/* -*- 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 Network simplex algorithm for finding a minimum cost flow.

#include <vector>

#include <limits>

#include <algorithm>

#include <lemon/graph_adaptor.h>

#include <lemon/graph_utils.h>

#include <lemon/smart_graph.h>

#include <lemon/math.h>

namespace lemon {

  /// \addtogroup min_cost_flow

  /// @{

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

  /// for finding a minimum cost flow.

  ///

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

  /// for finding a minimum cost flow.

  ///

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

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

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

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

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

  ///

  /// \warning

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

  /// - Supply values should be \e signed \e integers.

  /// - The value types of the maps should be convertible to each other.

  /// - \c CostMap::Value must be signed type.

  ///

  /// \note \ref NetworkSimplex provides five different pivot rule

  /// implementations that significantly affect the efficiency of the

  /// algorithm.

  /// By default "Block Search" pivot rule is used, which proved to be

  /// by far the most efficient according to our benchmark tests.

  /// However another pivot rule can be selected using \ref run()

  /// function with the proper parameter.

  ///

  /// \author Peter Kovacs

  template < typename Graph,

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

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

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

             typename SupplyMap = typename Graph::template NodeMap<int> >

  class NetworkSimplex

  {

    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<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 SGraph::template EdgeMap<bool> BoolEdgeMap;

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

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

    typedef std::vector<Edge> EdgeVector;

  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;

  public:

    /// Enum type to select the pivot rule used by \ref run().

    enum PivotRuleEnum {

      FIRST_ELIGIBLE_PIVOT,

      BEST_ELIGIBLE_PIVOT,

      BLOCK_SEARCH_PIVOT,

      CANDIDATE_LIST_PIVOT,

      ALTERING_LIST_PIVOT

    };

  private:

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

    ///

    /// 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:

      ///\e

      ReducedCostMap( const SGraph &gr,

                      const SCostMap &cost_map,

                      const SPotentialMap &pot_map ) :

        _gr(gr), _cost_map(cost_map), _pot_map(pot_map) {}

      ///\e

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

        return _cost_map[e] + _pot_map[_gr.source(e)]

                            - _pot_map[_gr.target(e)];

      }

    }; //class ReducedCostMap

  private:

    /// \brief Implementation of the "First Eligible" pivot rule for the

    /// \ref NetworkSimplex "network simplex" algorithm.

    ///

    /// This class implements the "First Eligible" pivot rule

    /// for the \ref NetworkSimplex "network simplex" algorithm.

    ///

    /// For more information see \ref NetworkSimplex::run().

    class FirstEligiblePivotRule

    {

    private:

      // References to the NetworkSimplex class

      NetworkSimplex &_ns;

      EdgeVector &_edges;

      int _next_edge;

    public:

      /// Constructor

      FirstEligiblePivotRule(NetworkSimplex &ns, EdgeVector &edges) :

        _ns(ns), _edges(edges), _next_edge(0) {}

      /// Find next entering edge

      bool findEnteringEdge() {

        Edge e;

        for (int i = _next_edge; i < int(_edges.size()); ++i) {

          e = _edges[i];

          if (_ns._state[e] * _ns._red_cost[e] < 0) {

            _ns._in_edge = e;

            _next_edge = i + 1;

            return true;

          }

        }

        for (int i = 0; i < _next_edge; ++i) {

          e = _edges[i];

          if (_ns._state[e] * _ns._red_cost[e] < 0) {

            _ns._in_edge = e;

            _next_edge = i + 1;

            return true;

          }

        }

        return false;

      }

    }; //class FirstEligiblePivotRule

    /// \brief Implementation of the "Best Eligible" pivot rule for the

    /// \ref NetworkSimplex "network simplex" algorithm.

    ///

    /// This class implements the "Best Eligible" pivot rule

    /// for the \ref NetworkSimplex "network simplex" algorithm.

    ///

    /// For more information see \ref NetworkSimplex::run().

    class BestEligiblePivotRule

    {

    private:

      // References to the NetworkSimplex class

      NetworkSimplex &_ns;

      EdgeVector &_edges;

    public:

      /// Constructor

      BestEligiblePivotRule(NetworkSimplex &ns, EdgeVector &edges) :

        _ns(ns), _edges(edges) {}

      /// Find next entering edge

      bool findEnteringEdge() {

        Cost min = 0;

        Edge e;

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

          e = _edges[i];

          if (_ns._state[e] * _ns._red_cost[e] < min) {

            min = _ns._state[e] * _ns._red_cost[e];

            _ns._in_edge = e;

          }

        }

        return min < 0;

      }

    }; //class BestEligiblePivotRule

    /// \brief Implementation of the "Block Search" pivot rule for the

    /// \ref NetworkSimplex "network simplex" algorithm.

    ///

    /// This class implements the "Block Search" pivot rule

    /// for the \ref NetworkSimplex "network simplex" algorithm.

    ///

    /// For more information see \ref NetworkSimplex::run().

    class BlockSearchPivotRule

    {

    private:

      // References to the NetworkSimplex class

      NetworkSimplex &_ns;

      EdgeVector &_edges;

      int _block_size;

      int _next_edge, _min_edge;

    public:

      /// Constructor

      BlockSearchPivotRule(NetworkSimplex &ns, EdgeVector &edges) :

        _ns(ns), _edges(edges), _next_edge(0), _min_edge(0)

      {

        // The main parameters of the pivot rule

        const double BLOCK_SIZE_FACTOR = 2.0;

        const int MIN_BLOCK_SIZE = 10;

        _block_size = std::max( int(BLOCK_SIZE_FACTOR * sqrt(_edges.size())),

                                MIN_BLOCK_SIZE );

      }

      /// Find next entering edge

      bool findEnteringEdge() {

        Cost curr, min = 0;

        Edge e;

        int cnt = _block_size;

        int i;

        for (i = _next_edge; i < int(_edges.size()); ++i) {

          e = _edges[i];

          if ((curr = _ns._state[e] * _ns._red_cost[e]) < min) {

            min = curr;

            _min_edge = i;

          }

          if (--cnt == 0) {

            if (min < 0) break;

            cnt = _block_size;

          }

        }

        if (min == 0 || cnt > 0) {

          for (i = 0; i < _next_edge; ++i) {

            e = _edges[i];

            if ((curr = _ns._state[e] * _ns._red_cost[e]) < min) {

              min = curr;

              _min_edge = i;

            }

            if (--cnt == 0) {

              if (min < 0) break;

              cnt = _block_size;

            }

          }

        }

        if (min >= 0) return false;

        _ns._in_edge = _edges[_min_edge];

        _next_edge = i;

        return true;

      }

    }; //class BlockSearchPivotRule

    /// \brief Implementation of the "Candidate List" pivot rule for the

    /// \ref NetworkSimplex "network simplex" algorithm.

    ///

    /// This class implements the "Candidate List" pivot rule

    /// for the \ref NetworkSimplex "network simplex" algorithm.

    ///

    /// For more information see \ref NetworkSimplex::run().

    class CandidateListPivotRule

    {

    private:

      // References to the NetworkSimplex class

      NetworkSimplex &_ns;

      EdgeVector &_edges;

      EdgeVector _candidates;

      int _list_length, _minor_limit;

      int _curr_length, _minor_count;

      int _next_edge, _min_edge;

    public:

      /// Constructor

      CandidateListPivotRule(NetworkSimplex &ns, EdgeVector &edges) :

        _ns(ns), _edges(edges), _next_edge(0), _min_edge(0)

      {

        // The main parameters of the pivot rule

        const double LIST_LENGTH_FACTOR = 1.0;

        const int MIN_LIST_LENGTH = 10;

        const double MINOR_LIMIT_FACTOR = 0.1;

        const int MIN_MINOR_LIMIT = 3;

        _list_length = std::max( int(LIST_LENGTH_FACTOR * sqrt(_edges.size())),

                                 MIN_LIST_LENGTH );

        _minor_limit = std::max( int(MINOR_LIMIT_FACTOR * _list_length),

                                 MIN_MINOR_LIMIT );

        _curr_length = _minor_count = 0;

        _candidates.resize(_list_length);

      }

      /// Find next entering edge

      bool findEnteringEdge() {

        Cost min, curr;

        if (_curr_length > 0 && _minor_count < _minor_limit) {

          // Minor iteration: select the best eligible edge from the

          // current candidate list

          ++_minor_count;

          Edge e;

          min = 0;

          for (int i = 0; i < _curr_length; ++i) {

            e = _candidates[i];

            curr = _ns._state[e] * _ns._red_cost[e];

            if (curr < min) {

              min = curr;

              _ns._in_edge = e;

            }

            if (curr >= 0) {

              _candidates[i--] = _candidates[--_curr_length];

            }

          }

          if (min < 0) return true;

        }

        // Major iteration: build a new candidate list

        Edge e;

        min = 0;

        _curr_length = 0;

        int i;

        for (i = _next_edge; i < int(_edges.size()); ++i) {

          e = _edges[i];

          if ((curr = _ns._state[e] * _ns._red_cost[e]) < 0) {

            _candidates[_curr_length++] = e;

            if (curr < min) {

              min = curr;

              _min_edge = i;

            }

            if (_curr_length == _list_length) break;

          }

        }

        if (_curr_length < _list_length) {

          for (i = 0; i < _next_edge; ++i) {

            e = _edges[i];

            if ((curr = _ns._state[e] * _ns._red_cost[e]) < 0) {

              _candidates[_curr_length++] = e;

              if (curr < min) {

                min = curr;

                _min_edge = i;

              }

              if (_curr_length == _list_length) break;

            }

          }

        }

        if (_curr_length == 0) return false;

        _minor_count = 1;

        _ns._in_edge = _edges[_min_edge];

        _next_edge = i;

        return true;

      }

    }; //class CandidateListPivotRule

    /// \brief Implementation of the "Altering Candidate List" pivot rule

    /// for the \ref NetworkSimplex "network simplex" algorithm.

    ///

    /// This class implements the "Altering Candidate List" pivot rule

    /// for the \ref NetworkSimplex "network simplex" algorithm.

    ///

    /// For more information see \ref NetworkSimplex::run().

    class AlteringListPivotRule

    {

    private:

      // References to the NetworkSimplex class

      NetworkSimplex &_ns;

      EdgeVector &_edges;

      EdgeVector _candidates;

      SCostMap _cand_cost;

      int _block_size, _head_length, _curr_length;

      int _next_edge;

      // Functor class to compare edges during sort of the candidate list

      class SortFunc

      {

      private:

        const SCostMap &_map;

      public:

        SortFunc(const SCostMap &map) : _map(map) {}

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

          return _map[e1] > _map[e2];

        }

      };

      SortFunc _sort_func;

    public:

      /// Constructor

      AlteringListPivotRule(NetworkSimplex &ns, EdgeVector &edges) :

        _ns(ns), _edges(edges), _cand_cost(_ns._graph),

        _next_edge(0), _sort_func(_cand_cost)

      {

        // The main parameters of the pivot rule

        const double BLOCK_SIZE_FACTOR = 1.5;

        const int MIN_BLOCK_SIZE = 10;

        const double HEAD_LENGTH_FACTOR = 0.1;

        const int MIN_HEAD_LENGTH = 3;

        _block_size = std::max( int(BLOCK_SIZE_FACTOR * sqrt(_edges.size())),

                                MIN_BLOCK_SIZE );

        _head_length = std::max( int(HEAD_LENGTH_FACTOR * _block_size),

                                 MIN_HEAD_LENGTH );

        _candidates.resize(_head_length + _block_size);

        _curr_length = 0;

      }

      /// Find next entering edge

      bool findEnteringEdge() {

        // Check the current candidate list

        Edge e;

        for (int ix = 0; ix < _curr_length; ++ix) {

          e = _candidates[ix];

          if ((_cand_cost[e] = _ns._state[e] * _ns._red_cost[e]) >= 0) {

            _candidates[ix--] = _candidates[--_curr_length];

          }

        }

        // Extend the list

        int cnt = _block_size;

        int last_edge = 0;

        int limit = _head_length;

        for (int i = _next_edge; i < int(_edges.size()); ++i) {

          e = _edges[i];

          if ((_cand_cost[e] = _ns._state[e] * _ns._red_cost[e]) < 0) {

            _candidates[_curr_length++] = e;

            last_edge = i;

          }

          if (--cnt == 0) {

            if (_curr_length > limit) break;

            limit = 0;

            cnt = _block_size;

          }

        }

        if (_curr_length <= limit) {

          for (int i = 0; i < _next_edge; ++i) {

            e = _edges[i];

            if ((_cand_cost[e] = _ns._state[e] * _ns._red_cost[e]) < 0) {

              _candidates[_curr_length++] = e;

              last_edge = i;

            }

            if (--cnt == 0) {

              if (_curr_length > limit) break;

              limit = 0;

              cnt = _block_size;

            }

          }

        }

        if (_curr_length == 0) return false;

        _next_edge = last_edge + 1;

        // Make heap of the candidate list (approximating a partial sort)

        make_heap( _candidates.begin(), _candidates.begin() + _curr_length,

                   _sort_func );

        // Pop the first element of the heap

        _ns._in_edge = _candidates[0];

        pop_heap( _candidates.begin(), _candidates.begin() + _curr_length,

                  _sort_func );

        _curr_length = std::min(_head_length, _curr_length - 1);

        return true;

      }

    }; //class AlteringListPivotRule

  private:

    // State constants for edges

    enum EdgeStateEnum {

      STATE_UPPER = -1,

      STATE_TREE  =  0,

      STATE_LOWER =  1

    };

  private:

    // 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;

    bool _valid_supply;

    // Edge map of the current flow

    SCapacityMap _flow;

    // Node map of the current potentials

    SPotentialMap _potential;

    // The depth node map of the spanning tree structure

    IntNodeMap _depth;

    // The parent node map of the spanning tree structure

    NodeNodeMap _parent;

    // The pred_edge node map of the spanning tree structure

    EdgeNodeMap _pred_edge;

    // The thread node map of the spanning tree structure

    NodeNodeMap _thread;

    // The forward node map of the spanning tree structure

    BoolNodeMap _forward;

    // The state edge map

    IntEdgeMap _state;

    // The artificial root node of the spanning tree structure

    Node _root;

    // The reduced cost map

    ReducedCostMap _red_cost;

    // The non-artifical edges

    EdgeVector _edges;

    // Members for handling the original graph

    FlowMap *_flow_result;

    PotentialMap *_potential_result;

    bool _local_flow;

    bool _local_potential;

    NodeRefMap _node_ref;

    EdgeRefMap _edge_ref;

    // 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;

  public :

    /// \brief General constructor (with lower bounds).

    ///

    /// General constructor (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(), _graph_ref(graph), _lower(&lower), _capacity(_graph),

      _cost(_graph), _supply(_graph), _flow(_graph),

      _potential(_graph), _depth(_graph), _parent(_graph),

      _pred_edge(_graph), _thread(_graph), _forward(_graph),

      _state(_graph), _red_cost(_graph, _cost, _potential),

      _flow_result(NULL), _potential_result(NULL),

      _local_flow(false), _local_potential(false),

      _node_ref(graph), _edge_ref(graph)

    {

      // Check 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;

      // Copy _graph_ref to _graph

      _graph.reserveNode(countNodes(_graph_ref) + 1);

      _graph.reserveEdge(countEdges(_graph_ref) + countNodes(_graph_ref));

      copyGraph(_graph, _graph_ref)

        .edgeMap(_capacity, capacity)

        .edgeMap(_cost, cost)

        .nodeMap(_supply, supply)

        .nodeRef(_node_ref)

        .edgeRef(_edge_ref)

        .run();

      // Remove non-zero lower bounds

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

        if (lower[e] != 0) {

        _capacity[_edge_ref[e]] = capacity[e] - lower[e];

          _supply[_node_ref[_graph_ref.source(e)]] -= lower[e];

          _supply[_node_ref[_graph_ref.target(e)]] += lower[e];

      }

      }

    }

    /// \brief General constructor (without lower bounds).

    ///

    /// General constructor (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(), _graph_ref(graph), _lower(NULL), _capacity(_graph),

      _cost(_graph), _supply(_graph), _flow(_graph),

      _potential(_graph), _depth(_graph), _parent(_graph),

      _pred_edge(_graph), _thread(_graph), _forward(_graph),

      _state(_graph), _red_cost(_graph, _cost, _potential),

      _flow_result(NULL), _potential_result(NULL),

      _local_flow(false), _local_potential(false),

      _node_ref(graph), _edge_ref(graph)

    {

      // Check 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;

      // Copy _graph_ref to _graph

      _graph.reserveNode(countNodes(_graph_ref) + 1);

      _graph.reserveEdge(countEdges(_graph_ref) + countNodes(_graph_ref));

      copyGraph(_graph, _graph_ref)

        .edgeMap(_capacity, capacity)

        .edgeMap(_cost, cost)

        .nodeMap(_supply, supply)

        .nodeRef(_node_ref)

        .edgeRef(_edge_ref)

        .run();

    }

    /// \brief Simple constructor (with lower bounds).

    ///

    /// Simple constructor (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(), _graph_ref(graph), _lower(&lower), _capacity(_graph),

      _cost(_graph), _supply(_graph, 0), _flow(_graph),

      _potential(_graph), _depth(_graph), _parent(_graph),

      _pred_edge(_graph), _thread(_graph), _forward(_graph),

      _state(_graph), _red_cost(_graph, _cost, _potential),

      _flow_result(NULL), _potential_result(NULL),

      _local_flow(false), _local_potential(false),

      _node_ref(graph), _edge_ref(graph)

    {

      // Copy _graph_ref to graph

      _graph.reserveNode(countNodes(_graph_ref) + 1);

      _graph.reserveEdge(countEdges(_graph_ref) + countNodes(_graph_ref));

      copyGraph(_graph, _graph_ref)

        .edgeMap(_capacity, capacity)

        .edgeMap(_cost, cost)

        .nodeRef(_node_ref)

        .edgeRef(_edge_ref)

        .run();

      // Remove non-zero lower bounds

      _supply[_node_ref[s]] =  flow_value;

      _supply[_node_ref[t]] = -flow_value;

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

        if (lower[e] != 0) {

        _capacity[_edge_ref[e]] = capacity[e] - lower[e];

          _supply[_node_ref[_graph_ref.source(e)]] -= lower[e];

          _supply[_node_ref[_graph_ref.target(e)]] += lower[e];

      }

      }

      _valid_supply = true;

    }

    /// \brief Simple constructor (without lower bounds).

    ///

    /// Simple constructor (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(), _graph_ref(graph), _lower(NULL), _capacity(_graph),

      _cost(_graph), _supply(_graph, 0), _flow(_graph),

      _potential(_graph), _depth(_graph), _parent(_graph),

      _pred_edge(_graph), _thread(_graph), _forward(_graph),

      _state(_graph), _red_cost(_graph, _cost, _potential),

      _flow_result(NULL), _potential_result(NULL),

      _local_flow(false), _local_potential(false),

      _node_ref(graph), _edge_ref(graph)

    {

      // Copy _graph_ref to graph

      _graph.reserveNode(countNodes(_graph_ref) + 1);

      _graph.reserveEdge(countEdges(_graph_ref) + countNodes(_graph_ref));

      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;

    }

    /// Destructor.

    ~NetworkSimplex() {

      if (_local_flow) delete _flow_result;

      if (_local_potential) delete _potential_result;

    }

    /// \brief Set the flow map.

    ///

    /// Set the flow map.

    ///

    /// \return \c (*this)

    NetworkSimplex& flowMap(FlowMap &map) {

      if (_local_flow) {

        delete _flow_result;

        _local_flow = false;

      }

      _flow_result = ↦

      return *this;

    }

    /// \brief Set the potential map.

    ///

    /// Set the potential map.

    ///

    /// \return \c (*this)

    NetworkSimplex& potentialMap(PotentialMap &map) {

      if (_local_potential) {

        delete _potential_result;

        _local_potential = false;

      }

      _potential_result = ↦

      return *this;

    }

    /// \name Execution control

    /// @{

    /// \brief Runs the algorithm.

    ///

    /// Runs the algorithm.

    ///

    /// \param pivot_rule The pivot rule that is used during the

    /// algorithm.

    ///

    /// The available pivot rules:

    ///

    /// - FIRST_ELIGIBLE_PIVOT The next eligible edge is selected in

    /// a wraparound fashion in every iteration

    /// (\ref FirstEligiblePivotRule).

    ///

    /// - BEST_ELIGIBLE_PIVOT The best eligible edge is selected in

    /// every iteration (\ref BestEligiblePivotRule).

    ///

    /// - BLOCK_SEARCH_PIVOT A specified number of edges are examined in

    /// every iteration in a wraparound fashion and the best eligible

    /// edge is selected from this block (\ref BlockSearchPivotRule).

    ///

    /// - CANDIDATE_LIST_PIVOT In a major iteration a candidate list is

    /// built from eligible edges in a wraparound fashion and in the

    /// following minor iterations the best eligible edge is selected

    /// from this list (\ref CandidateListPivotRule).

    ///

    /// - ALTERING_LIST_PIVOT It is a modified version of the

    /// "Candidate List" pivot rule. It keeps only the several best

    /// eligible edges from the former candidate list and extends this

    /// list in every iteration (\ref AlteringListPivotRule).

    ///

    /// According to our comprehensive benchmark tests the "Block Search"

    /// pivot rule proved to be the fastest and the most robust on

    /// various test inputs. Thus it is the default option.

    ///

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

    bool run(PivotRuleEnum pivot_rule = BLOCK_SEARCH_PIVOT) {

      return init() && start(pivot_rule);

    }

    /// @}

    /// \name Query Functions

    /// The results of the algorithm can be obtained using these

    /// functions.\n

    /// \ref lemon::NetworkSimplex::run() "run()" must be called before

    /// using them.

    /// @{

    /// \brief Return a const reference to the edge map storing the

    /// found flow.

    ///

    /// Return a const reference to the edge map storing the found flow.

    ///

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

    const FlowMap& flowMap() const {

      return *_flow_result;

    }

    /// \brief Return a const reference to the node map storing the

    /// found potentials (the dual solution).

    ///

    /// Return a const reference to the node map storing the found

    /// potentials (the dual solution).

    ///

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

    const PotentialMap& potentialMap() const {

      return *_potential_result;

    }

    /// \brief Return the flow on the given edge.

    ///

    /// Return the flow on the given edge.

    ///

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

    Capacity flow(const typename Graph::Edge& edge) const {

      return (*_flow_result)[edge];

    }

    /// \brief Return the potential of the given node.

    ///

    /// Return the potential of the given node.

    ///

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

    Cost potential(const typename Graph::Node& node) const {

      return (*_potential_result)[node];

    }

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

    ///

    /// Return 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;

    }

    /// @}

  private:

    // Extend the underlying graph and initialize all the node and edge maps

    bool init() {

      if (!_valid_supply) return false;

      // Initialize result maps

      if (!_flow_result) {

        _flow_result = new FlowMap(_graph_ref);

        _local_flow = true;

      }

      if (!_potential_result) {

        _potential_result = new PotentialMap(_graph_ref);

        _local_potential = true;

      }

      // Initialize the edge vector and the edge maps

      _edges.reserve(countEdges(_graph));

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

        _edges.push_back(e);

        _flow[e] = 0;

        _state[e] = STATE_LOWER;

      }

      // Add 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;

      // Add artificial edges to the graph and initialize the node maps

      // of the spanning tree data structure

      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;

        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] = STATE_TREE;

        _pred_edge[u] = e;

      }

      _thread[last] = _root;

      return true;

    }

    // Find the join node

    void findJoinNode() {

      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];

      }

      join = u;

    }

    // Find the leaving edge of the cycle and returns true if the 

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

    bool findLeavingEdge() {

      // Initialize first and second nodes according to the direction

      // of the cycle

      if (_state[_in_edge] == STATE_LOWER) {

        first  = _graph.source(_in_edge);

        second = _graph.target(_in_edge);

      } else {

        first  = _graph.target(_in_edge);

        second = _graph.source(_in_edge);