/*

   igraph library.

   Copyright (C) 2006-2012  Gabor Csardi <csardi.gabor@gmail.com>

   334 Harvard street, Cambridge, MA 02139 USA

   This program is free software; you can redistribute it and/or modify

   it under the terms of the GNU General Public License as published by

   the Free Software Foundation; either version 2 of the License, or

   (at your option) any later version.

   This program is distributed in the hope that it will be useful,

   but WITHOUT ANY WARRANTY; without even the implied warranty of

   MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE.  See the

   GNU General Public License for more details.

   You should have received a copy of the GNU General Public License

   along with this program; if not, write to the Free Software

   Foundation, Inc., 51 Franklin Street, Fifth Floor, Boston, MA

   02110-1301 USA

*/

#include "igraph_flow.h"

#include "igraph_adjlist.h"

#include "igraph_components.h"

#include "igraph_conversion.h"

#include "igraph_constants.h"

#include "igraph_constructors.h"

#include "igraph_cycles.h"

#include "igraph_dqueue.h"

#include "igraph_error.h"

#include "igraph_interface.h"

#include "igraph_progress.h"

#include "igraph_operators.h"

#include "igraph_structural.h"

#include "core/buckets.h"

#include "core/cutheap.h"

#include "core/interruption.h"

#include "flow/flow_internal.h"

#include "math/safe_intop.h"

/*

 * Some general remarks about the functions in this file.

 *

 * The following measures can be calculated:

 * ( 1) s-t maximum flow value, directed graph

 * ( 2) s-t maximum flow value, undirected graph

 * ( 3) s-t maximum flow, directed graph

 * ( 4) s-t maximum flow, undirected graph

 * ( 5) s-t minimum cut value, directed graph

 * ( 6) s-t minimum cut value, undirected graph

 * ( 7) minimum cut value, directed graph

 * ( 8) minimum cut value, undirected graph

 * ( 9) s-t minimum cut, directed graph

 * (10) s-t minimum cut, undirected graph

 * (11) minimum cut, directed graph

 * (12) minimum cut, undirected graph

 * (13) s-t edge connectivity, directed graph

 * (14) s-t edge connectivity, undirected graph

 * (15) edge connectivity, directed graph

 * (16) edge connectivity, undirected graph

 * (17) s-t vertex connectivity, directed graph

 * (18) s-t vertex connectivity, undirected graph

 * (19) vertex connectivity, directed graph

 * (20) vertex connectivity, undirected graph

 * (21) s-t number of edge disjoint paths, directed graph

 * (22) s-t number of edge disjoint paths, undirected graph

 * (23) s-t number of vertex disjoint paths, directed graph

 * (24) s-t number of vertex disjoint paths, undirected graph

 * (25) graph adhesion, directed graph

 * (26) graph adhesion, undirected graph

 * (27) graph cohesion, directed graph

 * (28) graph cohesion, undirected graph

 *

 * This is how they are calculated:

 * ( 1) igraph_maxflow_value, calls igraph_maxflow.

 * ( 2) igraph_maxflow_value, calls igraph_maxflow, this calls

 *      igraph_i_maxflow_undirected. This transforms the graph into a

 *      directed graph, including two mutual edges instead of every

 *      undirected edge, then igraph_maxflow is called again with the

 *      directed graph.

 * ( 3) igraph_maxflow, does the push-relabel algorithm, optionally

 *      calculates the cut, the partitions and the flow itself.

 * ( 4) igraph_maxflow calls igraph_i_maxflow_undirected, this converts

 *      the undirected graph into a directed one, adding two mutual edges

 *      for each undirected edge, then igraph_maxflow is called again,

 *      with the directed graph. After igraph_maxflow returns, we need

 *      to edit the flow (and the cut) to make it sense for the

 *      original graph.

 * ( 5) igraph_st_mincut_value, we just call igraph_maxflow_value

 * ( 6) igraph_st_mincut_value, we just call igraph_maxflow_value

 * ( 7) igraph_mincut_value, we call igraph_maxflow_value (|V|-1)*2

 *      times, from vertex 0 to all other vertices and from all other

 *      vertices to vertex 0

 * ( 8) We call igraph_i_mincut_value_undirected, that calls

 *      igraph_i_mincut_undirected with partition=partition2=cut=NULL

 *      The Stoer-Wagner algorithm is used.

 * ( 9) igraph_st_mincut, just calls igraph_maxflow.

 * (10) igraph_st_mincut, just calls igraph_maxflow.

 * (11) igraph_mincut, calls igraph_i_mincut_directed, which runs

 *      the maximum flow algorithm 2(|V|-1) times, from vertex zero to

 *      and from all other vertices and stores the smallest cut.

 * (12) igraph_mincut, igraph_i_mincut_undirected is called,

 *      this is the Stoer-Wagner algorithm

 * (13) We just call igraph_maxflow_value, back to (1)

 * (14) We just call igraph_maxflow_value, back to (2)

 * (15) We just call igraph_mincut_value (possibly after some basic

 *      checks). Back to (7)

 * (16) We just call igraph_mincut_value (possibly after some basic

 *      checks). Back to (8).

 * (17) We call igraph_i_st_vertex_connectivity_directed.

 *      That creates a new graph with 2*|V| vertices and smartly chosen

 *      edges, so that the s-t edge connectivity of this graph is the

 *      same as the s-t vertex connectivity of the original graph.

 *      So finally it calls igraph_maxflow_value, go to (1)

 * (18) We call igraph_i_st_vertex_connectivity_undirected.

 *      We convert the graph to a directed one,

 *      IGRAPH_TO_DIRECTED_MUTUAL method. Then we call

 *      igraph_i_st_vertex_connectivity_directed, see (17).

 * (19) We call igraph_i_vertex_connectivity_directed.

 *      That calls igraph_st_vertex_connectivity for all pairs of

 *      vertices. Back to (17).

 * (20) We call igraph_i_vertex_connectivity_undirected.

 *      That converts the graph into a directed one

 *      (IGRAPH_TO_DIRECTED_MUTUAL) and calls the directed version,

 *      igraph_i_vertex_connectivity_directed, see (19).

 * (21) igraph_edge_disjoint_paths, we just call igraph_maxflow_value, (1).

 * (22) igraph_edge_disjoint_paths, we just call igraph_maxflow_value, (2).

 * (23) igraph_vertex_disjoint_paths, if there is a connection between

 *      the two vertices, then we remove that (or all of them if there

 *      are many), as this could mess up vertex connectivity

 *      calculation. The we call

 *      igraph_i_st_vertex_connectivity_directed, see (19).

 * (24) igraph_vertex_disjoint_paths, if there is a connection between

 *      the two vertices, then we remove that (or all of them if there

 *      are many), as this could mess up vertex connectivity

 *      calculation. The we call

 *      igraph_i_st_vertex_connectivity_undirected, see (20).

 * (25) We just call igraph_edge_connectivity, see (15).

 * (26) We just call igraph_edge_connectivity, see (16).

 * (27) We just call igraph_vertex_connectivity, see (19).

 * (28) We just call igraph_vertex_connectivity, see (20).

 */

/*

 * This is an internal function that calculates the maximum flow value

 * on undirected graphs, either for an s-t vertex pair or for the

 * graph (i.e. all vertex pairs).

 *

 * It does it by converting the undirected graph to a corresponding

 * directed graph, including reciprocal directed edges instead of each

 * undirected edge.

 */

static igraph_error_t igraph_i_maxflow_undirected(

        const igraph_t *graph,

        igraph_real_t *value,

        igraph_vector_t *flow,

        igraph_vector_int_t *cut,

        igraph_vector_int_t *partition,

        igraph_vector_int_t *partition2,

        igraph_int_t source,

        igraph_int_t target,

        const igraph_vector_t *capacity,

        igraph_maxflow_stats_t *stats) {

    const igraph_int_t no_of_edges = igraph_ecount(graph);

    const igraph_int_t no_of_nodes = igraph_vcount(graph);

    igraph_vector_int_t edges;

    igraph_vector_t newcapacity;

    igraph_t newgraph;

    igraph_int_t size;

    /* We need to convert this to directed by hand, since we need to be

       sure that the edge IDs will be handled properly to build the new

       capacity vector. */

    IGRAPH_VECTOR_INT_INIT_FINALLY(&edges, 0);

    IGRAPH_VECTOR_INIT_FINALLY(&newcapacity, no_of_edges * 2);

    IGRAPH_SAFE_MULT(no_of_edges, 4, &size);

    IGRAPH_CHECK(igraph_vector_int_reserve(&edges, size));

    IGRAPH_CHECK(igraph_get_edgelist(graph, &edges, 0));

    IGRAPH_CHECK(igraph_vector_int_resize(&edges, size));

    for (igraph_int_t i = 0; i < no_of_edges; i++) {

        VECTOR(edges)[no_of_edges * 2 + i * 2] = VECTOR(edges)[i * 2 + 1];

        VECTOR(edges)[no_of_edges * 2 + i * 2 + 1] = VECTOR(edges)[i * 2];

        VECTOR(newcapacity)[i] = VECTOR(newcapacity)[no_of_edges + i] =

                                     capacity ? VECTOR(*capacity)[i] : 1.0;

    }

    IGRAPH_CHECK(igraph_create(&newgraph, &edges, no_of_nodes, IGRAPH_DIRECTED));

    IGRAPH_FINALLY(igraph_destroy, &newgraph);

    IGRAPH_CHECK(igraph_maxflow(&newgraph, value, flow, cut, partition,

                                partition2, source, target, &newcapacity, stats));

    if (cut) {

        igraph_int_t cs = igraph_vector_int_size(cut);

        for (igraph_int_t i = 0; i < cs; i++) {

            if (VECTOR(*cut)[i] >= no_of_edges) {

                VECTOR(*cut)[i] -= no_of_edges;

            }

        }

    }

    /* The flow has one non-zero value for each real-nonreal edge pair,

       by definition, we convert it to a positive-negative vector. If

       for an edge the flow is negative that means that it is going

       from the bigger vertex ID to the smaller one. For positive

       values the direction is the opposite. */

    if (flow) {

        for (igraph_int_t i = 0; i < no_of_edges; i++) {

            VECTOR(*flow)[i] -= VECTOR(*flow)[i + no_of_edges];

        }

        IGRAPH_CHECK(igraph_vector_resize(flow, no_of_edges));

    }

    igraph_destroy(&newgraph);

    igraph_vector_int_destroy(&edges);

    igraph_vector_destroy(&newcapacity);

    IGRAPH_FINALLY_CLEAN(3);

    return IGRAPH_SUCCESS;

}

#define FIRST(i)       (VECTOR(*first)[(i)])

#define LAST(i)        (VECTOR(*first)[(i)+1])

#define CURRENT(i)     (VECTOR(*current)[(i)])

#define RESCAP(i)      (VECTOR(*rescap)[(i)])

#define REV(i)         (VECTOR(*rev)[(i)])

#define HEAD(i)        (VECTOR(*to)[(i)])

#define EXCESS(i)      (VECTOR(*excess)[(i)])

#define DIST(i)        (VECTOR(*distance)[(i)])

#define DISCHARGE(v)   (igraph_i_mf_discharge((v), &current, &first, &rescap, \

                        &to, &distance, &excess,        \

                        no_of_nodes, source, target,    \

                        &buckets, &ibuckets,        \

                        &rev, stats, &npushsince,       \

                        &nrelabelsince))

#define PUSH(v,e,n)    (igraph_i_mf_push((v), (e), (n), current, rescap,      \

                        excess, target, source, buckets,     \

                        ibuckets, distance, rev, stats,      \

                        npushsince))

#define RELABEL(v)     (igraph_i_mf_relabel((v), no_of_nodes, distance,       \

                        first, rescap, to, current,       \

                        stats, nrelabelsince))

#define GAP(b)         (igraph_i_mf_gap((b), stats, buckets, ibuckets,        \

                                        no_of_nodes, distance))

#define BFS()          (igraph_i_mf_bfs(&bfsq, source, target, no_of_nodes,   \

                                        &buckets, &ibuckets, &distance,       \

                                        &first, &current, &to, &excess,       \

                                        &rescap, &rev))

static void igraph_i_mf_gap(igraph_int_t b, igraph_maxflow_stats_t *stats,

                            igraph_buckets_t *buckets, igraph_dbuckets_t *ibuckets,

                            igraph_int_t no_of_nodes,

                            igraph_vector_int_t *distance) {

    IGRAPH_UNUSED(buckets);

    igraph_int_t bo;

    (stats->nogap)++;

    for (bo = b + 1; bo <= no_of_nodes; bo++) {

        while (!igraph_dbuckets_empty_bucket(ibuckets, bo)) {

            igraph_int_t n = igraph_dbuckets_pop(ibuckets, bo);

            (stats->nogapnodes)++;

            DIST(n) = no_of_nodes;

        }

    }

}

static void igraph_i_mf_relabel(igraph_int_t v, igraph_int_t no_of_nodes,

                                igraph_vector_int_t *distance,

                                igraph_vector_int_t *first,

                                igraph_vector_t *rescap, igraph_vector_int_t *to,

                                igraph_vector_int_t *current,

                                igraph_maxflow_stats_t *stats, igraph_int_t *nrelabelsince) {

    igraph_int_t min = no_of_nodes;

    igraph_int_t k, l, min_edge = 0;

    (stats->norelabel)++; (*nrelabelsince)++;

    DIST(v) = no_of_nodes;

    for (k = FIRST(v), l = LAST(v); k < l; k++) {

        if (RESCAP(k) > 0 && DIST(HEAD(k)) < min) {

            min = DIST(HEAD(k));

            min_edge = k;

        }

    }

    min++;

    if (min < no_of_nodes) {

        DIST(v) = min;

        CURRENT(v) = min_edge;

    }

}

static void igraph_i_mf_push(igraph_int_t v, igraph_int_t e, igraph_int_t n,

                             igraph_vector_int_t *current,

                             igraph_vector_t *rescap, igraph_vector_t *excess,

                             igraph_int_t target, igraph_int_t source,

                             igraph_buckets_t *buckets, igraph_dbuckets_t *ibuckets,

                             igraph_vector_int_t *distance,

                             igraph_vector_int_t *rev, igraph_maxflow_stats_t *stats,

                             igraph_int_t *npushsince) {

    IGRAPH_UNUSED(current);

    IGRAPH_UNUSED(source);

    igraph_real_t delta =

        RESCAP(e) < EXCESS(v) ? RESCAP(e) : EXCESS(v);

    (stats->nopush)++; (*npushsince)++;

    if (EXCESS(n) == 0 && n != target) {

        igraph_dbuckets_delete(ibuckets, DIST(n), n);

        igraph_buckets_add(buckets, DIST(n), n);

    }

    RESCAP(e) -= delta;

    RESCAP(REV(e)) += delta;

    EXCESS(n) += delta;

    EXCESS(v) -= delta;

}

static void igraph_i_mf_discharge(igraph_int_t v,

                                  igraph_vector_int_t *current,

                                  igraph_vector_int_t *first,

                                  igraph_vector_t *rescap,

                                  igraph_vector_int_t *to,

                                  igraph_vector_int_t *distance,

                                  igraph_vector_t *excess,

                                  igraph_int_t no_of_nodes, igraph_int_t source,

                                  igraph_int_t target, igraph_buckets_t *buckets,

                                  igraph_dbuckets_t *ibuckets,

                                  igraph_vector_int_t *rev,

                                  igraph_maxflow_stats_t *stats,

                                  igraph_int_t *npushsince, igraph_int_t *nrelabelsince) {

    do {

        igraph_int_t i;

        igraph_int_t start = CURRENT(v);

        igraph_int_t stop = LAST(v);

        for (i = start; i < stop; i++) {

            if (RESCAP(i) > 0) {

                igraph_int_t nei = HEAD(i);

                if (DIST(v) == DIST(nei) + 1) {

                    PUSH((v), i, nei);

                    if (EXCESS(v) == 0) {

                        break;

                    }

                }

            }

        }

        if (i == stop) {

            igraph_int_t origdist = DIST(v);

            RELABEL(v);

            if (igraph_buckets_empty_bucket(buckets, origdist) &&

                igraph_dbuckets_empty_bucket(ibuckets, origdist)) {

                GAP(origdist);

            }

            if (DIST(v) == no_of_nodes) {

                break;

            }

        } else {

            CURRENT(v) = i;

            igraph_dbuckets_add(ibuckets, DIST(v), v);

            break;

        }

    } while (1);

}

static igraph_error_t igraph_i_mf_bfs(igraph_dqueue_int_t *bfsq,

                                      igraph_int_t source, igraph_int_t target,

                                      igraph_int_t no_of_nodes, igraph_buckets_t *buckets,

                                      igraph_dbuckets_t *ibuckets,

                                      igraph_vector_int_t *distance,

                                      igraph_vector_int_t *first, igraph_vector_int_t *current,

                                      igraph_vector_int_t *to, igraph_vector_t *excess,

                                      igraph_vector_t *rescap, igraph_vector_int_t *rev) {

    igraph_int_t k, l;

    IGRAPH_UNUSED(source);

    igraph_buckets_clear(buckets);

    igraph_dbuckets_clear(ibuckets);

    igraph_vector_int_fill(distance, no_of_nodes);

    DIST(target) = 0;

    IGRAPH_CHECK(igraph_dqueue_int_push(bfsq, target));

    while (!igraph_dqueue_int_empty(bfsq)) {

        igraph_int_t node = igraph_dqueue_int_pop(bfsq);

        igraph_int_t ndist = DIST(node) + 1;

        for (k = FIRST(node), l = LAST(node); k < l; k++) {

            if (RESCAP(REV(k)) > 0) {

                igraph_int_t nei = HEAD(k);

                if (DIST(nei) == no_of_nodes) {

                    DIST(nei) = ndist;

                    CURRENT(nei) = FIRST(nei);

                    if (EXCESS(nei) > 0) {

                        igraph_buckets_add(buckets, ndist, nei);

                    } else {

                        igraph_dbuckets_add(ibuckets, ndist, nei);

                    }

                    IGRAPH_CHECK(igraph_dqueue_int_push(bfsq, nei));

                }

            }

        }

    }

    return IGRAPH_SUCCESS;

}

/**

 * \function igraph_maxflow

 * \brief Maximum network flow between a pair of vertices.

 *

 * This function implements the Goldberg-Tarjan algorithm for

 * calculating value of the maximum flow in a directed or undirected

 * graph. The algorithm was given in Andrew V. Goldberg, Robert

 * E. Tarjan: A New Approach to the Maximum-Flow Problem, Journal of

 * the ACM, 35(4), 921-940, 1988

 * https://doi.org/10.1145/48014.61051.

 *

 * </para><para>

 * The input of the function is a graph, a vector

 * of real numbers giving the capacity of the edges and two vertices

 * of the graph, the source and the target. A flow is a function

 * assigning positive real numbers to the edges and satisfying two

 * requirements: (1) the flow value is less than the capacity of the

 * edge and (2) at each vertex except the source and the target, the

 * incoming flow (i.e. the sum of the flow on the incoming edges) is

 * the same as the outgoing flow (i.e. the sum of the flow on the

 * outgoing edges). The value of the flow is the incoming flow at the

 * target vertex. The maximum flow is the flow with the maximum

 * value.

 *

 * \param graph The input graph, either directed or undirected.

 * \param value Pointer to a real number, the value of the maximum

 *        will be placed here, unless it is a null pointer.

 * \param flow If not a null pointer, then it must be a pointer to an

 *        initialized vector. The vector will be resized, and the flow

 *        on each edge will be placed in it, in the order of the edge

 *        IDs. For undirected graphs this argument is bit trickier,

 *        since for these the flow direction is not predetermined by

 *        the edge direction. For these graphs the elements of the

 *        \p flow vector can be negative, this means that the flow

 *        goes from the bigger vertex ID to the smaller one. Positive

 *        values mean that the flow goes from the smaller vertex ID to

 *        the bigger one.

 * \param cut A null pointer or a pointer to an initialized vector.

 *        If not a null pointer, then the minimum cut corresponding to

 *        the maximum flow is stored here, i.e. all edge IDs that are

 *        part of the minimum cut are stored in the vector.

 * \param partition A null pointer or a pointer to an initialized

 *        vector. If not a null pointer, then the first partition of

 *        the minimum cut that corresponds to the maximum flow will be

 *        placed here. The first partition is always the one that

 *        contains the source vertex.

 * \param partition2 A null pointer or a pointer to an initialized

 *        vector. If not a null pointer, then the second partition of

 *        the minimum cut that corresponds to the maximum flow will be

 *        placed here. The second partition is always the one that

 *        contains the target vertex.

 * \param source The id of the source vertex.

 * \param target The id of the target vertex.

 * \param capacity Vector containing the capacity of the edges. If \c NULL, then

 *        every edge is considered to have capacity 1.0.

 * \param stats Counts of the number of different operations

 *        performed by the algorithm are stored here.

 * \return Error code.

 *

 * Time complexity: O(|V|^3). In practice it is much faster, but I

 * cannot prove a better lower bound for the data structure I've

 * used. In fact, this implementation runs much faster than the

 * \c hi_pr implementation discussed in

 * B. V. Cherkassky and A. V. Goldberg: On implementing the

 * push-relabel method for the maximum flow problem, (Algorithmica,

 * 19:390--410, 1997) on all the graph classes I've tried.

 *

 * \sa \ref igraph_mincut_value(), \ref igraph_edge_connectivity(),

 * \ref igraph_vertex_connectivity() for

 * properties based on the maximum flow.

 *

 * \example examples/simple/flow.c

 * \example examples/simple/flow2.c

 */

igraph_error_t igraph_maxflow(const igraph_t *graph, igraph_real_t *value,

                              igraph_vector_t *flow, igraph_vector_int_t *cut,

                              igraph_vector_int_t *partition, igraph_vector_int_t *partition2,

                              igraph_int_t source, igraph_int_t target,

                              const igraph_vector_t *capacity,

                              igraph_maxflow_stats_t *stats) {

    const igraph_int_t no_of_nodes = igraph_vcount(graph);

    const igraph_int_t no_of_orig_edges = igraph_ecount(graph);

    const igraph_int_t no_of_edges = 2 * no_of_orig_edges;

    igraph_vector_t rescap, excess;

    igraph_vector_int_t from, to, rev, distance;

    igraph_vector_int_t edges, rank;

    igraph_vector_int_t current, first;

    igraph_buckets_t buckets;

    igraph_dbuckets_t ibuckets;

    igraph_dqueue_int_t bfsq;

    igraph_int_t idx;

    igraph_int_t npushsince = 0, nrelabelsince = 0;

    igraph_maxflow_stats_t local_stats;   /* used if the user passed a null pointer for stats */

    if (stats == NULL) {

        stats = &local_stats;

    }

    if (!igraph_is_directed(graph)) {

        IGRAPH_CHECK(igraph_i_maxflow_undirected(graph, value, flow, cut,

                     partition, partition2, source,

                     target, capacity, stats));

        return IGRAPH_SUCCESS;

    }

    if (capacity && igraph_vector_size(capacity) != no_of_orig_edges) {

        IGRAPH_ERROR("Capacity vector must match number of edges in length.", IGRAPH_EINVAL);

    }

    if (source < 0 || source >= no_of_nodes || target < 0 || target >= no_of_nodes) {

        IGRAPH_ERROR("Invalid source or target vertex.", IGRAPH_EINVVID);

    }

    if (source == target) {

        IGRAPH_ERROR("Source and target vertices are the same.", IGRAPH_EINVAL);

    }

    stats->nopush = stats->norelabel = stats->nogap = stats->nogapnodes =

                                           stats->nobfs = 0;

    /*

     * The data structure:

     * - First of all, we consider every edge twice, first the edge

     *   itself, but also its opposite.

     * - (from, to) contain all edges (original + opposite), ordered by

     *   the id of the source vertex. During the algorithm we just need

     *   'to', so from is destroyed soon. We only need it in the

     *   beginning, to create the 'first' pointers.

     * - 'first' is a pointer vector for 'to', first[i] points to the

     *   first neighbor of vertex i and first[i+1]-1 is the last

     *   neighbor of vertex i. (Unless vertex i is isolate, in which

     *   case first[i]==first[i+1]).

     * - 'rev' contains a mapping from an edge to its opposite pair

     * - 'rescap' contains the residual capacities of the edges, this is

     *   initially equal to the capacity of the edges for the original

     *   edges and it is zero for the opposite edges.

     * - 'excess' contains the excess flow for the vertices. I.e. the flow

     *   that is coming in, but it is not going out.

     * - 'current' stores the next neighboring vertex to check, for every

     *   vertex, when excess flow is being pushed to neighbors.

     * - 'distance' stores the distance of the vertices from the source.

     * - 'rank' and 'edges' are only needed temporarily, for ordering and

     *   storing the edges.

     * - we use an igraph_buckets_t data structure ('buckets') to find

     *   the vertices with the highest 'distance' values quickly.

     *   This always contains the vertices that have a positive excess

     *   flow.

     */

#undef FIRST

#undef LAST

#undef CURRENT

#undef RESCAP

#undef REV

#undef HEAD

#undef EXCESS

#undef DIST

#define FIRST(i)       (VECTOR(first)[(i)])

#define LAST(i)        (VECTOR(first)[(i)+1])

#define CURRENT(i)     (VECTOR(current)[(i)])

#define RESCAP(i)      (VECTOR(rescap)[(i)])

#define REV(i)         (VECTOR(rev)[(i)])

#define HEAD(i)        (VECTOR(to)[(i)])

#define EXCESS(i)      (VECTOR(excess)[(i)])

#define DIST(i)        (VECTOR(distance)[(i)])

    IGRAPH_CHECK(igraph_dqueue_int_init(&bfsq, no_of_nodes));

    IGRAPH_FINALLY(igraph_dqueue_int_destroy, &bfsq);

    IGRAPH_VECTOR_INT_INIT_FINALLY(&to,       no_of_edges);

    IGRAPH_VECTOR_INT_INIT_FINALLY(&rev,      no_of_edges);

    IGRAPH_VECTOR_INIT_FINALLY(&rescap,       no_of_edges);

    IGRAPH_VECTOR_INIT_FINALLY(&excess,       no_of_nodes);

    IGRAPH_VECTOR_INT_INIT_FINALLY(&distance, no_of_nodes);

    IGRAPH_VECTOR_INT_INIT_FINALLY(&first,    no_of_nodes + 1);

    IGRAPH_VECTOR_INT_INIT_FINALLY(&rank,         no_of_edges);

    IGRAPH_VECTOR_INT_INIT_FINALLY(&from,     no_of_edges);

    IGRAPH_VECTOR_INT_INIT_FINALLY(&edges,        no_of_edges);

    /* Create the basic data structure */

    IGRAPH_CHECK(igraph_get_edgelist(graph, &edges, 0));

    IGRAPH_CHECK(igraph_i_vector_int_rank(&edges, &rank, no_of_nodes));

    for (igraph_int_t i = 0; i < no_of_edges; i += 2) {

        const igraph_int_t pos = VECTOR(rank)[i];

        const igraph_int_t pos2 = VECTOR(rank)[i + 1];

        VECTOR(from)[pos] = VECTOR(edges)[i];

        VECTOR(to)[pos]   = VECTOR(edges)[i + 1];

        VECTOR(from)[pos2] = VECTOR(edges)[i + 1];

        VECTOR(to)[pos2]   = VECTOR(edges)[i];

        VECTOR(rev)[pos] = pos2;

        VECTOR(rev)[pos2] = pos;

        VECTOR(rescap)[pos] = capacity ? VECTOR(*capacity)[i / 2] : 1.0;

        VECTOR(rescap)[pos2] = 0.0;

    }

    /* The first pointers. This is a but trickier, than one would

       think, because of the possible isolate vertices. */

    idx = -1;

    for (igraph_int_t i = 0; i <= VECTOR(from)[0]; i++) {

        idx++; VECTOR(first)[idx] = 0;

    }

    for (igraph_int_t i = 1; i < no_of_edges; i++) {

        const igraph_int_t n = (VECTOR(from)[i] -

                                    VECTOR(from)[ VECTOR(first)[idx] ]);

        for (igraph_int_t j = 0; j < n; j++) {

            idx++; VECTOR(first)[idx] = i;

        }

    }

    idx++;

    while (idx < no_of_nodes + 1) {

        VECTOR(first)[idx++] = no_of_edges;

    }

    igraph_vector_int_destroy(&from);

    igraph_vector_int_destroy(&edges);

    IGRAPH_FINALLY_CLEAN(2);

    if (!flow) {

        igraph_vector_int_destroy(&rank);

        IGRAPH_FINALLY_CLEAN(1);

    }

    /* And the current pointers, initially the same as the first */

    IGRAPH_VECTOR_INT_INIT_FINALLY(&current, no_of_nodes);

    for (igraph_int_t i = 0; i < no_of_nodes; i++) {

        VECTOR(current)[i] = VECTOR(first)[i];

    }

    /* OK, the graph is set up, initialization */

    IGRAPH_CHECK(igraph_buckets_init(&buckets, no_of_nodes + 1, no_of_nodes));

    IGRAPH_FINALLY(igraph_buckets_destroy, &buckets);

    IGRAPH_CHECK(igraph_dbuckets_init(&ibuckets, no_of_nodes + 1, no_of_nodes));

    IGRAPH_FINALLY(igraph_dbuckets_destroy, &ibuckets);

    /* Send as much flow as possible from the source to its neighbors */

    for (igraph_int_t i = FIRST(source), j = LAST(source); i < j; i++) {

        if (HEAD(i) != source) {

            const igraph_real_t delta = RESCAP(i);

            RESCAP(i) = 0;

            RESCAP(REV(i)) += delta;

            EXCESS(HEAD(i)) += delta;

        }

    }

    IGRAPH_CHECK(BFS());

    (stats->nobfs)++;

    while (!igraph_buckets_empty(&buckets)) {

        const igraph_int_t vertex = igraph_buckets_popmax(&buckets);

        DISCHARGE(vertex);

        if (npushsince > no_of_nodes / 2 && nrelabelsince > no_of_nodes) {

            (stats->nobfs)++;

            BFS();

            npushsince = nrelabelsince = 0;

        }

    }

    /* Store the result */

    if (value) {

        *value = EXCESS(target);

    }

    /* If we also need the minimum cut */

    if (cut || partition || partition2) {

        /* We need to find all vertices from which the target is reachable

           in the residual graph. We do a breadth-first search, going

           backwards. */

        igraph_dqueue_int_t Q;

        igraph_vector_bool_t added;

        igraph_int_t marked = 0;

        IGRAPH_CHECK(igraph_vector_bool_init(&added, no_of_nodes));

        IGRAPH_FINALLY(igraph_vector_bool_destroy, &added);

        IGRAPH_CHECK(igraph_dqueue_int_init(&Q, 100));

        IGRAPH_FINALLY(igraph_dqueue_int_destroy, &Q);

        IGRAPH_CHECK(igraph_dqueue_int_push(&Q, target));

        VECTOR(added)[target] = true;

        marked++;

        while (!igraph_dqueue_int_empty(&Q)) {

            const igraph_int_t actnode = igraph_dqueue_int_pop(&Q);

            for (igraph_int_t i = FIRST(actnode), j = LAST(actnode); i < j; i++) {

                igraph_int_t nei = HEAD(i);

                if (!VECTOR(added)[nei] && RESCAP(REV(i)) > 0.0) {

                    VECTOR(added)[nei] = true;

                    marked++;

                    IGRAPH_CHECK(igraph_dqueue_int_push(&Q, nei));

                }

            }

        }

        igraph_dqueue_int_destroy(&Q);

        IGRAPH_FINALLY_CLEAN(1);

        /* Now we marked each vertex that is on one side of the cut,

           check the crossing edges */

        if (cut) {

            igraph_vector_int_clear(cut);

            for (igraph_int_t i = 0; i < no_of_orig_edges; i++) {

                igraph_int_t f = IGRAPH_FROM(graph, i);

                igraph_int_t t = IGRAPH_TO(graph, i);

                if (!VECTOR(added)[f] && VECTOR(added)[t]) {

                    IGRAPH_CHECK(igraph_vector_int_push_back(cut, i));

                }

            }

        }

        if (partition2) {

            igraph_int_t x = 0;

            IGRAPH_CHECK(igraph_vector_int_resize(partition2, marked));

            for (igraph_int_t i = 0; i < no_of_nodes; i++) {

                if (VECTOR(added)[i]) {

                    VECTOR(*partition2)[x++] = i;

                }

            }

        }

        if (partition) {

            igraph_int_t x = 0;

            IGRAPH_CHECK(igraph_vector_int_resize(partition,

                                                  no_of_nodes - marked));

            for (igraph_int_t i = 0; i < no_of_nodes; i++) {

                if (!VECTOR(added)[i]) {

                    VECTOR(*partition)[x++] = i;

                }

            }

        }

        igraph_vector_bool_destroy(&added);

        IGRAPH_FINALLY_CLEAN(1);

    }

    if (flow) {

        /* Initialize the backward distances, with a breadth-first search

           from the source */

        igraph_dqueue_int_t Q;

        igraph_vector_int_t added; /* uses more than two values, cannot be bool */

        igraph_t flow_graph;

        igraph_vector_int_t flow_edges;

        igraph_bool_t dag;

        IGRAPH_CHECK(igraph_vector_int_init(&added, no_of_nodes));

        IGRAPH_FINALLY(igraph_vector_int_destroy, &added);

        IGRAPH_CHECK(igraph_dqueue_int_init(&Q, 100));

        IGRAPH_FINALLY(igraph_dqueue_int_destroy, &Q);

        IGRAPH_CHECK(igraph_dqueue_int_push(&Q, source));

        IGRAPH_CHECK(igraph_dqueue_int_push(&Q, 0));

        VECTOR(added)[source] = 1;

        while (!igraph_dqueue_int_empty(&Q)) {

            const igraph_int_t actnode = igraph_dqueue_int_pop(&Q);

            const igraph_int_t actdist = igraph_dqueue_int_pop(&Q);

            DIST(actnode) = actdist;

            for (igraph_int_t i = FIRST(actnode), j = LAST(actnode); i < j; i++) {

                const igraph_int_t nei = HEAD(i);

                if (!VECTOR(added)[nei] && RESCAP(REV(i)) > 0.0) {

                    VECTOR(added)[nei] = 1;

                    IGRAPH_CHECK(igraph_dqueue_int_push(&Q, nei));

                    IGRAPH_CHECK(igraph_dqueue_int_push(&Q, actdist + 1));

                }

            }

        } /* !igraph_dqueue_int_empty(&Q) */

        igraph_vector_int_destroy(&added);

        igraph_dqueue_int_destroy(&Q);

        IGRAPH_FINALLY_CLEAN(2);

        /* Reinitialize the buckets */

        igraph_buckets_clear(&buckets);

        for (igraph_int_t i = 0; i < no_of_nodes; i++) {

            if (EXCESS(i) > 0.0 && i != source && i != target) {

                igraph_buckets_add(&buckets, DIST(i), i);

            }

        }

        /* Now we return the flow to the source */

        while (!igraph_buckets_empty(&buckets)) {

            const igraph_int_t vertex = igraph_buckets_popmax(&buckets);

            /* DISCHARGE(vertex) comes here */

            do {

                igraph_int_t i, j;

                for (i = CURRENT(vertex), j = LAST(vertex); i < j; i++) {

                    if (RESCAP(i) > 0) {

                        igraph_int_t nei = HEAD(i);

                        if (DIST(vertex) == DIST(nei) + 1) {

                            igraph_real_t delta =

                                RESCAP(i) < EXCESS(vertex) ? RESCAP(i) : EXCESS(vertex);

                            RESCAP(i) -= delta;

                            RESCAP(REV(i)) += delta;

                            if (nei != source && EXCESS(nei) == 0.0 &&

                                DIST(nei) != no_of_nodes) {

                                igraph_buckets_add(&buckets, DIST(nei), nei);

                            }

                            EXCESS(nei) += delta;

                            EXCESS(vertex) -= delta;

                            if (EXCESS(vertex) == 0) {

                                break;

                            }

                        }

                    }

                }

                if (i == j) {

                    /* RELABEL(vertex) comes here */

                    igraph_int_t min;

                    igraph_int_t min_edge = 0;

                    DIST(vertex) = min = no_of_nodes;

                    for (igraph_int_t k = FIRST(vertex), l = LAST(vertex); k < l; k++) {

                        if (RESCAP(k) > 0) {

                            if (DIST(HEAD(k)) < min) {

                                min = DIST(HEAD(k));

                                min_edge = k;

                            }

                        }

                    }

                    min++;

                    if (min < no_of_nodes) {

                        DIST(vertex) = min;

                        CURRENT(vertex) = min_edge;

                        /* Vertex is still active */

                        igraph_buckets_add(&buckets, DIST(vertex), vertex);

                    }

                    /* TODO: gap heuristics here ??? */

                } else {

                    CURRENT(vertex) = FIRST(vertex);

                }

                break;

            } while (true);

        }

        /* We need to eliminate flow cycles now. Before that we check that

           there is a cycle in the flow graph.

           First we do a couple of DFSes from the source vertex to the

           target and factor out the paths we find. If there is no more

           path to the target, then all remaining flow must be in flow

           cycles, so we don't need it at all.

           Some details. 'stack' contains the whole path of the DFS, both

           the vertices and the edges, they are alternating in the stack.

           'current' helps finding the next outgoing edge of a vertex

           quickly, the next edge of 'v' is FIRST(v)+CURRENT(v). If this

           is LAST(v), then there are no more edges to try.

           The 'added' vector contains 0 if the vertex was not visited

           before, 1 if it is currently in 'stack', and 2 if it is not in

           'stack', but it was visited before. */

        IGRAPH_VECTOR_INT_INIT_FINALLY(&flow_edges, 0);

        for (igraph_int_t i = 0, j = 0; i < no_of_edges; i += 2, j++) {

            const igraph_int_t pos = VECTOR(rank)[i];

            if ((capacity ? VECTOR(*capacity)[j] : 1.0) > RESCAP(pos)) {

                IGRAPH_CHECK(igraph_vector_int_push_back(&flow_edges,

                             IGRAPH_FROM(graph, j)));

                IGRAPH_CHECK(igraph_vector_int_push_back(&flow_edges,

                             IGRAPH_TO(graph, j)));

            }

        }

        IGRAPH_CHECK(igraph_create(&flow_graph, &flow_edges, no_of_nodes,

                                   IGRAPH_DIRECTED));

        igraph_vector_int_destroy(&flow_edges);

        IGRAPH_FINALLY_CLEAN(1);

        IGRAPH_FINALLY(igraph_destroy, &flow_graph);

        IGRAPH_CHECK(igraph_is_dag(&flow_graph, &dag));

        igraph_destroy(&flow_graph);

        IGRAPH_FINALLY_CLEAN(1);

        if (!dag) {

            igraph_vector_int_t stack;

            igraph_vector_t mycap;

            IGRAPH_CHECK(igraph_vector_int_init(&stack, 0));

            IGRAPH_FINALLY(igraph_vector_int_destroy, &stack);

            IGRAPH_CHECK(igraph_vector_int_init(&added, no_of_nodes));

            IGRAPH_FINALLY(igraph_vector_int_destroy, &added);

            IGRAPH_VECTOR_INIT_FINALLY(&mycap, no_of_edges);

#define MYCAP(i)      (VECTOR(mycap)[(i)])

            for (igraph_int_t i = 0; i < no_of_edges; i += 2) {

                const igraph_int_t pos = VECTOR(rank)[i];

                const igraph_int_t pos2 = VECTOR(rank)[i + 1];

                MYCAP(pos) = (capacity ? VECTOR(*capacity)[i / 2] : 1.0) - RESCAP(pos);

                MYCAP(pos2) = 0.0;

            }

            do {

                igraph_vector_int_null(&current);

                igraph_vector_int_clear(&stack);

                igraph_vector_int_null(&added);

                IGRAPH_CHECK(igraph_vector_int_push_back(&stack, -1));

                IGRAPH_CHECK(igraph_vector_int_push_back(&stack, source));

                VECTOR(added)[source] = 1;

                while (!igraph_vector_int_empty(&stack) &&

                       igraph_vector_int_tail(&stack) != target) {

                    const igraph_int_t actnode = igraph_vector_int_tail(&stack);

                    igraph_int_t edge = FIRST(actnode) + CURRENT(actnode);

                    igraph_int_t nei;

                    while (edge < LAST(actnode) && MYCAP(edge) == 0.0) {

                        edge++;

                    }

                    nei = edge < LAST(actnode) ? HEAD(edge) : -1;

                    if (edge < LAST(actnode) && !VECTOR(added)[nei]) {

                        /* Go forward along next edge, if the vertex was not

                           visited before */

                        IGRAPH_CHECK(igraph_vector_int_push_back(&stack, edge));

                        IGRAPH_CHECK(igraph_vector_int_push_back(&stack, nei));

                        VECTOR(added)[nei] = 1;

                        CURRENT(actnode) += 1;

                    } else if (edge < LAST(actnode) && VECTOR(added)[nei] == 1) {

                        /* We found a flow cycle, factor it out. Go back in stack

                           until we find 'nei' again, determine the flow along the

                           cycle. */

                        igraph_real_t thisflow = MYCAP(edge);

                        for (igraph_int_t idx = igraph_vector_int_size(&stack) - 2;

                             idx >= 0 && VECTOR(stack)[idx + 1] != nei; idx -= 2) {

                            const igraph_int_t e = VECTOR(stack)[idx];

                            const igraph_real_t rcap = e >= 0 ? MYCAP(e) : MYCAP(edge);

                            if (rcap < thisflow) {

                                thisflow = rcap;

                            }

                        }