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[isl.git] / isl_scheduler.c

/*
 * Copyright 2011      INRIA Saclay
 * Copyright 2012-2014 Ecole Normale Superieure
 * Copyright 2015      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
 */

#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_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>

/*
 * 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".
 */

enum isl_edge_type {
        isl_edge_validity = 0,
        isl_edge_first = isl_edge_validity,
        isl_edge_coincidence,
        isl_edge_condition,
        isl_edge_conditional_validity,
        isl_edge_proximity,
        isl_edge_last = isl_edge_proximity
};

/* The constraints that need to be satisfied by a schedule on "domain".
 *
 * "context" specifies extra constraints on the parameters.
 *
 * "validity" constraints map domain elements i to domain elements
 * that should be scheduled after i.  (Hard constraint)
 * "proximity" constraints map domain elements i to domains elements
 * that should be scheduled as early as possible after i (or before i).
 * (Soft constraint)
 *
 * "condition" and "conditional_validity" constraints map possibly "tagged"
 * domain elements i -> s to "tagged" domain elements j -> t.
 * The elements of the "conditional_validity" constraints, but without the
 * tags (i.e., the elements i -> j) are treated as validity constraints,
 * except that during the construction of a tilable band,
 * the elements of the "conditional_validity" constraints may be violated
 * provided that all adjacent elements of the "condition" constraints
 * are local within the band.
 * A dependence is local within a band if domain and range are mapped
 * to the same schedule point by the band.
 */
struct isl_schedule_constraints {
        isl_union_set *domain;
        isl_set *context;

        isl_union_map *constraint[isl_edge_last + 1];
};

__isl_give isl_schedule_constraints *isl_schedule_constraints_copy(
        __isl_keep isl_schedule_constraints *sc)
{
        isl_ctx *ctx;
        isl_schedule_constraints *sc_copy;
        enum isl_edge_type i;

        ctx = isl_union_set_get_ctx(sc->domain);
        sc_copy = isl_calloc_type(ctx, struct isl_schedule_constraints);
        if (!sc_copy)
                return NULL;

        sc_copy->domain = isl_union_set_copy(sc->domain);
        sc_copy->context = isl_set_copy(sc->context);
        if (!sc_copy->domain || !sc_copy->context)
                return isl_schedule_constraints_free(sc_copy);

        for (i = isl_edge_first; i <= isl_edge_last; ++i) {
                sc_copy->constraint[i] = isl_union_map_copy(sc->constraint[i]);
                if (!sc_copy->constraint[i])
                        return isl_schedule_constraints_free(sc_copy);
        }

        return sc_copy;
}


/* Construct an isl_schedule_constraints object for computing a schedule
 * on "domain".  The initial object does not impose any constraints.
 */
__isl_give isl_schedule_constraints *isl_schedule_constraints_on_domain(
        __isl_take isl_union_set *domain)
{
        isl_ctx *ctx;
        isl_space *space;
        isl_schedule_constraints *sc;
        isl_union_map *empty;
        enum isl_edge_type i;

        if (!domain)
                return NULL;

        ctx = isl_union_set_get_ctx(domain);
        sc = isl_calloc_type(ctx, struct isl_schedule_constraints);
        if (!sc)
                goto error;

        space = isl_union_set_get_space(domain);
        sc->domain = domain;
        sc->context = isl_set_universe(isl_space_copy(space));
        empty = isl_union_map_empty(space);
        for (i = isl_edge_first; i <= isl_edge_last; ++i) {
                sc->constraint[i] = isl_union_map_copy(empty);
                if (!sc->constraint[i])
                        sc->domain = isl_union_set_free(sc->domain);
        }
        isl_union_map_free(empty);

        if (!sc->domain || !sc->context)
                return isl_schedule_constraints_free(sc);

        return sc;
error:
        isl_union_set_free(domain);
        return NULL;
}

/* Replace the context of "sc" by "context".
 */
__isl_give isl_schedule_constraints *isl_schedule_constraints_set_context(
        __isl_take isl_schedule_constraints *sc, __isl_take isl_set *context)
{
        if (!sc || !context)
                goto error;

        isl_set_free(sc->context);
        sc->context = context;

        return sc;
error:
        isl_schedule_constraints_free(sc);
        isl_set_free(context);
        return NULL;
}

/* Replace the validity constraints of "sc" by "validity".
 */
__isl_give isl_schedule_constraints *isl_schedule_constraints_set_validity(
        __isl_take isl_schedule_constraints *sc,
        __isl_take isl_union_map *validity)
{
        if (!sc || !validity)
                goto error;

        isl_union_map_free(sc->constraint[isl_edge_validity]);
        sc->constraint[isl_edge_validity] = validity;

        return sc;
error:
        isl_schedule_constraints_free(sc);
        isl_union_map_free(validity);
        return NULL;
}

/* Replace the coincidence constraints of "sc" by "coincidence".
 */
__isl_give isl_schedule_constraints *isl_schedule_constraints_set_coincidence(
        __isl_take isl_schedule_constraints *sc,
        __isl_take isl_union_map *coincidence)
{
        if (!sc || !coincidence)
                goto error;

        isl_union_map_free(sc->constraint[isl_edge_coincidence]);
        sc->constraint[isl_edge_coincidence] = coincidence;

        return sc;
error:
        isl_schedule_constraints_free(sc);
        isl_union_map_free(coincidence);
        return NULL;
}

/* Replace the proximity constraints of "sc" by "proximity".
 */
__isl_give isl_schedule_constraints *isl_schedule_constraints_set_proximity(
        __isl_take isl_schedule_constraints *sc,
        __isl_take isl_union_map *proximity)
{
        if (!sc || !proximity)
                goto error;

        isl_union_map_free(sc->constraint[isl_edge_proximity]);
        sc->constraint[isl_edge_proximity] = proximity;

        return sc;
error:
        isl_schedule_constraints_free(sc);
        isl_union_map_free(proximity);
        return NULL;
}

/* Replace the conditional validity constraints of "sc" by "condition"
 * and "validity".
 */
__isl_give isl_schedule_constraints *
isl_schedule_constraints_set_conditional_validity(
        __isl_take isl_schedule_constraints *sc,
        __isl_take isl_union_map *condition,
        __isl_take isl_union_map *validity)
{
        if (!sc || !condition || !validity)
                goto error;

        isl_union_map_free(sc->constraint[isl_edge_condition]);
        sc->constraint[isl_edge_condition] = condition;
        isl_union_map_free(sc->constraint[isl_edge_conditional_validity]);
        sc->constraint[isl_edge_conditional_validity] = validity;

        return sc;
error:
        isl_schedule_constraints_free(sc);
        isl_union_map_free(condition);
        isl_union_map_free(validity);
        return NULL;
}

__isl_null isl_schedule_constraints *isl_schedule_constraints_free(
        __isl_take isl_schedule_constraints *sc)
{
        enum isl_edge_type i;

        if (!sc)
                return NULL;

        isl_union_set_free(sc->domain);
        isl_set_free(sc->context);
        for (i = isl_edge_first; i <= isl_edge_last; ++i)
                isl_union_map_free(sc->constraint[i]);

        free(sc);

        return NULL;
}

isl_ctx *isl_schedule_constraints_get_ctx(
        __isl_keep isl_schedule_constraints *sc)
{
        return sc ? isl_union_set_get_ctx(sc->domain) : NULL;
}

/* Return the validity constraints of "sc".
 */
__isl_give isl_union_map *isl_schedule_constraints_get_validity(
        __isl_keep isl_schedule_constraints *sc)
{
        if (!sc)
                return NULL;

        return isl_union_map_copy(sc->constraint[isl_edge_validity]);
}

/* Return the coincidence constraints of "sc".
 */
__isl_give isl_union_map *isl_schedule_constraints_get_coincidence(
        __isl_keep isl_schedule_constraints *sc)
{
        if (!sc)
                return NULL;

        return isl_union_map_copy(sc->constraint[isl_edge_coincidence]);
}

/* Return the conditional validity constraints of "sc".
 */
__isl_give isl_union_map *isl_schedule_constraints_get_conditional_validity(
        __isl_keep isl_schedule_constraints *sc)
{
        if (!sc)
                return NULL;

        return
            isl_union_map_copy(sc->constraint[isl_edge_conditional_validity]);
}

/* Return the conditions for the conditional validity constraints of "sc".
 */
__isl_give isl_union_map *
isl_schedule_constraints_get_conditional_validity_condition(
        __isl_keep isl_schedule_constraints *sc)
{
        if (!sc)
                return NULL;

        return isl_union_map_copy(sc->constraint[isl_edge_condition]);
}

void isl_schedule_constraints_dump(__isl_keep isl_schedule_constraints *sc)
{
        if (!sc)
                return;

        fprintf(stderr, "domain: ");
        isl_union_set_dump(sc->domain);
        fprintf(stderr, "context: ");
        isl_set_dump(sc->context);
        fprintf(stderr, "validity: ");
        isl_union_map_dump(sc->constraint[isl_edge_validity]);
        fprintf(stderr, "proximity: ");
        isl_union_map_dump(sc->constraint[isl_edge_proximity]);
        fprintf(stderr, "coincidence: ");
        isl_union_map_dump(sc->constraint[isl_edge_coincidence]);
        fprintf(stderr, "condition: ");
        isl_union_map_dump(sc->constraint[isl_edge_condition]);
        fprintf(stderr, "conditional_validity: ");
        isl_union_map_dump(sc->constraint[isl_edge_conditional_validity]);
}

/* Align the parameters of the fields of "sc".
 */
static __isl_give isl_schedule_constraints *
isl_schedule_constraints_align_params(__isl_take isl_schedule_constraints *sc)
{
        isl_space *space;
        enum isl_edge_type i;

        if (!sc)
                return NULL;

        space = isl_union_set_get_space(sc->domain);
        space = isl_space_align_params(space, isl_set_get_space(sc->context));
        for (i = isl_edge_first; i <= isl_edge_last; ++i)
                space = isl_space_align_params(space,
                                    isl_union_map_get_space(sc->constraint[i]));

        for (i = isl_edge_first; i <= isl_edge_last; ++i) {
                sc->constraint[i] = isl_union_map_align_params(
                                    sc->constraint[i], isl_space_copy(space));
                if (!sc->constraint[i])
                        space = isl_space_free(space);
        }
        sc->context = isl_set_align_params(sc->context, isl_space_copy(space));
        sc->domain = isl_union_set_align_params(sc->domain, space);
        if (!sc->context || !sc->domain)
                return isl_schedule_constraints_free(sc);

        return sc;
}

/* Return the total number of isl_maps in the constraints of "sc".
 */
static __isl_give int isl_schedule_constraints_n_map(
        __isl_keep isl_schedule_constraints *sc)
{
        enum isl_edge_type i;
        int n = 0;

        for (i = isl_edge_first; i <= isl_edge_last; ++i)
                n += isl_union_map_n_map(sc->constraint[i]);

        return n;
}

/* Internal information about a node that is used during the construction
 * of a schedule.
 * space represents the space in which the domain lives
 * 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.
 * 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
 *
 * 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.
 */
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;
        int      start;
        int      nvar;
        int      nparam;

        int      scc;

        int     *coincident;
};

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

        return isl_space_is_equal(node->space, dim);
}

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
 * 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.
 */
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 validity : 1;
        unsigned coincidence : 1;
        unsigned proximity : 1;
        unsigned local : 1;
        unsigned condition : 1;
        unsigned conditional_validity : 1;

        int start;
        int end;
};

/* 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
 * it 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
 *
 * node_table contains pointers into the node array, hashed on the space
 *
 * 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
 */
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;
};

/* 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_hash(graph->node[i].space);
                entry = isl_hash_table_find(ctx, graph->node_table, hash,
                                            &node_has_space,
                                            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 *dim)
{
        struct isl_hash_table_entry *entry;
        uint32_t hash;

        hash = isl_space_get_hash(dim);
        entry = isl_hash_table_find(ctx, graph->node_table, hash,
                                    &node_has_space, dim, 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);
                        if (graph->root)
                                free(graph->node[i].coincident);
                }
        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;
}

/* Add the number of basic maps in "map" to *n.
 */
static isl_stat add_n_basic_map(__isl_take isl_map *map, void *user)
{
        int *n = user;

        *n += isl_map_n_basic_map(map);
        isl_map_free(map);

        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 int compute_max_row(struct isl_sched_graph *graph,
        __isl_keep isl_schedule_constraints *sc)
{
        enum isl_edge_type i;
        int n_edge;

        graph->n = 0;
        graph->maxvar = 0;
        if (isl_union_set_foreach_set(sc->domain, &init_n_maxvar, graph) < 0)
                return -1;
        n_edge = 0;
        for (i = isl_edge_first; i <= isl_edge_last; ++i)
                if (isl_union_map_foreach_map(sc->constraint[i],
                                                &add_n_basic_map, &n_edge) < 0)
                        return -1;
        graph->max_row = n_edge + graph->maxvar;

        return 0;
}

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

        if (!bset)
                return -1;

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

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

        return 0;
}

/* Add a new node to the graph representing the given space.
 * "nvar" is the (possibly compressed) number of variables and
 * may be smaller than then number of set variables in "space"
 * 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.
 */
static isl_stat add_node(struct isl_sched_graph *graph,
        __isl_take isl_space *space, 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;
        int *coincident;

        if (!space)
                return isl_stat_error;

        ctx = isl_space_get_ctx(space);
        nparam = isl_space_dim(space, isl_dim_param);
        if (!ctx->opt->schedule_parametric)
                nparam = 0;
        sched = isl_mat_alloc(ctx, 0, 1 + nparam + nvar);
        graph->node[graph->n].space = space;
        graph->node[graph->n].nvar = nvar;
        graph->node[graph->n].nparam = nparam;
        graph->node[graph->n].sched = sched;
        graph->node[graph->n].sched_map = NULL;
        coincident = isl_calloc_array(ctx, int, graph->max_row);
        graph->node[graph->n].coincident = coincident;
        graph->node[graph->n].compressed = compressed;
        graph->node[graph->n].hull = hull;
        graph->node[graph->n].compress = compress;
        graph->node[graph->n].decompress = decompress;
        graph->n++;

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

        return isl_stat_ok;
}

/* 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.
 */
static isl_stat extract_node(__isl_take isl_set *set, void *user)
{
        int nvar;
        int has_equality;
        isl_space *space;
        isl_basic_set *hull;
        isl_set *hull_set;
        isl_morph *morph;
        isl_multi_aff *compress, *decompress;
        struct isl_sched_graph *graph = user;

        space = isl_set_get_space(set);
        hull = isl_set_affine_hull(set);
        hull = isl_basic_set_remove_divs(hull);
        nvar = isl_space_dim(space, 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, space, nvar, 0, NULL, NULL, NULL);
        }

        morph = isl_basic_set_variable_compression(hull, isl_dim_set);
        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, space, nvar, 1, hull_set, compress, decompress);
error:
        isl_basic_set_free(hull);
        isl_space_free(space);
        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.
 * "type" is the type of the schedule constraint from which edge2 was
 * extracted.
 * 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(enum isl_edge_type type, struct isl_sched_edge *edge1,
        struct isl_sched_edge *edge2)
{
        edge1->validity |= edge2->validity;
        edge1->coincidence |= edge2->coincidence;
        edge1->proximity |= edge2->proximity;
        edge1->condition |= edge2->condition;
        edge1->conditional_validity |= edge2->conditional_validity;
        isl_map_free(edge2->map);

        if (type == isl_edge_condition) {
                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 (type == isl_edge_conditional_validity) {
                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 (type == isl_edge_condition && !edge1->tagged_condition)
                return -1;
        if (type == isl_edge_conditional_validity && !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;
}

/* 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;
        isl_space *dim;
        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));
                }
        }

        dim = isl_space_domain(isl_map_get_space(map));
        src = graph_find_node(ctx, graph, dim);
        isl_space_free(dim);
        dim = isl_space_range(isl_map_get_space(map));
        dst = graph_find_node(ctx, graph, dim);
        isl_space_free(dim);

        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].validity = 0;
        graph->edge[graph->n_edge].coincidence = 0;
        graph->edge[graph->n_edge].proximity = 0;
        graph->edge[graph->n_edge].condition = 0;
        graph->edge[graph->n_edge].local = 0;
        graph->edge[graph->n_edge].conditional_validity = 0;
        graph->edge[graph->n_edge].tagged_condition = NULL;
        graph->edge[graph->n_edge].tagged_validity = NULL;
        if (data->type == isl_edge_validity)
                graph->edge[graph->n_edge].validity = 1;
        if (data->type == isl_edge_coincidence)
                graph->edge[graph->n_edge].coincidence = 1;
        if (data->type == isl_edge_proximity)
                graph->edge[graph->n_edge].proximity = 1;
        if (data->type == isl_edge_condition) {
                graph->edge[graph->n_edge].condition = 1;
                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].conditional_validity = 1;
                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(data->type, edge, &graph->edge[graph->n_edge]) < 0)
                return -1;

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

/* 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 validity edges).
 * If weak is set, we consider the graph to be undirected and
 * we effectively compute the (weakly) connected components.
 * Additionally, we also consider other edges when weak is set.
 */
static int detect_ccs(isl_ctx *ctx, struct isl_sched_graph *graph, int weak)
{
        int i, n;
        struct isl_tarjan_graph *g = NULL;

        g = isl_tarjan_graph_init(ctx, graph->n,
                weak ? &node_follows_weak : &node_follows_strong, graph);
        if (!g)
                return -1;

        graph->weak = weak;
        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.
 */
static int detect_sccs(isl_ctx *ctx, struct isl_sched_graph *graph)
{
        return detect_ccs(ctx, graph, 0);
}

/* Apply Tarjan's algorithm to detect the (weakly) connected components
 * in the dependence graph.
 */
static int detect_wccs(isl_ctx *ctx, struct isl_sched_graph *graph)
{
        return detect_ccs(ctx, graph, 1);
}

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;

        if (isl_map_to_basic_set_has(graph->intra_hmap, map))
                return isl_map_to_basic_set_get(graph->intra_hmap, map);

        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;

        if (isl_map_to_basic_set_has(graph->inter_hmap, map))
                return isl_map_to_basic_set_get(graph->inter_hmap, map);

        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;
}

/* 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 int add_intra_validity_constraints(struct isl_sched_graph *graph,
        struct isl_sched_edge *edge)
{
        unsigned total;
        isl_map *map = isl_map_copy(edge->map);
        isl_ctx *ctx = isl_map_get_ctx(map);
        isl_space *dim;
        isl_dim_map *dim_map;
        isl_basic_set *coef;
        struct isl_sched_node *node = edge->src;

        coef = intra_coefficients(graph, node, map);

        dim = isl_space_domain(isl_space_unwrap(isl_basic_set_get_space(coef)));

        coef = isl_basic_set_transform_dims(coef, isl_dim_set,
                    isl_space_dim(dim, isl_dim_set), isl_mat_copy(node->cmap));
        if (!coef)
                goto error;

        total = isl_basic_set_total_dim(graph->lp);
        dim_map = isl_dim_map_alloc(ctx, total);
        isl_dim_map_range(dim_map, node->start + 2 * node->nparam + 1, 2,
                          isl_space_dim(dim, isl_dim_set), 1,
                          node->nvar, -1);
        isl_dim_map_range(dim_map, node->start + 2 * node->nparam + 2, 2,
                          isl_space_dim(dim, isl_dim_set), 1,
                          node->nvar, 1);
        graph->lp = isl_basic_set_extend_constraints(graph->lp,
                        coef->n_eq, coef->n_ineq);
        graph->lp = isl_basic_set_add_constraints_dim_map(graph->lp,
                                                           coef, dim_map);
        isl_space_free(dim);

        return 0;
error:
        isl_space_free(dim);
        return -1;
}

/* 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_j_n^- - (c_i_n^+ - c_i_n^-),
 *  c_j_x^+ - c_j_x^- - (c_i_x^+ - c_i_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 int add_inter_validity_constraints(struct isl_sched_graph *graph,
        struct isl_sched_edge *edge)
{
        unsigned total;
        isl_map *map = isl_map_copy(edge->map);
        isl_ctx *ctx = isl_map_get_ctx(map);
        isl_space *dim;
        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);

        dim = isl_space_domain(isl_space_unwrap(isl_basic_set_get_space(coef)));

        coef = isl_basic_set_transform_dims(coef, isl_dim_set,
                    isl_space_dim(dim, isl_dim_set), isl_mat_copy(src->cmap));
        coef = isl_basic_set_transform_dims(coef, isl_dim_set,
                    isl_space_dim(dim, isl_dim_set) + src->nvar,
                    isl_mat_copy(dst->cmap));
        if (!coef)
                goto error;

        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, 1);
        isl_dim_map_range(dim_map, dst->start + 1, 2, 1, 1, dst->nparam, -1);
        isl_dim_map_range(dim_map, dst->start + 2, 2, 1, 1, dst->nparam, 1);
        isl_dim_map_range(dim_map, dst->start + 2 * dst->nparam + 1, 2,
                          isl_space_dim(dim, isl_dim_set) + src->nvar, 1,
                          dst->nvar, -1);
        isl_dim_map_range(dim_map, dst->start + 2 * dst->nparam + 2, 2,
                          isl_space_dim(dim, isl_dim_set) + src->nvar, 1,
                          dst->nvar, 1);

        isl_dim_map_range(dim_map, src->start, 0, 0, 0, 1, -1);
        isl_dim_map_range(dim_map, src->start + 1, 2, 1, 1, src->nparam, 1);
        isl_dim_map_range(dim_map, src->start + 2, 2, 1, 1, src->nparam, -1);
        isl_dim_map_range(dim_map, src->start + 2 * src->nparam + 1, 2,
                          isl_space_dim(dim, isl_dim_set), 1,
                          src->nvar, 1);
        isl_dim_map_range(dim_map, src->start + 2 * src->nparam + 2, 2,
                          isl_space_dim(dim, isl_dim_set), 1,
                          src->nvar, -1);

        edge->start = graph->lp->n_ineq;
        graph->lp = isl_basic_set_extend_constraints(graph->lp,
                        coef->n_eq, coef->n_ineq);
        graph->lp = isl_basic_set_add_constraints_dim_map(graph->lp,
                                                           coef, dim_map);
        if (!graph->lp)
                goto error;
        isl_space_free(dim);
        edge->end = graph->lp->n_ineq;

        return 0;
error:
        isl_space_free(dim);
        return -1;
}

/* 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 int add_intra_proximity_constraints(struct isl_sched_graph *graph,
        struct isl_sched_edge *edge, int s, int local)
{
        unsigned total;
        unsigned nparam;
        isl_map *map = isl_map_copy(edge->map);
        isl_ctx *ctx = isl_map_get_ctx(map);
        isl_space *dim;
        isl_dim_map *dim_map;
        isl_basic_set *coef;
        struct isl_sched_node *node = edge->src;

        coef = intra_coefficients(graph, node, map);

        dim = isl_space_domain(isl_space_unwrap(isl_basic_set_get_space(coef)));

        coef = isl_basic_set_transform_dims(coef, isl_dim_set,
                    isl_space_dim(dim, isl_dim_set), isl_mat_copy(node->cmap));
        if (!coef)
                goto error;

        nparam = isl_space_dim(node->space, isl_dim_param);
        total = isl_basic_set_total_dim(graph->lp);
        dim_map = isl_dim_map_alloc(ctx, total);

        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);
        }
        isl_dim_map_range(dim_map, node->start + 2 * node->nparam + 1, 2,
                          isl_space_dim(dim, isl_dim_set), 1,
                          node->nvar, s);
        isl_dim_map_range(dim_map, node->start + 2 * node->nparam + 2, 2,
                          isl_space_dim(dim, isl_dim_set), 1,
                          node->nvar, -s);
        graph->lp = isl_basic_set_extend_constraints(graph->lp,
                        coef->n_eq, coef->n_ineq);
        graph->lp = isl_basic_set_add_constraints_dim_map(graph->lp,
                                                           coef, dim_map);
        isl_space_free(dim);

        return 0;
error:
        isl_space_free(dim);
        return -1;
}

/* 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_j_x+s*c_i_x)
 * with each coefficient (except m_0, c_j_0 and c_i_0)
 * 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, 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_j_x+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 int add_inter_proximity_constraints(struct isl_sched_graph *graph,
        struct isl_sched_edge *edge, int s, int local)
{
        unsigned total;
        unsigned nparam;
        isl_map *map = isl_map_copy(edge->map);
        isl_ctx *ctx = isl_map_get_ctx(map);
        isl_space *dim;
        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);

        dim = isl_space_domain(isl_space_unwrap(isl_basic_set_get_space(coef)));

        coef = isl_basic_set_transform_dims(coef, isl_dim_set,
                    isl_space_dim(dim, isl_dim_set), isl_mat_copy(src->cmap));
        coef = isl_basic_set_transform_dims(coef, isl_dim_set,
                    isl_space_dim(dim, isl_dim_set) + src->nvar,
                    isl_mat_copy(dst->cmap));
        if (!coef)
                goto error;

        nparam = isl_space_dim(src->space, isl_dim_param);
        total = isl_basic_set_total_dim(graph->lp);
        dim_map = isl_dim_map_alloc(ctx, total);

        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);
        }

        isl_dim_map_range(dim_map, dst->start, 0, 0, 0, 1, -s);
        isl_dim_map_range(dim_map, dst->start + 1, 2, 1, 1, dst->nparam, s);
        isl_dim_map_range(dim_map, dst->start + 2, 2, 1, 1, dst->nparam, -s);
        isl_dim_map_range(dim_map, dst->start + 2 * dst->nparam + 1, 2,
                          isl_space_dim(dim, isl_dim_set) + src->nvar, 1,
                          dst->nvar, s);
        isl_dim_map_range(dim_map, dst->start + 2 * dst->nparam + 2, 2,
                          isl_space_dim(dim, isl_dim_set) + 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, 2, 1, 1, src->nparam, -s);
        isl_dim_map_range(dim_map, src->start + 2, 2, 1, 1, src->nparam, s);
        isl_dim_map_range(dim_map, src->start + 2 * src->nparam + 1, 2,
                          isl_space_dim(dim, isl_dim_set), 1,
                          src->nvar, -s);
        isl_dim_map_range(dim_map, src->start + 2 * src->nparam + 2, 2,
                          isl_space_dim(dim, isl_dim_set), 1,
                          src->nvar, s);

        graph->lp = isl_basic_set_extend_constraints(graph->lp,
                        coef->n_eq, coef->n_ineq);
        graph->lp = isl_basic_set_add_constraints_dim_map(graph->lp,
                                                           coef, dim_map);
        isl_space_free(dim);

        return 0;
error:
        isl_space_free(dim);
        return -1;
}

/* 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 = edge->local || (edge->coincidence && use_coincidence);
                if (!edge->validity && !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 = edge->local || (edge->coincidence && use_coincidence);
                if (!edge->validity && !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 = edge->local || (edge->coincidence && use_coincidence);
                if (!edge->proximity && !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 (edge->validity || 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);
        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->rank < 0)
                return -1;
        return 0;
}

/* How many times should we count the constraints in "edge"?
 *
 * If carry is set, then we are counting the number of
 * (validity or conditional validity) constraints that will be added
 * in setup_carry_lp and we count each edge exactly once.
 *
 * Otherwise, 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 carry,
        int use_coincidence)
{
        if (carry && !edge->validity && !edge->conditional_validity)
                return 0;
        if (carry)
                return 1;
        if (edge->proximity || edge->local)
                return 2;
        if (use_coincidence && edge->coincidence)
                return 2;
        if (edge->validity)
                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 int 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 carry, int use_coincidence)
{
        isl_basic_set *coef;
        int f = edge_multiplicity(edge, carry, use_coincidence);

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

        if (edge->src == edge->dst)
                coef = intra_coefficients(graph, edge->src, map);
        else
                coef = inter_coefficients(graph, edge, map);
        if (!coef)
                return -1;
        *n_eq += f * coef->n_eq;
        *n_ineq += f * coef->n_ineq;
        isl_basic_set_free(coef);

        return 0;
}

/* 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,
                                            0, use_coincidence) < 0)
                        return -1;
        }

        return 0;
}

/* 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 (ctx->opt->schedule_max_coefficient == -1)
                return 0;

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

        return 0;
}

/* 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'.
 */
static int add_bound_coefficient_constraints(isl_ctx *ctx,
        struct isl_sched_graph *graph)
{
        int i, j, k;
        int max_coefficient;
        int total;

        max_coefficient = ctx->opt->schedule_max_coefficient;

        if (max_coefficient == -1)
                return 0;

        total = isl_basic_set_total_dim(graph->lp);

        for (i = 0; i < graph->n; ++i) {
                struct isl_sched_node *node = &graph->node[i];
                for (j = 0; j < 2 * node->nparam + 2 * node->nvar; ++j) {
                        int dim;
                        k = isl_basic_set_alloc_inequality(graph->lp);
                        if (k < 0)
                                return -1;
                        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_coefficient);
                }
        }

        return 0;
}

/* 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 positive and negative parts 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
 *              - positive and negative parts of 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 int setup_lp(isl_ctx *ctx, struct isl_sched_graph *graph,
        int use_coincidence)
{
        int i, j;
        int k;
        unsigned nparam;
        unsigned total;
        isl_space *dim;
        int parametric;
        int param_pos;
        int n_eq, n_ineq;
        int max_constant_term;

        max_constant_term = ctx->opt->schedule_max_constant_term;

        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 -1;
                node->start = total;
                total += 1 + 2 * (node->nparam + node->nvar);
        }

        if (count_constraints(graph, &n_eq, &n_ineq, use_coincidence) < 0)
                return -1;
        if (count_bound_coefficient_constraints(ctx, graph, &n_eq, &n_ineq) < 0)
                return -1;

        dim = isl_space_set_alloc(ctx, 0, total);
        isl_basic_set_free(graph->lp);
        n_eq += 2 + parametric;
        if (max_constant_term != -1)
                n_ineq += graph->n;

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

        k = isl_basic_set_alloc_equality(graph->lp);
        if (k < 0)
                return -1;
        isl_seq_clr(graph->lp->eq[k], 1 +  total);
        isl_int_set_si(graph->lp->eq[k][1], -1);
        for (i = 0; i < 2 * nparam; ++i)
                isl_int_set_si(graph->lp->eq[k][1 + param_pos + i], 1);

        if (parametric) {
                k = isl_basic_set_alloc_equality(graph->lp);
                if (k < 0)
                        return -1;
                isl_seq_clr(graph->lp->eq[k], 1 +  total);
                isl_int_set_si(graph->lp->eq[k][3], -1);
                for (i = 0; i < graph->n; ++i) {
                        int pos = 1 + graph->node[i].start + 1;

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

        k = isl_basic_set_alloc_equality(graph->lp);
        if (k < 0)
                return -1;
        isl_seq_clr(graph->lp->eq[k], 1 +  total);
        isl_int_set_si(graph->lp->eq[k][4], -1);
        for (i = 0; i < graph->n; ++i) {
                struct isl_sched_node *node = &graph->node[i];
                int pos = 1 + node->start + 1 + 2 * node->nparam;

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

        if (max_constant_term != -1)
                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 -1;
                        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_constant_term);
                }

        if (add_bound_coefficient_constraints(ctx, graph) < 0)
                return -1;
        if (add_all_validity_constraints(graph, use_coincidence) < 0)
                return -1;
        if (add_all_proximity_constraints(graph, use_coincidence) < 0)
                return -1;

        return 0;
}

/* 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 (!graph->edge[i].validity)
                        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->start + 1 + 2*(node->nparam+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;
}

/* 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 = isl_vec_alloc(sol->ctx, node->nvar);
                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;
                node->sched = isl_mat_set_element(node->sched, row, 0,
                                                  sol->el[1 + pos]);
                for (j = 0; j < node->nparam + node->nvar; ++j)
                        isl_int_sub(sol->el[1 + pos + 1 + 2 * j + 1],
                                    sol->el[1 + pos + 1 + 2 * j + 1],
                                    sol->el[1 + pos + 1 + 2 * j]);
                for (j = 0; j < node->nparam; ++j)
                        node->sched = isl_mat_set_element(node->sched,
                                        row, 1 + j, sol->el[1+pos+1+2*j+1]);
                for (j = 0; j < node->nvar; ++j)
                        isl_int_set(csol->el[j],
                                    sol->el[1+pos+1+2*(node->nparam+j)+1]);
                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;

        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 (!graph->edge[i].conditional_validity)
                        continue;
                if (graph->edge[i].validity)
                        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;

                graph->edge[i].validity = 1;
        }

        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 (!graph->edge[i].condition)
                        continue;
                if (graph->edge[i].local)
                        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->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 (edge->validity || edge->conditional_validity)
                                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].validity = edge->validity;
                dst->edge[dst->n_edge].proximity = edge->proximity;
                dst->edge[dst->n_edge].coincidence = edge->coincidence;
                dst->edge[dst->n_edge].condition = edge->condition;
                dst->edge[dst->n_edge].conditional_validity =
                                                edge->conditional_validity;
                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;
}

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.  The caller should precompute the number
 * of nodes and edges that satisfy these predicates and pass them along
 * as "n" and "n_edge".
 * 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 n, int n_edge,
        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 };
        int t;

        if (graph_alloc(ctx, &split, n, n_edge) < 0)
                goto error;
        if (copy_nodes(&split, graph, node_pred, data) < 0)
                goto error;
        if (graph_init_table(ctx, &split) < 0)
                goto error;
        for (t = 0; t <= isl_edge_last; ++t)
                split.max_edge[t] = graph->max_edge[t];
        if (graph_init_edge_tables(ctx, &split) < 0)
                goto error;
        if (copy_edges(ctx, &split, graph, edge_pred, data) < 0)
                goto error;
        split.n_row = graph->n_row;
        split.max_row = graph->max_row;
        split.n_total_row = graph->n_total_row;
        split.band_start = graph->band_start;

        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.
 *
 * 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 i, n, e1, e2;
        isl_ctx *ctx;
        isl_union_set_list *filters;

        if (!node)
                return NULL;

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

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

                if (before)
                        n++;
        }

        e1 = e2 = 0;
        for (i = 0; i < graph->n_edge; ++i) {
                if (graph->edge[i].dst->scc <= graph->src_scc)
                        e1++;
                if (graph->edge[i].src->scc > graph->src_scc)
                        e2++;
        }

        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, 0);
        node = isl_schedule_node_child(node, 0);

        node = compute_sub_schedule(node, ctx, graph, n, e1,
                                &node_scc_at_most, &edge_dst_scc_at_most,
                                graph->src_scc, 0);
        node = isl_schedule_node_parent(node);
        node = isl_schedule_node_next_sibling(node);
        node = isl_schedule_node_child(node, 0);
        node = compute_sub_schedule(node, ctx, graph, graph->n - n, e2,
                                &node_scc_at_least, &edge_src_scc_at_least,
                                graph->src_scc + 1, 0);
        node = isl_schedule_node_parent(node);
        node = isl_schedule_node_parent(node);

        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 constraints to graph->lp that force the dependence "map" (which
 * is part of the dependence relation of "edge")
 * to be respected and attempt to carry it, where the edge is one from
 * a node j to itself.  "pos" is the sequence number of the given map.
 * That is, add constraints that 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 R.
 * We obtain general constraints on coefficients (c_0, c_n, c_x)
 * of valid constraints for (y - x) and then plug in (-e_i, 0, c_j_x),
 * with each coefficient in c_j_x represented as a pair of non-negative
 * coefficients.
 */
static int add_intra_constraints(struct isl_sched_graph *graph,
        struct isl_sched_edge *edge, __isl_take isl_map *map, int pos)
{
        unsigned total;
        isl_ctx *ctx = isl_map_get_ctx(map);
        isl_space *dim;
        isl_dim_map *dim_map;
        isl_basic_set *coef;
        struct isl_sched_node *node = edge->src;

        coef = intra_coefficients(graph, node, map);
        if (!coef)
                return -1;

        dim = isl_space_domain(isl_space_unwrap(isl_basic_set_get_space(coef)));

        total = isl_basic_set_total_dim(graph->lp);
        dim_map = isl_dim_map_alloc(ctx, total);
        isl_dim_map_range(dim_map, 3 + pos, 0, 0, 0, 1, -1);
        isl_dim_map_range(dim_map, node->start + 2 * node->nparam + 1, 2,
                          isl_space_dim(dim, isl_dim_set), 1,
                          node->nvar, -1);
        isl_dim_map_range(dim_map, node->start + 2 * node->nparam + 2, 2,
                          isl_space_dim(dim, isl_dim_set), 1,
                          node->nvar, 1);
        graph->lp = isl_basic_set_extend_constraints(graph->lp,
                        coef->n_eq, coef->n_ineq);
        graph->lp = isl_basic_set_add_constraints_dim_map(graph->lp,
                                                           coef, dim_map);
        isl_space_free(dim);

        return 0;
}

/* Add constraints to graph->lp that force the dependence "map" (which
 * is part of the dependence relation of "edge")
 * to be respected and attempt to carry it, where the edge is one from
 * node j to node k.  "pos" is the sequence number of the given map.
 * That is, add constraints that 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 R.
 * We obtain general constraints on coefficients (c_0, c_n, c_x)
 * of valid constraints for R and then plug in
 * (-e_i + c_k_0 - c_j_0, c_k_n - c_j_n, c_k_x - c_j_x)
 * with each coefficient (except e_i, c_k_0 and c_j_0)
 * represented as a pair of non-negative coefficients.
 */
static int add_inter_constraints(struct isl_sched_graph *graph,
        struct isl_sched_edge *edge, __isl_take isl_map *map, int pos)
{
        unsigned total;
        isl_ctx *ctx = isl_map_get_ctx(map);
        isl_space *dim;
        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);
        if (!coef)
                return -1;

        dim = isl_space_domain(isl_space_unwrap(isl_basic_set_get_space(coef)));

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

        isl_dim_map_range(dim_map, 3 + pos, 0, 0, 0, 1, -1);

        isl_dim_map_range(dim_map, dst->start, 0, 0, 0, 1, 1);
        isl_dim_map_range(dim_map, dst->start + 1, 2, 1, 1, dst->nparam, -1);
        isl_dim_map_range(dim_map, dst->start + 2, 2, 1, 1, dst->nparam, 1);
        isl_dim_map_range(dim_map, dst->start + 2 * dst->nparam + 1, 2,
                          isl_space_dim(dim, isl_dim_set) + src->nvar, 1,
                          dst->nvar, -1);
        isl_dim_map_range(dim_map, dst->start + 2 * dst->nparam + 2, 2,
                          isl_space_dim(dim, isl_dim_set) + src->nvar, 1,
                          dst->nvar, 1);

        isl_dim_map_range(dim_map, src->start, 0, 0, 0, 1, -1);
        isl_dim_map_range(dim_map, src->start + 1, 2, 1, 1, src->nparam, 1);
        isl_dim_map_range(dim_map, src->start + 2, 2, 1, 1, src->nparam, -1);
        isl_dim_map_range(dim_map, src->start + 2 * src->nparam + 1, 2,
                          isl_space_dim(dim, isl_dim_set), 1,
                          src->nvar, 1);
        isl_dim_map_range(dim_map, src->start + 2 * src->nparam + 2, 2,
                          isl_space_dim(dim, isl_dim_set), 1,
                          src->nvar, -1);

        graph->lp = isl_basic_set_extend_constraints(graph->lp,
                        coef->n_eq, coef->n_ineq);
        graph->lp = isl_basic_set_add_constraints_dim_map(graph->lp,
                                                           coef, dim_map);
        isl_space_free(dim);

        return 0;
}

/* Add constraints to graph->lp that force all (conditional) validity
 * dependences to be respected and attempt to carry them.
 */
static int add_all_constraints(struct isl_sched_graph *graph)
{
        int i, j;
        int pos;

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

                if (!edge->validity && !edge->conditional_validity)
                        continue;

                for (j = 0; j < edge->map->n; ++j) {
                        isl_basic_map *bmap;
                        isl_map *map;

                        bmap = isl_basic_map_copy(edge->map->p[j]);
                        map = isl_map_from_basic_map(bmap);

                        if (edge->src == edge->dst &&
                            add_intra_constraints(graph, edge, map, pos) < 0)
                                return -1;
                        if (edge->src != edge->dst &&
                            add_inter_constraints(graph, edge, map, pos) < 0)
                                return -1;
                        ++pos;
                }
        }

        return 0;
}

/* Count the number of equality and inequality constraints
 * that will be added to the carry_lp problem.
 * We count each edge exactly once.
 */
static int count_all_constraints(struct isl_sched_graph *graph,
        int *n_eq, int *n_ineq)
{
        int i, j;

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

                        bmap = isl_basic_map_copy(edge->map->p[j]);
                        map = isl_map_from_basic_map(bmap);

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

        return 0;
}

/* Construct an LP problem for finding schedule coefficients
 * such that the schedule carries as many 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.
 * Note that if the dependence relation is a union of basic maps,
 * then we have to consider each basic map individually as it may only
 * be possible to carry the dependences expressed by some of those
 * basic maps and not all of them.
 * Below, we consider each of those basic maps as a separate "edge".
 *
 * 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 positive and negative parts 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
 *              - positive and negative parts of 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 int setup_carry_lp(isl_ctx *ctx, struct isl_sched_graph *graph)
{
        int i, j;
        int k;
        isl_space *dim;
        unsigned total;
        int n_eq, n_ineq;
        int n_edge;

        n_edge = 0;
        for (i = 0; i < graph->n_edge; ++i)
                n_edge += graph->edge[i].map->n;

        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 + 2 * (node->nparam + node->nvar);
        }

        if (count_all_constraints(graph, &n_eq, &n_ineq) < 0)
                return -1;

        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 -1;
        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);

        k = isl_basic_set_alloc_equality(graph->lp);
        if (k < 0)
                return -1;
        isl_seq_clr(graph->lp->eq[k], 1 +  total);
        isl_int_set_si(graph->lp->eq[k][2], -1);
        for (i = 0; i < graph->n; ++i) {
                int pos = 1 + graph->node[i].start + 1;

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

        k = isl_basic_set_alloc_equality(graph->lp);
        if (k < 0)
                return -1;
        isl_seq_clr(graph->lp->eq[k], 1 +  total);
        isl_int_set_si(graph->lp->eq[k][3], -1);
        for (i = 0; i < graph->n; ++i) {
                struct isl_sched_node *node = &graph->node[i];
                int pos = 1 + node->start + 1 + 2 * node->nparam;

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

        for (i = 0; i < n_edge; ++i) {
                k = isl_basic_set_alloc_inequality(graph->lp);
                if (k < 0)
                        return -1;
                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(graph) < 0)
                return -1;

        return 0;
}

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 orthogonal to
 * 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 i;
        int pos;
        int trivial;
        isl_ctx *ctx;
        isl_vec *node_sol;

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

        ctx = isl_vec_get_ctx(sol);
        node_sol = isl_vec_alloc(ctx, node->nvar);
        if (!node_sol)
                return -1;

        pos = 1 + node->start + 1 + 2 * node->nparam;

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

        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;
}

/* Construct a schedule row for each node such that as many dependences
 * as possible are carried and then continue with the next band.
 *
 * 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_dependences(
        __isl_take isl_schedule_node *node, struct isl_sched_graph *graph)
{
        int i;
        int n_edge;
        int trivial;
        isl_ctx *ctx;
        isl_vec *sol;
        isl_basic_set *lp;

        if (!node)
                return NULL;

        n_edge = 0;
        for (i = 0; i < graph->n_edge; ++i)
                n_edge += graph->edge[i].map->n;

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

        lp = isl_basic_set_copy(graph->lp);
        sol = isl_tab_basic_set_non_neg_lexmin(lp);
        if (!sol)
                return isl_schedule_node_free(node);

        if (sol->size == 0) {
                isl_vec_free(sol);
                isl_die(ctx, isl_error_internal,
                        "error in schedule construction",
                        return isl_schedule_node_free(node));
        }

        isl_int_divexact(sol->el[1], sol->el[1], sol->el[0]);
        if (isl_int_cmp_si(sol->el[1], n_edge) >= 0) {
                isl_vec_free(sol);
                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);
}

/* Topologically sort statements mapped to the same schedule iteration
 * and add insert a sequence node in front of "node"
 * corresponding to this order.
 *
 * 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)
{
        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)
                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 (graph->edge[i].validity ||
                    graph->edge[i].conditional_validity)
                        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 (graph->edge[i].condition)
                        graph->edge[i].local = 0;
}

/* 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 (graph->edge[i].condition)
                        any_condition = 1;
                if (graph->edge[i].conditional_validity)
                        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 (graph->edge[i].coincidence)
                        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_local_space *ls;
        isl_aff *aff;
        int row;

        row = isl_mat_rows(node->sched) - 1;
        ls = isl_local_space_from_space(isl_space_copy(node->space));
        aff = extract_schedule_row(ls, node, row);
        return isl_map_from_aff(aff);
}

/* 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 (!graph->edge[i].condition)
                        continue;
                if (graph->edge[i].local)
                        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;

                graph->edge[i].local = 1;

                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 (!graph->edge[i].conditional_validity)
                        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;
}

/* Compute a schedule for a connected dependence graph and return
 * the updated schedule node.
 *
 * 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.
 * If we can't find any more rows 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)
 * - try to carry as many 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 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.
 *
 * If we manage 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.
 *
 * 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_give isl_schedule_node *compute_schedule_wcc(
        __isl_take isl_schedule_node *node, struct isl_sched_graph *graph)
{
        int has_coincidence;
        int use_coincidence;
        int force_coincidence = 0;
        int check_conditional;
        int insert;
        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 (sort_sccs(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);

        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_schedule_node_free(node);
                sol = solve_lp(graph);
                if (!sol)
                        return isl_schedule_node_free(node);
                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;
                        }
                        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);
                        return carry_dependences(node, graph);
                }
                coincident = !has_coincidence || use_coincidence;
                if (update_schedule(graph, sol, 1, coincident) < 0)
                        return isl_schedule_node_free(node);

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

        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);
        if (insert)
                node = isl_schedule_node_parent(node);

        return node;
}

/* 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, i;
        int n, n_edge;
        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) {
                n = 0;
                for (i = 0; i < graph->n; ++i)
                        if (graph->node[i].scc == component)
                                n++;
                n_edge = 0;
                for (i = 0; i < graph->n_edge; ++i)
                        if (graph->edge[i].src->scc == component &&
                            graph->edge[i].dst->scc == component)
                                n_edge++;

                node = isl_schedule_node_child(node, component);
                node = isl_schedule_node_child(node, 0);
                node = compute_sub_schedule(node, ctx, graph, n, n_edge,
                                    &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.
 *
 * 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.
 * However, the returned schedule contains the original domain
 * (before this intersection).
 */
__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;
        struct isl_extract_edge_data data;
        enum isl_edge_type i;
        int r;

        sc = isl_schedule_constraints_align_params(sc);
        if (!sc)
                return NULL;

        graph.n = isl_union_set_n_set(sc->domain);
        if (graph.n == 0) {
                isl_union_set *domain = isl_union_set_copy(sc->domain);
                sched = isl_schedule_from_domain(domain);
                goto done;
        }
        if (graph_alloc(ctx, &graph, graph.n,
            isl_schedule_constraints_n_map(sc)) < 0)
                goto error;
        if (compute_max_row(&graph, sc) < 0)
                goto error;
        graph.root = 1;
        graph.n = 0;
        domain = isl_union_set_copy(sc->domain);
        domain = isl_union_set_intersect_params(domain,
                                                isl_set_copy(sc->context));
        r = isl_union_set_foreach_set(domain, &extract_node, &graph);
        isl_union_set_free(domain);
        if (r < 0)
                goto error;
        if (graph_init_table(ctx, &graph) < 0)
                goto error;
        for (i = isl_edge_first; i <= isl_edge_last; ++i)
                graph.max_edge[i] = isl_union_map_n_map(sc->constraint[i]);
        if (graph_init_edge_tables(ctx, &graph) < 0)
                goto error;
        graph.n_edge = 0;
        data.graph = &graph;
        for (i = isl_edge_first; i <= isl_edge_last; ++i) {
                data.type = i;
                if (isl_union_map_foreach_map(sc->constraint[i],
                                                &extract_edge, &data) < 0)
                        goto error;
        }

        node = isl_schedule_node_from_domain(isl_union_set_copy(sc->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);

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

        return sched;
error:
        graph_free(ctx, &graph);
        isl_schedule_constraints_free(sc);
        return NULL;
}

/* 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);
}