct constraints for the c_*_x themselves,
 * but for the coefficients of c_*_x written as a linear combination
 * of the columns in node->cmap.
 */
static isl_stat add_inter_validity_constraints(struct isl_sched_graph *graph,
        struct isl_sched_edge *edge)
{
        int offset;
        isl_map *map;
        isl_ctx *ctx;
        isl_dim_map *dim_map;
        isl_basic_set *coef;
        struct isl_sched_node *src = edge->src;
        struct isl_sched_node *dst = edge->dst;

        if (!graph->lp)
                return isl_stat_error;

        map = isl_map_copy(edge->map);
        ctx = isl_map_get_ctx(map);
        coef = inter_coefficients(graph, edge, map);

        offset = coef_var_offset(coef);

        coef = isl_basic_set_transform_dims(coef, isl_dim_set,
                                offset, isl_mat_copy(src->cmap));
        coef = isl_basic_set_transform_dims(coef, isl_dim_set,
                                offset + src->nvar, isl_mat_copy(dst->cmap));
        if (!coef)
                return isl_stat_error;

        dim_map = inter_dim_map(ctx, graph, src, dst, offset, 1);

        edge->start = graph->lp->n_ineq;
        graph->lp = add_constraints_dim_map(graph->lp, coef, dim_map);
        if (!graph->lp)
                return isl_stat_error;
        edge->end = graph->lp->n_ineq;

        return isl_stat_ok;
}

/* Add constraints to graph->lp that bound the dependence distance for the given
 * dependence from a node i to itself.
 * If s = 1, we add the constraint
 *
 *      c_i_x (y - x) <= m_0 + m_n n
 *
 * or
 *
 *      -c_i_x (y - x) + m_0 + m_n n >= 0
 *
 * for each (x,y) in R.
 * If s = -1, we add the constraint
 *
 *      -c_i_x (y - x) <= m_0 + m_n n
 *
 * or
 *
 *      c_i_x (y - x) + m_0 + m_n n >= 0
 *
 * for each (x,y) in R.
 * We obtain general constraints on coefficients (c_0, c_n, c_x)
 * of valid constraints for (y - x) and then plug in (m_0, m_n, -s * c_i_x),
 * with each coefficient (except m_0) represented as a pair of non-negative
 * coefficients.
 *
 * Actually, we do not construct constraints for the c_i_x themselves,
 * but for the coefficients of c_i_x written as a linear combination
 * of the columns in node->cmap.
 *
 *
 * If "local" is set, then we add constraints
 *
 *      c_i_x (y - x) <= 0
 *
 * or
 *
 *      -c_i_x (y - x) <= 0
 *
 * instead, forcing the dependence distance to be (less than or) equal to 0.
 * That is, we plug in (0, 0, -s * c_i_x),
 * Note that dependences marked local are treated as validity constraints
 * by add_all_validity_constraints and therefore also have
 * their distances bounded by 0 from below.
 */
static isl_stat add_intra_proximity_constraints(struct isl_sched_graph *graph,
        struct isl_sched_edge *edge, int s, int local)
{
        int offset;
        unsigned nparam;
        isl_map *map = isl_map_copy(edge->map);
        isl_ctx *ctx = isl_map_get_ctx(map);
        isl_dim_map *dim_map;
        isl_basic_set *coef;
        struct isl_sched_node *node = edge->src;

        coef = intra_coefficients(graph, node, map);

        offset = coef_var_offset(coef);

        coef = isl_basic_set_transform_dims(coef, isl_dim_set,
                                            offset, isl_mat_copy(node->cmap));
        if (!coef)
                return isl_stat_error;

        nparam = isl_space_dim(node->space, isl_dim_param);
        dim_map = intra_dim_map(ctx, graph, node, offset, -s);

        if (!local) {
                isl_dim_map_range(dim_map, 1, 0, 0, 0, 1, 1);
                isl_dim_map_range(dim_map, 4, 2, 1, 1, nparam, -1);
                isl_dim_map_range(dim_map, 5, 2, 1, 1, nparam, 1);
        }
        graph->lp = add_constraints_dim_map(graph->lp, coef, dim_map);

        return isl_stat_ok;
}

/* Add constraints to graph->lp that bound the dependence distance for the given
 * dependence from node i to node j.
 * If s = 1, we add the constraint
 *
 *      (c_j_0 + c_j_n n + c_j_x y) - (c_i_0 + c_i_n n + c_i_x x)
 *              <= m_0 + m_n n
 *
 * or
 *
 *      -(c_j_0 + c_j_n n + c_j_x y) + (c_i_0 + c_i_n n + c_i_x x) +
 *              m_0 + m_n n >= 0
 *
 * for each (x,y) in R.
 * If s = -1, we add the constraint
 *
 *      -((c_j_0 + c_j_n n + c_j_x y) - (c_i_0 + c_i_n n + c_i_x x))
 *              <= m_0 + m_n n
 *
 * or
 *
 *      (c_j_0 + c_j_n n + c_j_x y) - (c_i_0 + c_i_n n + c_i_x x) +
 *              m_0 + m_n n >= 0
 *
 * for each (x,y) in R.
 * We obtain general constraints on coefficients (c_0, c_n, c_x, c_y)
 * of valid constraints for R and then plug in
 * (m_0 - s*c_j_0 + s*c_i_0, m_n - s*c_j_n + s*c_i_n,
 *  s*c_i_x, -s*c_j_x)
 * with each coefficient (except m_0, c_*_0 and c_*_n)
 * represented as a pair of non-negative coefficients.
 *
 * Actually, we do not construct constraints for the c_*_x themselves,
 * but for the coefficients of c_*_x written as a linear combination
 * of the columns in node->cmap.
 *
 *
 * If "local" is set (and s = 1), then we add constraints
 *
 *      (c_j_0 + c_j_n n + c_j_x y) - (c_i_0 + c_i_n n + c_i_x x) <= 0
 *
 * or
 *
 *      -((c_j_0 + c_j_n n + c_j_x y) + (c_i_0 + c_i_n n + c_i_x x)) >= 0
 *
 * instead, forcing the dependence distance to be (less than or) equal to 0.
 * That is, we plug in
 * (-s*c_j_0 + s*c_i_0, -s*c_j_n + s*c_i_n, s*c_i_x, -s*c_j_x).
 * Note that dependences marked local are treated as validity constraints
 * by add_all_validity_constraints and therefore also have
 * their distances bounded by 0 from below.
 */
static isl_stat add_inter_proximity_constraints(struct isl_sched_graph *graph,
        struct isl_sched_edge *edge, int s, int local)
{
        int offset;
        unsigned nparam;
        isl_map *map = isl_map_copy(edge->map);
        isl_ctx *ctx = isl_map_get_ctx(map);
        isl_dim_map *dim_map;
        isl_basic_set *coef;
        struct isl_sched_node *src = edge->src;
        struct isl_sched_node *dst = edge->dst;

        coef = inter_coefficients(graph, edge, map);

        offset = coef_var_offset(coef);

        coef = isl_basic_set_transform_dims(coef, isl_dim_set,
                                offset, isl_mat_copy(src->cmap));
        coef = isl_basic_set_transform_dims(coef, isl_dim_set,
                                offset + src->nvar, isl_mat_copy(dst->cmap));
        if (!coef)
                return isl_stat_error;

        nparam = isl_space_dim(src->space, isl_dim_param);
        dim_map = inter_dim_map(ctx, graph, src, dst, offset, -s);

        if (!local) {
                isl_dim_map_range(dim_map, 1, 0, 0, 0, 1, 1);
                isl_dim_map_range(dim_map, 4, 2, 1, 1, nparam, -1);
                isl_dim_map_range(dim_map, 5, 2, 1, 1, nparam, 1);
        }

        graph->lp = add_constraints_dim_map(graph->lp, coef, dim_map);

        return isl_stat_ok;
}

/* Add all validity constraints to graph->lp.
 *
 * An edge that is forced to be local needs to have its dependence
 * distances equal to zero.  We take care of bounding them by 0 from below
 * here.  add_all_proximity_constraints takes care of bounding them by 0
 * from above.
 *
 * If "use_coincidence" is set, then we treat coincidence edges as local edges.
 * Otherwise, we ignore them.
 */
static int add_all_validity_constraints(struct isl_sched_graph *graph,
        int use_coincidence)
{
        int i;

        for (i = 0; i < graph->n_edge; ++i) {
                struct isl_sched_edge *edge = &graph->edge[i];
                int local;

                local = is_local(edge) ||
                        (is_coincidence(edge) && use_coincidence);
                if (!is_validity(edge) && !local)
                        continue;
                if (edge->src != edge->dst)
                        continue;
                if (add_intra_validity_constraints(graph, edge) < 0)
                        return -1;
        }

        for (i = 0; i < graph->n_edge; ++i) {
                struct isl_sched_edge *edge = &graph->edge[i];
                int local;

                local = is_local(edge) ||
                        (is_coincidence(edge) && use_coincidence);
                if (!is_validity(edge) && !local)
                        continue;
                if (edge->src == edge->dst)
                        continue;
                if (add_inter_validity_constraints(graph, edge) < 0)
                        return -1;
        }

        return 0;
}

/* Add constraints to graph->lp that bound the dependence distance
 * for all dependence relations.
 * If a given proximity dependence is identical to a validity
 * dependence, then the dependence distance is already bounded
 * from below (by zero), so we only need to bound the distance
 * from above.  (This includes the case of "local" dependences
 * which are treated as validity dependence by add_all_validity_constraints.)
 * Otherwise, we need to bound the distance both from above and from below.
 *
 * If "use_coincidence" is set, then we treat coincidence edges as local edges.
 * Otherwise, we ignore them.
 */
static int add_all_proximity_constraints(struct isl_sched_graph *graph,
        int use_coincidence)
{
        int i;

        for (i = 0; i < graph->n_edge; ++i) {
                struct isl_sched_edge *edge = &graph->edge[i];
                int local;

                local = is_local(edge) ||
                        (is_coincidence(edge) && use_coincidence);
                if (!is_proximity(edge) && !local)
                        continue;
                if (edge->src == edge->dst &&
                    add_intra_proximity_constraints(graph, edge, 1, local) < 0)
                        return -1;
                if (edge->src != edge->dst &&
                    add_inter_proximity_constraints(graph, edge, 1, local) < 0)
                        return -1;
                if (is_validity(edge) || local)
                        continue;
                if (edge->src == edge->dst &&
                    add_intra_proximity_constraints(graph, edge, -1, 0) < 0)
                        return -1;
                if (edge->src != edge->dst &&
                    add_inter_proximity_constraints(graph, edge, -1, 0) < 0)
                        return -1;
        }

        return 0;
}

/* Compute a basis for the rows in the linear part of the schedule
 * and extend this basis to a full basis.  The remaining rows
 * can then be used to force linear independence from the rows
 * in the schedule.
 *
 * In particular, given the schedule rows S, we compute
 *
 *      S   = H Q
 *      S U = H
 *
 * with H the Hermite normal form of S.  That is, all but the
 * first rank columns of H are zero and so each row in S is
 * a linear combination of the first rank rows of Q.
 * The matrix Q is then transposed because we will write the
 * coefficients of the next schedule row as a column vector s
 * and express this s as a linear combination s = Q c of the
 * computed basis.
 * Similarly, the matrix U is transposed such that we can
 * compute the coefficients c = U s from a schedule row s.
 */
static int node_update_cmap(struct isl_sched_node *node)
{
        isl_mat *H, *U, *Q;
        int n_row = isl_mat_rows(node->sched);

        H = isl_mat_sub_alloc(node->sched, 0, n_row,
                              1 + node->nparam, node->nvar);

        H = isl_mat_left_hermite(H, 0, &U, &Q);
        isl_mat_free(node->cmap);
        isl_mat_free(node->cinv);
        isl_mat_free(node->ctrans);
        node->ctrans = isl_mat_copy(Q);
        node->cmap = isl_mat_transpose(Q);
        node->cinv = isl_mat_transpose(U);
        node->rank = isl_mat_initial_non_zero_cols(H);
        isl_mat_free(H);

        if (!node->cmap || !node->cinv || !node->ctrans || node->rank < 0)
                return -1;
        return 0;
}

/* Is "edge" marked as a validity or a conditional validity edge?
 */
static int is_any_validity(struct isl_sched_edge *edge)
{
        return is_validity(edge) || is_conditional_validity(edge);
}

/* How many times should we count the constraints in "edge"?
 *
 * We count as follows
 * validity             -> 1 (>= 0)
 * validity+proximity   -> 2 (>= 0 and upper bound)
 * proximity            -> 2 (lower and upper bound)
 * local(+any)          -> 2 (>= 0 and <= 0)
 *
 * If an edge is only marked conditional_validity then it counts
 * as zero since it is only checked afterwards.
 *
 * If "use_coincidence" is set, then we treat coincidence edges as local edges.
 * Otherwise, we ignore them.
 */
static int edge_multiplicity(struct isl_sched_edge *edge, int use_coincidence)
{
        if (is_proximity(edge) || is_local(edge))
                return 2;
        if (use_coincidence && is_coincidence(edge))
                return 2;
        if (is_validity(edge))
                return 1;
        return 0;
}

/* Count the number of equality and inequality constraints
 * that will be added for the given map.
 *
 * "use_coincidence" is set if we should take into account coincidence edges.
 */
static isl_stat count_map_constraints(struct isl_sched_graph *graph,
        struct isl_sched_edge *edge, __isl_take isl_map *map,
        int *n_eq, int *n_ineq, int use_coincidence)
{
        isl_basic_set *coef;
        int f = edge_multiplicity(edge, use_coincidence);

        if (f == 0) {
                isl_map_free(map);
                return isl_stat_ok;
        }

        if (edge->src == edge->dst)
                coef = intra_coefficients(graph, edge->src, map);
        else
                coef = inter_coefficients(graph, edge, map);
        if (!coef)
                return isl_stat_error;
        *n_eq += f * isl_basic_set_n_equality(coef);
        *n_ineq += f * isl_basic_set_n_inequality(coef);
        isl_basic_set_free(coef);

        return isl_stat_ok;
}

/* Count the number of equality and inequality constraints
 * that will be added to the main lp problem.
 * We count as follows
 * validity             -> 1 (>= 0)
 * validity+proximity   -> 2 (>= 0 and upper bound)
 * proximity            -> 2 (lower and upper bound)
 * local(+any)          -> 2 (>= 0 and <= 0)
 *
 * If "use_coincidence" is set, then we treat coincidence edges as local edges.
 * Otherwise, we ignore them.
 */
static int count_constraints(struct isl_sched_graph *graph,
        int *n_eq, int *n_ineq, int use_coincidence)
{
        int i;

        *n_eq = *n_ineq = 0;
        for (i = 0; i < graph->n_edge; ++i) {
                struct isl_sched_edge *edge = &graph->edge[i];
                isl_map *map = isl_map_copy(edge->map);

                if (count_map_constraints(graph, edge, map, n_eq, n_ineq,
                                            use_coincidence) < 0)
                        return -1;
        }

        return 0;
}

/* Count the number of constraints that will be added by
 * add_bound_constant_constraints to bound the values of the constant terms
 * and increment *n_eq and *n_ineq accordingly.
 *
 * In practice, add_bound_constant_constraints only adds inequalities.
 */
static isl_stat count_bound_constant_constraints(isl_ctx *ctx,
        struct isl_sched_graph *graph, int *n_eq, int *n_ineq)
{
        if (isl_options_get_schedule_max_constant_term(ctx) == -1)
                return isl_stat_ok;

        *n_ineq += graph->n;

        return isl_stat_ok;
}

/* Add constraints to bound the values of the constant terms in the schedule,
 * if requested by the user.
 *
 * The maximal value of the constant terms is defined by the option
 * "schedule_max_constant_term".
 *
 * Within each node, the coefficients have the following order:
 *      - c_i_0
 *      - c_i_n (if parametric)
 *      - positive and negative parts of c_i_x
 */
static isl_stat add_bound_constant_constraints(isl_ctx *ctx,
        struct isl_sched_graph *graph)
{
        int i, k;
        int max;
        int total;

        max = isl_options_get_schedule_max_constant_term(ctx);
        if (max == -1)
                return isl_stat_ok;

        total = isl_basic_set_dim(graph->lp, isl_dim_set);

        for (i = 0; i < graph->n; ++i) {
                struct isl_sched_node *node = &graph->node[i];
                k = isl_basic_set_alloc_inequality(graph->lp);
                if (k < 0)
                        return isl_stat_error;
                isl_seq_clr(graph->lp->ineq[k], 1 + total);
                isl_int_set_si(graph->lp->ineq[k][1 + node->start], -1);
                isl_int_set_si(graph->lp->ineq[k][0], max);
        }

        return isl_stat_ok;
}

/* Count the number of constraints that will be added by
 * add_bound_coefficient_constraints and increment *n_eq and *n_ineq
 * accordingly.
 *
 * In practice, add_bound_coefficient_constraints only adds inequalities.
 */
static int count_bound_coefficient_constraints(isl_ctx *ctx,
        struct isl_sched_graph *graph, int *n_eq, int *n_ineq)
{
        int i;

        if (isl_options_get_schedule_max_coefficient(ctx) == -1 &&
            !isl_options_get_schedule_treat_coalescing(ctx))
                return 0;

        for (i = 0; i < graph->n; ++i)
                *n_ineq += graph->node[i].nparam + 2 * graph->node[i].nvar;

        return 0;
}

/* Add constraints to graph->lp that bound the values of
 * the parameter schedule coefficients of "node" to "max" and
 * the variable schedule coefficients to the corresponding entry
 * in node->max.
 * In either case, a negative value means that no bound needs to be imposed.
 *
 * For parameter coefficients, this amounts to adding a constraint
 *
 *      c_n <= max
 *
 * i.e.,
 *
 *      -c_n + max >= 0
 *
 * The variables coefficients are, however, not represented directly.
 * Instead, the variables coefficients c_x are written as a linear
 * combination c_x = cmap c_z of some other coefficients c_z,
 * which are in turn encoded as c_z = c_z^+ - c_z^-.
 * Let a_j be the elements of row i of node->cmap, then
 *
 *      -max_i <= c_x_i <= max_i
 *
 * is encoded as
 *
 *      -max_i <= \sum_j a_j (c_z_j^+ - c_z_j^-) <= max_i
 *
 * or
 *
 *      -\sum_j a_j (c_z_j^+ - c_z_j^-) + max_i >= 0
 *      \sum_j a_j (c_z_j^+ - c_z_j^-) + max_i >= 0
 */
static isl_stat node_add_coefficient_constraints(isl_ctx *ctx,
        struct isl_sched_graph *graph, struct isl_sched_node *node, int max)
{
        int i, j, k;
        int total;
        isl_vec *ineq;

        total = isl_basic_set_dim(graph->lp, isl_dim_set);

        for (j = 0; j < node->nparam; ++j) {
                int dim;

                if (max < 0)
                        continue;

                k = isl_basic_set_alloc_inequality(graph->lp);
                if (k < 0)
                        return isl_stat_error;
                dim = 1 + node->start + 1 + j;
                isl_seq_clr(graph->lp->ineq[k], 1 + total);
                isl_int_set_si(graph->lp->ineq[k][dim], -1);
                isl_int_set_si(graph->lp->ineq[k][0], max);
        }

        ineq = isl_vec_alloc(ctx, 1 + total);
        ineq = isl_vec_clr(ineq);
        if (!ineq)
                return isl_stat_error;
        for (i = 0; i < node->nvar; ++i) {
                int pos = 1 + node_var_coef_offset(node);

                if (isl_int_is_neg(node->max->el[i]))
                        continue;

                for (j = 0; j < node->nvar; ++j) {
                        isl_int_set(ineq->el[pos + 2 * j],
                                        node->cmap->row[i][j]);
                        isl_int_neg(ineq->el[pos + 2 * j + 1],
                                        node->cmap->row[i][j]);
                }
                isl_int_set(ineq->el[0], node->max->el[i]);

                k = isl_basic_set_alloc_inequality(graph->lp);
                if (k < 0)
                        goto error;
                isl_seq_cpy(graph->lp->ineq[k], ineq->el, 1 + total);

                isl_seq_neg(ineq->el + pos, ineq->el + pos, 2 * node->nvar);
                k = isl_basic_set_alloc_inequality(graph->lp);
                if (k < 0)
                        goto error;
                isl_seq_cpy(graph->lp->ineq[k], ineq->el, 1 + total);
        }
        isl_vec_free(ineq);

        return isl_stat_ok;
error:
        isl_vec_free(ineq);
        return isl_stat_error;
}

/* Add constraints that bound the values of the variable and parameter
 * coefficients of the schedule.
 *
 * The maximal value of the coefficients is defined by the option
 * 'schedule_max_coefficient' and the entries in node->max.
 * These latter entries are only set if either the schedule_max_coefficient
 * option or the schedule_treat_coalescing option is set.
 */
static isl_stat add_bound_coefficient_constraints(isl_ctx *ctx,
        struct isl_sched_graph *graph)
{
        int i;
        int max;

        max = isl_options_get_schedule_max_coefficient(ctx);

        if (max == -1 && !isl_options_get_schedule_treat_coalescing(ctx))
                return isl_stat_ok;

        for (i = 0; i < graph->n; ++i) {
                struct isl_sched_node *node = &graph->node[i];

                if (node_add_coefficient_constraints(ctx, graph, node, max) < 0)
                        return isl_stat_error;
        }

        return isl_stat_ok;
}

/* Add a constraint to graph->lp that equates the value at position
 * "sum_pos" to the sum of the "n" values starting at "first".
 */
static isl_stat add_sum_constraint(struct isl_sched_graph *graph,
        int sum_pos, int first, int n)
{
        int i, k;
        int total;

        total = isl_basic_set_dim(graph->lp, isl_dim_set);

        k = isl_basic_set_alloc_equality(graph->lp);
        if (k < 0)
                return isl_stat_error;
        isl_seq_clr(graph->lp->eq[k], 1 + total);
        isl_int_set_si(graph->lp->eq[k][1 + sum_pos], -1);
        for (i = 0; i < n; ++i)
                isl_int_set_si(graph->lp->eq[k][1 + first + i], 1);

        return isl_stat_ok;
}

/* Add a constraint to graph->lp that equates the value at position
 * "sum_pos" to the sum of the parameter coefficients of all nodes.
 *
 * Within each node, the coefficients have the following order:
 *      - c_i_0
 *      - c_i_n (if parametric)
 *      - positive and negative parts of c_i_x
 */
static isl_stat add_param_sum_constraint(struct isl_sched_graph *graph,
        int sum_pos)
{
        int i, j, k;
        int total;

        total = isl_basic_set_dim(graph->lp, isl_dim_set);

        k = isl_basic_set_alloc_equality(graph->lp);
        if (k < 0)
                return isl_stat_error;
        isl_seq_clr(graph->lp->eq[k], 1 + total);
        isl_int_set_si(graph->lp->eq[k][1 + sum_pos], -1);
        for (i = 0; i < graph->n; ++i) {
                int pos = 1 + graph->node[i].start + 1;

                for (j = 0; j < graph->node[i].nparam; ++j)
                        isl_int_set_si(graph->lp->eq[k][pos + j], 1);
        }

        return isl_stat_ok;
}

/* Add a constraint to graph->lp that equates the value at position
 * "sum_pos" to the sum of the variable coefficients of all nodes.
 *
 * Within each node, the coefficients have the following order:
 *      - c_i_0
 *      - c_i_n (if parametric)
 *      - positive and negative parts of c_i_x
 */
static isl_stat add_var_sum_constraint(struct isl_sched_graph *graph,
        int sum_pos)
{
        int i, j, k;
        int total;

        total = isl_basic_set_dim(graph->lp, isl_dim_set);

        k = isl_basic_set_alloc_equality(graph->lp);
        if (k < 0)
                return isl_stat_error;
        isl_seq_clr(graph->lp->eq[k], 1 + total);
        isl_int_set_si(graph->lp->eq[k][1 + sum_pos], -1);
        for (i = 0; i < graph->n; ++i) {
                struct isl_sched_node *node = &graph->node[i];
                int pos = 1 + node_var_coef_offset(node);

                for (j = 0; j < 2 * node->nvar; ++j)
                        isl_int_set_si(graph->lp->eq[k][pos + j], 1);
        }

        return isl_stat_ok;
}

/* Construct an ILP problem for finding schedule coefficients
 * that result in non-negative, but small dependence distances
 * over all dependences.
 * In particular, the dependence distances over proximity edges
 * are bounded by m_0 + m_n n and we compute schedule coefficients
 * with small values (preferably zero) of m_n and m_0.
 *
 * All variables of the ILP are non-negative.  The actual coefficients
 * may be negative, so each coefficient is represented as the difference
 * of two non-negative variables.  The negative part always appears
 * immediately before the positive part.
 * Other than that, the variables have the following order
 *
 *      - sum of positive and negative parts of m_n coefficients
 *      - m_0
 *      - sum of all c_n coefficients
 *              (unconstrained when computing non-parametric schedules)
 *      - sum of positive and negative parts of all c_x coefficients
 *      - positive and negative parts of m_n coefficients
 *      - for each node
 *              - c_i_0
 *              - c_i_n (if parametric)
 *              - positive and negative parts of c_i_x
 *
 * The c_i_x are not represented directly, but through the columns of
 * node->cmap.  That is, the computed values are for variable t_i_x
 * such that c_i_x = Q t_i_x with Q equal to node->cmap.
 *
 * The constraints are those from the edges plus two or three equalities
 * to express the sums.
 *
 * If "use_coincidence" is set, then we treat coincidence edges as local edges.
 * Otherwise, we ignore them.
 */
static isl_stat setup_lp(isl_ctx *ctx, struct isl_sched_graph *graph,
        int use_coincidence)
{
        int i;
        unsigned nparam;
        unsigned total;
        isl_space *space;
        int parametric;
        int param_pos;
        int n_eq, n_ineq;

        parametric = ctx->opt->schedule_parametric;
        nparam = isl_space_dim(graph->node[0].space, isl_dim_param);
        param_pos = 4;
        total = param_pos + 2 * nparam;
        for (i = 0; i < graph->n; ++i) {
                struct isl_sched_node *node = &graph->node[graph->sorted[i]];
                if (node_update_cmap(node) < 0)
                        return isl_stat_error;
                node->start = total;
                total += 1 + node->nparam + 2 * node->nvar;
        }

        if (count_constraints(graph, &n_eq, &n_ineq, use_coincidence) < 0)
                return isl_stat_error;
        if (count_bound_constant_constraints(ctx, graph, &n_eq, &n_ineq) < 0)
                return isl_stat_error;
        if (count_bound_coefficient_constraints(ctx, graph, &n_eq, &n_ineq) < 0)
                return isl_stat_error;

        space = isl_space_set_alloc(ctx, 0, total);
        isl_basic_set_free(graph->lp);
        n_eq += 2 + parametric;

        graph->lp = isl_basic_set_alloc_space(space, 0, n_eq, n_ineq);

        if (add_sum_constraint(graph, 0, param_pos, 2 * nparam) < 0)
                return isl_stat_error;
        if (parametric && add_param_sum_constraint(graph, 2) < 0)
                return isl_stat_error;
        if (add_var_sum_constraint(graph, 3) < 0)
                return isl_stat_error;
        if (add_bound_constant_constraints(ctx, graph) < 0)
                return isl_stat_error;
        if (add_bound_coefficient_constraints(ctx, graph) < 0)
                return isl_stat_error;
        if (add_all_validity_constraints(graph, use_coincidence) < 0)
                return isl_stat_error;
        if (add_all_proximity_constraints(graph, use_coincidence) < 0)
                return isl_stat_error;

        return isl_stat_ok;
}

/* Analyze the conflicting constraint found by
 * isl_tab_basic_set_non_trivial_lexmin.  If it corresponds to the validity
 * constraint of one of the edges between distinct nodes, living, moreover
 * in distinct SCCs, then record the source and sink SCC as this may
 * be a good place to cut between SCCs.
 */
static int check_conflict(int con, void *user)
{
        int i;
        struct isl_sched_graph *graph = user;

        if (graph->src_scc >= 0)
                return 0;

        con -= graph->lp->n_eq;

        if (con >= graph->lp->n_ineq)
                return 0;

        for (i = 0; i < graph->n_edge; ++i) {
                if (!is_validity(&graph->edge[i]))
                        continue;
                if (graph->edge[i].src == graph->edge[i].dst)
                        continue;
                if (graph->edge[i].src->scc == graph->edge[i].dst->scc)
                        continue;
                if (graph->edge[i].start > con)
                        continue;
                if (graph->edge[i].end <= con)
                        continue;
                graph->src_scc = graph->edge[i].src->scc;
                graph->dst_scc = graph->edge[i].dst->scc;
        }

        return 0;
}

/* Check whether the next schedule row of the given node needs to be
 * non-trivial.  Lower-dimensional domains may have some trivial rows,
 * but as soon as the number of remaining required non-trivial rows
 * is as large as the number or remaining rows to be computed,
 * all remaining rows need to be non-trivial.
 */
static int needs_row(struct isl_sched_graph *graph, struct isl_sched_node *node)
{
        return node->nvar - node->rank >= graph->maxvar - graph->n_row;
}

/* Solve the ILP problem constructed in setup_lp.
 * For each node such that all the remaining rows of its schedule
 * need to be non-trivial, we construct a non-triviality region.
 * This region imposes that the next row is independent of previous rows.
 * In particular the coefficients c_i_x are represented by t_i_x
 * variables with c_i_x = Q t_i_x and Q a unimodular matrix such that
 * its first columns span the rows of the previously computed part
 * of the schedule.  The non-triviality region enforces that at least
 * one of the remaining components of t_i_x is non-zero, i.e.,
 * that the new schedule row depends on at least one of the remaining
 * columns of Q.
 */
static __isl_give isl_vec *solve_lp(struct isl_sched_graph *graph)
{
        int i;
        isl_vec *sol;
        isl_basic_set *lp;

        for (i = 0; i < graph->n; ++i) {
                struct isl_sched_node *node = &graph->node[i];
                int skip = node->rank;
                graph->region[i].pos = node_var_coef_offset(node) + 2 * skip;
                if (needs_row(graph, node))
                        graph->region[i].len = 2 * (node->nvar - skip);
                else
                        graph->region[i].len = 0;
        }
        lp = isl_basic_set_copy(graph->lp);
        sol = isl_tab_basic_set_non_trivial_lexmin(lp, 2, graph->n,
                                       graph->region, &check_conflict, graph);
        return sol;
}

/* Extract the coefficients for the variables of "node" from "sol".
 *
 * Within each node, the coefficients have the following order:
 *      - c_i_0
 *      - c_i_n (if parametric)
 *      - positive and negative parts of c_i_x
 *
 * The c_i_x^- appear before their c_i_x^+ counterpart.
 *
 * Return c_i_x = c_i_x^+ - c_i_x^-
 */
static __isl_give isl_vec *extract_var_coef(struct isl_sched_node *node,
        __isl_keep isl_vec *sol)
{
        int i;
        int pos;
        isl_vec *csol;

        if (!sol)
                return NULL;
        csol = isl_vec_alloc(isl_vec_get_ctx(sol), node->nvar);
        if (!csol)
                return NULL;

        pos = 1 + node_var_coef_offset(node);
        for (i = 0; i < node->nvar; ++i)
                isl_int_sub(csol->el[i],
                            sol->el[pos + 2 * i + 1], sol->el[pos + 2 * i]);

        return csol;
}

/* Update the schedules of all nodes based on the given solution
 * of the LP problem.
 * The new row is added to the current band.
 * All possibly negative coefficients are encoded as a difference
 * of two non-negative variables, so we need to perform the subtraction
 * here.  Moreover, if use_cmap is set, then the solution does
 * not refer to the actual coefficients c_i_x, but instead to variables
 * t_i_x such that c_i_x = Q t_i_x and Q is equal to node->cmap.
 * In this case, we then also need to perform this multiplication
 * to obtain the values of c_i_x.
 *
 * If coincident is set, then the caller guarantees that the new
 * row satisfies the coincidence constraints.
 */
static int update_schedule(struct isl_sched_graph *graph,
        __isl_take isl_vec *sol, int use_cmap, int coincident)
{
        int i, j;
        isl_vec *csol = NULL;

        if (!sol)
                goto error;
        if (sol->size == 0)
                isl_die(sol->ctx, isl_error_internal,
                        "no solution found", goto error);
        if (graph->n_total_row >= graph->max_row)
                isl_die(sol->ctx, isl_error_internal,
                        "too many schedule rows", goto error);

        for (i = 0; i < graph->n; ++i) {
                struct isl_sched_node *node = &graph->node[i];
                int pos = node->start;
                int row = isl_mat_rows(node->sched);

                isl_vec_free(csol);
                csol = extract_var_coef(node, sol);
                if (!csol)
                        goto error;

                isl_map_free(node->sched_map);
                node->sched_map = NULL;
                node->sched = isl_mat_add_rows(node->sched, 1);
                if (!node->sched)
                        goto error;
                for (j = 0; j < 1 + node->nparam; ++j)
                        node->sched = isl_mat_set_element(node->sched,
                                                row, j, sol->el[1 + pos + j]);
                if (use_cmap)
                        csol = isl_mat_vec_product(isl_mat_copy(node->cmap),
                                                   csol);
                if (!csol)
                        goto error;
                for (j = 0; j < node->nvar; ++j)
                        node->sched = isl_mat_set_element(node->sched,
                                        row, 1 + node->nparam + j, csol->el[j]);
                node->coincident[graph->n_total_row] = coincident;
        }
        isl_vec_free(sol);
        isl_vec_free(csol);

        graph->n_row++;
        graph->n_total_row++;

        return 0;
error:
        isl_vec_free(sol);
        isl_vec_free(csol);
        return -1;
}

/* Convert row "row" of node->sched into an isl_aff living in "ls"
 * and return this isl_aff.
 */
static __isl_give isl_aff *extract_schedule_row(__isl_take isl_local_space *ls,
        struct isl_sched_node *node, int row)
{
        int j;
        isl_int v;
        isl_aff *aff;

        isl_int_init(v);

        aff = isl_aff_zero_on_domain(ls);
        isl_mat_get_element(node->sched, row, 0, &v);
        aff = isl_aff_set_constant(aff, v);
        for (j = 0; j < node->nparam; ++j) {
                isl_mat_get_element(node->sched, row, 1 + j, &v);
                aff = isl_aff_set_coefficient(aff, isl_dim_param, j, v);
        }
        for (j = 0; j < node->nvar; ++j) {
                isl_mat_get_element(node->sched, row, 1 + node->nparam + j, &v);
                aff = isl_aff_set_coefficient(aff, isl_dim_in, j, v);
        }

        isl_int_clear(v);

        return aff;
}

/* Convert the "n" rows starting at "first" of node->sched into a multi_aff
 * and return this multi_aff.
 *
 * The result is defined over the uncompressed node domain.
 */
static __isl_give isl_multi_aff *node_extract_partial_schedule_multi_aff(
        struct isl_sched_node *node, int first, int n)
{
        int i;
        isl_space *space;
        isl_local_space *ls;
        isl_aff *aff;
        isl_multi_aff *ma;
        int nrow;

        if (!node)
                return NULL;
        nrow = isl_mat_rows(node->sched);
        if (node->compressed)
                space = isl_multi_aff_get_domain_space(node->decompress);
        else
                space = isl_space_copy(node->space);
        ls = isl_local_space_from_space(isl_space_copy(space));
        space = isl_space_from_domain(space);
        space = isl_space_add_dims(space, isl_dim_out, n);
        ma = isl_multi_aff_zero(space);

        for (i = first; i < first + n; ++i) {
                aff = extract_schedule_row(isl_local_space_copy(ls), node, i);
                ma = isl_multi_aff_set_aff(ma, i - first, aff);
        }

        isl_local_space_free(ls);

        if (node->compressed)
                ma = isl_multi_aff_pullback_multi_aff(ma,
                                        isl_multi_aff_copy(node->compress));

        return ma;
}

/* Convert node->sched into a multi_aff and return this multi_aff.
 *
 * The result is defined over the uncompressed node domain.
 */
static __isl_give isl_multi_aff *node_extract_schedule_multi_aff(
        struct isl_sched_node *node)
{
        int nrow;

        nrow = isl_mat_rows(node->sched);
        return node_extract_partial_schedule_multi_aff(node, 0, nrow);
}

/* Convert node->sched into a map and return this map.
 *
 * The result is cached in node->sched_map, which needs to be released
 * whenever node->sched is updated.
 * It is defined over the uncompressed node domain.
 */
static __isl_give isl_map *node_extract_schedule(struct isl_sched_node *node)
{
        if (!node->sched_map) {
                isl_multi_aff *ma;

                ma = node_extract_schedule_multi_aff(node);
                node->sched_map = isl_map_from_multi_aff(ma);
        }

        return isl_map_copy(node->sched_map);
}

/* Construct a map that can be used to update a dependence relation
 * based on the current schedule.
 * That is, construct a map expressing that source and sink
 * are executed within the same iteration of the current schedule.
 * This map can then be intersected with the dependence relation.
 * This is not the most efficient way, but this shouldn't be a critical
 * operation.
 */
static __isl_give isl_map *specializer(struct isl_sched_node *src,
        struct isl_sched_node *dst)
{
        isl_map *src_sched, *dst_sched;

        src_sched = node_extract_schedule(src);
        dst_sched = node_extract_schedule(dst);
        return isl_map_apply_range(src_sched, isl_map_reverse(dst_sched));
}

/* Intersect the domains of the nested relations in domain and range
 * of "umap" with "map".
 */
static __isl_give isl_union_map *intersect_domains(
        __isl_take isl_union_map *umap, __isl_keep isl_map *map)
{
        isl_union_set *uset;

        umap = isl_union_map_zip(umap);
        uset = isl_union_set_from_set(isl_map_wrap(isl_map_copy(map)));
        umap = isl_union_map_intersect_domain(umap, uset);
        umap = isl_union_map_zip(umap);
        return umap;
}

/* Update the dependence relation of the given edge based
 * on the current schedule.
 * If the dependence is carried completely by the current schedule, then
 * it is removed from the edge_tables.  It is kept in the list of edges
 * as otherwise all edge_tables would have to be recomputed.
 */
static int update_edge(struct isl_sched_graph *graph,
        struct isl_sched_edge *edge)
{
        int empty;
        isl_map *id;

        id = specializer(edge->src, edge->dst);
        edge->map = isl_map_intersect(edge->map, isl_map_copy(id));
        if (!edge->map)
                goto error;

        if (edge->tagged_condition) {
                edge->tagged_condition =
                        intersect_domains(edge->tagged_condition, id);
                if (!edge->tagged_condition)
                        goto error;
        }
        if (edge->tagged_validity) {
                edge->tagged_validity =
                        intersect_domains(edge->tagged_validity, id);
                if (!edge->tagged_validity)
                        goto error;
        }

        empty = isl_map_plain_is_empty(edge->map);
        if (empty < 0)
                goto error;
        if (empty)
                graph_remove_edge(graph, edge);

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

/* Does the domain of "umap" intersect "uset"?
 */
static int domain_intersects(__isl_keep isl_union_map *umap,
        __isl_keep isl_union_set *uset)
{
        int empty;

        umap = isl_union_map_copy(umap);
        umap = isl_union_map_intersect_domain(umap, isl_union_set_copy(uset));
        empty = isl_union_map_is_empty(umap);
        isl_union_map_free(umap);

        return empty < 0 ? -1 : !empty;
}

/* Does the range of "umap" intersect "uset"?
 */
static int range_intersects(__isl_keep isl_union_map *umap,
        __isl_keep isl_union_set *uset)
{
        int empty;

        umap = isl_union_map_copy(umap);
        umap = isl_union_map_intersect_range(umap, isl_union_set_copy(uset));
        empty = isl_union_map_is_empty(umap);
        isl_union_map_free(umap);

        return empty < 0 ? -1 : !empty;
}

/* Are the condition dependences of "edge" local with respect to
 * the current schedule?
 *
 * That is, are domain and range of the condition dependences mapped
 * to the same point?
 *
 * In other words, is the condition false?
 */
static int is_condition_false(struct isl_sched_edge *edge)
{
        isl_union_map *umap;
        isl_map *map, *sched, *test;
        int empty, local;

        empty = isl_union_map_is_empty(edge->tagged_condition);
        if (empty < 0 || empty)
                return empty;

        umap = isl_union_map_copy(edge->tagged_condition);
        umap = isl_union_map_zip(umap);
        umap = isl_union_set_unwrap(isl_union_map_domain(umap));
        map = isl_map_from_union_map(umap);

        sched = node_extract_schedule(edge->src);
        map = isl_map_apply_domain(map, sched);
        sched = node_extract_schedule(edge->dst);
        map = isl_map_apply_range(map, sched);

        test = isl_map_identity(isl_map_get_space(map));
        local = isl_map_is_subset(map, test);
        isl_map_free(map);
        isl_map_free(test);

        return local;
}

/* For each conditional validity constraint that is adjacent
 * to a condition with domain in condition_source or range in condition_sink,
 * turn it into an unconditional validity constraint.
 */
static int unconditionalize_adjacent_validity(struct isl_sched_graph *graph,
        __isl_take isl_union_set *condition_source,
        __isl_take isl_union_set *condition_sink)
{
        int i;

        condition_source = isl_union_set_coalesce(condition_source);
        condition_sink = isl_union_set_coalesce(condition_sink);

        for (i = 0; i < graph->n_edge; ++i) {
                int adjacent;
                isl_union_map *validity;

                if (!is_conditional_validity(&graph->edge[i]))
                        continue;
                if (is_validity(&graph->edge[i]))
                        continue;

                validity = graph->edge[i].tagged_validity;
                adjacent = domain_intersects(validity, condition_sink);
                if (adjacent >= 0 && !adjacent)
                        adjacent = range_intersects(validity, condition_source);
                if (adjacent < 0)
                        goto error;
                if (!adjacent)
                        continue;

                set_validity(&graph->edge[i]);
        }

        isl_union_set_free(condition_source);
        isl_union_set_free(condition_sink);
        return 0;
error:
        isl_union_set_free(condition_source);
        isl_union_set_free(condition_sink);
        return -1;
}

/* Update the dependence relations of all edges based on the current schedule
 * and enforce conditional validity constraints that are adjacent
 * to satisfied condition constraints.
 *
 * First check if any of the condition constraints are satisfied
 * (i.e., not local to the outer schedule) and keep track of
 * their domain and range.
 * Then update all dependence relations (which removes the non-local
 * constraints).
 * Finally, if any condition constraints turned out to be satisfied,
 * then turn all adjacent conditional validity constraints into
 * unconditional validity constraints.
 */
static int update_edges(isl_ctx *ctx, struct isl_sched_graph *graph)
{
        int i;
        int any = 0;
        isl_union_set *source, *sink;

        source = isl_union_set_empty(isl_space_params_alloc(ctx, 0));
        sink = isl_union_set_empty(isl_space_params_alloc(ctx, 0));
        for (i = 0; i < graph->n_edge; ++i) {
                int local;
                isl_union_set *uset;
                isl_union_map *umap;

                if (!is_condition(&graph->edge[i]))
                        continue;
                if (is_local(&graph->edge[i]))
                        continue;
                local = is_condition_false(&graph->edge[i]);
                if (local < 0)
                        goto error;
                if (local)
                        continue;

                any = 1;

                umap = isl_union_map_copy(graph->edge[i].tagged_condition);
                uset = isl_union_map_domain(umap);
                source = isl_union_set_union(source, uset);

                umap = isl_union_map_copy(graph->edge[i].tagged_condition);
                uset = isl_union_map_range(umap);
                sink = isl_union_set_union(sink, uset);
        }

        for (i = graph->n_edge - 1; i >= 0; --i) {
                if (update_edge(graph, &graph->edge[i]) < 0)
                        goto error;
        }

        if (any)
                return unconditionalize_adjacent_validity(graph, source, sink);

        isl_union_set_free(source);
        isl_union_set_free(sink);
        return 0;
error:
        isl_union_set_free(source);
        isl_union_set_free(sink);
        return -1;
}

static void next_band(struct isl_sched_graph *graph)
{
        graph->band_start = graph->n_total_row;
}

/* Return the union of the universe domains of the nodes in "graph"
 * that satisfy "pred".
 */
static __isl_give isl_union_set *isl_sched_graph_domain(isl_ctx *ctx,
        struct isl_sched_graph *graph,
        int (*pred)(struct isl_sched_node *node, int data), int data)
{
        int i;
        isl_set *set;
        isl_union_set *dom;

        for (i = 0; i < graph->n; ++i)
                if (pred(&graph->node[i], data))
                        break;

        if (i >= graph->n)
                isl_die(ctx, isl_error_internal,
                        "empty component", return NULL);

        set = isl_set_universe(isl_space_copy(graph->node[i].space));
        dom = isl_union_set_from_set(set);

        for (i = i + 1; i < graph->n; ++i) {
                if (!pred(&graph->node[i], data))
                        continue;
                set = isl_set_universe(isl_space_copy(graph->node[i].space));
                dom = isl_union_set_union(dom, isl_union_set_from_set(set));
        }

        return dom;
}

/* Return a list of unions of universe domains, where each element
 * in the list corresponds to an SCC (or WCC) indexed by node->scc.
 */
static __isl_give isl_union_set_list *extract_sccs(isl_ctx *ctx,
        struct isl_sched_graph *graph)
{
        int i;
        isl_union_set_list *filters;

        filters = isl_union_set_list_alloc(ctx, graph->scc);
        for (i = 0; i < graph->scc; ++i) {
                isl_union_set *dom;

                dom = isl_sched_graph_domain(ctx, graph, &node_scc_exactly, i);
                filters = isl_union_set_list_add(filters, dom);
        }

        return filters;
}

/* Return a list of two unions of universe domains, one for the SCCs up
 * to and including graph->src_scc and another for the other SCCs.
 */
static __isl_give isl_union_set_list *extract_split(isl_ctx *ctx,
        struct isl_sched_graph *graph)
{
        isl_union_set *dom;
        isl_union_set_list *filters;

        filters = isl_union_set_list_alloc(ctx, 2);
        dom = isl_sched_graph_domain(ctx, graph,
                                        &node_scc_at_most, graph->src_scc);
        filters = isl_union_set_list_add(filters, dom);
        dom = isl_sched_graph_domain(ctx, graph,
                                        &node_scc_at_least, graph->src_scc + 1);
        filters = isl_union_set_list_add(filters, dom);

        return filters;
}

/* Copy nodes that satisfy node_pred from the src dependence graph
 * to the dst dependence graph.
 */
static int copy_nodes(struct isl_sched_graph *dst, struct isl_sched_graph *src,
        int (*node_pred)(struct isl_sched_node *node, int data), int data)
{
        int i;

        dst->n = 0;
        for (i = 0; i < src->n; ++i) {
                int j;

                if (!node_pred(&src->node[i], data))
                        continue;

                j = dst->n;
                dst->node[j].space = isl_space_copy(src->node[i].space);
                dst->node[j].compressed = src->node[i].compressed;
                dst->node[j].hull = isl_set_copy(src->node[i].hull);
                dst->node[j].compress =
                        isl_multi_aff_copy(src->node[i].compress);
                dst->node[j].decompress =
                        isl_multi_aff_copy(src->node[i].decompress);
                dst->node[j].nvar = src->node[i].nvar;
                dst->node[j].nparam = src->node[i].nparam;
                dst->node[j].sched = isl_mat_copy(src->node[i].sched);
                dst->node[j].sched_map = isl_map_copy(src->node[i].sched_map);
                dst->node[j].coincident = src->node[i].coincident;
                dst->node[j].sizes = isl_multi_val_copy(src->node[i].sizes);
                dst->node[j].max = isl_vec_copy(src->node[i].max);
                dst->n++;

                if (!dst->node[j].space || !dst->node[j].sched)
                        return -1;
                if (dst->node[j].compressed &&
                    (!dst->node[j].hull || !dst->node[j].compress ||
                     !dst->node[j].decompress))
                        return -1;
        }

        return 0;
}

/* Copy non-empty edges that satisfy edge_pred from the src dependence graph
 * to the dst dependence graph.
 * If the source or destination node of the edge is not in the destination
 * graph, then it must be a backward proximity edge and it should simply
 * be ignored.
 */
static int copy_edges(isl_ctx *ctx, struct isl_sched_graph *dst,
        struct isl_sched_graph *src,
        int (*edge_pred)(struct isl_sched_edge *edge, int data), int data)
{
        int i;
        enum isl_edge_type t;

        dst->n_edge = 0;
        for (i = 0; i < src->n_edge; ++i) {
                struct isl_sched_edge *edge = &src->edge[i];
                isl_map *map;
                isl_union_map *tagged_condition;
                isl_union_map *tagged_validity;
                struct isl_sched_node *dst_src, *dst_dst;

                if (!edge_pred(edge, data))
                        continue;

                if (isl_map_plain_is_empty(edge->map))
                        continue;

                dst_src = graph_find_node(ctx, dst, edge->src->space);
                dst_dst = graph_find_node(ctx, dst, edge->dst->space);
                if (!dst_src || !dst_dst) {
                        if (is_validity(edge) || is_conditional_validity(edge))
                                isl_die(ctx, isl_error_internal,
                                        "backward (conditional) validity edge",
                                        return -1);
                        continue;
                }

                map = isl_map_copy(edge->map);
                tagged_condition = isl_union_map_copy(edge->tagged_condition);
                tagged_validity = isl_union_map_copy(edge->tagged_validity);

                dst->edge[dst->n_edge].src = dst_src;
                dst->edge[dst->n_edge].dst = dst_dst;
                dst->edge[dst->n_edge].map = map;
                dst->edge[dst->n_edge].tagged_condition = tagged_condition;
                dst->edge[dst->n_edge].tagged_validity = tagged_validity;
                dst->edge[dst->n_edge].types = edge->types;
                dst->n_edge++;

                if (edge->tagged_condition && !tagged_condition)
                        return -1;
                if (edge->tagged_validity && !tagged_validity)
                        return -1;

                for (t = isl_edge_first; t <= isl_edge_last; ++t) {
                        if (edge !=
                            graph_find_edge(src, t, edge->src, edge->dst))
                                continue;
                        if (graph_edge_table_add(ctx, dst, t,
                                            &dst->edge[dst->n_edge - 1]) < 0)
                                return -1;
                }
        }

        return 0;
}

/* Compute the maximal number of variables over all nodes.
 * This is the maximal number of linearly independent schedule
 * rows that we need to compute.
 * Just in case we end up in a part of the dependence graph
 * with only lower-dimensional domains, we make sure we will
 * compute the required amount of extra linearly independent rows.
 */
static int compute_maxvar(struct isl_sched_graph *graph)
{
        int i;

        graph->maxvar = 0;
        for (i = 0; i < graph->n; ++i) {
                struct isl_sched_node *node = &graph->node[i];
                int nvar;

                if (node_update_cmap(node) < 0)
                        return -1;
                nvar = node->nvar + graph->n_row - node->rank;
                if (nvar > graph->maxvar)
                        graph->maxvar = nvar;
        }

        return 0;
}

/* Extract the subgraph of "graph" that consists of the node satisfying
 * "node_pred" and the edges satisfying "edge_pred" and store
 * the result in "sub".
 */
static int extract_sub_graph(isl_ctx *ctx, struct isl_sched_graph *graph,
        int (*node_pred)(struct isl_sched_node *node, int data),
        int (*edge_pred)(struct isl_sched_edge *edge, int data),
        int data, struct isl_sched_graph *sub)
{
        int i, n = 0, n_edge = 0;
        int t;

        for (i = 0; i < graph->n; ++i)
                if (node_pred(&graph->node[i], data))
                        ++n;
        for (i = 0; i < graph->n_edge; ++i)
                if (edge_pred(&graph->edge[i], data))
                        ++n_edge;
        if (graph_alloc(ctx, sub, n, n_edge) < 0)
                return -1;
        if (copy_nodes(sub, graph, node_pred, data) < 0)
                return -1;
        if (graph_init_table(ctx, sub) < 0)
                return -1;
        for (t = 0; t <= isl_edge_last; ++t)
                sub->max_edge[t] = graph->max_edge[t];
        if (graph_init_edge_tables(ctx, sub) < 0)
                return -1;
        if (copy_edges(ctx, sub, graph, edge_pred, data) < 0)
                return -1;
        sub->n_row = graph->n_row;
        sub->max_row = graph->max_row;
        sub->n_total_row = graph->n_total_row;
        sub->band_start = graph->band_start;

        return 0;
}

static __isl_give isl_schedule_node *compute_schedule(isl_schedule_node *node,
        struct isl_sched_graph *graph);
static __isl_give isl_schedule_node *compute_schedule_wcc(
        isl_schedule_node *node, struct isl_sched_graph *graph);

/* Compute a schedule for a subgraph of "graph".  In particular, for
 * the graph composed of nodes that satisfy node_pred and edges that
 * that satisfy edge_pred.
 * If the subgraph is known to consist of a single component, then wcc should
 * be set and then we call compute_schedule_wcc on the constructed subgraph.
 * Otherwise, we call compute_schedule, which will check whether the subgraph
 * is connected.
 *
 * The schedule is inserted at "node" and the updated schedule node
 * is returned.
 */
static __isl_give isl_schedule_node *compute_sub_schedule(
        __isl_take isl_schedule_node *node, isl_ctx *ctx,
        struct isl_sched_graph *graph,
        int (*node_pred)(struct isl_sched_node *node, int data),
        int (*edge_pred)(struct isl_sched_edge *edge, int data),
        int data, int wcc)
{
        struct isl_sched_graph split = { 0 };

        if (extract_sub_graph(ctx, graph, node_pred, edge_pred, data,
                                &split) < 0)
                goto error;

        if (wcc)
                node = compute_schedule_wcc(node, &split);
        else
                node = compute_schedule(node, &split);

        graph_free(ctx, &split);
        return node;
error:
        graph_free(ctx, &split);
        return isl_schedule_node_free(node);
}

static int edge_scc_exactly(struct isl_sched_edge *edge, int scc)
{
        return edge->src->scc == scc && edge->dst->scc == scc;
}

static int edge_dst_scc_at_most(struct isl_sched_edge *edge, int scc)
{
        return edge->dst->scc <= scc;
}

static int edge_src_scc_at_least(struct isl_sched_edge *edge, int scc)
{
        return edge->src->scc >= scc;
}

/* Reset the current band by dropping all its schedule rows.
 */
static int reset_band(struct isl_sched_graph *graph)
{
        int i;
        int drop;

        drop = graph->n_total_row - graph->band_start;
        graph->n_total_row -= drop;
        graph->n_row -= drop;

        for (i = 0; i < graph->n; ++i) {
                struct isl_sched_node *node = &graph->node[i];

                isl_map_free(node->sched_map);
                node->sched_map = NULL;

                node->sched = isl_mat_drop_rows(node->sched,
                                                graph->band_start, drop);

                if (!node->sched)
                        return -1;
        }

        return 0;
}

/* Split the current graph into two parts and compute a schedule for each
 * part individually.  In particular, one part consists of all SCCs up
 * to and including graph->src_scc, while the other part contains the other
 * SCCs.  The split is enforced by a sequence node inserted at position "node"
 * in the schedule tree.  Return the updated schedule node.
 * If either of these two parts consists of a sequence, then it is spliced
 * into the sequence containing the two parts.
 *
 * The current band is reset. It would be possible to reuse
 * the previously computed rows as the first rows in the next
 * band, but recomputing them may result in better rows as we are looking
 * at a smaller part of the dependence graph.
 */
static __isl_give isl_schedule_node *compute_split_schedule(
        __isl_take isl_schedule_node *node, struct isl_sched_graph *graph)
{
        int is_seq;
        isl_ctx *ctx;
        isl_union_set_list *filters;

        if (!node)
                return NULL;

        if (reset_band(graph) < 0)
                return isl_schedule_node_free(node);

        next_band(graph);

        ctx = isl_schedule_node_get_ctx(node);
        filters = extract_split(ctx, graph);
        node = isl_schedule_node_insert_sequence(node, filters);
        node = isl_schedule_node_child(node, 1);
        node = isl_schedule_node_child(node, 0);

        node = compute_sub_schedule(node, ctx, graph,
                                &node_scc_at_least, &edge_src_scc_at_least,
                                graph->src_scc + 1, 0);
        is_seq = isl_schedule_node_get_type(node) == isl_schedule_node_sequence;
        node = isl_schedule_node_parent(node);
        node = isl_schedule_node_parent(node);
        if (is_seq)
                node = isl_schedule_node_sequence_splice_child(node, 1);
        node = isl_schedule_node_child(node, 0);
        node = isl_schedule_node_child(node, 0);
        node = compute_sub_schedule(node, ctx, graph,
                                &node_scc_at_most, &edge_dst_scc_at_most,
                                graph->src_scc, 0);
        is_seq = isl_schedule_node_get_type(node) == isl_schedule_node_sequence;
        node = isl_schedule_node_parent(node);
        node = isl_schedule_node_parent(node);
        if (is_seq)
                node = isl_schedule_node_sequence_splice_child(node, 0);

        return node;
}

/* Insert a band node at position "node" in the schedule tree corresponding
 * to the current band in "graph".  Mark the band node permutable
 * if "permutable" is set.
 * The partial schedules and the coincidence property are extracted
 * from the graph nodes.
 * Return the updated schedule node.
 */
static __isl_give isl_schedule_node *insert_current_band(
        __isl_take isl_schedule_node *node, struct isl_sched_graph *graph,
        int permutable)
{
        int i;
        int start, end, n;
        isl_multi_aff *ma;
        isl_multi_pw_aff *mpa;
        isl_multi_union_pw_aff *mupa;

        if (!node)
                return NULL;

        if (graph->n < 1)
                isl_die(isl_schedule_node_get_ctx(node), isl_error_internal,
                        "graph should have at least one node",
                        return isl_schedule_node_free(node));

        start = graph->band_start;
        end = graph->n_total_row;
        n = end - start;

        ma = node_extract_partial_schedule_multi_aff(&graph->node[0], start, n);
        mpa = isl_multi_pw_aff_from_multi_aff(ma);
        mupa = isl_multi_union_pw_aff_from_multi_pw_aff(mpa);

        for (i = 1; i < graph->n; ++i) {
                isl_multi_union_pw_aff *mupa_i;

                ma = node_extract_partial_schedule_multi_aff(&graph->node[i],
                                                                start, n);
                mpa = isl_multi_pw_aff_from_multi_aff(ma);
                mupa_i = isl_multi_union_pw_aff_from_multi_pw_aff(mpa);
                mupa = isl_multi_union_pw_aff_union_add(mupa, mupa_i);
        }
        node = isl_schedule_node_insert_partial_schedule(node, mupa);

        for (i = 0; i < n; ++i)
                node = isl_schedule_node_band_member_set_coincident(node, i,
                                        graph->node[0].coincident[start + i]);
        node = isl_schedule_node_band_set_permutable(node, permutable);

        return node;
}

/* Update the dependence relations based on the current schedule,
 * add the current band to "node" and then continue with the computation
 * of the next band.
 * Return the updated schedule node.
 */
static __isl_give isl_schedule_node *compute_next_band(
        __isl_take isl_schedule_node *node,
        struct isl_sched_graph *graph, int permutable)
{
        isl_ctx *ctx;

        if (!node)
                return NULL;

        ctx = isl_schedule_node_get_ctx(node);
        if (update_edges(ctx, graph) < 0)
                return isl_schedule_node_free(node);
        node = insert_current_band(node, graph, permutable);
        next_band(graph);

        node = isl_schedule_node_child(node, 0);
        node = compute_schedule(node, graph);
        node = isl_schedule_node_parent(node);

        return node;
}

/* Add the constraints "coef" derived from an edge from "node" to itself
 * to graph->lp in order to respect the dependences and to try and carry them.
 * "pos" is the sequence number of the edge that needs to be carried.
 * "coef" represents general constraints on coefficients (c_0, c_n, c_x)
 * of valid constraints for (y - x) with x and y instances of the node.
 *
 * The constraints added to graph->lp need to enforce
 *
 *      (c_j_0 + c_j_n n + c_j_x y) - (c_j_0 + c_j_n n + c_j_x x)
 *      = c_j_x (y - x) >= e_i
 *
 * for each (x,y) in the dependence relation of the edge.
 * That is, (-e_i, 0, c_j_x) needs to be plugged in for (c_0, c_n, c_x),
 * taking into account that each coefficient in c_j_x is represented
 * as a pair of non-negative coefficients.
 */
static isl_stat add_intra_constraints(struct isl_sched_graph *graph,
        struct isl_sched_node *node, __isl_take isl_basic_set *coef, int pos)
{
        int offset;
        isl_ctx *ctx;
        isl_dim_map *dim_map;

        if (!coef)
                return isl_stat_error;

        ctx = isl_basic_set_get_ctx(coef);
        offset = coef_var_offset(coef);
        dim_map = intra_dim_map(ctx, graph, node, offset, 1);
        isl_dim_map_range(dim_map, 3 + pos, 0, 0, 0, 1, -1);
        graph->lp = add_constraints_dim_map(graph->lp, coef, dim_map);

        return isl_stat_ok;
}

/* Add the constraints "coef" derived from an edge from "src" to "dst"
 * to graph->lp in order to respect the dependences and to try and carry them.
 * "pos" is the sequence number of the edge that needs to be carried.
 * "coef" represents general constraints on coefficients (c_0, c_n, c_x, c_y)
 * of valid constraints for (x, y) with x and y instances of "src" and "dst".
 *
 * The constraints added to graph->lp need to enforce
 *
 *      (c_k_0 + c_k_n n + c_k_x y) - (c_j_0 + c_j_n n + c_j_x x) >= e_i
 *
 * for each (x,y) in the dependence relation of the edge.
 * That is,
 * (-e_i + c_k_0 - c_j_0, c_k_n - c_j_n, -c_j_x, c_k_x)
 * needs to be plugged in for (c_0, c_n, c_x, c_y),
 * taking into account that each coefficient in c_j_x and c_k_x is represented
 * as a pair of non-negative coefficients.
 */
static isl_stat add_inter_constraints(struct isl_sched_graph *graph,
        struct isl_sched_node *src, struct isl_sched_node *dst,
        __isl_take isl_basic_set *coef, int pos)
{
        int offset;
        isl_ctx *ctx;
        isl_dim_map *dim_map;

        if (!coef)
                return isl_stat_error;

        ctx = isl_basic_set_get_ctx(coef);
        offset = coef_var_offset(coef);
        dim_map = inter_dim_map(ctx, graph, src, dst, offset, 1);
        isl_dim_map_range(dim_map, 3 + pos, 0, 0, 0, 1, -1);
        graph->lp = add_constraints_dim_map(graph->lp, coef, dim_map);

        return isl_stat_ok;
}

/* Data structure collecting information used during the construction
 * of an LP for carrying dependences.
 *
 * "intra" is a sequence of coefficient constraints for intra-node edges.
 * "inter" is a sequence of coefficient constraints for inter-node edges.
 */
struct isl_carry {
        isl_basic_set_list *intra;
        isl_basic_set_list *inter;
};

/* Free all the data stored in "carry".
 */
static void isl_carry_clear(struct isl_carry *carry)
{
        isl_basic_set_list_free(carry->intra);
        isl_basic_set_list_free(carry->inter);
}

/* Return a pointer to the node in "graph" that lives in "space".
 * If the requested node has been compressed, then "space"
 * corresponds to the compressed space.
 *
 * First try and see if "space" is the space of an uncompressed node.
 * If so, return that node.
 * Otherwise, "space" was constructed by construct_compressed_id and
 * contains a user pointer pointing to the node in the tuple id.
 */
static struct isl_sched_node *graph_find_compressed_node(isl_ctx *ctx,
        struct isl_sched_graph *graph, __isl_keep isl_space *space)
{
        isl_id *id;
        struct isl_sched_node *node;

        if (!space)
                return NULL;

        node = graph_find_node(ctx, graph, space);
        if (node)
                return node;

        id = isl_space_get_tuple_id(space, isl_dim_set);
        node = isl_id_get_user(id);
        isl_id_free(id);

        if (!node)
                return NULL;

        if (!(node >= &graph->node[0] && node < &graph->node[graph->n]))
                isl_die(ctx, isl_error_internal,
                        "space points to invalid node", return NULL);

        return node;
}

/* Internal data structure for add_all_constraints.
 *
 * "graph" is the schedule constraint graph for which an LP problem
 * is being constructed.
 * "pos" is the position of the next edge that needs to be carried.
 */
struct isl_add_all_constraints_data {
        isl_ctx *ctx;
        struct isl_sched_graph *graph;
        int pos;
};

/* Add the constraints "coef" derived from an edge from a node to itself
 * to data->graph->lp in order to respect the dependences and
 * to try and carry them.
 *
 * The space of "coef" is of the form
 *
 *      coefficients[[c_cst, c_n] -> S[c_x]]
 *
 * with S[c_x] the (compressed) space of the node.
 * Extract the node from the space and call add_intra_constraints.
 */
static isl_stat lp_add_intra(__isl_take isl_basic_set *coef, void *user)
{
        struct isl_add_all_constraints_data *data = user;
        isl_space *space;
        struct isl_sched_node *node;

        space = isl_basic_set_get_space(coef);
        space = isl_space_range(isl_space_unwrap(space));
        node = graph_find_compressed_node(data->ctx, data->graph, space);
        isl_space_free(space);
        return add_intra_constraints(data->graph, node, coef, data->pos++);
}

/* Add the constraints "coef" derived from an edge from a node j
 * to a node k to data->graph->lp in order to respect the dependences and
 * to try and carry them.
 *
 * The space of "coef" is of the form
 *
 *      coefficients[[c_cst, c_n] -> [S_j[c_x] -> S_k[c_y]]]
 *
 * with S_j[c_x] and S_k[c_y] the (compressed) spaces of the nodes.
 * Extract the nodes from the space and call add_inter_constraints.
 */
static isl_stat lp_add_inter(__isl_take isl_basic_set *coef, void *user)
{
        struct isl_add_all_constraints_data *data = user;
        isl_space *space, *dom;
        struct isl_sched_node *src, *dst;

        space = isl_basic_set_get_space(coef);
        space = isl_space_unwrap(isl_space_range(isl_space_unwrap(space)));
        dom = isl_space_domain(isl_space_copy(space));
        src = graph_find_compressed_node(data->ctx, data->graph, dom);
        isl_space_free(dom);
        space = isl_space_range(space);
        dst = graph_find_compressed_node(data->ctx, data->graph, space);
        isl_space_free(space);

        return add_inter_constraints(data->graph, src, dst, coef, data->pos++);
}

/* Add constraints to graph->lp that force all (conditional) validity
 * dependences to be respected and attempt to carry them.
 * "intra" is the sequence of coefficient constraints for intra-node edges.
 * "inter" is the sequence of coefficient constraints for inter-node edges.
 */
static isl_stat add_all_constraints(isl_ctx *ctx, struct isl_sched_graph *graph,
        __isl_keep isl_basic_set_list *intra,
        __isl_keep isl_basic_set_list *inter)
{
        struct isl_add_all_constraints_data data = { ctx, graph };

        data.pos = 0;
        if (isl_basic_set_list_foreach(intra, &lp_add_intra, &data) < 0)
                return isl_stat_error;
        if (isl_basic_set_list_foreach(inter, &lp_add_inter, &data) < 0)
                return isl_stat_error;
        return isl_stat_ok;
}

/* Internal data structure for count_all_constraints
 * for keeping track of the number of equality and inequality constraints.
 */
struct isl_sched_count {
        int n_eq;
        int n_ineq;
};

/* Add the number of equality and inequality constraints of "bset"
 * to data->n_eq and data->n_ineq.
 */
static isl_stat bset_update_count(__isl_take isl_basic_set *bset, void *user)
{
        struct isl_sched_count *data = user;

        data->n_eq += isl_basic_set_n_equality(bset);
        data->n_ineq += isl_basic_set_n_inequality(bset);
        isl_basic_set_free(bset);

        return isl_stat_ok;
}

/* Count the number of equality and inequality constraints
 * that will be added to the carry_lp problem.
 * We count each edge exactly once.
 * "intra" is the sequence of coefficient constraints for intra-node edges.
 * "inter" is the sequence of coefficient constraints for inter-node edges.
 */
static isl_stat count_all_constraints(__isl_keep isl_basic_set_list *intra,
        __isl_keep isl_basic_set_list *inter, int *n_eq, int *n_ineq)
{
        struct isl_sched_count data;

        data.n_eq = data.n_ineq = 0;
        if (isl_basic_set_list_foreach(inter, &bset_update_count, &data) < 0)
                return isl_stat_error;
        if (isl_basic_set_list_foreach(intra, &bset_update_count, &data) < 0)
                return isl_stat_error;

        *n_eq = data.n_eq;
        *n_ineq = data.n_ineq;

        return isl_stat_ok;
}

/* Construct an LP problem for finding schedule coefficients
 * such that the schedule carries as many validity dependences as possible.
 * In particular, for each dependence i, we bound the dependence distance
 * from below by e_i, with 0 <= e_i <= 1 and then maximize the sum
 * of all e_i's.  Dependences with e_i = 0 in the solution are simply
 * respected, while those with e_i > 0 (in practice e_i = 1) are carried.
 * "intra" is the sequence of coefficient constraints for intra-node edges.
 * "inter" is the sequence of coefficient constraints for inter-node edges.
 * "n_edge" is the total number of edges.
 *
 * All variables of the LP are non-negative.  The actual coefficients
 * may be negative, so each coefficient is represented as the difference
 * of two non-negative variables.  The negative part always appears
 * immediately before the positive part.
 * Other than that, the variables have the following order
 *
 *      - sum of (1 - e_i) over all edges
 *      - sum of all c_n coefficients
 *              (unconstrained when computing non-parametric schedules)
 *      - sum of positive and negative parts of all c_x coefficients
 *      - for each edge
 *              - e_i
 *      - for each node
 *              - c_i_0
 *              - c_i_n (if parametric)
 *              - positive and negative parts of c_i_x
 *
 * The constraints are those from the (validity) edges plus three equalities
 * to express the sums and n_edge inequalities to express e_i <= 1.
 */
static isl_stat setup_carry_lp(isl_ctx *ctx, struct isl_sched_graph *graph,
        int n_edge, __isl_keep isl_basic_set_list *intra,
        __isl_keep isl_basic_set_list *inter)
{
        int i;
        int k;
        isl_space *dim;
        unsigned total;
        int n_eq, n_ineq;

        total = 3 + n_edge;
        for (i = 0; i < graph->n; ++i) {
                struct isl_sched_node *node = &graph->node[graph->sorted[i]];
                node->start = total;
                total += 1 + node->nparam + 2 * node->nvar;
        }

        if (count_all_constraints(intra, inter, &n_eq, &n_ineq) < 0)
                return isl_stat_error;

        dim = isl_space_set_alloc(ctx, 0, total);
        isl_basic_set_free(graph->lp);
        n_eq += 3;
        n_ineq += n_edge;
        graph->lp = isl_basic_set_alloc_space(dim, 0, n_eq, n_ineq);
        graph->lp = isl_basic_set_set_rational(graph->lp);

        k = isl_basic_set_alloc_equality(graph->lp);
        if (k < 0)
                return isl_stat_error;
        isl_seq_clr(graph->lp->eq[k], 1 + total);
        isl_int_set_si(graph->lp->eq[k][0], -n_edge);
        isl_int_set_si(graph->lp->eq[k][1], 1);
        for (i = 0; i < n_edge; ++i)
                isl_int_set_si(graph->lp->eq[k][4 + i], 1);

        if (add_param_sum_constraint(graph, 1) < 0)
                return isl_stat_error;
        if (add_var_sum_constraint(graph, 2) < 0)
                return isl_stat_error;

        for (i = 0; i < n_edge; ++i) {
                k = isl_basic_set_alloc_inequality(graph->lp);
                if (k < 0)
                        return isl_stat_error;
                isl_seq_clr(graph->lp->ineq[k], 1 + total);
                isl_int_set_si(graph->lp->ineq[k][4 + i], -1);
                isl_int_set_si(graph->lp->ineq[k][0], 1);
        }

        if (add_all_constraints(ctx, graph, intra, inter) < 0)
                return isl_stat_error;

        return isl_stat_ok;
}

static __isl_give isl_schedule_node *compute_component_schedule(
        __isl_take isl_schedule_node *node, struct isl_sched_graph *graph,
        int wcc);

/* Comparison function for sorting the statements based on
 * the corresponding value in "r".
 */
static int smaller_value(const void *a, const void *b, void *data)
{
        isl_vec *r = data;
        const int *i1 = a;
        const int *i2 = b;

        return isl_int_cmp(r->el[*i1], r->el[*i2]);
}

/* If the schedule_split_scaled option is set and if the linear
 * parts of the scheduling rows for all nodes in the graphs have
 * a non-trivial common divisor, then split off the remainder of the
 * constant term modulo this common divisor from the linear part.
 * Otherwise, insert a band node directly and continue with
 * the construction of the schedule.
 *
 * If a non-trivial common divisor is found, then
 * the linear part is reduced and the remainder is enforced
 * by a sequence node with the children placed in the order
 * of this remainder.
 * In particular, we assign an scc index based on the remainder and
 * then rely on compute_component_schedule to insert the sequence and
 * to continue the schedule construction on each part.
 */
static __isl_give isl_schedule_node *split_scaled(
        __isl_take isl_schedule_node *node, struct isl_sched_graph *graph)
{
        int i;
        int row;
        int scc;
        isl_ctx *ctx;
        isl_int gcd, gcd_i;
        isl_vec *r;
        int *order;

        if (!node)
                return NULL;

        ctx = isl_schedule_node_get_ctx(node);
        if (!ctx->opt->schedule_split_scaled)
                return compute_next_band(node, graph, 0);
        if (graph->n <= 1)
                return compute_next_band(node, graph, 0);

        isl_int_init(gcd);
        isl_int_init(gcd_i);

        isl_int_set_si(gcd, 0);

        row = isl_mat_rows(graph->node[0].sched) - 1;

        for (i = 0; i < graph->n; ++i) {
                struct isl_sched_node *node = &graph->node[i];
                int cols = isl_mat_cols(node->sched);

                isl_seq_gcd(node->sched->row[row] + 1, cols - 1, &gcd_i);
                isl_int_gcd(gcd, gcd, gcd_i);
        }

        isl_int_clear(gcd_i);

        if (isl_int_cmp_si(gcd, 1) <= 0) {
                isl_int_clear(gcd);
                return compute_next_band(node, graph, 0);
        }

        r = isl_vec_alloc(ctx, graph->n);
        order = isl_calloc_array(ctx, int, graph->n);
        if (!r || !order)
                goto error;

        for (i = 0; i < graph->n; ++i) {
                struct isl_sched_node *node = &graph->node[i];

                order[i] = i;
                isl_int_fdiv_r(r->el[i], node->sched->row[row][0], gcd);
                isl_int_fdiv_q(node->sched->row[row][0],
                               node->sched->row[row][0], gcd);
                isl_int_mul(node->sched->row[row][0],
                            node->sched->row[row][0], gcd);
                node->sched = isl_mat_scale_down_row(node->sched, row, gcd);
                if (!node->sched)
                        goto error;
        }

        if (isl_sort(order, graph->n, sizeof(order[0]), &smaller_value, r) < 0)
                goto error;

        scc = 0;
        for (i = 0; i < graph->n; ++i) {
                if (i > 0 && isl_int_ne(r->el[order[i - 1]], r->el[order[i]]))
                        ++scc;
                graph->node[order[i]].scc = scc;
        }
        graph->scc = ++scc;
        graph->weak = 0;

        isl_int_clear(gcd);
        isl_vec_free(r);
        free(order);

        if (update_edges(ctx, graph) < 0)
                return isl_schedule_node_free(node);
        node = insert_current_band(node, graph, 0);
        next_band(graph);

        node = isl_schedule_node_child(node, 0);
        node = compute_component_schedule(node, graph, 0);
        node = isl_schedule_node_parent(node);

        return node;
error:
        isl_vec_free(r);
        free(order);
        isl_int_clear(gcd);
        return isl_schedule_node_free(node);
}

/* Is the schedule row "sol" trivial on node "node"?
 * That is, is the solution zero on the dimensions linearly independent of
 * the previously found solutions?
 * Return 1 if the solution is trivial, 0 if it is not and -1 on error.
 *
 * Each coefficient is represented as the difference between
 * two non-negative values in "sol".  "sol" has been computed
 * in terms of the original iterators (i.e., without use of cmap).
 * We construct the schedule row s and write it as a linear
 * combination of (linear combinations of) previously computed schedule rows.
 * s = Q c or c = U s.
 * If the final entries of c are all zero, then the solution is trivial.
 */
static int is_trivial(struct isl_sched_node *node, __isl_keep isl_vec *sol)
{
        int trivial;
        isl_vec *node_sol;

        if (!sol)
                return -1;
        if (node->nvar == node->rank)
                return 0;

        node_sol = extract_var_coef(node, sol);
        node_sol = isl_mat_vec_product(isl_mat_copy(node->cinv), node_sol);
        if (!node_sol)
                return -1;

        trivial = isl_seq_first_non_zero(node_sol->el + node->rank,
                                        node->nvar - node->rank) == -1;

        isl_vec_free(node_sol);

        return trivial;
}

/* Is the schedule row "sol" trivial on any node where it should
 * not be trivial?
 * "sol" has been computed in terms of the original iterators
 * (i.e., without use of cmap).
 * Return 1 if any solution is trivial, 0 if they are not and -1 on error.
 */
static int is_any_trivial(struct isl_sched_graph *graph,
        __isl_keep isl_vec *sol)
{
        int i;

        for (i = 0; i < graph->n; ++i) {
                struct isl_sched_node *node = &graph->node[i];
                int trivial;

                if (!needs_row(graph, node))
                        continue;
                trivial = is_trivial(node, sol);
                if (trivial < 0 || trivial)
                        return trivial;
        }

        return 0;
}

/* Does the schedule represented by "sol" perform loop coalescing on "node"?
 * If so, return the position of the coalesced dimension.
 * Otherwise, return node->nvar or -1 on error.
 *
 * In particular, look for pairs of coefficients c_i and c_j such that
 * |c_j/c_i| >= size_i, i.e., |c_j| >= |c_i * size_i|.
 * If any such pair is found, then return i.
 * If size_i is infinity, then no check on c_i needs to be performed.
 */
static int find_node_coalescing(struct isl_sched_node *node,
        __isl_keep isl_vec *sol)
{
        int i, j;
        isl_int max;
        isl_vec *csol;

        if (node->nvar <= 1)
                return node->nvar;

        csol = extract_var_coef(node, sol);
        if (!csol)
                return -1;
        isl_int_init(max);
        for (i = 0; i < node->nvar; ++i) {
                isl_val *v;

                if (isl_int_is_zero(csol->el[i]))
                        continue;
                v = isl_multi_val_get_val(node->sizes, i);
                if (!v)
                        goto error;
                if (!isl_val_is_int(v)) {
                        isl_val_free(v);
                        continue;
                }
                isl_int_mul(max, v->n, csol->el[i]);
                isl_val_free(v);

                for (j = 0; j < node->nvar; ++j) {
                        if (j == i)
                                continue;
                        if (isl_int_abs_ge(csol->el[j], max))
                                break;
                }
                if (j < node->nvar)
                        break;
        }

        isl_int_clear(max);
        isl_vec_free(csol);
        return i;
error:
        isl_int_clear(max);
        isl_vec_free(csol);
        return -1;
}

/* Force the schedule coefficient at position "pos" of "node" to be zero
 * in "tl".
 * The coefficient is encoded as the difference between two non-negative
 * variables.  Force these two variables to have the same value.
 */
static __isl_give isl_tab_lexmin *zero_out_node_coef(
        __isl_take isl_tab_lexmin *tl, struct isl_sched_node *node, int pos)
{
        int dim;
        isl_ctx *ctx;
        isl_vec *eq;

        ctx = isl_space_get_ctx(node->space);
        dim = isl_tab_lexmin_dim(tl);
        if (dim < 0)
                return isl_tab_lexmin_free(tl);
        eq = isl_vec_alloc(ctx, 1 + dim);
        eq = isl_vec_clr(eq);
        if (!eq)
                return isl_tab_lexmin_free(tl);

        pos = 1 + node_var_coef_offset(node) + 2 * pos;
        isl_int_set_si(eq->el[pos], 1);
        isl_int_set_si(eq->el[pos + 1], -1);
        tl = isl_tab_lexmin_add_eq(tl, eq->el);
        isl_vec_free(eq);

        return tl;
}

/* Return the lexicographically smallest rational point in the basic set
 * from which "tl" was constructed, double checking that this input set
 * was not empty.
 */
static __isl_give isl_vec *non_empty_solution(__isl_keep isl_tab_lexmin *tl)
{
        isl_vec *sol;

        sol = isl_tab_lexmin_get_solution(tl);
        if (!sol)
                return NULL;
        if (sol->size == 0)
                isl_die(isl_vec_get_ctx(sol), isl_error_internal,
                        "error in schedule construction",
                        return isl_vec_free(sol));
        return sol;
}

/* Does the solution "sol" of the LP problem constructed by setup_carry_lp
 * carry any of the "n_edge" groups of dependences?
 * The value in the first position is the sum of (1 - e_i) over all "n_edge"
 * edges, with 0 <= e_i <= 1 equal to 1 when the dependences represented
 * by the edge are carried by the solution.
 * If the sum of the (1 - e_i) is smaller than "n_edge" then at least
 * one of those is carried.
 *
 * Note that despite the fact that the problem is solved using a rational
 * solver, the solution is guaranteed to be integral.
 * Specifically, the dependence distance lower bounds e_i (and therefore
 * also their sum) are integers.  See Lemma 5 of [1].
 *
 * Any potential denominator of the sum is cleared by this function.
 * The denominator is not relevant for any of the other elements
 * in the solution.
 *
 * [1] P. Feautrier, Some Efficient Solutions to the Affine Scheduling
 *     Problem, Part II: Multi-Dimensional Time.
 *     In Intl. Journal of Parallel Programming, 1992.
 */
static int carries_dependences(__isl_keep isl_vec *sol, int n_edge)
{
        isl_int_divexact(sol->el[1], sol->el[1], sol->el[0]);
        isl_int_set_si(sol->el[0], 1);
        return isl_int_cmp_si(sol->el[1], n_edge) < 0;
}

/* Return the lexicographically smallest rational point in "lp",
 * assuming that all variables are non-negative and performing some
 * additional sanity checks.
 * In particular, "lp" should not be empty by construction.
 * Double check that this is the case.
 * If dependences are not carried for any of the "n_edge" edges,
 * then return an empty vector.
 *
 * If the schedule_treat_coalescing option is set and
 * if the computed schedule performs loop coalescing on a given node,
 * i.e., if it is of the form
 *
 *      c_i i + c_j j + ...
 *
 * with |c_j/c_i| >= size_i, then force the coefficient c_i to be zero
 * to cut out this solution.  Repeat this process until no more loop
 * coalescing occurs or until no more dependences can be carried.
 * In the latter case, revert to the previously computed solution.
 */
static __isl_give isl_vec *non_neg_lexmin(struct isl_sched_graph *graph,
        __isl_take isl_basic_set *lp, int n_edge)
{
        int i, pos;
        isl_ctx *ctx;
        isl_tab_lexmin *tl;
        isl_vec *sol, *prev = NULL;
        int treat_coalescing;

        if (!lp)
                return NULL;
        ctx = isl_basic_set_get_ctx(lp);
        treat_coalescing = isl_options_get_schedule_treat_coalescing(ctx);
        tl = isl_tab_lexmin_from_basic_set(lp);

        do {
                sol = non_empty_solution(tl);
                if (!sol)
                        goto error;

                if (!carries_dependences(sol, n_edge)) {
                        if (!prev)
                                prev = isl_vec_alloc(ctx, 0);
                        isl_vec_free(sol);
                        sol = prev;
                        break;
                }
                prev = isl_vec_free(prev);
                if (!treat_coalescing)
                        break;
                for (i = 0; i < graph->n; ++i) {
                        struct isl_sched_node *node = &graph->node[i];

                        pos = find_node_coalescing(node, sol);
                        if (pos < 0)
                                goto error;
                        if (pos < node->nvar)
                                break;
                }
                if (i < graph->n) {
                        prev = sol;
                        tl = zero_out_node_coef(tl, &graph->node[i], pos);
                }
        } while (i < graph->n);

        isl_tab_lexmin_free(tl);

        return sol;
error:
        isl_tab_lexmin_free(tl);
        isl_vec_free(prev);
        isl_vec_free(sol);
        return NULL;
}

/* If "edge" is an edge from a node to itself, then add the corresponding
 * dependence relation to "umap".
 * If "node" has been compressed, then the dependence relation
 * is also compressed first.
 */
static __isl_give isl_union_map *add_intra(__isl_take isl_union_map *umap,
        struct isl_sched_edge *edge)
{
        isl_map *map;
        struct isl_sched_node *node = edge->src;

        if (edge->src != edge->dst)
                return umap;

        map = isl_map_copy(edge->map);
        if (node->compressed) {
                map = isl_map_preimage_domain_multi_aff(map,
                                    isl_multi_aff_copy(node->decompress));
                map = isl_map_preimage_range_multi_aff(map,
                                    isl_multi_aff_copy(node->decompress));
        }
        umap = isl_union_map_add_map(umap, map);
        return umap;
}

/* If "edge" is an edge from a node to another node, then add the corresponding
 * dependence relation to "umap".
 * If the source or destination nodes of "edge" have been compressed,
 * then the dependence relation is also compressed first.
 */
static __isl_give isl_union_map *add_inter(__isl_take isl_union_map *umap,
        struct isl_sched_edge *edge)
{
        isl_map *map;

        if (edge->src == edge->dst)
                return umap;

        map = isl_map_copy(edge->map);
        if (edge->src->compressed)
                map = isl_map_preimage_domain_multi_aff(map,
                                    isl_multi_aff_copy(edge->src->decompress));
        if (edge->dst->compressed)
                map = isl_map_preimage_range_multi_aff(map,
                                    isl_multi_aff_copy(edge->dst->decompress));
        umap = isl_union_map_add_map(umap, map);
        return umap;
}

/* For each (conditional) validity edge in "graph",
 * add the corresponding dependence relation using "add"
 * to a collection of dependence relations and return the result.
 * If "coincidence" is set, then coincidence edges are considered as well.
 */
static __isl_give isl_union_map *collect_validity(struct isl_sched_graph *graph,
        __isl_give isl_union_map *(*add)(__isl_take isl_union_map *umap,
                struct isl_sched_edge *edge), int coincidence)
{
        int i;
        isl_space *space;
        isl_union_map *umap;

        space = isl_space_copy(graph->node[0].space);
        umap = isl_union_map_empty(space);

        for (i = 0; i < graph->n_edge; ++i) {
                struct isl_sched_edge *edge = &graph->edge[i];

                if (!is_any_validity(edge) &&
                    (!coincidence || !is_coincidence(edge)))
                        continue;

                umap = add(umap, edge);
        }

        return umap;
}

/* For each dependence relation on a (conditional) validity edge
 * from a node to itself,
 * construct the set of coefficients of valid constraints for elements
 * in that dependence relation and collect the results.
 * If "coincidence" is set, then coincidence edges are considered as well.
 *
 * In particular, for each dependence relation R, constraints
 * on coefficients (c_0, c_n, c_x) are constructed such that
 *
 *      c_0 + c_n n + c_x d >= 0 for each d in delta R = { y - x | (x,y) in R }
 *
 * This computation is essentially the same as that performed
 * by intra_coefficients, except that it operates on multiple
 * edges together.
 *
 * Note that if a dependence relation is a union of basic maps,
 * then each basic map needs to be treated individually as it may only
 * be possible to carry the dependences expressed by some of those
 * basic maps and not all of them.
 * The collected validity constraints are therefore not coalesced and
 * it is assumed that they are not coalesced automatically.
 * Duplicate basic maps can be removed, however.
 * In particular, if the same basic map appears as a disjunct
 * in multiple edges, then it only needs to be carried once.
 */
static __isl_give isl_basic_set_list *collect_intra_validity(
        struct isl_sched_graph *graph, int coincidence)
{
        isl_union_map *intra;
        isl_union_set *delta;
        isl_basic_set_list *list;

        intra = collect_validity(graph, &add_intra, coincidence);
        delta = isl_union_map_deltas(intra);
        delta = isl_union_set_remove_divs(delta);
        list = isl_union_set_get_basic_set_list(delta);
        isl_union_set_free(delta);

        return isl_basic_set_list_coefficients(list);
}

/* For each dependence relation on a (conditional) validity edge
 * from a node to some other node,
 * construct the set of coefficients of valid constraints for elements
 * in that dependence relation and collect the results.
 * If "coincidence" is set, then coincidence edges are considered as well.
 *
 * In particular, for each dependence relation R, constraints
 * on coefficients (c_0, c_n, c_x, c_y) are constructed such that
 *
 *      c_0 + c_n n + c_x x + c_y y >= 0 for each (x,y) in R
 *
 * This computation is essentially the same as that performed
 * by inter_coefficients, except that it operates on multiple
 * edges together.
 *
 * Note that if a dependence relation is a union of basic maps,
 * then each basic map needs to be treated individually as it may only
 * be possible to carry the dependences expressed by some of those
 * basic maps and not all of them.
 * The collected validity constraints are therefore not coalesced and
 * it is assumed that they are not coalesced automatically.
 * Duplicate basic maps can be removed, however.
 * In particular, if the same basic map appears as a disjunct
 * in multiple edges, then it only needs to be carried once.
 */
static __isl_give isl_basic_set_list *collect_inter_validity(
        struct isl_sched_graph *graph, int coincidence)
{
        isl_union_map *inter;
        isl_union_set *wrap;
        isl_basic_set_list *list;

        inter = collect_validity(graph, &add_inter, coincidence);
        inter = isl_union_map_remove_divs(inter);
        wrap = isl_union_map_wrap(inter);
        list = isl_union_set_get_basic_set_list(wrap);
        isl_union_set_free(wrap);
        return isl_basic_set_list_coefficients(list);
}

/* Construct an LP problem for finding schedule coefficients
 * such that the schedule carries as many of the validity dependences
 * as possible and
 * return the lexicographically smallest non-trivial solution.
 * If "coincidence" is set, then try and carry coincidence edges as well.
 *
 * The variable "n_edge" stores the number of groups that should be carried.
 * If none of the "n_edge" groups can be carried
 * then return an empty vector.
 * If, moreover, "n_edge" is zero, then the LP problem does not even
 * need to be constructed.
 */
static __isl_give isl_vec *compute_carrying_sol(isl_ctx *ctx,
        struct isl_sched_graph *graph, int coincidence)
{
        int n_intra, n_inter;
        int n_edge;
        isl_basic_set *lp;
        struct isl_carry carry = { 0 };

        carry.intra = collect_intra_validity(graph, coincidence);
        carry.inter = collect_inter_validity(graph, coincidence);
        if (!carry.intra || !carry.inter)
                goto error;
        n_intra = isl_basic_set_list_n_basic_set(carry.intra);
        n_inter = isl_basic_set_list_n_basic_set(carry.inter);
        n_edge = n_intra + n_inter;
        if (n_edge == 0) {
                isl_carry_clear(&carry);
                return isl_vec_alloc(ctx, 0);
        }

        if (setup_carry_lp(ctx, graph, n_edge, carry.intra, carry.inter) < 0)
                goto error;

        isl_carry_clear(&carry);
        lp = isl_basic_set_copy(graph->lp);
        return non_neg_lexmin(graph, lp, n_edge);
error:
        isl_carry_clear(&carry);
        return NULL;
}

/* Construct a schedule row for each node such that as many validity dependences
 * as possible are carried and then continue with the next band.
 * If "coincidence" is set, then try and carry coincidence edges as well.
 *
 * If there are no validity dependences, then no dependence can be carried and
 * the procedure is guaranteed to fail.  If there is more than one component,
 * then try computing a schedule on each component separately
 * to prevent or at least postpone this failure.
 *
 * If a schedule row is computed, then check that dependences are carried
 * for at least one of the edges.
 *
 * If the computed schedule row turns out to be trivial on one or
 * more nodes where it should not be trivial, then we throw it away
 * and try again on each component separately.
 *
 * If there is only one component, then we accept the schedule row anyway,
 * but we do not consider it as a complete row and therefore do not
 * increment graph->n_row.  Note that the ranks of the nodes that
 * do get a non-trivial schedule part will get updated regardless and
 * graph->maxvar is computed based on these ranks.  The test for
 * whether more schedule rows are required in compute_schedule_wcc
 * is therefore not affected.
 *
 * Insert a band corresponding to the schedule row at position "node"
 * of the schedule tree and continue with the construction of the schedule.
 * This insertion and the continued construction is performed by split_scaled
 * after optionally checking for non-trivial common divisors.
 */
static __isl_give isl_schedule_node *carry(__isl_take isl_schedule_node *node,
        struct isl_sched_graph *graph, int coincidence)
{
        int trivial;
        isl_ctx *ctx;
        isl_vec *sol;

        if (!node)
                return NULL;

        ctx = isl_schedule_node_get_ctx(node);
        sol = compute_carrying_sol(ctx, graph, coincidence);
        if (!sol)
                return isl_schedule_node_free(node);
        if (sol->size == 0) {
                isl_vec_free(sol);
                if (graph->scc > 1)
                        return compute_component_schedule(node, graph, 1);
                isl_die(ctx, isl_error_unknown, "unable to carry dependences",
                        return isl_schedule_node_free(node));
        }

        trivial = is_any_trivial(graph, sol);
        if (trivial < 0) {
                sol = isl_vec_free(sol);
        } else if (trivial && graph->scc > 1) {
                isl_vec_free(sol);
                return compute_component_schedule(node, graph, 1);
        }

        if (update_schedule(graph, sol, 0, 0) < 0)
                return isl_schedule_node_free(node);
        if (trivial)
                graph->n_row--;

        return split_scaled(node, graph);
}

/* Construct a schedule row for each node such that as many validity dependences
 * as possible are carried and then continue with the next band.
 */
static __isl_give isl_schedule_node *carry_dependences(
        __isl_take isl_schedule_node *node, struct isl_sched_graph *graph)
{
        return carry(node, graph, 0);
}

/* Construct a schedule row for each node such that as many validity or
 * coincidence dependences as possible are carried and
 * then continue with the next band.
 */
static __isl_give isl_schedule_node *carry_coincidence(
        __isl_take isl_schedule_node *node, struct isl_sched_graph *graph)
{
        return carry(node, graph, 1);
}

/* Topologically sort statements mapped to the same schedule iteration
 * and add insert a sequence node in front of "node"
 * corresponding to this order.
 * If "initialized" is set, then it may be assumed that compute_maxvar
 * has been called on the current band.  Otherwise, call
 * compute_maxvar if and before carry_dependences gets called.
 *
 * If it turns out to be impossible to sort the statements apart,
 * because different dependences impose different orderings
 * on the statements, then we extend the schedule such that
 * it carries at least one more dependence.
 */
static __isl_give isl_schedule_node *sort_statements(
        __isl_take isl_schedule_node *node, struct isl_sched_graph *graph,
        int initialized)
{
        isl_ctx *ctx;
        isl_union_set_list *filters;

        if (!node)
                return NULL;

        ctx = isl_schedule_node_get_ctx(node);
        if (graph->n < 1)
                isl_die(ctx, isl_error_internal,
                        "graph should have at least one node",
                        return isl_schedule_node_free(node));

        if (graph->n == 1)
                return node;

        if (update_edges(ctx, graph) < 0)
                return isl_schedule_node_free(node);

        if (graph->n_edge == 0)
                return node;

        if (detect_sccs(ctx, graph) < 0)
                return isl_schedule_node_free(node);

        next_band(graph);
        if (graph->scc < graph->n) {
                if (!initialized && compute_maxvar(graph) < 0)
                        return isl_schedule_node_free(node);
                return carry_dependences(node, graph);
        }

        filters = extract_sccs(ctx, graph);
        node = isl_schedule_node_insert_sequence(node, filters);

        return node;
}

/* Are there any (non-empty) (conditional) validity edges in the graph?
 */
static int has_validity_edges(struct isl_sched_graph *graph)
{
        int i;

        for (i = 0; i < graph->n_edge; ++i) {
                int empty;

                empty = isl_map_plain_is_empty(graph->edge[i].map);
                if (empty < 0)
                        return -1;
                if (empty)
                        continue;
                if (is_any_validity(&graph->edge[i]))
                        return 1;
        }

        return 0;
}

/* Should we apply a Feautrier step?
 * That is, did the user request the Feautrier algorithm and are
 * there any validity dependences (left)?
 */
static int need_feautrier_step(isl_ctx *ctx, struct isl_sched_graph *graph)
{
        if (ctx->opt->schedule_algorithm != ISL_SCHEDULE_ALGORITHM_FEAUTRIER)
                return 0;

        return has_validity_edges(graph);
}

/* Compute a schedule for a connected dependence graph using Feautrier's
 * multi-dimensional scheduling algorithm and return the updated schedule node.
 *
 * The original algorithm is described in [1].
 * The main idea is to minimize the number of scheduling dimensions, by
 * trying to satisfy as many dependences as possible per scheduling dimension.
 *
 * [1] P. Feautrier, Some Efficient Solutions to the Affine Scheduling
 *     Problem, Part II: Multi-Dimensional Time.
 *     In Intl. Journal of Parallel Programming, 1992.
 */
static __isl_give isl_schedule_node *compute_schedule_wcc_feautrier(
        isl_schedule_node *node, struct isl_sched_graph *graph)
{
        return carry_dependences(node, graph);
}

/* Turn off the "local" bit on all (condition) edges.
 */
static void clear_local_edges(struct isl_sched_graph *graph)
{
        int i;

        for (i = 0; i < graph->n_edge; ++i)
                if (is_condition(&graph->edge[i]))
                        clear_local(&graph->edge[i]);
}

/* Does "graph" have both condition and conditional validity edges?
 */
static int need_condition_check(struct isl_sched_graph *graph)
{
        int i;
        int any_condition = 0;
        int any_conditional_validity = 0;

        for (i = 0; i < graph->n_edge; ++i) {
                if (is_condition(&graph->edge[i]))
                        any_condition = 1;
                if (is_conditional_validity(&graph->edge[i]))
                        any_conditional_validity = 1;
        }

        return any_condition && any_conditional_validity;
}

/* Does "graph" contain any coincidence edge?
 */
static int has_any_coincidence(struct isl_sched_graph *graph)
{
        int i;

        for (i = 0; i < graph->n_edge; ++i)
                if (is_coincidence(&graph->edge[i]))
                        return 1;

        return 0;
}

/* Extract the final schedule row as a map with the iteration domain
 * of "node" as domain.
 */
static __isl_give isl_map *final_row(struct isl_sched_node *node)
{
        isl_multi_aff *ma;
        int row;

        row = isl_mat_rows(node->sched) - 1;
        ma = node_extract_partial_schedule_multi_aff(node, row, 1);
        return isl_map_from_multi_aff(ma);
}

/* Is the conditional validity dependence in the edge with index "edge_index"
 * violated by the latest (i.e., final) row of the schedule?
 * That is, is i scheduled after j
 * for any conditional validity dependence i -> j?
 */
static int is_violated(struct isl_sched_graph *graph, int edge_index)
{
        isl_map *src_sched, *dst_sched, *map;
        struct isl_sched_edge *edge = &graph->edge[edge_index];
        int empty;

        src_sched = final_row(edge->src);
        dst_sched = final_row(edge->dst);
        map = isl_map_copy(edge->map);
        map = isl_map_apply_domain(map, src_sched);
        map = isl_map_apply_range(map, dst_sched);
        map = isl_map_order_gt(map, isl_dim_in, 0, isl_dim_out, 0);
        empty = isl_map_is_empty(map);
        isl_map_free(map);

        if (empty < 0)
                return -1;

        return !empty;
}

/* Does "graph" have any satisfied condition edges that
 * are adjacent to the conditional validity constraint with
 * domain "conditional_source" and range "conditional_sink"?
 *
 * A satisfied condition is one that is not local.
 * If a condition was forced to be local already (i.e., marked as local)
 * then there is no need to check if it is in fact local.
 *
 * Additionally, mark all adjacent condition edges found as local.
 */
static int has_adjacent_true_conditions(struct isl_sched_graph *graph,
        __isl_keep isl_union_set *conditional_source,
        __isl_keep isl_union_set *conditional_sink)
{
        int i;
        int any = 0;

        for (i = 0; i < graph->n_edge; ++i) {
                int adjacent, local;
                isl_union_map *condition;

                if (!is_condition(&graph->edge[i]))
                        continue;
                if (is_local(&graph->edge[i]))
                        continue;

                condition = graph->edge[i].tagged_condition;
                adjacent = domain_intersects(condition, conditional_sink);
                if (adjacent >= 0 && !adjacent)
                        adjacent = range_intersects(condition,
                                                        conditional_source);
                if (adjacent < 0)
                        return -1;
                if (!adjacent)
                        continue;

                set_local(&graph->edge[i]);

                local = is_condition_false(&graph->edge[i]);
                if (local < 0)
                        return -1;
                if (!local)
                        any = 1;
        }

        return any;
}

/* Are there any violated conditional validity dependences with
 * adjacent condition dependences that are not local with respect
 * to the current schedule?
 * That is, is the conditional validity constraint violated?
 *
 * Additionally, mark all those adjacent condition dependences as local.
 * We also mark those adjacent condition dependences that were not marked
 * as local before, but just happened to be local already.  This ensures
 * that they remain local if the schedule is recomputed.
 *
 * We first collect domain and range of all violated conditional validity
 * dependences and then check if there are any adjacent non-local
 * condition dependences.
 */
static int has_violated_conditional_constraint(isl_ctx *ctx,
        struct isl_sched_graph *graph)
{
        int i;
        int any = 0;
        isl_union_set *source, *sink;

        source = isl_union_set_empty(isl_space_params_alloc(ctx, 0));
        sink = isl_union_set_empty(isl_space_params_alloc(ctx, 0));
        for (i = 0; i < graph->n_edge; ++i) {
                isl_union_set *uset;
                isl_union_map *umap;
                int violated;

                if (!is_conditional_validity(&graph->edge[i]))
                        continue;

                violated = is_violated(graph, i);
                if (violated < 0)
                        goto error;
                if (!violated)
                        continue;

                any = 1;

                umap = isl_union_map_copy(graph->edge[i].tagged_validity);
                uset = isl_union_map_domain(umap);
                source = isl_union_set_union(source, uset);
                source = isl_union_set_coalesce(source);

                umap = isl_union_map_copy(graph->edge[i].tagged_validity);
                uset = isl_union_map_range(umap);
                sink = isl_union_set_union(sink, uset);
                sink = isl_union_set_coalesce(sink);
        }

        if (any)
                any = has_adjacent_true_conditions(graph, source, sink);

        isl_union_set_free(source);
        isl_union_set_free(sink);
        return any;
error:
        isl_union_set_free(source);
        isl_union_set_free(sink);
        return -1;
}

/* Examine the current band (the rows between graph->band_start and
 * graph->n_total_row), deciding whether to drop it or add it to "node"
 * and then continue with the computation of the next band, if any.
 * If "initialized" is set, then it may be assumed that compute_maxvar
 * has been called on the current band.  Otherwise, call
 * compute_maxvar if and before carry_dependences gets called.
 *
 * The caller keeps looking for a new row as long as
 * graph->n_row < graph->maxvar.  If the latest attempt to find
 * such a row failed (i.e., we still have graph->n_row < graph->maxvar),
 * then we either
 * - split between SCCs and start over (assuming we found an interesting
 *      pair of SCCs between which to split)
 * - continue with the next band (assuming the current band has at least
 *      one row)
 * - if outer coincidence needs to be enforced, then try to carry as many
 *      validity or coincidence dependences as possible and
 *      continue with the next band
 * - try to carry as many validity dependences as possible and
 *      continue with the next band
 * In each case, we first insert a band node in the schedule tree
 * if any rows have been computed.
 *
 * If the caller managed to complete the schedule, we insert a band node
 * (if any schedule rows were computed) and we finish off by topologically
 * sorting the statements based on the remaining dependences.
 */
static __isl_give isl_schedule_node *compute_schedule_finish_band(
        __isl_take isl_schedule_node *node, struct isl_sched_graph *graph,
        int initialized)
{
        int insert;

        if (!node)
                return NULL;

        if (graph->n_row < graph->maxvar) {
                isl_ctx *ctx;
                int empty = graph->n_total_row == graph->band_start;

                ctx = isl_schedule_node_get_ctx(node);
                if (!ctx->opt->schedule_maximize_band_depth && !empty)
                        return compute_next_band(node, graph, 1);
                if (graph->src_scc >= 0)
                        return compute_split_schedule(node, graph);
                if (!empty)
                        return compute_next_band(node, graph, 1);
                if (!initialized && compute_maxvar(graph) < 0)
                        return isl_schedule_node_free(node);
                if (isl_options_get_schedule_outer_coincidence(ctx))
                        return carry_coincidence(node, graph);
                return carry_dependences(node, graph);
        }

        insert = graph->n_total_row > graph->band_start;
        if (insert) {
                node = insert_current_band(node, graph, 1);
                node = isl_schedule_node_child(node, 0);
        }
        node = sort_statements(node, graph, initialized);
        if (insert)
                node = isl_schedule_node_parent(node);

        return node;
}

/* Construct a band of schedule rows for a connected dependence graph.
 * The caller is responsible for determining the strongly connected
 * components and calling compute_maxvar first.
 *
 * We try to find a sequence of as many schedule rows as possible that result
 * in non-negative dependence distances (independent of the previous rows
 * in the sequence, i.e., such that the sequence is tilable), with as
 * many of the initial rows as possible satisfying the coincidence constraints.
 * The computation stops if we can't find any more rows or if we have found
 * all the rows we wanted to find.
 *
 * If ctx->opt->schedule_outer_coincidence is set, then we force the
 * outermost dimension to satisfy the coincidence constraints.  If this
 * turns out to be impossible, we fall back on the general scheme above
 * and try to carry as many dependences as possible.
 *
 * If "graph" contains both condition and conditional validity dependences,
 * then we need to check that that the conditional schedule constraint
 * is satisfied, i.e., there are no violated conditional validity dependences
 * that are adjacent to any non-local condition dependences.
 * If there are, then we mark all those adjacent condition dependences
 * as local and recompute the current band.  Those dependences that
 * are marked local will then be forced to be local.
 * The initial computation is performed with no dependences marked as local.
 * If we are lucky, then there will be no violated conditional validity
 * dependences adjacent to any non-local condition dependences.
 * Otherwise, we mark some additional condition dependences as local and
 * recompute.  We continue this process until there are no violations left or
 * until we are no longer able to compute a schedule.
 * Since there are only a finite number of dependences,
 * there will only be a finite number of iterations.
 */
static isl_stat compute_schedule_wcc_band(isl_ctx *ctx,
        struct isl_sched_graph *graph)
{
        int has_coincidence;
        int use_coincidence;
        int force_coincidence = 0;
        int check_conditional;

        if (sort_sccs(graph) < 0)
                return isl_stat_error;

        clear_local_edges(graph);
        check_conditional = need_condition_check(graph);
        has_coincidence = has_any_coincidence(graph);

        if (ctx->opt->schedule_outer_coincidence)
                force_coincidence = 1;

        use_coincidence = has_coincidence;
        while (graph->n_row < graph->maxvar) {
                isl_vec *sol;
                int violated;
                int coincident;

                graph->src_scc = -1;
                graph->dst_scc = -1;

                if (setup_lp(ctx, graph, use_coincidence) < 0)
                        return isl_stat_error;
                sol = solve_lp(graph);
                if (!sol)
                        return isl_stat_error;
                if (sol->size == 0) {
                        int empty = graph->n_total_row == graph->band_start;

                        isl_vec_free(sol);
                        if (use_coincidence && (!force_coincidence || !empty)) {
                                use_coincidence = 0;
                                continue;
                        }
                        return isl_stat_ok;
                }
                coincident = !has_coincidence || use_coincidence;
                if (update_schedule(graph, sol, 1, coincident) < 0)
                        return isl_stat_error;

                if (!check_conditional)
                        continue;
                violated = has_violated_conditional_constraint(ctx, graph);
                if (violated < 0)
                        return isl_stat_error;
                if (!violated)
                        continue;
                if (reset_band(graph) < 0)
                        return isl_stat_error;
                use_coincidence = has_coincidence;
        }

        return isl_stat_ok;
}

/* Compute a schedule for a connected dependence graph by considering
 * the graph as a whole and return the updated schedule node.
 *
 * The actual schedule rows of the current band are computed by
 * compute_schedule_wcc_band.  compute_schedule_finish_band takes
 * care of integrating the band into "node" and continuing
 * the computation.
 */
static __isl_give isl_schedule_node *compute_schedule_wcc_whole(
        __isl_take isl_schedule_node *node, struct isl_sched_graph *graph)
{
        isl_ctx *ctx;

        if (!node)
                return NULL;

        ctx = isl_schedule_node_get_ctx(node);
        if (compute_schedule_wcc_band(ctx, graph) < 0)
                return isl_schedule_node_free(node);

        return compute_schedule_finish_band(node, graph, 1);
}

/* Clustering information used by compute_schedule_wcc_clustering.
 *
 * "n" is the number of SCCs in the original dependence graph
 * "scc" is an array of "n" elements, each representing an SCC
 * of the original dependence graph.  All entries in the same cluster
 * have the same number of schedule rows.
 * "scc_cluster" maps each SCC index to the cluster to which it belongs,
 * where each cluster is represented by the index of the first SCC
 * in the cluster.  Initially, each SCC belongs to a cluster containing
 * only that SCC.
 *
 * "scc_in_merge" is used by merge_clusters_along_edge to keep
 * track of which SCCs need to be merged.
 *
 * "cluster" contains the merged clusters of SCCs after the clustering
 * has completed.
 *
 * "scc_node" is a temporary data structure used inside copy_partial.
 * For each SCC, it keeps track of the number of nodes in the SCC
 * that have already been copied.
 */
struct isl_clustering {
        int n;
        struct isl_sched_graph *scc;
        struct isl_sched_graph *cluster;
        int *scc_cluster;
        int *scc_node;
        int *scc_in_merge;
};

/* Initialize the clustering data structure "c" from "graph".
 *
 * In particular, allocate memory, extract the SCCs from "graph"
 * into c->scc, initialize scc_cluster and construct
 * a band of schedule rows for each SCC.
 * Within each SCC, there is only one SCC by definition.
 * Each SCC initially belongs to a cluster containing only that SCC.
 */
static isl_stat clustering_init(isl_ctx *ctx, struct isl_clustering *c,
        struct isl_sched_graph *graph)
{
        int i;

        c->n = graph->scc;
        c->scc = isl_calloc_array(ctx, struct isl_sched_graph, c->n);
        c->cluster = isl_calloc_array(ctx, struct isl_sched_graph, c->n);
        c->scc_cluster = isl_calloc_array(ctx, int, c->n);
        c->scc_node = isl_calloc_array(ctx, int, c->n);
        c->scc_in_merge = isl_calloc_array(ctx, int, c->n);
        if (!c->scc || !c->cluster ||
            !c->scc_cluster || !c->scc_node || !c->scc_in_merge)
                return isl_stat_error;

        for (i = 0; i < c->n; ++i) {
                if (extract_sub_graph(ctx, graph, &node_scc_exactly,
                                        &edge_scc_exactly, i, &c->scc[i]) < 0)
                        return isl_stat_error;
                c->scc[i].scc = 1;
                if (compute_maxvar(&c->scc[i]) < 0)
                        return isl_stat_error;
                if (compute_schedule_wcc_band(ctx, &c->scc[i]) < 0)
                        return isl_stat_error;
                c->scc_cluster[i] = i;
        }

        return isl_stat_ok;
}

/* Free all memory allocated for "c".
 */
static void clustering_free(isl_ctx *ctx, struct isl_clustering *c)
{
        int i;

        if (c->scc)
                for (i = 0; i < c->n; ++i)
                        graph_free(ctx, &c->scc[i]);
        free(c->scc);
        if (c->cluster)
                for (i = 0; i < c->n; ++i)
                        graph_free(ctx, &c->cluster[i]);
        free(c->cluster);
        free(c->scc_cluster);
        free(c->scc_node);
        free(c->scc_in_merge);
}

/* Should we refrain from merging the cluster in "graph" with
 * any other cluster?
 * In particular, is its current schedule band empty and incomplete.
 */
static int bad_cluster(struct isl_sched_graph *graph)
{
        return graph->n_row < graph->maxvar &&
                graph->n_total_row == graph->band_start;
}

/* Is "edge" a proximity edge with a non-empty dependence relation?
 */
static isl_bool is_non_empty_proximity(struct isl_sched_edge *edge)
{
        if (!is_proximity(edge))
                return isl_bool_false;
        return isl_bool_not(isl_map_plain_is_empty(edge->map));
}

/* Return the index of an edge in "graph" that can be used to merge
 * two clusters in "c".
 * Return graph->n_edge if no such edge can be found.
 * Return -1 on error.
 *
 * In particular, return a proximity edge between two clusters
 * that is not marked "no_merge" and such that neither of the
 * two clusters has an incomplete, empty band.
 *
 * If there are multiple such edges, then try and find the most
 * appropriate edge to use for merging.  In particular, pick the edge
 * with the greatest weight.  If there are multiple of those,
 * then pick one with the shortest distance between
 * the two cluster representatives.
 */
static int find_proximity(struct isl_sched_graph *graph,
        struct isl_clustering *c)
{
        int i, best = graph->n_edge, best_dist, best_weight;

        for (i = 0; i < graph->n_edge; ++i) {
                struct isl_sched_edge *edge = &graph->edge[i];
                int dist, weight;
                isl_bool prox;

                prox = is_non_empty_proximity(edge);
                if (prox < 0)
                        return -1;
                if (!prox)
                        continue;
                if (edge->no_merge)
                        continue;
                if (bad_cluster(&c->scc[edge->src->scc]) ||
                    bad_cluster(&c->scc[edge->dst->scc]))
                        continue;
                dist = c->scc_cluster[edge->dst->scc] -
                        c->scc_cluster[edge->src->scc];
                if (dist == 0)
                        continue;
                weight = edge->weight;
                if (best < graph->n_edge) {
                        if (best_weight > weight)
                                continue;
                        if (best_weight == weight && best_dist <= dist)
                                continue;
                }
                best = i;
                best_dist = dist;
                best_weight = weight;
        }

        return best;
}

/* Internal data structure used in mark_merge_sccs.
 *
 * "graph" is the dependence graph in which a strongly connected
 * component is constructed.
 * "scc_cluster" maps each SCC index to the cluster to which it belongs.
 * "src" and "dst" are the indices of the nodes that are being merged.
 */
struct isl_mark_merge_sccs_data {
        struct isl_sched_graph *graph;
        int *scc_cluster;
        int src;
        int dst;
};

/* Check whether the cluster containing node "i" depends on the cluster
 * containing node "j".  If "i" and "j" belong to the same cluster,
 * then they are taken to depend on each other to ensure that
 * the resulting strongly connected component consists of complete
 * clusters.  Furthermore, if "i" and "j" are the two nodes that
 * are being merged, then they are taken to depend on each other as well.
 * Otherwise, check if there is a (conditional) validity dependence
 * from node[j] to node[i], forcing node[i] to follow node[j].
 */
static isl_bool cluster_follows(int i, int j, void *user)
{
        struct isl_mark_merge_sccs_data *data = user;
        struct isl_sched_graph *graph = data->graph;
        int *scc_cluster = data->scc_cluster;

        if (data->src == i && data->dst == j)
                return isl_bool_true;
        if (data->src == j && data->dst == i)
                return isl_bool_true;
        if (scc_cluster[graph->node[i].scc] == scc_cluster[graph->node[j].scc])
                return isl_bool_true;

        return graph_has_validity_edge(graph, &graph->node[j], &graph->node[i]);
}

/* Mark all SCCs that belong to either of the two clusters in "c"
 * connected by the edge in "graph" with index "edge", or to any
 * of the intermediate clusters.
 * The marking is recorded in c->scc_in_merge.
 *
 * The given edge has been selected for merging two clusters,
 * meaning that there is at least a proximity edge between the two nodes.
 * However, there may also be (indirect) validity dependences
 * between the two nodes.  When merging the two clusters, all clusters
 * containing one or more of the intermediate nodes along the
 * indirect validity dependences need to be merged in as well.
 *
 * First collect all such nodes by computing the strongly connected
 * component (SCC) containing the two nodes connected by the edge, where
 * the two nodes are considered to depend on each other to make
 * sure they end up in the same SCC.  Similarly, each node is considered
 * to depend on every other node in the same cluster to ensure
 * that the SCC consists of complete clusters.
 *
 * Then the original SCCs that contain any of these nodes are marked
 * in c->scc_in_merge.
 */
static isl_stat mark_merge_sccs(isl_ctx *ctx, struct isl_sched_graph *graph,
        int edge, struct isl_clustering *c)
{
        struct isl_mark_merge_sccs_data data;
        struct isl_tarjan_graph *g;
        int i;

        for (i = 0; i < c->n; ++i)
                c->scc_in_merge[i] = 0;

        data.graph = graph;
        data.scc_cluster = c->scc_cluster;
        data.src = graph->edge[edge].src - graph->node;
        data.dst = graph->edge[edge].dst - graph->node;

        g = isl_tarjan_graph_component(ctx, graph->n, data.dst,
                                        &cluster_follows, &data);
        if (!g)
                goto error;

        i = g->op;
        if (i < 3)
                isl_die(ctx, isl_error_internal,
                        "expecting at least two nodes in component",
                        goto error);
        if (g->order[--i] != -1)
                isl_die(ctx, isl_error_internal,
                        "expecting end of component marker", goto error);

        for (--i; i >= 0 && g->order[i] != -1; --i) {
                int scc = graph->node[g->order[i]].scc;
                c->scc_in_merge[scc] = 1;
        }

        isl_tarjan_graph_free(g);
        return isl_stat_ok;
error:
        isl_tarjan_graph_free(g);
        return isl_stat_error;
}

/* Construct the identifier "cluster_i".
 */
static __isl_give isl_id *cluster_id(isl_ctx *ctx, int i)
{
        char name[40];

        snprintf(name, sizeof(name), "cluster_%d", i);
        return isl_id_alloc(ctx, name, NULL);
}

/* Construct the space of the cluster with index "i" containing
 * the strongly connected component "scc".
 *
 * In particular, construct a space called cluster_i with dimension equal
 * to the number of schedule rows in the current band of "scc".
 */
static __isl_give isl_space *cluster_space(struct isl_sched_graph *scc, int i)
{
        int nvar;
        isl_space *space;
        isl_id *id;

        nvar = scc->n_total_row - scc->band_start;
        space = isl_space_copy(scc->node[0].space);
        space = isl_space_params(space);
        space = isl_space_set_from_params(space);
        space = isl_space_add_dims(space, isl_dim_set, nvar);
        id = cluster_id(isl_space_get_ctx(space), i);
        space = isl_space_set_tuple_id(space, isl_dim_set, id);

        return space;
}

/* Collect the domain of the graph for merging clusters.
 *
 * In particular, for each cluster with first SCC "i", construct
 * a set in the space called cluster_i with dimension equal
 * to the number of schedule rows in the current band of the cluster.
 */
static __isl_give isl_union_set *collect_domain(isl_ctx *ctx,
        struct isl_sched_graph *graph, struct isl_clustering *c)
{
        int i;
        isl_space *space;
        isl_union_set *domain;

        space = isl_space_params_alloc(ctx, 0);
        domain = isl_union_set_empty(space);

        for (i = 0; i < graph->scc; ++i) {
                isl_space *space;

                if (!c->scc_in_merge[i])
                        continue;
                if (c->scc_cluster[i] != i)
                        continue;
                space = cluster_space(&c->scc[i], i);
                domain = isl_union_set_add_set(domain, isl_set_universe(space));
        }

        return domain;
}

/* Construct a map from the original instances to the corresponding
 * cluster instance in the current bands of the clusters in "c".
 */
static __isl_give isl_union_map *collect_cluster_map(isl_ctx *ctx,
        struct isl_sched_graph *graph, struct isl_clustering *c)
{
        int i, j;
        isl_space *space;
        isl_union_map *cluster_map;

        space = isl_space_params_alloc(ctx, 0);
        cluster_map = isl_union_map_empty(space);
        for (i = 0; i < graph->scc; ++i) {
                int start, n;
                isl_id *id;

                if (!c->scc_in_merge[i])
                        continue;

                id = cluster_id(ctx, c->scc_cluster[i]);
                start = c->scc[i].band_start;
                n = c->scc[i].n_total_row - start;
                for (j = 0; j < c->scc[i].n; ++j) {
                        isl_multi_aff *ma;
                        isl_map *map;
                        struct isl_sched_node *node = &c->scc[i].node[j];

                        ma = node_extract_partial_schedule_multi_aff(node,
                                                                    start, n);
                        ma = isl_multi_aff_set_tuple_id(ma, isl_dim_out,
                                                            isl_id_copy(id));
                        map = isl_map_from_multi_aff(ma);
                        cluster_map = isl_union_map_add_map(cluster_map, map);
                }
                isl_id_free(id);
        }

        return cluster_map;
}

/* Add "umap" to the schedule constraints "sc" of all types of "edge"
 * that are not isl_edge_condition or isl_edge_conditional_validity.
 */
static __isl_give isl_schedule_constraints *add_non_conditional_constraints(
        struct isl_sched_edge *edge, __isl_keep isl_union_map *umap,
        __isl_take isl_schedule_constraints *sc)
{
        enum isl_edge_type t;

        if (!sc)
                return NULL;

        for (t = isl_edge_first; t <= isl_edge_last; ++t) {
                if (t == isl_edge_condition ||
                    t == isl_edge_conditional_validity)
                        continue;
                if (!is_type(edge, t))
                        continue;
                sc = isl_schedule_constraints_add(sc, t,
                                                    isl_union_map_copy(umap));
        }

        return sc;
}

/* Add schedule constraints of types isl_edge_condition and
 * isl_edge_conditional_validity to "sc" by applying "umap" to
 * the domains of the wrapped relations in domain and range
 * of the corresponding tagged constraints of "edge".
 */
static __isl_give isl_schedule_constraints *add_conditional_constraints(
        struct isl_sched_edge *edge, __isl_keep isl_union_map *umap,
        __isl_take isl_schedule_constraints *sc)
{
        enum isl_edge_type t;
        isl_union_map *tagged;

        for (t = isl_edge_condition; t <= isl_edge_conditional_validity; ++t) {
                if (!is_type(edge, t))
                        continue;
                if (t == isl_edge_condition)
                        tagged = isl_union_map_copy(edge->tagged_condition);
                else
                        tagged = isl_union_map_copy(edge->tagged_validity);
                tagged = isl_union_map_zip(tagged);
                tagged = isl_union_map_apply_domain(tagged,
                                        isl_union_map_copy(umap));
                tagged = isl_union_map_zip(tagged);
                sc = isl_schedule_constraints_add(sc, t, tagged);
                if (!sc)
                        return NULL;
        }

        return sc;
}

/* Given a mapping "cluster_map" from the original instances to
 * the cluster instances, add schedule constraints on the clusters
 * to "sc" corresponding to the original constraints represented by "edge".
 *
 * For non-tagged dependence constraints, the cluster constraints
 * are obtained by applying "cluster_map" to the edge->map.
 *
 * For tagged dependence constraints, "cluster_map" needs to be applied
 * to the domains of the wrapped relations in domain and range
 * of the tagged dependence constraints.  Pick out the mappings
 * from these domains from "cluster_map" and construct their product.
 * This mapping can then be applied to the pair of domains.
 */
static __isl_give isl_schedule_constraints *collect_edge_constraints(
        struct isl_sched_edge *edge, __isl_keep isl_union_map *cluster_map,
        __isl_take isl_schedule_constraints *sc)
{
        isl_union_map *umap;
        isl_space *space;
        isl_union_set *uset;
        isl_union_map *umap1, *umap2;

        if (!sc)
                return NULL;

        umap = isl_union_map_from_map(isl_map_copy(edge->map));
        umap = isl_union_map_apply_domain(umap,
                                isl_union_map_copy(cluster_map));
        umap = isl_union_map_apply_range(umap,
                                isl_union_map_copy(cluster_map));
        sc = add_non_conditional_constraints(edge, umap, sc);
        isl_union_map_free(umap);

        if (!sc || (!is_condition(edge) && !is_conditional_validity(edge)))
                return sc;

        space = isl_space_domain(isl_map_get_space(edge->map));
        uset = isl_union_set_from_set(isl_set_universe(space));
        umap1 = isl_union_map_copy(cluster_map);
        umap1 = isl_union_map_intersect_domain(umap1, uset);
        space = isl_space_range(isl_map_get_space(edge->map));
        uset = isl_union_set_from_set(isl_set_universe(space));
        umap2 = isl_union_map_copy(cluster_map);
        umap2 = isl_union_map_intersect_domain(umap2, uset);
        umap = isl_union_map_product(umap1, umap2);

        sc = add_conditional_constraints(edge, umap, sc);

        isl_union_map_free(umap);
        return sc;
}

/* Given a mapping "cluster_map" from the original instances to
 * the cluster instances, add schedule constraints on the clusters
 * to "sc" corresponding to all edges in "graph" between nodes that
 * belong to SCCs that are marked for merging in "scc_in_merge".
 */
static __isl_give isl_schedule_constraints *collect_constraints(
        struct isl_sched_graph *graph, int *scc_in_merge,
        __isl_keep isl_union_map *cluster_map,
        __isl_take isl_schedule_constraints *sc)
{
        int i;

        for (i = 0; i < graph->n_edge; ++i) {
                struct isl_sched_edge *edge = &graph->edge[i];

                if (!scc_in_merge[edge->src->scc])
                        continue;
                if (!scc_in_merge[edge->dst->scc])
                        continue;
                sc = collect_edge_constraints(edge, cluster_map, sc);
        }

        return sc;
}

/* Construct a dependence graph for scheduling clusters with respect
 * to each other and store the result in "merge_graph".
 * In particular, the nodes of the graph correspond to the schedule
 * dimensions of the current bands of those clusters that have been
 * marked for merging in "c".
 *
 * First construct an isl_schedule_constraints object for this domain
 * by transforming the edges in "graph" to the domain.
 * Then initialize a dependence graph for scheduling from these
 * constraints.
 */
static isl_stat init_merge_graph(isl_ctx *ctx, struct isl_sched_graph *graph,
        struct isl_clustering *c, struct isl_sched_graph *merge_graph)
{
        isl_union_set *domain;
        isl_union_map *cluster_map;
        isl_schedule_constraints *sc;
        isl_stat r;

        domain = collect_domain(ctx, graph, c);
        sc = isl_schedule_constraints_on_domain(domain);
        if (!sc)
                return isl_stat_error;
        cluster_map = collect_cluster_map(ctx, graph, c);
        sc = collect_constraints(graph, c->scc_in_merge, cluster_map, sc);
        isl_union_map_free(cluster_map);

        r = graph_init(merge_graph, sc);

        isl_schedule_constraints_free(sc);

        return r;
}

/* Compute the maximal number of remaining schedule rows that still need
 * to be computed for the nodes that belong to clusters with the maximal
 * dimension for the current band (i.e., the band that is to be merged).
 * Only clusters that are about to be merged are considered.
 * "maxvar" is the maximal dimension for the current band.
 * "c" contains information about the clusters.
 *
 * Return the maximal number of remaining schedule rows or -1 on error.
 */
static int compute_maxvar_max_slack(int maxvar, struct isl_clustering *c)
{
        int i, j;
        int max_slack;

        max_slack = 0;
        for (i = 0; i < c->n; ++i) {
                int nvar;
                struct isl_sched_graph *scc;

                if (!c->scc_in_merge[i])
                        continue;
                scc = &c->scc[i];
                nvar = scc->n_total_row - scc->band_start;
                if (nvar != maxvar)
                        continue;
                for (j = 0; j < scc->n; ++j) {
                        struct isl_sched_node *node = &scc->node[j];
                        int slack;

                        if (node_update_cmap(node) < 0)
                                return -1;
                        slack = node->nvar - node->rank;
                        if (slack > max_slack)
                                max_slack = slack;
                }
        }

        return max_slack;
}

/* If there are any clusters where the dimension of the current band
 * (i.e., the band that is to be merged) is smaller than "maxvar" and
 * if there are any nodes in such a cluster where the number
 * of remaining schedule rows that still need to be computed
 * is greater than "max_slack", then return the smallest current band
 * dimension of all these clusters.  Otherwise return the original value
 * of "maxvar".  Return -1 in case of any error.
 * Only clusters that are about to be merged are considered.
 * "c" contains information about the clusters.
 */
static int limit_maxvar_to_slack(int maxvar, int max_slack,
        struct isl_clustering *c)
{
        int i, j;

        for (i = 0; i < c->n; ++i) {
                int nvar;
                struct isl_sched_graph *scc;

                if (!c->scc_in_merge[i])
                        continue;
                scc = &c->scc[i];
                nvar = scc->n_total_row - scc->band_start;
                if (nvar >= maxvar)
                        continue;
                for (j = 0; j < scc->n; ++j) {
                        struct isl_sched_node *node = &scc->node[j];
                        int slack;

                        if (node_update_cmap(node) < 0)
                                return -1;
                        slack = node->nvar - node->rank;
                        if (slack > max_slack) {
                                maxvar = nvar;
                                break;
                        }
                }
        }

        return maxvar;
}

/* Adjust merge_graph->maxvar based on the number of remaining schedule rows
 * that still need to be computed.  In particular, if there is a node
 * in a cluster where the dimension of the current band is smaller
 * than merge_graph->maxvar, but the number of remaining schedule rows
 * is greater than that of any node in a cluster with the maximal
 * dimension for the current band (i.e., merge_graph->maxvar),
 * then adjust merge_graph->maxvar to the (smallest) current band dimension
 * of those clusters.  Without this adjustment, the total number of
 * schedule dimensions would be increased, resulting in a skewed view
 * of the number of coincident dimensions.
 * "c" contains information about the clusters.
 *
 * If the maximize_band_depth option is set and merge_graph->maxvar is reduced,
 * then there is no point in attempting any merge since it will be rejected
 * anyway.  Set merge_graph->maxvar to zero in such cases.
 */
static isl_stat adjust_maxvar_to_slack(isl_ctx *ctx,
        struct isl_sched_graph *merge_graph, struct isl_clustering *c)
{
        int max_slack, maxvar;

        max_slack = compute_maxvar_max_slack(merge_graph->maxvar, c);
        if (max_slack < 0)
                return isl_stat_error;
        maxvar = limit_maxvar_to_slack(merge_graph->maxvar, max_slack, c);
        if (maxvar < 0)
                return isl_stat_error;

        if (maxvar < merge_graph->maxvar) {
                if (isl_options_get_schedule_maximize_band_depth(ctx))
                        merge_graph->maxvar = 0;
                else
                        merge_graph->maxvar = maxvar;
        }

        return isl_stat_ok;
}

/* Return the number of coincident dimensions in the current band of "graph",
 * where the nodes of "graph" are assumed to be scheduled by a single band.
 */
static int get_n_coincident(struct isl_sched_graph *graph)
{
        int i;

        for (i = graph->band_start; i < graph->n_total_row; ++i)
                if (!graph->node[0].coincident[i])
                        break;

        return i - graph->band_start;
}

/* Should the clusters be merged based on the cluster schedule
 * in the current (and only) band of "merge_graph", given that
 * coincidence should be maximized?
 *
 * If the number of coincident schedule dimensions in the merged band
 * would be less than the maximal number of coincident schedule dimensions
 * in any of the merged clusters, then the clusters should not be merged.
 */
static isl_bool ok_to_merge_coincident(struct isl_clustering *c,
        struct isl_sched_graph *merge_graph)
{
        int i;
        int n_coincident;
        int max_coincident;

        max_coincident = 0;
        for (i = 0; i < c->n; ++i) {
                if (!c->scc_in_merge[i])
                        continue;
                n_coincident = get_n_coincident(&c->scc[i]);
                if (n_coincident > max_coincident)
                        max_coincident = n_coincident;
        }

        n_coincident = get_n_coincident(merge_graph);

        return n_coincident >= max_coincident;
}

/* Return the transformation on "node" expressed by the current (and only)
 * band of "merge_graph" applied to the clusters in "c".
 *
 * First find the representation of "node" in its SCC in "c" and
 * extract the transformation expressed by the current band.
 * Then extract the transformation applied by "merge_graph"
 * to the cluster to which this SCC belongs.
 * Combine the two to obtain the complete transformation on the node.
 *
 * Note that the range of the first transformation is an anonymous space,
 * while the domain of the second is named "cluster_X".  The range
 * of the former therefore needs to be adjusted before the two
 * can be combined.
 */
static __isl_give isl_map *extract_node_transformation(isl_ctx *ctx,
        struct isl_sched_node *node, struct isl_clustering *c,
        struct isl_sched_graph *merge_graph)
{
        struct isl_sched_node *scc_node, *cluster_node;
        int start, n;
        isl_id *id;
        isl_space *space;
        isl_multi_aff *ma, *ma2;

        scc_node = graph_find_node(ctx, &c->scc[node->scc], node->space);
        start = c->scc[node->scc].band_start;
        n = c->scc[node->scc].n_total_row - start;
        ma = node_extract_partial_schedule_multi_aff(scc_node, start, n);
        space = cluster_space(&c->scc[node->scc], c->scc_cluster[node->scc]);
        cluster_node = graph_find_node(ctx, merge_graph, space);
        if (space && !cluster_node)
                isl_die(ctx, isl_error_internal, "unable to find cluster",
                        space = isl_space_free(space));
        id = isl_space_get_tuple_id(space, isl_dim_set);
        ma = isl_multi_aff_set_tuple_id(ma, isl_dim_out, id);
        isl_space_free(space);
        n = merge_graph->n_total_row;
        ma2 = node_extract_partial_schedule_multi_aff(cluster_node, 0, n);
        ma = isl_multi_aff_pullback_multi_aff(ma2, ma);

        return isl_map_from_multi_aff(ma);
}

/* Give a set of distances "set", are they bounded by a small constant
 * in direction "pos"?
 * In practice, check if they are bounded by 2 by checking that there
 * are no elements with a value greater than or equal to 3 or
 * smaller than or equal to -3.
 */
static isl_bool distance_is_bounded(__isl_keep isl_set *set, int pos)
{
        isl_bool bounded;
        isl_set *test;

        if (!set)
                return isl_bool_error;

        test = isl_set_copy(set);
        test = isl_set_lower_bound_si(test, isl_dim_set, pos, 3);
        bounded = isl_set_is_empty(test);
        isl_set_free(test);

        if (bounded < 0 || !bounded)
                return bounded;

        test = isl_set_copy(set);
        test = isl_set_upper_bound_si(test, isl_dim_set, pos, -3);
        bounded = isl_set_is_empty(test);
        isl_set_free(test);

        return bounded;
}

/* Does the set "set" have a fixed (but possible parametric) value
 * at dimension "pos"?
 */
static isl_bool has_single_value(__isl_keep isl_set *set, int pos)
{
        int n;
        isl_bool single;

        if (!set)
                return isl_bool_error;
        set = isl_set_copy(set);
        n = isl_set_dim(set, isl_dim_set);
        set = isl_set_project_out(set, isl_dim_set, pos + 1, n - (pos + 1));
        set = isl_set_project_out(set, isl_dim_set, 0, pos);
        single = isl_set_is_singleton(set);
        isl_set_free(set);

        return single;
}

/* Does "map" have a fixed (but possible parametric) value
 * at dimension "pos" of either its domain or its range?
 */
static isl_bool has_singular_src_or_dst(__isl_keep isl_map *map, int pos)
{
        isl_set *set;
        isl_bool single;

        set = isl_map_domain(isl_map_copy(map));
        single = has_single_value(set, pos);
        isl_set_free(set);

        if (single < 0 || single)
                return single;

        set = isl_map_range(isl_map_copy(map));
        single = has_single_value(set, pos);
        isl_set_free(set);

        return single;
}

/* Does the edge "edge" from "graph" have bounded dependence distances
 * in the merged graph "merge_graph" of a selection of clusters in "c"?
 *
 * Extract the complete transformations of the source and destination
 * nodes of the edge, apply them to the edge constraints and
 * compute the differences.  Finally, check if these differences are bounded
 * in each direction.
 *
 * If the dimension of the band is greater than the number of
 * dimensions that can be expected to be optimized by the edge
 * (based on its weight), then also allow the differences to be unbounded
 * in the remaining dimensions, but only if either the source or
 * the destination has a fixed value in that direction.
 * This allows a statement that produces values that are used by
 * several instances of another statement to be merged with that
 * other statement.
 * However, merging such clusters will introduce an inherently
 * large proximity distance inside the merged cluster, meaning
 * that proximity distances will no longer be optimized in
 * subsequent merges.  These merges are therefore only allowed
 * after all other possible merges have been tried.
 * The first time such a merge is encountered, the weight of the edge
 * is replaced by a negative weight.  The second time (i.e., after
 * all merges over edges with a non-negative weight have been tried),
 * the merge is allowed.
 */
static isl_bool has_bounded_distances(isl_ctx *ctx, struct isl_sched_edge *edge,
        struct isl_sched_graph *graph, struct isl_clustering *c,
        struct isl_sched_graph *merge_graph)
{
        int i, n, n_slack;
        isl_bool bounded;
        isl_map *map, *t;
        isl_set *dist;

        map = isl_map_copy(edge->map);
        t = extract_node_transformation(ctx, edge->src, c, merge_graph);
        map = isl_map_apply_domain(map, t);
        t = extract_node_transformation(ctx, edge->dst, c, merge_graph);
        map = isl_map_apply_range(map, t);
        dist = isl_map_deltas(isl_map_copy(map));

        bounded = isl_bool_true;
        n = isl_set_dim(dist, isl_dim_set);
        n_slack = n - edge->weight;
        if (edge->weight < 0)
                n_slack -= graph->max_weight + 1;
        for (i = 0; i < n; ++i) {
                isl_bool bounded_i, singular_i;

                bounded_i = distance_is_bounded(dist, i);
                if (bounded_i < 0)
                        goto error;
                if (bounded_i)
                        continue;
                if (edge->weight >= 0)
                        bounded = isl_bool_false;
                n_slack--;
                if (n_slack < 0)
                        break;
                singular_i = has_singular_src_or_dst(map, i);
                if (singular_i < 0)
                        goto error;
                if (singular_i)
                        continue;
                bounded = isl_bool_false;
                break;
        }
        if (!bounded && i >= n && edge->weight >= 0)
                edge->weight -= graph->max_weight + 1;
        isl_map_free(map);
        isl_set_free(dist);

        return bounded;
error:
        isl_map_free(map);
        isl_set_free(dist);
        return isl_bool_error;
}

/* Should the clusters be merged based on the cluster schedule
 * in the current (and only) band of "merge_graph"?
 * "graph" is the original dependence graph, while "c" records
 * which SCCs are involved in the latest merge.
 *
 * In particular, is there at least one proximity constraint
 * that is optimized by the merge?
 *
 * A proximity constraint is considered to be optimized
 * if the dependence distances are small.
 */
static isl_bool ok_to_merge_proximity(isl_ctx *ctx,
        struct isl_sched_graph *graph, struct isl_clustering *c,
        struct isl_sched_graph *merge_graph)
{
        int i;

        for (i = 0; i < graph->n_edge; ++i) {
                struct isl_sched_edge *edge = &graph->edge[i];
                isl_bool bounded;

                if (!is_proximity(edge))
                        continue;
                if (!c->scc_in_merge[edge->src->scc])
                        continue;
                if (!c->scc_in_merge[edge->dst->scc])
                        continue;
                if (c->scc_cluster[edge->dst->scc] ==
                    c->scc_cluster[edge->src->scc])
                        continue;
                bounded = has_bounded_distances(ctx, edge, graph, c,
                                                merge_graph);
                if (bounded < 0 || bounded)
                        return bounded;
        }

        return isl_bool_false;
}

/* Should the clusters be merged based on the cluster schedule
 * in the current (and only) band of "merge_graph"?
 * "graph" is the original dependence graph, while "c" records
 * which SCCs are involved in the latest merge.
 *
 * If the current band is empty, then the clusters should not be merged.
 *
 * If the band depth should be maximized and the merge schedule
 * is incomplete (meaning that the dimension of some of the schedule
 * bands in the original schedule will be reduced), then the clusters
 * should not be merged.
 *
 * If the schedule_maximize_coincidence option is set, then check that
 * the number of coincident schedule dimensions is not reduced.
 *
 * Finally, only allow the merge if at least one proximity
 * constraint is optimized.
 */
static isl_bool ok_to_merge(isl_ctx *ctx, struct isl_sched_graph *graph,
        struct isl_clustering *c, struct isl_sched_graph *merge_graph)
{
        if (merge_graph->n_total_row == merge_graph->band_start)
                return isl_bool_false;

        if (isl_options_get_schedule_maximize_band_depth(ctx) &&
            merge_graph->n_total_row < merge_graph->maxvar)
                return isl_bool_false;

        if (isl_options_get_schedule_maximize_coincidence(ctx)) {
                isl_bool ok;

                ok = ok_to_merge_coincident(c, merge_graph);
                if (ok < 0 || !ok)
                        return ok;
        }

        return ok_to_merge_proximity(ctx, graph, c, merge_graph);
}

/* Apply the schedule in "t_node" to the "n" rows starting at "first"
 * of the schedule in "node" and return the result.
 *
 * That is, essentially compute
 *
 *      T * N(first:first+n-1)
 *
 * taking into account the constant term and the parameter coefficients
 * in "t_node".
 */
static __isl_give isl_mat *node_transformation(isl_ctx *ctx,
        struct isl_sched_node *t_node, struct isl_sched_node *node,
        int first, int n)
{
        int i, j;
        isl_mat *t;
        int n_row, n_col, n_param, n_var;

        n_param = node->nparam;
        n_var = node->nvar;
        n_row = isl_mat_rows(t_node->sched);
        n_col = isl_mat_cols(node->sched);
        t = isl_mat_alloc(ctx, n_row, n_col);
        if (!t)
                return NULL;
        for (i = 0; i < n_row; ++i) {
                isl_seq_cpy(t->row[i], t_node->sched->row[i], 1 + n_param);
                isl_seq_clr(t->row[i] + 1 + n_param, n_var);
                for (j = 0; j < n; ++j)
                        isl_seq_addmul(t->row[i],
                                        t_node->sched->row[i][1 + n_param + j],
                                        node->sched->row[first + j],
                                        1 + n_param + n_var);
        }
        return t;
}

/* Apply the cluster schedule in "t_node" to the current band
 * schedule of the nodes in "graph".
 *
 * In particular, replace the rows starting at band_start
 * by the result of applying the cluster schedule in "t_node"
 * to the original rows.
 *
 * The coincidence of the schedule is determined by the coincidence
 * of the cluster schedule.
 */
static isl_stat transform(isl_ctx *ctx, struct isl_sched_graph *graph,
        struct isl_sched_node *t_node)
{
        int i, j;
        int n_new;
        int start, n;

        start = graph->band_start;
        n = graph->n_total_row - start;

        n_new = isl_mat_rows(t_node->sched);
        for (i = 0; i < graph->n; ++i) {
                struct isl_sched_node *node = &graph->node[i];
                isl_mat *t;

                t = node_transformation(ctx, t_node, node, start, n);
                node->sched = isl_mat_drop_rows(node->sched, start, n);
                node->sched = isl_mat_concat(node->sched, t);
                node->sched_map = isl_map_free(node->sched_map);
                if (!node->sched)
                        return isl_stat_error;
                for (j = 0; j < n_new; ++j)
                        node->coincident[start + j] = t_node->coincident[j];
        }
        graph->n_total_row -= n;
        graph->n_row -= n;
        graph->n_total_row += n_new;
        graph->n_row += n_new;

        return isl_stat_ok;
}

/* Merge the clusters marked for merging in "c" into a single
 * cluster using the cluster schedule in the current band of "merge_graph".
 * The representative SCC for the new cluster is the SCC with
 * the smallest index.
 *
 * The current band schedule of each SCC in the new cluster is obtained
 * by applying the schedule of the corresponding original cluster
 * to the original band schedule.
 * All SCCs in the new cluster have the same number of schedule rows.
 */
static isl_stat merge(isl_ctx *ctx, struct isl_clustering *c,
        struct isl_sched_graph *merge_graph)
{
        int i;
        int cluster = -1;
        isl_space *space;

        for (i = 0; i < c->n; ++i) {
                struct isl_sched_node *node;

                if (!c->scc_in_merge[i])
                        continue;
                if (cluster < 0)
                        cluster = i;
                space = cluster_space(&c->scc[i], c->scc_cluster[i]);
                if (!space)
                        return isl_stat_error;
                node = graph_find_node(ctx, merge_graph, space);
                isl_space_free(space);
                if (!node)
                        isl_die(ctx, isl_error_internal,
                                "unable to find cluster",
                                return isl_stat_error);
                if (transform(ctx, &c->scc[i], node) < 0)
                        return isl_stat_error;
                c->scc_cluster[i] = cluster;
        }

        return isl_stat_ok;
}

/* Try and merge the clusters of SCCs marked in c->scc_in_merge
 * by scheduling the current cluster bands with respect to each other.
 *
 * Construct a dependence graph with a space for each cluster and
 * with the coordinates of each space corresponding to the schedule
 * dimensions of the current band of that cluster.
 * Construct a cluster schedule in this cluster dependence graph and
 * apply it to the current cluster bands if it is applicable
 * according to ok_to_merge.
 *
 * If the number of remaining schedule dimensions in a cluster
 * with a non-maximal current schedule dimension is greater than
 * the number of remaining schedule dimensions in clusters
 * with a maximal current schedule dimension, then restrict
 * the number of rows to be computed in the cluster schedule
 * to the minimal such non-maximal current schedule dimension.
 * Do this by adjusting merge_graph.maxvar.
 *
 * Return isl_bool_true if the clusters have effectively been merged
 * into a single cluster.
 *
 * Note that since the standard scheduling algorithm minimizes the maximal
 * distance over proximity constraints, the proximity constraints between
 * the merged clusters may not be optimized any further than what is
 * sufficient to bring the distances within the limits of the internal
 * proximity constraints inside the individual clusters.
 * It may therefore make sense to perform an additional translation step
 * to bring the clusters closer to each other, while maintaining
 * the linear part of the merging schedule found using the standard
 * scheduling algorithm.
 */
static isl_bool try_merge(isl_ctx *ctx, struct isl_sched_graph *graph,
        struct isl_clustering *c)
{
        struct isl_sched_graph merge_graph = { 0 };
        isl_bool merged;

        if (init_merge_graph(ctx, graph, c, &merge_graph) < 0)
                goto error;

        if (compute_maxvar(&merge_graph) < 0)
                goto error;
        if (adjust_maxvar_to_slack(ctx, &merge_graph,c) < 0)
                goto error;
        if (compute_schedule_wcc_band(ctx, &merge_graph) < 0)
                goto error;
        merged = ok_to_merge(ctx, graph, c, &merge_graph);
        if (merged && merge(ctx, c, &merge_graph) < 0)
                goto error;

        graph_free(ctx, &merge_graph);
        return merged;
error:
        graph_free(ctx, &merge_graph);
        return isl_bool_error;
}

/* Is there any edge marked "no_merge" between two SCCs that are
 * about to be merged (i.e., that are set in "scc_in_merge")?
 * "merge_edge" is the proximity edge along which the clusters of SCCs
 * are going to be merged.
 *
 * If there is any edge between two SCCs with a negative weight,
 * while the weight of "merge_edge" is non-negative, then this
 * means that the edge was postponed.  "merge_edge" should then
 * also be postponed since merging along the edge with negative weight should
 * be postponed until all edges with non-negative weight have been tried.
 * Replace the weight of "merge_edge" by a negative weight as well and
 * tell the caller not to attempt a merge.
 */
static int any_no_merge(struct isl_sched_graph *graph, int *scc_in_merge,
        struct isl_sched_edge *merge_edge)
{
        int i;

        for (i = 0; i < graph->n_edge; ++i) {
                struct isl_sched_edge *edge = &graph->edge[i];

                if (!scc_in_merge[edge->src->scc])
                        continue;
                if (!scc_in_merge[edge->dst->scc])
                        continue;
                if (edge->no_merge)
                        return 1;
                if (merge_edge->weight >= 0 && edge->weight < 0) {
                        merge_edge->weight -= graph->max_weight + 1;
                        return 1;
                }
        }

        return 0;
}

/* Merge the two clusters in "c" connected by the edge in "graph"
 * with index "edge" into a single cluster.
 * If it turns out to be impossible to merge these two clusters,
 * then mark the edge as "no_merge" such that it will not be
 * considered again.
 *
 * First mark all SCCs that need to be merged.  This includes the SCCs
 * in the two clusters, but it may also include the SCCs
 * of intermediate clusters.
 * If there is already a no_merge edge between any pair of such SCCs,
 * then simply mark the current edge as no_merge as well.
 * Likewise, if any of those edges was postponed by has_bounded_distances,
 * then postpone the current edge as well.
 * Otherwise, try and merge the clusters and mark "edge" as "no_merge"
 * if the clusters did not end up getting merged, unless the non-merge
 * is due to the fact that the edge was postponed.  This postponement
 * can be recognized by a change in weight (from non-negative to negative).
 */
static isl_stat merge_clusters_along_edge(isl_ctx *ctx,
        struct isl_sched_graph *graph, int edge, struct isl_clustering *c)
{
        isl_bool merged;
        int edge_weight = graph->edge[edge].weight;

        if (mark_merge_sccs(ctx, graph, edge, c) < 0)
                return isl_stat_error;

        if (any_no_merge(graph, c->scc_in_merge, &graph->edge[edge]))
                merged = isl_bool_false;
        else
                merged = try_merge(ctx, graph, c);
        if (merged < 0)
                return isl_stat_error;
        if (!merged && edge_weight == graph->edge[edge].weight)
                graph->edge[edge].no_merge = 1;

        return isl_stat_ok;
}

/* Does "node" belong to the cluster identified by "cluster"?
 */
static int node_cluster_exactly(struct isl_sched_node *node, int cluster)
{
        return node->cluster == cluster;
}

/* Does "edge" connect two nodes belonging to the cluster
 * identified by "cluster"?
 */
static int edge_cluster_exactly(struct isl_sched_edge *edge, int cluster)
{
        return edge->src->cluster == cluster && edge->dst->cluster == cluster;
}

/* Swap the schedule of "node1" and "node2".
 * Both nodes have been derived from the same node in a common parent graph.
 * Since the "coincident" field is shared with that node
 * in the parent graph, there is no need to also swap this field.
 */
static void swap_sched(struct isl_sched_node *node1,
        struct isl_sched_node *node2)
{
        isl_mat *sched;
        isl_map *sched_map;

        sched = node1->sched;
        node1->sched = node2->sched;
        node2->sched = sched;

        sched_map = node1->sched_map;
        node1->sched_map = node2->sched_map;
        node2->sched_map = sched_map;
}

/* Copy the current band schedule from the SCCs that form the cluster
 * with index "pos" to the actual cluster at position "pos".
 * By construction, the index of the first SCC that belongs to the cluster
 * is also "pos".
 *
 * The order of the nodes inside both the SCCs and the cluster
 * is assumed to be same as the order in the original "graph".
 *
 * Since the SCC graphs will no longer be used after this function,
 * the schedules are actually swapped rather than copied.
 */
static isl_stat copy_partial(struct isl_sched_graph *graph,
        struct isl_clustering *c, int pos)
{
        int i, j;

        c->cluster[pos].n_total_row = c->scc[pos].n_total_row;
        c->cluster[pos].n_row = c->scc[pos].n_row;
        c->cluster[pos].maxvar = c->scc[pos].maxvar;
        j = 0;
        for (i = 0; i < graph->n; ++i) {
                int k;
                int s;

                if (graph->node[i].cluster != pos)
                        continue;
                s = graph->node[i].scc;
                k = c->scc_node[s]++;
                swap_sched(&c->cluster[pos].node[j], &c->scc[s].node[k]);
                if (c->scc[s].maxvar > c->cluster[pos].maxvar)
                        c->cluster[pos].maxvar = c->scc[s].maxvar;
                ++j;
        }

        return isl_stat_ok;
}

/* Is there a (conditional) validity dependence from node[j] to node[i],
 * forcing node[i] to follow node[j] or do the nodes belong to the same
 * cluster?
 */
static isl_bool node_follows_strong_or_same_cluster(int i, int j, void *user)
{
        struct isl_sched_graph *graph = user;

        if (graph->node[i].cluster == graph->node[j].cluster)
                return isl_bool_true;
        return graph_has_validity_edge(graph, &graph->node[j], &graph->node[i]);
}

/* Extract the merged clusters of SCCs in "graph", sort them, and
 * store them in c->clusters.  Update c->scc_cluster accordingly.
 *
 * First keep track of the cluster containing the SCC to which a node
 * belongs in the node itself.
 * Then extract the clusters into c->clusters, copying the current
 * band schedule from the SCCs that belong to the cluster.
 * Do this only once per cluster.
 *
 * Finally, topologically sort the clusters and update c->scc_cluster
 * to match the new scc numbering.  While the SCCs were originally
 * sorted already, some SCCs that depend on some other SCCs may
 * have been merged with SCCs that appear before these other SCCs.
 * A reordering may therefore be required.
 */
static isl_stat extract_clusters(isl_ctx *ctx, struct isl_sched_graph *graph,
        struct isl_clustering *c)
{
        int i;

        for (i = 0; i < graph->n; ++i)
                graph->node[i].cluster = c->scc_cluster[graph->node[i].scc];

        for (i = 0; i < graph->scc; ++i) {
                if (c->scc_cluster[i] != i)
                        continue;
                if (extract_sub_graph(ctx, graph, &node_cluster_exactly,
                                &edge_cluster_exactly, i, &c->cluster[i]) < 0)
                        return isl_stat_error;
                c->cluster[i].src_scc = -1;
                c->cluster[i].dst_scc = -1;
                if (copy_partial(graph, c, i) < 0)
                        return isl_stat_error;
        }

        if (detect_ccs(ctx, graph, &node_follows_strong_or_same_cluster) < 0)
                return isl_stat_error;
        for (i = 0; i < graph->n; ++i)
                c->scc_cluster[graph->node[i].scc] = graph->node[i].cluster;

        return isl_stat_ok;
}

/* Compute weights on the proximity edges of "graph" that can
 * be used by find_proximity to find the most appropriate
 * proximity edge to use to merge two clusters in "c".
 * The weights are also used by has_bounded_distances to determine
 * whether the merge should be allowed.
 * Store the maximum of the computed weights in graph->max_weight.
 *
 * The computed weight is a measure for the number of remaining schedule
 * dimensions that can still be completely aligned.
 * In particular, compute the number of equalities between
 * input dimensions and output dimensions in the proximity constraints.
 * The directions that are already handled by outer schedule bands
 * are projected out prior to determining this number.
 *
 * Edges that will never be considered by find_proximity are ignored.
 */
static isl_stat compute_weights(struct isl_sched_graph *graph,
        struct isl_clustering *c)
{
        int i;

        graph->max_weight = 0;

        for (i = 0; i < graph->n_edge; ++i) {
                struct isl_sched_edge *edge = &graph->edge[i];
                struct isl_sched_node *src = edge->src;
                struct isl_sched_node *dst = edge->dst;
                isl_basic_map *hull;
                isl_bool prox;
                int n_in, n_out;

                prox = is_non_empty_proximity(edge);
                if (prox < 0)
                        return isl_stat_error;
                if (!prox)
                        continue;
                if (bad_cluster(&c->scc[edge->src->scc]) ||
                    bad_cluster(&c->scc[edge->dst->scc]))
                        continue;
                if (c->scc_cluster[edge->dst->scc] ==
                    c->scc_cluster[edge->src->scc])
                        continue;

                hull = isl_map_affine_hull(isl_map_copy(edge->map));
                hull = isl_basic_map_transform_dims(hull, isl_dim_in, 0,
                                                    isl_mat_copy(src->ctrans));
                hull = isl_basic_map_transform_dims(hull, isl_dim_out, 0,
                                                    isl_mat_copy(dst->ctrans));
                hull = isl_basic_map_project_out(hull,
                                                isl_dim_in, 0, src->rank);
                hull = isl_basic_map_project_out(hull,
                                                isl_dim_out, 0, dst->rank);
                hull = isl_basic_map_remove_divs(hull);
                n_in = isl_basic_map_dim(hull, isl_dim_in);
                n_out = isl_basic_map_dim(hull, isl_dim_out);
                hull = isl_basic_map_drop_constraints_not_involving_dims(hull,
                                                        isl_dim_in, 0, n_in);
                hull = isl_basic_map_drop_constraints_not_involving_dims(hull,
                                                        isl_dim_out, 0, n_out);
                if (!hull)
                        return isl_stat_error;
                edge->weight = isl_basic_map_n_equality(hull);
                isl_basic_map_free(hull);

                if (edge->weight > graph->max_weight)
                        graph->max_weight = edge->weight;
        }

        return isl_stat_ok;
}

/* Call compute_schedule_finish_band on each of the clusters in "c"
 * in their topological order.  This order is determined by the scc
 * fields of the nodes in "graph".
 * Combine the results in a sequence expressing the topological order.
 *
 * If there is only one cluster left, then there is no need to introduce
 * a sequence node.  Also, in this case, the cluster necessarily contains
 * the SCC at position 0 in the original graph and is therefore also
 * stored in the first cluster of "c".
 */
static __isl_give isl_schedule_node *finish_bands_clustering(
        __isl_take isl_schedule_node *node, struct isl_sched_graph *graph,
        struct isl_clustering *c)
{
        int i;
        isl_ctx *ctx;
        isl_union_set_list *filters;

        if (graph->scc == 1)
                return compute_schedule_finish_band(node, &c->cluster[0], 0);

        ctx = isl_schedule_node_get_ctx(node);

        filters = extract_sccs(ctx, graph);
        node = isl_schedule_node_insert_sequence(node, filters);

        for (i = 0; i < graph->scc; ++i) {
                int j = c->scc_cluster[i];
                node = isl_schedule_node_child(node, i);
                node = isl_schedule_node_child(node, 0);
                node = compute_schedule_finish_band(node, &c->cluster[j], 0);
                node = isl_schedule_node_parent(node);
                node = isl_schedule_node_parent(node);
        }

        return node;
}

/* Compute a schedule for a connected dependence graph by first considering
 * each strongly connected component (SCC) in the graph separately and then
 * incrementally combining them into clusters.
 * Return the updated schedule node.
 *
 * Initially, each cluster consists of a single SCC, each with its
 * own band schedule.  The algorithm then tries to merge pairs
 * of clusters along a proximity edge until no more suitable
 * proximity edges can be found.  During this merging, the schedule
 * is maintained in the individual SCCs.
 * After the merging is completed, the full resulting clusters
 * are extracted and in finish_bands_clustering,
 * compute_schedule_finish_band is called on each of them to integrate
 * the band into "node" and to continue the computation.
 *
 * compute_weights initializes the weights that are used by find_proximity.
 */
static __isl_give isl_schedule_node *compute_schedule_wcc_clustering(
        __isl_take isl_schedule_node *node, struct isl_sched_graph *graph)
{
        isl_ctx *ctx;
        struct isl_clustering c;
        int i;

        ctx = isl_schedule_node_get_ctx(node);

        if (clustering_init(ctx, &c, graph) < 0)
                goto error;

        if (compute_weights(graph, &c) < 0)
                goto error;

        for (;;) {
                i = find_proximity(graph, &c);
                if (i < 0)
                        goto error;
                if (i >= graph->n_edge)
                        break;
                if (merge_clusters_along_edge(ctx, graph, i, &c) < 0)
                        goto error;
        }

        if (extract_clusters(ctx, graph, &c) < 0)
                goto error;

        node = finish_bands_clustering(node, graph, &c);

        clustering_free(ctx, &c);
        return node;
error:
        clustering_free(ctx, &c);
        return isl_schedule_node_free(node);
}

/* Compute a schedule for a connected dependence graph and return
 * the updated schedule node.
 *
 * If Feautrier's algorithm is selected, we first recursively try to satisfy
 * as many validity dependences as possible. When all validity dependences
 * are satisfied we extend the schedule to a full-dimensional schedule.
 *
 * Call compute_schedule_wcc_whole or compute_schedule_wcc_clustering
 * depending on whether the user has selected the option to try and
 * compute a schedule for the entire (weakly connected) component first.
 * If there is only a single strongly connected component (SCC), then
 * there is no point in trying to combine SCCs
 * in compute_schedule_wcc_clustering, so compute_schedule_wcc_whole
 * is called instead.
 */
static __isl_give isl_schedule_node *compute_schedule_wcc(
        __isl_take isl_schedule_node *node, struct isl_sched_graph *graph)
{
        isl_ctx *ctx;

        if (!node)
                return NULL;

        ctx = isl_schedule_node_get_ctx(node);
        if (detect_sccs(ctx, graph) < 0)
                return isl_schedule_node_free(node);

        if (compute_maxvar(graph) < 0)
                return isl_schedule_node_free(node);

        if (need_feautrier_step(ctx, graph))
                return compute_schedule_wcc_feautrier(node, graph);

        if (graph->scc <= 1 || isl_options_get_schedule_whole_component(ctx))
                return compute_schedule_wcc_whole(node, graph);
        else
                return compute_schedule_wcc_clustering(node, graph);
}

/* Compute a schedule for each group of nodes identified by node->scc
 * separately and then combine them in a sequence node (or as set node
 * if graph->weak is set) inserted at position "node" of the schedule tree.
 * Return the updated schedule node.
 *
 * If "wcc" is set then each of the groups belongs to a single
 * weakly connected component in the dependence graph so that
 * there is no need for compute_sub_schedule to look for weakly
 * connected components.
 */
static __isl_give isl_schedule_node *compute_component_schedule(
        __isl_take isl_schedule_node *node, struct isl_sched_graph *graph,
        int wcc)
{
        int component;
        isl_ctx *ctx;
        isl_union_set_list *filters;

        if (!node)
                return NULL;
        ctx = isl_schedule_node_get_ctx(node);

        filters = extract_sccs(ctx, graph);
        if (graph->weak)
                node = isl_schedule_node_insert_set(node, filters);
        else
                node = isl_schedule_node_insert_sequence(node, filters);

        for (component = 0; component < graph->scc; ++component) {
                node = isl_schedule_node_child(node, component);
                node = isl_schedule_node_child(node, 0);
                node = compute_sub_schedule(node, ctx, graph,
                                    &node_scc_exactly,
                                    &edge_scc_exactly, component, wcc);
                node = isl_schedule_node_parent(node);
                node = isl_schedule_node_parent(node);
        }

        return node;
}

/* Compute a schedule for the given dependence graph and insert it at "node".
 * Return the updated schedule node.
 *
 * We first check if the graph is connected (through validity and conditional
 * validity dependences) and, if not, compute a schedule
 * for each component separately.
 * If the schedule_serialize_sccs option is set, then we check for strongly
 * connected components instead and compute a separate schedule for
 * each such strongly connected component.
 */
static __isl_give isl_schedule_node *compute_schedule(isl_schedule_node *node,
        struct isl_sched_graph *graph)
{
        isl_ctx *ctx;

        if (!node)
                return NULL;

        ctx = isl_schedule_node_get_ctx(node);
        if (isl_options_get_schedule_serialize_sccs(ctx)) {
                if (detect_sccs(ctx, graph) < 0)
                        return isl_schedule_node_free(node);
        } else {
                if (detect_wccs(ctx, graph) < 0)
                        return isl_schedule_node_free(node);
        }

        if (graph->scc > 1)
                return compute_component_schedule(node, graph, 1);

        return compute_schedule_wcc(node, graph);
}

/* Compute a schedule on sc->domain that respects the given schedule
 * constraints.
 *
 * In particular, the schedule respects all the validity dependences.
 * If the default isl scheduling algorithm is used, it tries to minimize
 * the dependence distances over the proximity dependences.
 * If Feautrier's scheduling algorithm is used, the proximity dependence
 * distances are only minimized during the extension to a full-dimensional
 * schedule.
 *
 * If there are any condition and conditional validity dependences,
 * then the conditional validity dependences may be violated inside
 * a tilable band, provided they have no adjacent non-local
 * condition dependences.
 */
__isl_give isl_schedule *isl_schedule_constraints_compute_schedule(
        __isl_take isl_schedule_constraints *sc)
{
        isl_ctx *ctx = isl_schedule_constraints_get_ctx(sc);
        struct isl_sched_graph graph = { 0 };
        isl_schedule *sched;
        isl_schedule_node *node;
        isl_union_set *domain;

        sc = isl_schedule_constraints_align_params(sc);

        domain = isl_schedule_constraints_get_domain(sc);
        if (isl_union_set_n_set(domain) == 0) {
                isl_schedule_constraints_free(sc);
                return isl_schedule_from_domain(domain);
        }

        if (graph_init(&graph, sc) < 0)
                domain = isl_union_set_free(domain);

        node = isl_schedule_node_from_domain(domain);
        node = isl_schedule_node_child(node, 0);
        if (graph.n > 0)
                node = compute_schedule(node, &graph);
        sched = isl_schedule_node_get_schedule(node);
        isl_schedule_node_free(node);

        graph_free(ctx, &graph);
        isl_schedule_constraints_free(sc);

        return sched;
}

/* Compute a schedule for the given union of domains that respects
 * all the validity dependences and minimizes
 * the dependence distances over the proximity dependences.
 *
 * This function is kept for backward compatibility.
 */
__isl_give isl_schedule *isl_union_set_compute_schedule(
        __isl_take isl_union_set *domain,
        __isl_take isl_union_map *validity,
        __isl_take isl_union_map *proximity)
{
        isl_schedule_constraints *sc;

        sc = isl_schedule_constraints_on_domain(domain);
        sc = isl_schedule_constraints_set_validity(sc, validity);
        sc = isl_schedule_constraints_set_proximity(sc, proximity);

        return isl_schedule_constraints_compute_schedule(sc);
}