isl_union_set_compute_schedule: also keep track of proximity edges in hash table

[isl.git] / isl_schedule.c

/*
 * Copyright 2011      INRIA Saclay
 *
 * Use of this software is governed by the GNU LGPLv2.1 license
 *
 * Written by Sven Verdoolaege, INRIA Saclay - Ile-de-France,
 * Parc Club Orsay Universite, ZAC des vignes, 4 rue Jacques Monod,
 * 91893 Orsay, France
 */

#include <isl_ctx_private.h>
#include <isl_map_private.h>
#include <isl_space_private.h>
#include <isl/hash.h>
#include <isl/constraint.h>
#include <isl/schedule.h>
#include <isl_mat_private.h>
#include <isl/set.h>
#include <isl/seq.h>
#include <isl_tab.h>
#include <isl_dim_map.h>
#include <isl_hmap_map_basic_set.h>
#include <isl_qsort.h>
#include <isl_schedule_private.h>
#include <isl_band_private.h>
#include <isl_list_private.h>
#include <isl_options_private.h>

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


/* Internal information about a node that is used during the construction
 * of a schedule.
 * dim represents the space in which the domain lives
 * sched is a matrix representation of the schedule being constructed
 *      for this node
 * sched_map is an isl_map representation of the same (partial) schedule
 *      sched_map may be NULL
 * 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
 * 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
 *
 * scc is the index of SCC (or WCC) this node belongs to
 *
 * band contains the band index for each of the rows of the schedule.
 * band_id is used to differentiate between separate bands at the same
 * level within the same parent band, i.e., bands that are separated
 * by the parent band or bands that are independent of each other.
 * zero contains a boolean for each of the rows of the schedule,
 * indicating whether the corresponding scheduling dimension results
 * in zero dependence distances within its band and with respect
 * to the proximity edges.
 *
 * index, min_index and on_stack are used during the SCC detection
 * index represents the order in which nodes are visited.
 * min_index is the index of the root of a (sub)component.
 * on_stack indicates whether the node is currently on the stack.
 */
struct isl_sched_node {
        isl_space *dim;
        isl_mat *sched;
        isl_map *sched_map;
        int      rank;
        isl_mat *cmap;
        int      start;
        int      nvar;
        int      nparam;

        int      scc;

        int     *band;
        int     *band_id;
        int     *zero;

        /* scc detection */
        int      index;
        int      min_index;
        int      on_stack;
};

static int node_has_dim(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->dim, dim);
}

/* 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
 * src is the source node
 * dst is the sink node
 * validity is set if the edge is used to ensure correctness
 * proximity is set if the edge is used to minimize dependence distances
 *
 * 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;

        struct isl_sched_node *src;
        struct isl_sched_node *dst;

        int validity;
        int proximity;

        int start;
        int end;
};

enum isl_edge_type {
        isl_edge_validity = 0,
        isl_edge_proximity,
        isl_edge_last = isl_edge_proximity
};

/* 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
 *
 * n is the number of nodes
 * node is the list of nodes
 * maxvar is the maximal number of variables over all nodes
 * 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
 * n_band is the current number of completed bands
 * 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 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, sp, index and stack are used during the detection of SCCs
 * scc is the number of the next SCC
 * stack contains the nodes on the path from the root to the current node
 * sp is the stack pointer
 * index is the index of the last node visited
 */
struct isl_sched_graph {
        isl_hmap_map_basic_set *intra_hmap;
        isl_hmap_map_basic_set *inter_hmap;

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

        int *sorted;

        int n_band;
        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;

        /* scc detection */
        int scc;
        int sp;
        int index;
        int *stack;
};

/* 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].dim);
                entry = isl_hash_table_find(ctx, graph->node_table, hash,
                                            &node_has_dim,
                                            graph->node[i].dim, 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_dim, 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 int 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 -1;
        entry->data = edge;

        return 0;
}

/* 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->dim);
        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 give type
 * between the given two nodes.
 */
static int 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;
        int 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 -1;

        return !empty;
}

/* If there is an edge from the given source to the given destination
 * of any type then return this edge.
 * Otherwise, return NULL.
 */
static struct isl_sched_edge *graph_find_any_edge(struct isl_sched_graph *graph,
        struct isl_sched_node *src, struct isl_sched_node *dst)
{
        int i;
        struct isl_sched_edge *edge;

        for (i = 0; i <= isl_edge_last; ++i) {
                edge = graph_find_edge(graph, i, src, dst);
                if (edge)
                        return edge;
        }

        return NULL;
}

/* 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);
        int i;

        for (i = 0; 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 a validity edge
 * between the given two nodes.
 */
static int graph_has_validity_edge(struct isl_sched_graph *graph,
        struct isl_sched_node *src, struct isl_sched_node *dst)
{
        return graph_has_edge(graph, isl_edge_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->stack = isl_alloc_array(ctx, int, graph->n);
        graph->edge = isl_calloc_array(ctx,
                                        struct isl_sched_edge, graph->n_edge);

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

        if (!graph->node || !graph->region || !graph->stack || !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_hmap_map_basic_set_free(ctx, graph->intra_hmap);
        isl_hmap_map_basic_set_free(ctx, graph->inter_hmap);

        for (i = 0; i < graph->n; ++i) {
                isl_space_free(graph->node[i].dim);
                isl_mat_free(graph->node[i].sched);
                isl_map_free(graph->node[i].sched_map);
                isl_mat_free(graph->node[i].cmap);
                if (graph->root) {
                        free(graph->node[i].band);
                        free(graph->node[i].band_id);
                        free(graph->node[i].zero);
                }
        }
        free(graph->node);
        free(graph->sorted);
        for (i = 0; i < graph->n_edge; ++i)
                isl_map_free(graph->edge[i].map);
        free(graph->edge);
        free(graph->region);
        free(graph->stack);
        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);
}

/* Add a new node to the graph representing the given set.
 */
static int extract_node(__isl_take isl_set *set, void *user)
{
        int nvar, nparam;
        isl_ctx *ctx;
        isl_space *dim;
        isl_mat *sched;
        struct isl_sched_graph *graph = user;
        int *band, *band_id, *zero;

        ctx = isl_set_get_ctx(set);
        dim = isl_set_get_space(set);
        isl_set_free(set);
        nvar = isl_space_dim(dim, isl_dim_set);
        nparam = isl_space_dim(dim, isl_dim_param);
        if (!ctx->opt->schedule_parametric)
                nparam = 0;
        sched = isl_mat_alloc(ctx, 0, 1 + nparam + nvar);
        graph->node[graph->n].dim = dim;
        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;
        band = isl_alloc_array(ctx, int, graph->n_edge + nvar);
        graph->node[graph->n].band = band;
        band_id = isl_calloc_array(ctx, int, graph->n_edge + nvar);
        graph->node[graph->n].band_id = band_id;
        zero = isl_calloc_array(ctx, int, graph->n_edge + nvar);
        graph->node[graph->n].zero = zero;
        graph->n++;

        if (!sched || !band || !band_id || !zero)
                return -1;

        return 0;
}

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

/* 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.
 */
static int 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;
        int is_equal;

        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);
                return 0;
        }

        graph->edge[graph->n_edge].src = src;
        graph->edge[graph->n_edge].dst = dst;
        graph->edge[graph->n_edge].map = map;
        if (data->type == isl_edge_validity) {
                graph->edge[graph->n_edge].validity = 1;
                graph->edge[graph->n_edge].proximity = 0;
        }
        if (data->type == isl_edge_proximity) {
                graph->edge[graph->n_edge].validity = 0;
                graph->edge[graph->n_edge].proximity = 1;
        }
        graph->n_edge++;

        edge = graph_find_any_edge(graph, src, dst);
        if (!edge)
                return graph_edge_table_add(ctx, graph, data->type,
                                    &graph->edge[graph->n_edge - 1]);
        is_equal = isl_map_plain_is_equal(map, edge->map);
        if (is_equal < 0)
                return -1;
        if (!is_equal)
                return graph_edge_table_add(ctx, graph, data->type,
                                    &graph->edge[graph->n_edge - 1]);

        graph->n_edge--;
        edge->validity |= graph->edge[graph->n_edge].validity;
        edge->proximity |= graph->edge[graph->n_edge].proximity;
        isl_map_free(map);

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

/* Check whether there is a validity dependence from src to dst,
 * forcing dst to follow src.
 */
static int node_follows(struct isl_sched_graph *graph, 
        struct isl_sched_node *dst, struct isl_sched_node *src)
{
        return graph_has_validity_edge(graph, src, dst);
}

/* Perform Tarjan's algorithm for computing the strongly connected components
 * in the dependence graph (only validity edges).
 * If directed is not set, we consider the graph to be undirected and
 * we effectively compute the (weakly) connected components.
 */
static int detect_sccs_tarjan(struct isl_sched_graph *g, int i, int directed)
{
        int j;

        g->node[i].index = g->index;
        g->node[i].min_index = g->index;
        g->node[i].on_stack = 1;
        g->index++;
        g->stack[g->sp++] = i;

        for (j = g->n - 1; j >= 0; --j) {
                int f;

                if (j == i)
                        continue;
                if (g->node[j].index >= 0 &&
                        (!g->node[j].on_stack ||
                         g->node[j].index > g->node[i].min_index))
                        continue;
                
                f = node_follows(g, &g->node[i], &g->node[j]);
                if (f < 0)
                        return -1;
                if (!f && !directed) {
                        f = node_follows(g, &g->node[j], &g->node[i]);
                        if (f < 0)
                                return -1;
                }
                if (!f)
                        continue;
                if (g->node[j].index < 0) {
                        detect_sccs_tarjan(g, j, directed);
                        if (g->node[j].min_index < g->node[i].min_index)
                                g->node[i].min_index = g->node[j].min_index;
                } else if (g->node[j].index < g->node[i].min_index)
                        g->node[i].min_index = g->node[j].index;
        }

        if (g->node[i].index != g->node[i].min_index)
                return 0;

        do {
                j = g->stack[--g->sp];
                g->node[j].on_stack = 0;
                g->node[j].scc = g->scc;
        } while (j != i);
        g->scc++;

        return 0;
}

static int detect_ccs(struct isl_sched_graph *graph, int directed)
{
        int i;

        graph->index = 0;
        graph->sp = 0;
        graph->scc = 0;
        for (i = graph->n - 1; i >= 0; --i)
                graph->node[i].index = -1;

        for (i = graph->n - 1; i >= 0; --i) {
                if (graph->node[i].index >= 0)
                        continue;
                if (detect_sccs_tarjan(graph, i, directed) < 0)
                        return -1;
        }

        return 0;
}

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

/* Apply Tarjan's algorithm to detect the (weakly) connected components
 * in the dependence graph.
 */
static int detect_wccs(struct isl_sched_graph *graph)
{
        return detect_ccs(graph, 0);
}

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 void sort_sccs(struct isl_sched_graph *graph)
{
        isl_quicksort(graph->sorted, graph->n, sizeof(int), &cmp_scc, graph);
}

/* Given a dependence relation R from a 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).
 */
static __isl_give isl_basic_set *intra_coefficients(
        struct isl_sched_graph *graph, __isl_take isl_map *map)
{
        isl_ctx *ctx = isl_map_get_ctx(map);
        isl_set *delta;
        isl_basic_set *coef;

        if (isl_hmap_map_basic_set_has(ctx, graph->intra_hmap, map))
                return isl_hmap_map_basic_set_get(ctx, graph->intra_hmap, map);

        delta = isl_set_remove_divs(isl_map_deltas(isl_map_copy(map)));
        coef = isl_set_coefficients(delta);
        isl_hmap_map_basic_set_set(ctx, graph->intra_hmap, map,
                                        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
 *
 */
static __isl_give isl_basic_set *inter_coefficients(
        struct isl_sched_graph *graph, __isl_take isl_map *map)
{
        isl_ctx *ctx = isl_map_get_ctx(map);
        isl_set *set;
        isl_basic_set *coef;

        if (isl_hmap_map_basic_set_has(ctx, graph->inter_hmap, map))
                return isl_hmap_map_basic_set_get(ctx, graph->inter_hmap, map);

        set = isl_map_wrap(isl_map_remove_divs(isl_map_copy(map)));
        coef = isl_set_coefficients(set);
        isl_hmap_map_basic_set_set(ctx, graph->inter_hmap, map,
                                        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, 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));

        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;
}

/* 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, 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));

        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);
        isl_space_free(dim);
        edge->end = graph->lp->n_ineq;

        return 0;
}

/* 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.
 */
static int add_intra_proximity_constraints(struct isl_sched_graph *graph,
        struct isl_sched_edge *edge, int s)
{
        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, 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));

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

/* 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.
 */
static int add_inter_proximity_constraints(struct isl_sched_graph *graph,
        struct isl_sched_edge *edge, int s)
{
        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, 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));

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

        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;
}

static int add_all_validity_constraints(struct isl_sched_graph *graph)
{
        int i;

        for (i = 0; i < graph->n_edge; ++i) {
                struct isl_sched_edge *edge= &graph->edge[i];
                if (!edge->validity)
                        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];
                if (!edge->validity)
                        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.
 * Otherwise, we need to bound the distance both from above and from below.
 */
static int add_all_proximity_constraints(struct isl_sched_graph *graph)
{
        int i;

        for (i = 0; i < graph->n_edge; ++i) {
                struct isl_sched_edge *edge= &graph->edge[i];
                if (!edge->proximity)
                        continue;
                if (edge->src == edge->dst &&
                    add_intra_proximity_constraints(graph, edge, 1) < 0)
                        return -1;
                if (edge->src != edge->dst &&
                    add_inter_proximity_constraints(graph, edge, 1) < 0)
                        return -1;
                if (edge->validity)
                        continue;
                if (edge->src == edge->dst &&
                    add_intra_proximity_constraints(graph, edge, -1) < 0)
                        return -1;
                if (edge->src != edge->dst &&
                    add_inter_proximity_constraints(graph, edge, -1) < 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
 *
 * with H the Hermite normal form of S.  That is, all but the
 * first rank columns of Q 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.
 */
static int node_update_cmap(struct isl_sched_node *node)
{
        isl_mat *H, *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, NULL, &Q);
        isl_mat_free(node->cmap);
        node->cmap = isl_mat_transpose(Q);
        node->rank = isl_mat_initial_non_zero_cols(H);
        isl_mat_free(H);

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

/* Count the number of equality and inequality constraints
 * that will be added for the given map.
 * If carry is set, then we are counting the number of (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)
 */
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)
{
        isl_basic_set *coef;
        int f = carry ? 1 : edge->proximity ? 2 : 1;

        if (carry && !edge->validity) {
                isl_map_free(map);
                return 0;
        }

        if (edge->src == edge->dst)
                coef = intra_coefficients(graph, map);
        else
                coef = inter_coefficients(graph, 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)
 */
static int count_constraints(struct isl_sched_graph *graph,
        int *n_eq, int *n_ineq)
{
        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) < 0)
                        return -1;
        }

        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 force_zero is set, then we add equalities to ensure that
 * the sum of the m_n coefficients and m_0 are both zero.
 */
static int setup_lp(isl_ctx *ctx, struct isl_sched_graph *graph,
        int force_zero)
{
        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;
        int max_coefficient;

        max_constant_term = ctx->opt->schedule_max_constant_term;
        max_coefficient = ctx->opt->schedule_max_coefficient;

        parametric = ctx->opt->schedule_parametric;
        nparam = isl_space_dim(graph->node[0].dim, 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) < 0)
                return -1;

        dim = isl_space_set_alloc(ctx, 0, total);
        isl_basic_set_free(graph->lp);
        n_eq += 2 + parametric + force_zero;
        if (max_constant_term != -1)
                n_ineq += graph->n;
        if (max_coefficient != -1)
                for (i = 0; i < graph->n; ++i)
                        n_ineq += 2 * graph->node[i].nparam +
                                  2 * graph->node[i].nvar;

        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);
        if (!force_zero)
                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 (force_zero) {
                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);
        }

        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) < 0)
                return -1;
        if (add_all_proximity_constraints(graph) < 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 check_zero is set, then the first two coordinates of sol are
 * assumed to correspond to the dependence distance.  If these two
 * coordinates are zero, then the corresponding scheduling dimension
 * is marked as being zero distance.
 */
static int update_schedule(struct isl_sched_graph *graph,
        __isl_take isl_vec *sol, int use_cmap, int check_zero)
{
        int i, j;
        int zero = 0;
        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 (check_zero)
                zero = isl_int_is_zero(sol->el[1]) &&
                           isl_int_is_zero(sol->el[2]);

        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->band[graph->n_total_row] = graph->n_band;
                node->zero[graph->n_total_row] = zero;
        }
        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 node->sched into a map and return this map.
 * We simply add equality constraints that express each output variable
 * as the affine combination of parameters and input variables specified
 * by the schedule matrix.
 *
 * The result is cached in node->sched_map, which needs to be released
 * whenever node->sched is updated.
 */
static __isl_give isl_map *node_extract_schedule(struct isl_sched_node *node)
{
        int i, j;
        isl_space *dim;
        isl_local_space *ls;
        isl_basic_map *bmap;
        isl_constraint *c;
        int nrow, ncol;
        isl_int v;

        if (node->sched_map)
                return isl_map_copy(node->sched_map);

        nrow = isl_mat_rows(node->sched);
        ncol = isl_mat_cols(node->sched) - 1;
        dim = isl_space_from_domain(isl_space_copy(node->dim));
        dim = isl_space_add_dims(dim, isl_dim_out, nrow);
        bmap = isl_basic_map_universe(isl_space_copy(dim));
        ls = isl_local_space_from_space(dim);

        isl_int_init(v);

        for (i = 0; i < nrow; ++i) {
                c = isl_equality_alloc(isl_local_space_copy(ls));
                isl_constraint_set_coefficient_si(c, isl_dim_out, i, -1);
                isl_mat_get_element(node->sched, i, 0, &v);
                isl_constraint_set_constant(c, v);
                for (j = 0; j < node->nparam; ++j) {
                        isl_mat_get_element(node->sched, i, 1 + j, &v);
                        isl_constraint_set_coefficient(c, isl_dim_param, j, v);
                }
                for (j = 0; j < node->nvar; ++j) {
                        isl_mat_get_element(node->sched,
                                            i, 1 + node->nparam + j, &v);
                        isl_constraint_set_coefficient(c, isl_dim_in, j, v);
                }
                bmap = isl_basic_map_add_constraint(bmap, c);
        }

        isl_int_clear(v);

        isl_local_space_free(ls);

        node->sched_map = isl_map_from_basic_map(bmap);
        return isl_map_copy(node->sched_map);
}

/* Update the given dependence relation based on the current schedule.
 * That is, intersect the dependence relation with a map expressing
 * that source and sink are executed within the same iteration of
 * the current schedule.
 * This is not the most efficient way, but this shouldn't be a critical
 * operation.
 */
static __isl_give isl_map *specialize(__isl_take isl_map *map,
        struct isl_sched_node *src, struct isl_sched_node *dst)
{
        isl_map *src_sched, *dst_sched, *id;

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

/* Update the dependence relations of all edges based on the current schedule.
 * If a 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_edges(isl_ctx *ctx, struct isl_sched_graph *graph)
{
        int i;

        for (i = graph->n_edge - 1; i >= 0; --i) {
                struct isl_sched_edge *edge = &graph->edge[i];
                edge->map = specialize(edge->map, edge->src, edge->dst);
                if (!edge->map)
                        return -1;

                if (isl_map_plain_is_empty(edge->map))
                        graph_remove_edge(graph, edge);
        }

        return 0;
}

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

/* Topologically sort statements mapped to the same schedule iteration
 * and add a row to the schedule corresponding to this order.
 */
static int sort_statements(isl_ctx *ctx, struct isl_sched_graph *graph)
{
        int i, j;

        if (graph->n <= 1)
                return 0;

        if (update_edges(ctx, graph) < 0)
                return -1;

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

        if (detect_sccs(graph) < 0)
                return -1;

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

                isl_map_free(node->sched_map);
                node->sched_map = NULL;
                node->sched = isl_mat_add_rows(node->sched, 1);
                if (!node->sched)
                        return -1;
                node->sched = isl_mat_set_element_si(node->sched, row, 0,
                                                     node->scc);
                for (j = 1; j < cols; ++j)
                        node->sched = isl_mat_set_element_si(node->sched,
                                                             row, j, 0);
                node->band[graph->n_total_row] = graph->n_band;
        }

        graph->n_total_row++;
        next_band(graph);

        return 0;
}

/* Construct an isl_schedule based on the computed schedule stored
 * in graph and with parameters specified by dim.
 */
static __isl_give isl_schedule *extract_schedule(struct isl_sched_graph *graph,
        __isl_take isl_space *dim)
{
        int i;
        isl_ctx *ctx;
        isl_schedule *sched = NULL;
                
        if (!dim)
                return NULL;

        ctx = isl_space_get_ctx(dim);
        sched = isl_calloc(ctx, struct isl_schedule,
                           sizeof(struct isl_schedule) +
                           (graph->n - 1) * sizeof(struct isl_schedule_node));
        if (!sched)
                goto error;

        sched->ref = 1;
        sched->n = graph->n;
        sched->n_band = graph->n_band;
        sched->n_total_row = graph->n_total_row;

        for (i = 0; i < sched->n; ++i) {
                int r, b;
                int *band_end, *band_id, *zero;

                band_end = isl_alloc_array(ctx, int, graph->n_band);
                band_id = isl_alloc_array(ctx, int, graph->n_band);
                zero = isl_alloc_array(ctx, int, graph->n_total_row);
                sched->node[i].sched = node_extract_schedule(&graph->node[i]);
                sched->node[i].band_end = band_end;
                sched->node[i].band_id = band_id;
                sched->node[i].zero = zero;
                if (!band_end || !band_id || !zero)
                        goto error;

                for (r = 0; r < graph->n_total_row; ++r)
                        zero[r] = graph->node[i].zero[r];
                for (r = b = 0; r < graph->n_total_row; ++r) {
                        if (graph->node[i].band[r] == b)
                                continue;
                        band_end[b++] = r;
                        if (graph->node[i].band[r] == -1)
                                break;
                }
                if (r == graph->n_total_row)
                        band_end[b++] = r;
                sched->node[i].n_band = b;
                for (--b; b >= 0; --b)
                        band_id[b] = graph->node[i].band_id[b];
        }

        sched->dim = dim;

        return sched;
error:
        isl_space_free(dim);
        isl_schedule_free(sched);
        return NULL;
}

/* 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) {
                if (!node_pred(&src->node[i], data))
                        continue;
                dst->node[dst->n].dim = isl_space_copy(src->node[i].dim);
                dst->node[dst->n].nvar = src->node[i].nvar;
                dst->node[dst->n].nparam = src->node[i].nparam;
                dst->node[dst->n].sched = isl_mat_copy(src->node[i].sched);
                dst->node[dst->n].sched_map =
                        isl_map_copy(src->node[i].sched_map);
                dst->node[dst->n].band = src->node[i].band;
                dst->node[dst->n].band_id = src->node[i].band_id;
                dst->node[dst->n].zero = src->node[i].zero;
                dst->n++;
        }

        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;
        int t;

        dst->n_edge = 0;
        for (i = 0; i < src->n_edge; ++i) {
                struct isl_sched_edge *edge = &src->edge[i];
                isl_map *map;
                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->dim);
                dst_dst = graph_find_node(ctx, dst, edge->dst->dim);
                if (!dst_src || !dst_dst) {
                        if (edge->validity)
                                isl_die(ctx, isl_error_internal,
                                        "backward validity edge", return -1);
                        continue;
                }

                map = isl_map_copy(edge->map);

                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].validity = edge->validity;
                dst->edge[dst->n_edge].proximity = edge->proximity;
                dst->n_edge++;

                for (t = 0; 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;
}

/* Given a "src" dependence graph that contains the nodes from "dst"
 * that satisfy node_pred, copy the schedule computed in "src"
 * for those nodes back to "dst".
 */
static int copy_schedule(struct isl_sched_graph *dst,
        struct isl_sched_graph *src,
        int (*node_pred)(struct isl_sched_node *node, int data), int data)
{
        int i;

        src->n = 0;
        for (i = 0; i < dst->n; ++i) {
                if (!node_pred(&dst->node[i], data))
                        continue;
                isl_mat_free(dst->node[i].sched);
                isl_map_free(dst->node[i].sched_map);
                dst->node[i].sched = isl_mat_copy(src->node[src->n].sched);
                dst->node[i].sched_map =
                        isl_map_copy(src->node[src->n].sched_map);
                src->n++;
        }

        dst->n_total_row = src->n_total_row;
        dst->n_band = src->n_band;

        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 int compute_schedule(isl_ctx *ctx, struct isl_sched_graph *graph);
static int compute_schedule_wcc(isl_ctx *ctx, 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.
 */
static int compute_sub_schedule(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.n_total_row = graph->n_total_row;
        split.n_band = graph->n_band;
        split.band_start = graph->band_start;

        if (wcc && compute_schedule_wcc(ctx, &split) < 0)
                goto error;
        if (!wcc && compute_schedule(ctx, &split) < 0)
                goto error;

        copy_schedule(graph, &split, node_pred, data);

        graph_free(ctx, &split);
        return 0;
error:
        graph_free(ctx, &split);
        return -1;
}

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;
}

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;
}

/* Pad the schedules of all nodes with zero rows such that in the end
 * they all have graph->n_total_row rows.
 * The extra rows don't belong to any band, so they get assigned band number -1.
 */
static int pad_schedule(struct isl_sched_graph *graph)
{
        int i, j;

        for (i = 0; i < graph->n; ++i) {
                struct isl_sched_node *node = &graph->node[i];
                int row = isl_mat_rows(node->sched);
                if (graph->n_total_row > row) {
                        isl_map_free(node->sched_map);
                        node->sched_map = NULL;
                }
                node->sched = isl_mat_add_zero_rows(node->sched,
                                                    graph->n_total_row - row);
                if (!node->sched)
                        return -1;
                for (j = row; j < graph->n_total_row; ++j)
                        node->band[j] = -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 in the schedule by constant rows with two different
 * values (0 and 1).  These constant rows replace the previously computed rows
 * in the current band.
 * It would be possible to reuse them 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.
 * compute_split_schedule is only called when no zero-distance schedule row
 * could be found on the entire graph, so we wark the splitting row as
 * non zero-distance.
 *
 * The band_id of the second group is set to n, where n is the number
 * of nodes in the first group.  This ensures that the band_ids over
 * the two groups remain disjoint, even if either or both of the two
 * groups contain independent components.
 */
static int compute_split_schedule(isl_ctx *ctx, struct isl_sched_graph *graph)
{
        int i, j, n, e1, e2;
        int n_total_row, orig_total_row;
        int n_band, orig_band;
        int drop;

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

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

                if (before)
                        n++;

                isl_map_free(node->sched_map);
                node->sched_map = NULL;
                node->sched = isl_mat_drop_rows(node->sched,
                                                graph->band_start, drop);
                node->sched = isl_mat_add_rows(node->sched, 1);
                if (!node->sched)
                        return -1;
                node->sched = isl_mat_set_element_si(node->sched, row, 0,
                                                     !before);
                for (j = 1; j < cols; ++j)
                        node->sched = isl_mat_set_element_si(node->sched,
                                                             row, j, 0);
                node->band[graph->n_total_row] = graph->n_band;
                node->zero[graph->n_total_row] = 0;
        }

        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++;
        }

        graph->n_total_row++;
        next_band(graph);

        for (i = 0; i < graph->n; ++i) {
                struct isl_sched_node *node = &graph->node[i];
                if (node->scc > graph->src_scc)
                        node->band_id[graph->n_band] = n;
        }

        orig_total_row = graph->n_total_row;
        orig_band = graph->n_band;
        if (compute_sub_schedule(ctx, graph, n, e1,
                                &node_scc_at_most, &edge_dst_scc_at_most,
                                graph->src_scc, 0) < 0)
                return -1;
        n_total_row = graph->n_total_row;
        graph->n_total_row = orig_total_row;
        n_band = graph->n_band;
        graph->n_band = orig_band;
        if (compute_sub_schedule(ctx, graph, graph->n - n, e2,
                                &node_scc_at_least, &edge_src_scc_at_least,
                                graph->src_scc + 1, 0) < 0)
                return -1;
        if (n_total_row > graph->n_total_row)
                graph->n_total_row = n_total_row;
        if (n_band > graph->n_band)
                graph->n_band = n_band;

        return pad_schedule(graph);
}

/* Compute the next band of the schedule after updating the dependence
 * relations based on the the current schedule.
 */
static int compute_next_band(isl_ctx *ctx, struct isl_sched_graph *graph)
{
        if (update_edges(ctx, graph) < 0)
                return -1;
        next_band(graph);
                
        return compute_schedule(ctx, graph);
}

/* 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, map);

        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, map);

        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 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)
                        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)
                                    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.  Dependence 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 off 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;
}

/* If the schedule_split_scaled option is set and if the linear
 * parts of the scheduling rows for all nodes in the graphs have
 * non-trivial common divisor, then split off the constant term
 * from the linear part.
 * The constant term is then placed in a separate band and
 * the linear part is reduced.
 */
static int split_scaled(isl_ctx *ctx, struct isl_sched_graph *graph)
{
        int i;
        int row;
        isl_int gcd, gcd_i;

        if (!ctx->opt->schedule_split_scaled)
                return 0;
        if (graph->n <= 1)
                return 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 0;
        }

        next_band(graph);

        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_add_zero_rows(node->sched, 1);
                if (!node->sched)
                        goto error;
                isl_int_fdiv_r(node->sched->row[row + 1][0],
                               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;
                node->band[graph->n_total_row] = graph->n_band;
        }

        graph->n_total_row++;

        isl_int_clear(gcd);
        return 0;
error:
        isl_int_clear(gcd);
        return -1;
}

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

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

        if (setup_carry_lp(ctx, graph) < 0)
                return -1;

        lp = isl_basic_set_copy(graph->lp);
        sol = isl_tab_basic_set_non_neg_lexmin(lp);
        if (!sol)
                return -1;

        if (sol->size == 0) {
                isl_vec_free(sol);
                isl_die(ctx, isl_error_internal,
                        "error in schedule construction", return -1);
        }

        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 -1);
        }

        if (update_schedule(graph, sol, 0, 0) < 0)
                return -1;

        if (split_scaled(ctx, graph) < 0)
                return -1;

        return compute_next_band(ctx, graph);
}

/* Are there any (non-empty) 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)
                        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.
 * 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 int compute_schedule_wcc_feautrier(isl_ctx *ctx,
        struct isl_sched_graph *graph)
{
        return carry_dependences(ctx, graph);
}

/* Compute a schedule for a connected dependence graph.
 * 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).
 * 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
 *
 * 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 finish off by topologically
 * sorting the statements based on the remaining dependences.
 *
 * If ctx->opt->schedule_outer_zero_distance is set, then we force the
 * outermost dimension in the current band to be zero distance.  If this
 * turns out to be impossible, we fall back on the general scheme above
 * and try to carry as many dependences as possible.
 */
static int compute_schedule_wcc(isl_ctx *ctx, struct isl_sched_graph *graph)
{
        int force_zero = 0;

        if (detect_sccs(graph) < 0)
                return -1;
        sort_sccs(graph);

        if (compute_maxvar(graph) < 0)
                return -1;

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

        if (ctx->opt->schedule_outer_zero_distance)
                force_zero = 1;

        while (graph->n_row < graph->maxvar) {
                isl_vec *sol;

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

                if (setup_lp(ctx, graph, force_zero) < 0)
                        return -1;
                sol = solve_lp(graph);
                if (!sol)
                        return -1;
                if (sol->size == 0) {
                        isl_vec_free(sol);
                        if (!ctx->opt->schedule_maximize_band_depth &&
                            graph->n_total_row > graph->band_start)
                                return compute_next_band(ctx, graph);
                        if (graph->src_scc >= 0)
                                return compute_split_schedule(ctx, graph);
                        if (graph->n_total_row > graph->band_start)
                                return compute_next_band(ctx, graph);
                        return carry_dependences(ctx, graph);
                }
                if (update_schedule(graph, sol, 1, 1) < 0)
                        return -1;
                force_zero = 0;
        }

        if (graph->n_total_row > graph->band_start)
                next_band(graph);
        return sort_statements(ctx, graph);
}

/* Add a row to the schedules that separates the SCCs and move
 * to the next band.
 */
static int split_on_scc(struct isl_sched_graph *graph)
{
        int i;

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

                isl_map_free(node->sched_map);
                node->sched_map = NULL;
                node->sched = isl_mat_add_zero_rows(node->sched, 1);
                node->sched = isl_mat_set_element_si(node->sched, row, 0,
                                                     node->scc);
                if (!node->sched)
                        return -1;
                node->band[graph->n_total_row] = graph->n_band;
        }

        graph->n_total_row++;
        next_band(graph);

        return 0;
}

/* Compute a schedule for each component (identified by node->scc)
 * of the dependence graph separately and then combine the results.
 * Depending on the setting of schedule_fuse, a component may be
 * either weakly or strongly connected.
 *
 * The band_id is adjusted such that each component has a separate id.
 * Note that the band_id may have already been set to a value different
 * from zero by compute_split_schedule.
 */
static int compute_component_schedule(isl_ctx *ctx,
        struct isl_sched_graph *graph)
{
        int wcc, i;
        int n, n_edge;
        int n_total_row, orig_total_row;
        int n_band, orig_band;

        if (ctx->opt->schedule_fuse == ISL_SCHEDULE_FUSE_MIN)
                split_on_scc(graph);

        n_total_row = 0;
        orig_total_row = graph->n_total_row;
        n_band = 0;
        orig_band = graph->n_band;
        for (i = 0; i < graph->n; ++i)
                graph->node[i].band_id[graph->n_band] += graph->node[i].scc;
        for (wcc = 0; wcc < graph->scc; ++wcc) {
                n = 0;
                for (i = 0; i < graph->n; ++i)
                        if (graph->node[i].scc == wcc)
                                n++;
                n_edge = 0;
                for (i = 0; i < graph->n_edge; ++i)
                        if (graph->edge[i].src->scc == wcc &&
                            graph->edge[i].dst->scc == wcc)
                                n_edge++;

                if (compute_sub_schedule(ctx, graph, n, n_edge,
                                    &node_scc_exactly,
                                    &edge_scc_exactly, wcc, 1) < 0)
                        return -1;
                if (graph->n_total_row > n_total_row)
                        n_total_row = graph->n_total_row;
                graph->n_total_row = orig_total_row;
                if (graph->n_band > n_band)
                        n_band = graph->n_band;
                graph->n_band = orig_band;
        }

        graph->n_total_row = n_total_row;
        graph->n_band = n_band;

        return pad_schedule(graph);
}

/* Compute a schedule for the given dependence graph.
 * We first check if the graph is connected (through validity dependences)
 * and, if not, compute a schedule for each component separately.
 * If schedule_fuse is set to minimal fusion, then we check for strongly
 * connected components instead and compute a separate schedule for
 * each such strongly connected component.
 */
static int compute_schedule(isl_ctx *ctx, struct isl_sched_graph *graph)
{
        if (ctx->opt->schedule_fuse == ISL_SCHEDULE_FUSE_MIN) {
                if (detect_sccs(graph) < 0)
                        return -1;
        } else {
                if (detect_wccs(graph) < 0)
                        return -1;
        }

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

        return compute_schedule_wcc(ctx, graph);
}

/* Compute a schedule for the given union of domains that 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.
 */
__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_ctx *ctx = isl_union_set_get_ctx(domain);
        isl_space *dim;
        struct isl_sched_graph graph = { 0 };
        isl_schedule *sched;
        struct isl_extract_edge_data data;

        domain = isl_union_set_align_params(domain,
                                            isl_union_map_get_space(validity));
        domain = isl_union_set_align_params(domain,
                                            isl_union_map_get_space(proximity));
        dim = isl_union_set_get_space(domain);
        validity = isl_union_map_align_params(validity, isl_space_copy(dim));
        proximity = isl_union_map_align_params(proximity, dim);

        if (!domain)
                goto error;

        graph.n = isl_union_set_n_set(domain);
        if (graph.n == 0)
                goto empty;
        if (graph_alloc(ctx, &graph, graph.n,
            isl_union_map_n_map(validity) + isl_union_map_n_map(proximity)) < 0)
                goto error;
        graph.root = 1;
        graph.n = 0;
        if (isl_union_set_foreach_set(domain, &extract_node, &graph) < 0)
                goto error;
        if (graph_init_table(ctx, &graph) < 0)
                goto error;
        graph.max_edge[isl_edge_validity] = isl_union_map_n_map(validity);
        graph.max_edge[isl_edge_proximity] = isl_union_map_n_map(proximity);
        if (graph_init_edge_tables(ctx, &graph) < 0)
                goto error;
        graph.n_edge = 0;
        data.graph = &graph;
        data.type = isl_edge_validity;
        if (isl_union_map_foreach_map(validity, &extract_edge, &data) < 0)
                goto error;
        data.type = isl_edge_proximity;
        if (isl_union_map_foreach_map(proximity, &extract_edge, &data) < 0)
                goto error;

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

empty:
        sched = extract_schedule(&graph, isl_union_set_get_space(domain));

        graph_free(ctx, &graph);
        isl_union_set_free(domain);
        isl_union_map_free(validity);
        isl_union_map_free(proximity);

        return sched;
error:
        graph_free(ctx, &graph);
        isl_union_set_free(domain);
        isl_union_map_free(validity);
        isl_union_map_free(proximity);
        return NULL;
}

void *isl_schedule_free(__isl_take isl_schedule *sched)
{
        int i;
        if (!sched)
                return NULL;

        if (--sched->ref > 0)
                return NULL;

        for (i = 0; i < sched->n; ++i) {
                isl_map_free(sched->node[i].sched);
                free(sched->node[i].band_end);
                free(sched->node[i].band_id);
                free(sched->node[i].zero);
        }
        isl_space_free(sched->dim);
        isl_band_list_free(sched->band_forest);
        free(sched);
        return NULL;
}

isl_ctx *isl_schedule_get_ctx(__isl_keep isl_schedule *schedule)
{
        return schedule ? isl_space_get_ctx(schedule->dim) : NULL;
}

__isl_give isl_union_map *isl_schedule_get_map(__isl_keep isl_schedule *sched)
{
        int i;
        isl_union_map *umap;

        if (!sched)
                return NULL;

        umap = isl_union_map_empty(isl_space_copy(sched->dim));
        for (i = 0; i < sched->n; ++i)
                umap = isl_union_map_add_map(umap,
                                            isl_map_copy(sched->node[i].sched));

        return umap;
}

static __isl_give isl_band_list *construct_band_list(
        __isl_keep isl_schedule *schedule, __isl_keep isl_band *parent,
        int band_nr, int *parent_active, int n_active);

/* Construct an isl_band structure for the band in the given schedule
 * with sequence number band_nr for the n_active nodes marked by active.
 * If the nodes don't have a band with the given sequence number,
 * then a band without members is created.
 *
 * Because of the way the schedule is constructed, we know that
 * the position of the band inside the schedule of a node is the same
 * for all active nodes.
 */
static __isl_give isl_band *construct_band(__isl_keep isl_schedule *schedule,
        __isl_keep isl_band *parent,
        int band_nr, int *active, int n_active)
{
        int i, j;
        isl_ctx *ctx = isl_schedule_get_ctx(schedule);
        isl_band *band;
        unsigned start, end;

        band = isl_calloc_type(ctx, isl_band);
        if (!band)
                return NULL;

        band->ref = 1;
        band->schedule = schedule;
        band->parent = parent;

        for (i = 0; i < schedule->n; ++i)
                if (active[i] && schedule->node[i].n_band > band_nr + 1)
                        break;

        if (i < schedule->n) {
                band->children = construct_band_list(schedule, band,
                                                band_nr + 1, active, n_active);
                if (!band->children)
                        goto error;
        }

        for (i = 0; i < schedule->n; ++i)
                if (active[i])
                        break;

        if (i >= schedule->n)
                isl_die(ctx, isl_error_internal,
                        "band without active statements", goto error);

        start = band_nr ? schedule->node[i].band_end[band_nr - 1] : 0;
        end = band_nr < schedule->node[i].n_band ?
                schedule->node[i].band_end[band_nr] : start;
        band->n = end - start;

        band->zero = isl_alloc_array(ctx, int, band->n);
        if (!band->zero)
                goto error;

        for (j = 0; j < band->n; ++j)
                band->zero[j] = schedule->node[i].zero[start + j];

        band->map = isl_union_map_empty(isl_space_copy(schedule->dim));
        for (i = 0; i < schedule->n; ++i) {
                isl_map *map;
                unsigned n_out;

                if (!active[i])
                        continue;

                map = isl_map_copy(schedule->node[i].sched);
                n_out = isl_map_dim(map, isl_dim_out);
                map = isl_map_project_out(map, isl_dim_out, end, n_out - end);
                map = isl_map_project_out(map, isl_dim_out, 0, start);
                band->map = isl_union_map_union(band->map,
                                                isl_union_map_from_map(map));
        }
        if (!band->map)
                goto error;

        return band;
error:
        isl_band_free(band);
        return NULL;
}

/* Construct a list of bands that start at the same position (with
 * sequence number band_nr) in the schedules of the nodes that
 * were active in the parent band.
 *
 * A separate isl_band structure is created for each band_id
 * and for each node that does not have a band with sequence
 * number band_nr.  In the latter case, a band without members
 * is created.
 * This ensures that if a band has any children, then each node
 * that was active in the band is active in exactly one of the children.
 */
static __isl_give isl_band_list *construct_band_list(
        __isl_keep isl_schedule *schedule, __isl_keep isl_band *parent,
        int band_nr, int *parent_active, int n_active)
{
        int i, j;
        isl_ctx *ctx = isl_schedule_get_ctx(schedule);
        int *active;
        int n_band;
        isl_band_list *list;

        n_band = 0;
        for (i = 0; i < n_active; ++i) {
                for (j = 0; j < schedule->n; ++j) {
                        if (!parent_active[j])
                                continue;
                        if (schedule->node[j].n_band <= band_nr)
                                continue;
                        if (schedule->node[j].band_id[band_nr] == i) {
                                n_band++;
                                break;
                        }
                }
        }
        for (j = 0; j < schedule->n; ++j)
                if (schedule->node[j].n_band <= band_nr)
                        n_band++;

        if (n_band == 1) {
                isl_band *band;
                list = isl_band_list_alloc(ctx, n_band);
                band = construct_band(schedule, parent, band_nr,
                                        parent_active, n_active);
                return isl_band_list_add(list, band);
        }

        active = isl_alloc_array(ctx, int, schedule->n);
        if (!active)
                return NULL;

        list = isl_band_list_alloc(ctx, n_band);

        for (i = 0; i < n_active; ++i) {
                int n = 0;
                isl_band *band;

                for (j = 0; j < schedule->n; ++j) {
                        active[j] = parent_active[j] &&
                                        schedule->node[j].n_band > band_nr &&
                                        schedule->node[j].band_id[band_nr] == i;
                        if (active[j])
                                n++;
                }
                if (n == 0)
                        continue;

                band = construct_band(schedule, parent, band_nr, active, n);

                list = isl_band_list_add(list, band);
        }
        for (i = 0; i < schedule->n; ++i) {
                isl_band *band;
                if (!parent_active[i])
                        continue;
                if (schedule->node[i].n_band > band_nr)
                        continue;
                for (j = 0; j < schedule->n; ++j)
                        active[j] = j == i;
                band = construct_band(schedule, parent, band_nr, active, 1);
                list = isl_band_list_add(list, band);
        }

        free(active);

        return list;
}

/* Construct a band forest representation of the schedule and
 * return the list of roots.
 */
static __isl_give isl_band_list *construct_forest(
        __isl_keep isl_schedule *schedule)
{
        int i;
        isl_ctx *ctx = isl_schedule_get_ctx(schedule);
        isl_band_list *forest;
        int *active;

        active = isl_alloc_array(ctx, int, schedule->n);
        if (!active)
                return NULL;

        for (i = 0; i < schedule->n; ++i)
                active[i] = 1;

        forest = construct_band_list(schedule, NULL, 0, active, schedule->n);

        free(active);

        return forest;
}

/* Return the roots of a band forest representation of the schedule.
 */
__isl_give isl_band_list *isl_schedule_get_band_forest(
        __isl_keep isl_schedule *schedule)
{
        if (!schedule)
                return NULL;
        if (!schedule->band_forest)
                schedule->band_forest = construct_forest(schedule);
        return isl_band_list_dup(schedule->band_forest);
}

static __isl_give isl_printer *print_band_list(__isl_take isl_printer *p,
        __isl_keep isl_band_list *list);

static __isl_give isl_printer *print_band(__isl_take isl_printer *p,
        __isl_keep isl_band *band)
{
        isl_band_list *children;

        p = isl_printer_start_line(p);
        p = isl_printer_print_union_map(p, band->map);
        p = isl_printer_end_line(p);

        if (!isl_band_has_children(band))
                return p;

        children = isl_band_get_children(band);

        p = isl_printer_indent(p, 4);
        p = print_band_list(p, children);
        p = isl_printer_indent(p, -4);

        isl_band_list_free(children);

        return p;
}

static __isl_give isl_printer *print_band_list(__isl_take isl_printer *p,
        __isl_keep isl_band_list *list)
{
        int i, n;

        n = isl_band_list_n_band(list);
        for (i = 0; i < n; ++i) {
                isl_band *band;
                band = isl_band_list_get_band(list, i);
                p = print_band(p, band);
                isl_band_free(band);
        }

        return p;
}

__isl_give isl_printer *isl_printer_print_schedule(__isl_take isl_printer *p,
        __isl_keep isl_schedule *schedule)
{
        isl_band_list *forest;

        forest = isl_schedule_get_band_forest(schedule);

        p = print_band_list(p, forest);

        isl_band_list_free(forest);

        return p;
}

void isl_schedule_dump(__isl_keep isl_schedule *schedule)
{
        isl_printer *printer;

        if (!schedule)
                return;

        printer = isl_printer_to_file(isl_schedule_get_ctx(schedule), stderr);
        printer = isl_printer_print_schedule(printer, schedule);

        isl_printer_free(printer);
}