/* -*- C++ -*-

 *

 * This file is a part of LEMON, a generic C++ optimization library

 *

 * Copyright (C) 2003-2008

 * Egervary Jeno Kombinatorikus Optimalizalasi Kutatocsoport

 * (Egervary Research Group on Combinatorial Optimization, EGRES).

 *

 * Permission to use, modify and distribute this software is granted

 * provided that this copyright notice appears in all copies. For

 * precise terms see the accompanying LICENSE file.

 *

 * This software is provided "AS IS" with no warranty of any kind,

 * express or implied, and with no claim as to its suitability for any

 * purpose.

 *

 */

#ifndef LEMON_NETWORK_SIMPLEX_H

#define LEMON_NETWORK_SIMPLEX_H

/// \ingroup min_cost_flow

///

/// \file

/// \brief The network simplex algorithm for finding a minimum cost flow.

#include <limits>

#include <lemon/graph_adaptor.h>

#include <lemon/graph_utils.h>

#include <lemon/smart_graph.h>

/// \brief The pivot rule used in the algorithm.

//#define FIRST_ELIGIBLE_PIVOT

//#define BEST_ELIGIBLE_PIVOT

#define EDGE_BLOCK_PIVOT

//#define CANDIDATE_LIST_PIVOT

//#define SORTED_LIST_PIVOT

//#define _DEBUG_ITER_

// State constant for edges at their lower bounds.

#define LOWER   1

// State constant for edges in the spanning tree.

#define TREE    0

// State constant for edges at their upper bounds.

#define UPPER   -1

#ifdef EDGE_BLOCK_PIVOT

  #include <cmath>

  #define MIN_BLOCK_SIZE        10

#endif

#ifdef CANDIDATE_LIST_PIVOT

  #include <vector>

  #define LIST_LENGTH_DIV       50

  #define MINOR_LIMIT_DIV       200

#endif

#ifdef SORTED_LIST_PIVOT

  #include <vector>

  #include <algorithm>

  #define LIST_LENGTH_DIV       100

  #define LOWER_DIV             4

#endif

namespace lemon {

  /// \addtogroup min_cost_flow

  /// @{

  /// \brief Implementation of the network simplex algorithm for

  /// finding a minimum cost flow.

  ///

  /// \ref NetworkSimplex implements the network simplex algorithm for

  /// finding a minimum cost flow.

  ///

  /// \param Graph The directed graph type the algorithm runs on.

  /// \param LowerMap The type of the lower bound map.

  /// \param CapacityMap The type of the capacity (upper bound) map.

  /// \param CostMap The type of the cost (length) map.

  /// \param SupplyMap The type of the supply map.

  ///

  /// \warning

  /// - Edge capacities and costs should be non-negative integers.

  ///   However \c CostMap::Value should be signed type.

  /// - Supply values should be signed integers.

  /// - \c LowerMap::Value must be convertible to

  ///   \c CapacityMap::Value and \c CapacityMap::Value must be

  ///   convertible to \c SupplyMap::Value.

  ///

  /// \author Peter Kovacs

  template < typename Graph,

             typename LowerMap = typename Graph::template EdgeMap<int>,

             typename CapacityMap = LowerMap,

             typename CostMap = typename Graph::template EdgeMap<int>,

             typename SupplyMap = typename Graph::template NodeMap

                                  <typename CapacityMap::Value> >

  class NetworkSimplex

  {

    typedef typename LowerMap::Value Lower;

    typedef typename CapacityMap::Value Capacity;

    typedef typename CostMap::Value Cost;

    typedef typename SupplyMap::Value Supply;

    typedef SmartGraph SGraph;

    GRAPH_TYPEDEFS(typename SGraph);

    typedef typename SGraph::template EdgeMap<Lower> SLowerMap;

    typedef typename SGraph::template EdgeMap<Capacity> SCapacityMap;

    typedef typename SGraph::template EdgeMap<Cost> SCostMap;

    typedef typename SGraph::template NodeMap<Supply> SSupplyMap;

    typedef typename SGraph::template NodeMap<Cost> SPotentialMap;

    typedef typename SGraph::template NodeMap<int> IntNodeMap;

    typedef typename SGraph::template NodeMap<bool> BoolNodeMap;

    typedef typename SGraph::template NodeMap<Node> NodeNodeMap;

    typedef typename SGraph::template NodeMap<Edge> EdgeNodeMap;

    typedef typename SGraph::template EdgeMap<int> IntEdgeMap;

    typedef typename Graph::template NodeMap<Node> NodeRefMap;

    typedef typename Graph::template EdgeMap<Edge> EdgeRefMap;

  public:

    /// The type of the flow map.

    typedef typename Graph::template EdgeMap<Capacity> FlowMap;

    /// The type of the potential map.

    typedef typename Graph::template NodeMap<Cost> PotentialMap;

  protected:

    /// Map adaptor class for handling reduced edge costs.

    class ReducedCostMap : public MapBase<Edge, Cost>

    {

    private:

      const SGraph &gr;

      const SCostMap &cost_map;

      const SPotentialMap &pot_map;

    public:

      ReducedCostMap( const SGraph &_gr,

                      const SCostMap &_cm,

                      const SPotentialMap &_pm ) :

        gr(_gr), cost_map(_cm), pot_map(_pm) {}

      Cost operator[](const Edge &e) const {

        return cost_map[e] - pot_map[gr.source(e)] + pot_map[gr.target(e)];

      }

    }; //class ReducedCostMap

  protected:

    /// The directed graph the algorithm runs on.

    SGraph graph;

    /// The original graph.

    const Graph &graph_ref;

    /// The original lower bound map.

    const LowerMap *lower;

    /// The capacity map.

    SCapacityMap capacity;

    /// The cost map.

    SCostMap cost;

    /// The supply map.

    SSupplyMap supply;

    /// The reduced cost map.

    ReducedCostMap red_cost;

    bool valid_supply;

    /// The edge map of the current flow.

    SCapacityMap flow;

    /// The potential node map.

    SPotentialMap potential;

    FlowMap flow_result;

    PotentialMap potential_result;

    /// Node reference for the original graph.

    NodeRefMap node_ref;

    /// Edge reference for the original graph.

    EdgeRefMap edge_ref;

    /// The \c depth node map of the spanning tree structure.

    IntNodeMap depth;

    /// The \c parent node map of the spanning tree structure.

    NodeNodeMap parent;

    /// The \c pred_edge node map of the spanning tree structure.

    EdgeNodeMap pred_edge;

    /// The \c thread node map of the spanning tree structure.

    NodeNodeMap thread;

    /// The \c forward node map of the spanning tree structure.

    BoolNodeMap forward;

    /// The state edge map.

    IntEdgeMap state;

    /// The root node of the starting spanning tree.

    Node root;

    // The entering edge of the current pivot iteration.

    Edge in_edge;

    // Temporary nodes used in the current pivot iteration.

    Node join, u_in, v_in, u_out, v_out;

    Node right, first, second, last;

    Node stem, par_stem, new_stem;

    // The maximum augment amount along the found cycle in the current

    // pivot iteration.

    Capacity delta;

#ifdef EDGE_BLOCK_PIVOT

    /// The size of blocks for the "Edge Block" pivot rule.

    int block_size;

#endif

#ifdef CANDIDATE_LIST_PIVOT

    /// \brief The list of candidate edges for the "Candidate List"

    /// pivot rule.

    std::vector<Edge> candidates;

    /// \brief The maximum length of the edge list for the

    /// "Candidate List" pivot rule.

    int list_length;

    /// \brief The maximum number of minor iterations between two major

    /// itarations.

    int minor_limit;

    /// \brief The number of minor iterations.

    int minor_count;

#endif

#ifdef SORTED_LIST_PIVOT

    /// \brief The list of candidate edges for the "Sorted List"

    /// pivot rule.

    std::vector<Edge> candidates;

    /// \brief The maximum length of the edge list for the

    /// "Sorted List" pivot rule.

    int list_length;

    int list_index;

#endif

  public :

    /// \brief General constructor of the class (with lower bounds).

    ///

    /// General constructor of the class (with lower bounds).

    ///

    /// \param _graph The directed graph the algorithm runs on.

    /// \param _lower The lower bounds of the edges.

    /// \param _capacity The capacities (upper bounds) of the edges.

    /// \param _cost The cost (length) values of the edges.

    /// \param _supply The supply values of the nodes (signed).

    NetworkSimplex( const Graph &_graph,

                    const LowerMap &_lower,

                    const CapacityMap &_capacity,

                    const CostMap &_cost,

                    const SupplyMap &_supply ) :

      graph_ref(_graph), lower(&_lower), capacity(graph), cost(graph),

      supply(graph), flow(graph), flow_result(_graph), potential(graph),

      potential_result(_graph), depth(graph), parent(graph),

      pred_edge(graph), thread(graph), forward(graph), state(graph),

      node_ref(graph_ref), edge_ref(graph_ref),

      red_cost(graph, cost, potential)

    {

      // Checking the sum of supply values

      Supply sum = 0;

      for (typename Graph::NodeIt n(graph_ref); n != INVALID; ++n)

        sum += _supply[n];

      if (!(valid_supply = sum == 0)) return;

      // Copying graph_ref to graph

      graph.reserveNode(countNodes(graph_ref) + 1);

      graph.reserveEdge(countEdges(graph_ref) + countNodes(graph_ref));

      copyGraph(graph, graph_ref)

        .edgeMap(cost, _cost)

        .nodeRef(node_ref)

        .edgeRef(edge_ref)

        .run();

      // Removing non-zero lower bounds

      for (typename Graph::EdgeIt e(graph_ref); e != INVALID; ++e) {

        capacity[edge_ref[e]] = _capacity[e] - _lower[e];

      }

      for (typename Graph::NodeIt n(graph_ref); n != INVALID; ++n) {

        Supply s = _supply[n];

        for (typename Graph::InEdgeIt e(graph_ref, n); e != INVALID; ++e)

          s += _lower[e];

        for (typename Graph::OutEdgeIt e(graph_ref, n); e != INVALID; ++e)

          s -= _lower[e];

        supply[node_ref[n]] = s;

      }

    }

    /// \brief General constructor of the class (without lower bounds).

    ///

    /// General constructor of the class (without lower bounds).

    ///

    /// \param _graph The directed graph the algorithm runs on.

    /// \param _capacity The capacities (upper bounds) of the edges.

    /// \param _cost The cost (length) values of the edges.

    /// \param _supply The supply values of the nodes (signed).

    NetworkSimplex( const Graph &_graph,

                    const CapacityMap &_capacity,

                    const CostMap &_cost,

                    const SupplyMap &_supply ) :

      graph_ref(_graph), lower(NULL), capacity(graph), cost(graph),

      supply(graph), flow(graph), flow_result(_graph), potential(graph),

      potential_result(_graph), depth(graph), parent(graph),

      pred_edge(graph), thread(graph), forward(graph), state(graph),

      node_ref(graph_ref), edge_ref(graph_ref),

      red_cost(graph, cost, potential)

    {

      // Checking the sum of supply values

      Supply sum = 0;

      for (typename Graph::NodeIt n(graph_ref); n != INVALID; ++n)

        sum += _supply[n];

      if (!(valid_supply = sum == 0)) return;

      // Copying graph_ref to graph

      copyGraph(graph, graph_ref)

        .edgeMap(capacity, _capacity)

        .edgeMap(cost, _cost)

        .nodeMap(supply, _supply)

        .nodeRef(node_ref)

        .edgeRef(edge_ref)

        .run();

    }

    /// \brief Simple constructor of the class (with lower bounds).

    ///

    /// Simple constructor of the class (with lower bounds).

    ///

    /// \param _graph The directed graph the algorithm runs on.

    /// \param _lower The lower bounds of the edges.

    /// \param _capacity The capacities (upper bounds) of the edges.

    /// \param _cost The cost (length) values of the edges.

    /// \param _s The source node.

    /// \param _t The target node.

    /// \param _flow_value The required amount of flow from node \c _s

    /// to node \c _t (i.e. the supply of \c _s and the demand of \c _t).

    NetworkSimplex( const Graph &_graph,

                    const LowerMap &_lower,

                    const CapacityMap &_capacity,

                    const CostMap &_cost,

                    typename Graph::Node _s,

                    typename Graph::Node _t,

                    typename SupplyMap::Value _flow_value ) :

      graph_ref(_graph), lower(&_lower), capacity(graph), cost(graph),

      supply(graph), flow(graph), flow_result(_graph), potential(graph),

      potential_result(_graph), depth(graph), parent(graph),

      pred_edge(graph), thread(graph), forward(graph), state(graph),

      node_ref(graph_ref), edge_ref(graph_ref),

      red_cost(graph, cost, potential)

    {

      // Copying graph_ref to graph

      copyGraph(graph, graph_ref)

        .edgeMap(cost, _cost)

        .nodeRef(node_ref)

        .edgeRef(edge_ref)

        .run();

      // Removing non-zero lower bounds

      for (typename Graph::EdgeIt e(graph_ref); e != INVALID; ++e) {

        capacity[edge_ref[e]] = _capacity[e] - _lower[e];

      }

      for (typename Graph::NodeIt n(graph_ref); n != INVALID; ++n) {

        Supply s = 0;

        if (n == _s) s =  _flow_value;

        if (n == _t) s = -_flow_value;

        for (typename Graph::InEdgeIt e(graph_ref, n); e != INVALID; ++e)

          s += _lower[e];

        for (typename Graph::OutEdgeIt e(graph_ref, n); e != INVALID; ++e)

          s -= _lower[e];

        supply[node_ref[n]] = s;

      }

      valid_supply = true;

    }

    /// \brief Simple constructor of the class (without lower bounds).

    ///

    /// Simple constructor of the class (without lower bounds).

    ///

    /// \param _graph The directed graph the algorithm runs on.

    /// \param _capacity The capacities (upper bounds) of the edges.

    /// \param _cost The cost (length) values of the edges.

    /// \param _s The source node.

    /// \param _t The target node.

    /// \param _flow_value The required amount of flow from node \c _s

    /// to node \c _t (i.e. the supply of \c _s and the demand of \c _t).

    NetworkSimplex( const Graph &_graph,

                    const CapacityMap &_capacity,

                    const CostMap &_cost,

                    typename Graph::Node _s,

                    typename Graph::Node _t,

                    typename SupplyMap::Value _flow_value ) :

      graph_ref(_graph), lower(NULL), capacity(graph), cost(graph),

      supply(graph, 0), flow(graph), flow_result(_graph), potential(graph),

      potential_result(_graph), depth(graph), parent(graph),

      pred_edge(graph), thread(graph), forward(graph), state(graph),

      node_ref(graph_ref), edge_ref(graph_ref),

      red_cost(graph, cost, potential)

    {

      // Copying graph_ref to graph

      copyGraph(graph, graph_ref)

        .edgeMap(capacity, _capacity)

        .edgeMap(cost, _cost)

        .nodeRef(node_ref)

        .edgeRef(edge_ref)

        .run();

      supply[node_ref[_s]] =  _flow_value;

      supply[node_ref[_t]] = -_flow_value;

      valid_supply = true;

    }

    /// \brief Runs the algorithm.

    ///

    /// Runs the algorithm.

    ///

    /// \return \c true if a feasible flow can be found.

    bool run() {

      return init() && start();

    }

    /// \brief Returns a const reference to the flow map.

    ///

    /// Returns a const reference to the flow map.

    ///

    /// \pre \ref run() must be called before using this function.

    const FlowMap& flowMap() const {

      return flow_result;

    }

    /// \brief Returns a const reference to the potential map (the dual

    /// solution).

    ///

    /// Returns a const reference to the potential map (the dual

    /// solution).

    ///

    /// \pre \ref run() must be called before using this function.

    const PotentialMap& potentialMap() const {

      return potential_result;

    }

    /// \brief Returns the total cost of the found flow.

    ///

    /// Returns the total cost of the found flow. The complexity of the

    /// function is \f$ O(e) \f$.

    ///

    /// \pre \ref run() must be called before using this function.

    Cost totalCost() const {

      Cost c = 0;

      for (typename Graph::EdgeIt e(graph_ref); e != INVALID; ++e)

        c += flow_result[e] * cost[edge_ref[e]];

      return c;

    }

  protected:

    /// \brief Extends the underlaying graph and initializes all the

    /// node and edge maps.

    bool init() {

      if (!valid_supply) return false;

      // Initializing state and flow maps

      for (EdgeIt e(graph); e != INVALID; ++e) {

        flow[e] = 0;

        state[e] = LOWER;

      }

      // Adding an artificial root node to the graph

      root = graph.addNode();

      parent[root] = INVALID;

      pred_edge[root] = INVALID;

      depth[root] = 0;

      supply[root] = 0;

      potential[root] = 0;

      // Adding artificial edges to the graph and initializing the node

      // maps of the spanning tree data structure

      Supply sum = 0;

      Node last = root;

      Edge e;

      Cost max_cost = std::numeric_limits<Cost>::max() / 4;

      for (NodeIt u(graph); u != INVALID; ++u) {

        if (u == root) continue;

        thread[last] = u;

        last = u;

        parent[u] = root;

        depth[u] = 1;

        sum += supply[u];

        if (supply[u] >= 0) {

          e = graph.addEdge(u, root);

          flow[e] = supply[u];

          forward[u] = true;

          potential[u] = max_cost;

        } else {

          e = graph.addEdge(root, u);

          flow[e] = -supply[u];

          forward[u] = false;

          potential[u] = -max_cost;

        }

        cost[e] = max_cost;

        capacity[e] = std::numeric_limits<Capacity>::max();

        state[e] = TREE;

        pred_edge[u] = e;

      }

      thread[last] = root;

#ifdef EDGE_BLOCK_PIVOT

      // Initializing block_size for the edge block pivot rule

      int edge_num = countEdges(graph);

      block_size = 2 * int(sqrt(countEdges(graph)));

      if (block_size < MIN_BLOCK_SIZE) block_size = MIN_BLOCK_SIZE;

#endif

#ifdef CANDIDATE_LIST_PIVOT

      int edge_num = countEdges(graph);

      minor_count = 0;

      list_length = edge_num / LIST_LENGTH_DIV;

      minor_limit = edge_num / MINOR_LIMIT_DIV;

#endif

#ifdef SORTED_LIST_PIVOT

      int edge_num = countEdges(graph);

      list_index = 0;

      list_length = edge_num / LIST_LENGTH_DIV;

#endif

      return sum == 0;

    }

#ifdef FIRST_ELIGIBLE_PIVOT

    /// \brief Finds entering edge according to the "First Eligible"

    /// pivot rule.

    bool findEnteringEdge(EdgeIt &next_edge) {

      for (EdgeIt e = next_edge; e != INVALID; ++e) {

        if (state[e] * red_cost[e] < 0) {

          in_edge = e;

          next_edge = ++e;

          return true;

        }

      }

      for (EdgeIt e(graph); e != next_edge; ++e) {

        if (state[e] * red_cost[e] < 0) {

          in_edge = e;

          next_edge = ++e;

          return true;

        }

      }

      return false;

    }

#endif

#ifdef BEST_ELIGIBLE_PIVOT

    /// \brief Finds entering edge according to the "Best Eligible"

    /// pivot rule.

    bool findEnteringEdge() {

      Cost min = 0;

      for (EdgeIt e(graph); e != INVALID; ++e) {

        if (state[e] * red_cost[e] < min) {

          min = state[e] * red_cost[e];

          in_edge = e;

        }

      }

      return min < 0;

    }

#endif

#ifdef EDGE_BLOCK_PIVOT

    /// \brief Finds entering edge according to the "Edge Block"

    /// pivot rule.

    bool findEnteringEdge(EdgeIt &next_edge) {

      // Performing edge block selection

      Cost curr, min = 0;

      EdgeIt min_edge(graph);

      int cnt = 0;

      for (EdgeIt e = next_edge; e != INVALID; ++e) {

        if ((curr = state[e] * red_cost[e]) < min) {

          min = curr;

          min_edge = e;

        }

        if (++cnt == block_size) {

          if (min < 0) break;

          cnt = 0;

        }

      }

      if (!(min < 0)) {

        for (EdgeIt e(graph); e != next_edge; ++e) {

          if ((curr = state[e] * red_cost[e]) < min) {

            min = curr;

            min_edge = e;

          }

          if (++cnt == block_size) {

            if (min < 0) break;

            cnt = 0;

          }

        }

      }

      in_edge = min_edge;

      if ((next_edge = ++min_edge) == INVALID)

        next_edge = EdgeIt(graph);

      return min < 0;

    }

#endif

#ifdef CANDIDATE_LIST_PIVOT

    /// \brief Finds entering edge according to the "Candidate List"

    /// pivot rule.

    bool findEnteringEdge() {

      typedef typename std::vector<Edge>::iterator ListIt;

      if (minor_count >= minor_limit || candidates.size() == 0) {

        // Major iteration

        candidates.clear();

        for (EdgeIt e(graph); e != INVALID; ++e) {

          if (state[e] * red_cost[e] < 0) {

            candidates.push_back(e);

            if (candidates.size() == list_length) break;

          }

        }

        if (candidates.size() == 0) return false;

      }

      // Minor iteration

      ++minor_count;

      Cost min = 0;

      Edge e;

      for (int i = 0; i < candidates.size(); ++i) {

        e = candidates[i];

        if (state[e] * red_cost[e] < min) {

          min = state[e] * red_cost[e];

          in_edge = e;

        }

      }

      return true;

    }

#endif

#ifdef SORTED_LIST_PIVOT

    /// \brief Functor class to compare edges during sort of the

    /// candidate list.

    class SortFunc

    {

    private:

      const IntEdgeMap &st;

      const ReducedCostMap &rc;

    public:

      SortFunc(const IntEdgeMap &_st, const ReducedCostMap &_rc) :

        st(_st), rc(_rc) {}

      bool operator()(const Edge &e1, const Edge &e2) {

        return st[e1] * rc[e1] < st[e2] * rc[e2];

      }

    };

    /// \brief Finds entering edge according to the "Sorted List"

    /// pivot rule.

    bool findEnteringEdge() {

      static SortFunc sort_func(state, red_cost);

      // Minor iteration

      while (list_index < candidates.size()) {

        in_edge = candidates[list_index++];

        if (state[in_edge] * red_cost[in_edge] < 0) return true;

      }

      // Major iteration

      candidates.clear();

      Cost curr, min = 0;

      for (EdgeIt e(graph); e != INVALID; ++e) {

        if ((curr = state[e] * red_cost[e]) < min / LOWER_DIV) {

          candidates.push_back(e);

          if (curr < min) min = curr;

          if (candidates.size() == list_length) break;

        }

      }

      if (candidates.size() == 0) return false;

      sort(candidates.begin(), candidates.end(), sort_func);

      in_edge = candidates[0];

      list_index = 1;

      return true;

    }

#endif

    /// \brief Finds the join node.

    Node findJoinNode() {

      // Finding the join node

      Node u = graph.source(in_edge);

      Node v = graph.target(in_edge);

      while (u != v) {

        if (depth[u] == depth[v]) {

          u = parent[u];

          v = parent[v];

        }

        else if (depth[u] > depth[v]) u = parent[u];

        else v = parent[v];

      }

      return u;

    }

    /// \brief Finds the leaving edge of the cycle. Returns \c true if

    /// the leaving edge is not the same as the entering edge.

    bool findLeavingEdge() {

      // Initializing first and second nodes according to the direction

      // of the cycle

      if (state[in_edge] == LOWER) {

        first = graph.source(in_edge);

        second  = graph.target(in_edge);

      } else {

        first = graph.target(in_edge);

        second  = graph.source(in_edge);

      }

      delta = capacity[in_edge];

      bool result = false;

      Capacity d;

      Edge e;

      // Searching the cycle along the path form the first node to the

      // root node

      for (Node u = first; u != join; u = parent[u]) {

        e = pred_edge[u];

        d = forward[u] ? flow[e] : capacity[e] - flow[e];

        if (d < delta) {

          delta = d;

          u_out = u;

          u_in = first;

          v_in = second;

          result = true;

        }

      }

      // Searching the cycle along the path form the second node to the

      // root node

      for (Node u = second; u != join; u = parent[u]) {

        e = pred_edge[u];

        d = forward[u] ? capacity[e] - flow[e] : flow[e];

        if (d <= delta) {

          delta = d;

          u_out = u;

          u_in = second;

          v_in = first;

          result = true;

        }

      }

      return result;

    }

    /// \brief Changes \c flow and \c state edge maps.

    void changeFlows(bool change) {

      // Augmenting along the cycle

      if (delta > 0) {

        Capacity val = state[in_edge] * delta;

        flow[in_edge] += val;

        for (Node u = graph.source(in_edge); u != join; u = parent[u]) {

          flow[pred_edge[u]] += forward[u] ? -val : val;

        }

        for (Node u = graph.target(in_edge); u != join; u = parent[u]) {

          flow[pred_edge[u]] += forward[u] ? val : -val;

        }

      }

      // Updating the state of the entering and leaving edges

      if (change) {

        state[in_edge] = TREE;

        state[pred_edge[u_out]] =

          (flow[pred_edge[u_out]] == 0) ? LOWER : UPPER;

      } else {

        state[in_edge] = -state[in_edge];

      }

    }

    /// \brief Updates \c thread and \c parent node maps.

    void updateThreadParent() {

      Node u;

      v_out = parent[u_out];

      // Handling the case when join and v_out coincide

      bool par_first = false;

      if (join == v_out) {

        for (u = join; u != u_in && u != v_in; u = thread[u]) ;

        if (u == v_in) {

          par_first = true;

          while (thread[u] != u_out) u = thread[u];

          first = u;

        }

      }

      // Finding the last successor of u_in (u) and the node after it

      // (right) according to the thread index

      for (u = u_in; depth[thread[u]] > depth[u_in]; u = thread[u]) ;

      right = thread[u];

      if (thread[v_in] == u_out) {

        for (last = u; depth[last] > depth[u_out]; last = thread[last]) ;

        if (last == u_out) last = thread[last];

      }

      else last = thread[v_in];

      // Updating stem nodes

      thread[v_in] = stem = u_in;

      par_stem = v_in;

      while (stem != u_out) {

        thread[u] = new_stem = parent[stem];

        // Finding the node just before the stem node (u) according to

        // the original thread index

        for (u = new_stem; thread[u] != stem; u = thread[u]) ;

        thread[u] = right;

        // Changing the parent node of stem and shifting stem and

        // par_stem nodes

        parent[stem] = par_stem;

        par_stem = stem;

        stem = new_stem;

        // Finding the last successor of stem (u) and the node after it

        // (right) according to the thread index

        for (u = stem; depth[thread[u]] > depth[stem]; u = thread[u]) ;

        right = thread[u];

      }

      parent[u_out] = par_stem;

      thread[u] = last;

      if (join == v_out && par_first) {

        if (first != v_in) thread[first] = right;

      } else {

        for (u = v_out; thread[u] != u_out; u = thread[u]) ;

        thread[u] = right;

      }

    }

    /// \brief Updates \c pred_edge and \c forward node maps.

    void updatePredEdge() {

      Node u = u_out, v;

      while (u != u_in) {

        v = parent[u];

        pred_edge[u] = pred_edge[v];

        forward[u] = !forward[v];

        u = v;

      }

      pred_edge[u_in] = in_edge;

      forward[u_in] = (u_in == graph.source(in_edge));

    }

    /// \brief Updates \c depth and \c potential node maps.

    void updateDepthPotential() {

      depth[u_in] = depth[v_in] + 1;

      potential[u_in] = forward[u_in] ?

        potential[v_in] + cost[pred_edge[u_in]] :

        potential[v_in] - cost[pred_edge[u_in]];

      Node u = thread[u_in], v;

      while (true) {

        v = parent[u];

        if (v == INVALID) break;

        depth[u] = depth[v] + 1;

        potential[u] = forward[u] ?

          potential[v] + cost[pred_edge[u]] :

          potential[v] - cost[pred_edge[u]];

        if (depth[u] <= depth[v_in]) break;

        u = thread[u];

      }

    }

    /// \brief Executes the algorithm.

    bool start() {

      // Processing pivots

#ifdef _DEBUG_ITER_

      int iter_num = 0;

#endif

#if defined(FIRST_ELIGIBLE_PIVOT) || defined(EDGE_BLOCK_PIVOT)

      EdgeIt next_edge(graph);

      while (findEnteringEdge(next_edge))

#else

      while (findEnteringEdge())

#endif

      {

        join = findJoinNode();

        bool change = findLeavingEdge();

        changeFlows(change);

        if (change) {

          updateThreadParent();

          updatePredEdge();

          updateDepthPotential();

        }

#ifdef _DEBUG_ITER_

        ++iter_num;

#endif

      }

#ifdef _DEBUG_ITER_

      std::cout << "Network Simplex algorithm finished. " << iter_num

                << " pivot iterations performed." << std::endl;

#endif

      // Checking if the flow amount equals zero on all the

      // artificial edges

      for (InEdgeIt e(graph, root); e != INVALID; ++e)

        if (flow[e] > 0) return false;

      for (OutEdgeIt e(graph, root); e != INVALID; ++e)

        if (flow[e] > 0) return false;

      // Copying flow values to flow_result

      if (lower) {

        for (typename Graph::EdgeIt e(graph_ref); e != INVALID; ++e)

          flow_result[e] = (*lower)[e] + flow[edge_ref[e]];

      } else {

        for (typename Graph::EdgeIt e(graph_ref); e != INVALID; ++e)

          flow_result[e] = flow[edge_ref[e]];

      }

      // Copying potential values to potential_result

      for (typename Graph::NodeIt n(graph_ref); n != INVALID; ++n)

        potential_result[n] = potential[node_ref[n]];

      return true;

    }

  }; //class NetworkSimplex

  ///@}

} //namespace lemon

#endif //LEMON_NETWORK_SIMPLEX_H