/* -*- mode: C++; indent-tabs-mode: nil; -*-

 *

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

 *

 * Copyright (C) 2003-2009

 * 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/math.h>

namespace lemon {

  /// \addtogroup min_cost_flow

  /// @{

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

  /// for finding a \ref min_cost_flow "minimum cost flow".

  ///

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

  /// for finding a \ref min_cost_flow "minimum cost flow".

  ///

  /// \tparam Digraph The digraph 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

  /// - Arc 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.

#ifdef DOXYGEN

  template < typename Digraph,

             typename LowerMap,

             typename CapacityMap,

             typename CostMap,

             typename SupplyMap >

#else

  template < typename Digraph,

             typename LowerMap = typename Digraph::template ArcMap<int>,

             typename CapacityMap = typename Digraph::template ArcMap<int>,

             typename CostMap = typename Digraph::template ArcMap<int>,

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

#endif

  class NetworkSimplex

  {

    TEMPLATE_DIGRAPH_TYPEDEFS(Digraph);

    typedef typename CapacityMap::Value Capacity;

    typedef typename CostMap::Value Cost;

    typedef typename SupplyMap::Value Supply;

    typedef std::vector<Arc> ArcVector;

    typedef std::vector<Node> NodeVector;

    typedef std::vector<int> IntVector;

    typedef std::vector<bool> BoolVector;

    typedef std::vector<Capacity> CapacityVector;

    typedef std::vector<Cost> CostVector;

    typedef std::vector<Supply> SupplyVector;

  public:

    /// The type of the flow map

    typedef typename Digraph::template ArcMap<Capacity> FlowMap;

    /// The type of the potential map

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

  public:

    /// Enum type for selecting 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:

    // State constants for arcs

    enum ArcStateEnum {

      STATE_UPPER = -1,

      STATE_TREE  =  0,

      STATE_LOWER =  1

    };

  private:

    // References for the original data

    const Digraph &_orig_graph;

    const LowerMap *_orig_lower;

    const CapacityMap &_orig_cap;

    const CostMap &_orig_cost;

    const SupplyMap *_orig_supply;

    Node _orig_source;

    Node _orig_target;

    Capacity _orig_flow_value;

    // Result maps

    FlowMap *_flow_result;

    PotentialMap *_potential_result;

    bool _local_flow;

    bool _local_potential;

    // Data structures for storing the graph

    ArcVector _arc;

    NodeVector _node;

    IntNodeMap _node_id;

    IntVector _source;

    IntVector _target;

    // The number of nodes and arcs in the original graph

    int _node_num;

    int _arc_num;

    // Node and arc maps

    CapacityVector _cap;

    CostVector _cost;

    CostVector _supply;

    CapacityVector _flow;

    CostVector _pi;

    // Node and arc maps for the spanning tree structure

    IntVector _depth;

    IntVector _parent;

    IntVector _pred;

    IntVector _thread;

    BoolVector _forward;

    IntVector _state;

    // The root node

    int _root;

    // The entering arc in the current pivot iteration

    int _in_arc;

    // Temporary data used in the current pivot iteration

    int join, u_in, v_in, u_out, v_out;

    int right, first, second, last;

    int stem, par_stem, new_stem;

    Capacity delta;

  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

      const ArcVector &_arc;

      const IntVector  &_source;

      const IntVector  &_target;

      const CostVector &_cost;

      const IntVector  &_state;

      const CostVector &_pi;

      int &_in_arc;

      int _arc_num;

      // Pivot rule data

      int _next_arc;

    public:

      /// Constructor

      FirstEligiblePivotRule(NetworkSimplex &ns) :

        _arc(ns._arc), _source(ns._source), _target(ns._target),

        _cost(ns._cost), _state(ns._state), _pi(ns._pi),

        _in_arc(ns._in_arc), _arc_num(ns._arc_num), _next_arc(0)

      {}

      /// Find next entering arc

      bool findEnteringArc() {

        Cost c;

        for (int e = _next_arc; e < _arc_num; ++e) {

          c = _state[e] * (_cost[e] + _pi[_source[e]] - _pi[_target[e]]);

          if (c < 0) {

            _in_arc = e;

            _next_arc = e + 1;

            return true;

          }

        }

        for (int e = 0; e < _next_arc; ++e) {

          c = _state[e] * (_cost[e] + _pi[_source[e]] - _pi[_target[e]]);

          if (c < 0) {

            _in_arc = e;

            _next_arc = e + 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

      const ArcVector &_arc;

      const IntVector  &_source;

      const IntVector  &_target;

      const CostVector &_cost;

      const IntVector  &_state;

      const CostVector &_pi;

      int &_in_arc;

      int _arc_num;

    public:

      /// Constructor

      BestEligiblePivotRule(NetworkSimplex &ns) :

        _arc(ns._arc), _source(ns._source), _target(ns._target),

        _cost(ns._cost), _state(ns._state), _pi(ns._pi),

        _in_arc(ns._in_arc), _arc_num(ns._arc_num)

      {}

      /// Find next entering arc

      bool findEnteringArc() {

        Cost c, min = 0;

        for (int e = 0; e < _arc_num; ++e) {

          c = _state[e] * (_cost[e] + _pi[_source[e]] - _pi[_target[e]]);

          if (c < min) {

            min = c;

            _in_arc = 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

      const ArcVector &_arc;

      const IntVector  &_source;

      const IntVector  &_target;

      const CostVector &_cost;

      const IntVector  &_state;

      const CostVector &_pi;

      int &_in_arc;

      int _arc_num;

      // Pivot rule data

      int _block_size;

      int _next_arc;

    public:

      /// Constructor

      BlockSearchPivotRule(NetworkSimplex &ns) :

        _arc(ns._arc), _source(ns._source), _target(ns._target),

        _cost(ns._cost), _state(ns._state), _pi(ns._pi),

        _in_arc(ns._in_arc), _arc_num(ns._arc_num), _next_arc(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(_arc_num)),

                                MIN_BLOCK_SIZE );

      }

      /// Find next entering arc

      bool findEnteringArc() {

        Cost c, min = 0;

        int cnt = _block_size;

        int e, min_arc = _next_arc;

        for (e = _next_arc; e < _arc_num; ++e) {

          c = _state[e] * (_cost[e] + _pi[_source[e]] - _pi[_target[e]]);

          if (c < min) {

            min = c;

            min_arc = e;

          }

          if (--cnt == 0) {

            if (min < 0) break;

            cnt = _block_size;

          }

        }

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

          for (e = 0; e < _next_arc; ++e) {

            c = _state[e] * (_cost[e] + _pi[_source[e]] - _pi[_target[e]]);

            if (c < min) {

              min = c;

              min_arc = e;

            }

            if (--cnt == 0) {

              if (min < 0) break;

              cnt = _block_size;

            }

          }

        }

        if (min >= 0) return false;

        _in_arc = min_arc;

        _next_arc = e;

        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

      const ArcVector &_arc;

      const IntVector  &_source;

      const IntVector  &_target;

      const CostVector &_cost;

      const IntVector  &_state;

      const CostVector &_pi;

      int &_in_arc;

      int _arc_num;

      // Pivot rule data

      IntVector _candidates;

      int _list_length, _minor_limit;

      int _curr_length, _minor_count;

      int _next_arc;

    public:

      /// Constructor

      CandidateListPivotRule(NetworkSimplex &ns) :

        _arc(ns._arc), _source(ns._source), _target(ns._target),

        _cost(ns._cost), _state(ns._state), _pi(ns._pi),

        _in_arc(ns._in_arc), _arc_num(ns._arc_num), _next_arc(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(_arc_num)),

                                 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 arc

      bool findEnteringArc() {

        Cost min, c;

        int e, min_arc = _next_arc;

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

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

          // current candidate list

          ++_minor_count;

          min = 0;

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

            e = _candidates[i];

            c = _state[e] * (_cost[e] + _pi[_source[e]] - _pi[_target[e]]);

            if (c < min) {

              min = c;

              min_arc = e;

            }

            if (c >= 0) {

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

            }

          }

          if (min < 0) {

            _in_arc = min_arc;

            return true;

          }

        }

        // Major iteration: build a new candidate list

        min = 0;

        _curr_length = 0;

        for (e = _next_arc; e < _arc_num; ++e) {

          c = _state[e] * (_cost[e] + _pi[_source[e]] - _pi[_target[e]]);

          if (c < 0) {

            _candidates[_curr_length++] = e;

            if (c < min) {

              min = c;

              min_arc = e;

            }

            if (_curr_length == _list_length) break;

          }

        }

        if (_curr_length < _list_length) {

          for (e = 0; e < _next_arc; ++e) {

            c = _state[e] * (_cost[e] + _pi[_source[e]] - _pi[_target[e]]);

            if (c < 0) {

              _candidates[_curr_length++] = e;

              if (c < min) {

                min = c;

                min_arc = e;

              }

              if (_curr_length == _list_length) break;

            }

          }

        }

        if (_curr_length == 0) return false;

        _minor_count = 1;

        _in_arc = min_arc;

        _next_arc = e;

        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

      const ArcVector &_arc;

      const IntVector  &_source;

      const IntVector  &_target;

      const CostVector &_cost;

      const IntVector  &_state;

      const CostVector &_pi;

      int &_in_arc;

      int _arc_num;

      // Pivot rule data

      int _block_size, _head_length, _curr_length;

      int _next_arc;

      IntVector _candidates;

      CostVector _cand_cost;

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

      class SortFunc

      {

      private:

        const CostVector &_map;

      public:

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

        bool operator()(int left, int right) {

          return _map[left] > _map[right];

        }

      };

      SortFunc _sort_func;

    public:

      /// Constructor

      AlteringListPivotRule(NetworkSimplex &ns) :

        _arc(ns._arc), _source(ns._source), _target(ns._target),

        _cost(ns._cost), _state(ns._state), _pi(ns._pi),

        _in_arc(ns._in_arc), _arc_num(ns._arc_num),

        _next_arc(0), _cand_cost(ns._arc_num), _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(_arc_num)),

                                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 arc

      bool findEnteringArc() {

        // Check the current candidate list

        int e;

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

          e = _candidates[i];

          _cand_cost[e] = _state[e] *

            (_cost[e] + _pi[_source[e]] - _pi[_target[e]]);

          if (_cand_cost[e] >= 0) {

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

          }

        }

        // Extend the list

        int cnt = _block_size;

        int last_edge = 0;

        int limit = _head_length;

        for (int e = _next_arc; e < _arc_num; ++e) {

          _cand_cost[e] = _state[e] *

            (_cost[e] + _pi[_source[e]] - _pi[_target[e]]);

          if (_cand_cost[e] < 0) {

            _candidates[_curr_length++] = e;

            last_edge = e;

          }

          if (--cnt == 0) {

            if (_curr_length > limit) break;

            limit = 0;

            cnt = _block_size;

          }

        }

        if (_curr_length <= limit) {

          for (int e = 0; e < _next_arc; ++e) {

            _cand_cost[e] = _state[e] *

              (_cost[e] + _pi[_source[e]] - _pi[_target[e]]);

            if (_cand_cost[e] < 0) {

              _candidates[_curr_length++] = e;

              last_edge = e;

            }

            if (--cnt == 0) {

              if (_curr_length > limit) break;

              limit = 0;

              cnt = _block_size;

            }

          }

        }

        if (_curr_length == 0) return false;

        _next_arc = 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

        _in_arc = _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

  public:

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

    ///

    /// General constructor (with lower bounds).

    ///

    /// \param digraph The digraph the algorithm runs on.

    /// \param lower The lower bounds of the arcs.

    /// \param capacity The capacities (upper bounds) of the arcs.

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

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

    NetworkSimplex( const Digraph &digraph,

                    const LowerMap &lower,

                    const CapacityMap &capacity,

                    const CostMap &cost,

                    const SupplyMap &supply ) :

      _orig_graph(digraph), _orig_lower(&lower), _orig_cap(capacity),

      _orig_cost(cost), _orig_supply(&supply),

      _flow_result(NULL), _potential_result(NULL),

      _local_flow(false), _local_potential(false),

      _node_id(digraph)

    {}

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

    ///

    /// General constructor (without lower bounds).

    ///

    /// \param digraph The digraph the algorithm runs on.

    /// \param capacity The capacities (upper bounds) of the arcs.

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

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

    NetworkSimplex( const Digraph &digraph,

                    const CapacityMap &capacity,

                    const CostMap &cost,

                    const SupplyMap &supply ) :

      _orig_graph(digraph), _orig_lower(NULL), _orig_cap(capacity),

      _orig_cost(cost), _orig_supply(&supply),

      _flow_result(NULL), _potential_result(NULL),

      _local_flow(false), _local_potential(false),

      _node_id(digraph)

    {}

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

    ///

    /// Simple constructor (with lower bounds).

    ///

    /// \param digraph The digraph the algorithm runs on.

    /// \param lower The lower bounds of the arcs.

    /// \param capacity The capacities (upper bounds) of the arcs.

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

    /// \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 Digraph &digraph,

                    const LowerMap &lower,

                    const CapacityMap &capacity,

                    const CostMap &cost,

                    Node s, Node t,

                    Capacity flow_value ) :

      _orig_graph(digraph), _orig_lower(&lower), _orig_cap(capacity),

      _orig_cost(cost), _orig_supply(NULL),

      _orig_source(s), _orig_target(t), _orig_flow_value(flow_value),

      _flow_result(NULL), _potential_result(NULL),

      _local_flow(false), _local_potential(false),

      _node_id(digraph)

    {}

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

    ///

    /// Simple constructor (without lower bounds).

    ///

    /// \param digraph The digraph the algorithm runs on.

    /// \param capacity The capacities (upper bounds) of the arcs.

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

    /// \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 Digraph &digraph,

                    const CapacityMap &capacity,

                    const CostMap &cost,

                    Node s, Node t,

                    Capacity flow_value ) :

      _orig_graph(digraph), _orig_lower(NULL), _orig_cap(capacity),

      _orig_cost(cost), _orig_supply(NULL),

      _orig_source(s), _orig_target(t), _orig_flow_value(flow_value),

      _flow_result(NULL), _potential_result(NULL),

      _local_flow(false), _local_potential(false),

      _node_id(digraph)

    {}

    /// Destructor.

    ~NetworkSimplex() {

      if (_local_flow) delete _flow_result;

      if (_local_potential) delete _potential_result;

    }

    /// \brief Set the flow map.

    ///

    /// This function sets the flow map.

    ///

    /// \return <tt>(*this)</tt>

    NetworkSimplex& flowMap(FlowMap &map) {

      if (_local_flow) {

        delete _flow_result;

        _local_flow = false;

      }

      _flow_result = ↦

      return *this;

    }

    /// \brief Set the potential map.

    ///

    /// This function sets the potential map.

    ///

    /// \return <tt>(*this)</tt>

    NetworkSimplex& potentialMap(PotentialMap &map) {

      if (_local_potential) {

        delete _potential_result;

        _local_potential = false;

      }

      _potential_result = ↦

      return *this;

    }

    /// \name Execution control

    /// The algorithm can be executed using the

    /// \ref lemon::NetworkSimplex::run() "run()" function.

    /// @{

    /// \brief Run the algorithm.

    ///

    /// This function 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 arc is selected in

    /// a wraparound fashion in every iteration

    /// (\ref FirstEligiblePivotRule).

    ///

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

    /// every iteration (\ref BestEligiblePivotRule).

    ///

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

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

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

    ///

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

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

    /// following minor iterations the best eligible arc 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 arcs 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 flow map.

    ///

    /// This function returns a const reference to an arc 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 potential map

    /// (the dual solution).

    ///

    /// This function returns a const reference to a 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 arc.

    ///

    /// This function returns the flow on the given arc.

    ///

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

    Capacity flow(const Arc& arc) const {

      return (*_flow_result)[arc];

    }

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

    ///

    /// This function returns the potential of the given node.

    ///

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

    Cost potential(const Node& node) const {

      return (*_potential_result)[node];

    }

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

    ///

    /// This function 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 (ArcIt e(_orig_graph); e != INVALID; ++e)

        c += (*_flow_result)[e] * _orig_cost[e];

      return c;

    }

    /// @}

  private:

    // Initialize internal data structures

    bool init() {

      // Initialize result maps

      if (!_flow_result) {

        _flow_result = new FlowMap(_orig_graph);

        _local_flow = true;

      }

      if (!_potential_result) {

        _potential_result = new PotentialMap(_orig_graph);

        _local_potential = true;

      }

      // Initialize vectors

      _node_num = countNodes(_orig_graph);

      _arc_num = countArcs(_orig_graph);

      int all_node_num = _node_num + 1;

      int all_edge_num = _arc_num + _node_num;

      _arc.resize(_arc_num);

      _node.reserve(_node_num);

      _source.resize(all_edge_num);

      _target.resize(all_edge_num);

      _cap.resize(all_edge_num);

      _cost.resize(all_edge_num);

      _supply.resize(all_node_num);

      _flow.resize(all_edge_num, 0);

      _pi.resize(all_node_num, 0);

      _depth.resize(all_node_num);

      _parent.resize(all_node_num);

      _pred.resize(all_node_num);

      _thread.resize(all_node_num);

      _forward.resize(all_node_num);

      _state.resize(all_edge_num, STATE_LOWER);

      // Initialize node related data

      bool valid_supply = true;

      if (_orig_supply) {

        Supply sum = 0;

        int i = 0;

        for (NodeIt n(_orig_graph); n != INVALID; ++n, ++i) {

          _node.push_back(n);

          _node_id[n] = i;

          _supply[i] = (*_orig_supply)[n];

          sum += _supply[i];

        }

        valid_supply = (sum == 0);

      } else {

        int i = 0;

        for (NodeIt n(_orig_graph); n != INVALID; ++n, ++i) {

          _node.push_back(n);

          _node_id[n] = i;

          _supply[i] = 0;

        }

        _supply[_node_id[_orig_source]] =  _orig_flow_value;

        _supply[_node_id[_orig_target]] = -_orig_flow_value;

      }

      if (!valid_supply) return false;

      // Set data for the artificial root node

      _root = _node_num;

      _depth[_root] = 0;

      _parent[_root] = -1;

      _pred[_root] = -1;

      _thread[_root] = 0;

      _supply[_root] = 0;

      _pi[_root] = 0;

      // Store the arcs in a mixed order

      int k = std::max(int(sqrt(_arc_num)), 10);

      int i = 0;

      for (ArcIt e(_orig_graph); e != INVALID; ++e) {

        _arc[i] = e;

        if ((i += k) >= _arc_num) i = (i % k) + 1;

      }

      // Initialize arc maps

      for (int i = 0; i != _arc_num; ++i) {

        Arc e = _arc[i];

        _source[i] = _node_id[_orig_graph.source(e)];

        _target[i] = _node_id[_orig_graph.target(e)];

        _cost[i] = _orig_cost[e];

        _cap[i] = _orig_cap[e];

      }

      // Remove non-zero lower bounds

      if (_orig_lower) {

        for (int i = 0; i != _arc_num; ++i) {

          Capacity c = (*_orig_lower)[_arc[i]];

          if (c != 0) {

            _cap[i] -= c;

            _supply[_source[i]] -= c;

            _supply[_target[i]] += c;

          }

        }

      }

      // Add artificial arcs and initialize the spanning tree data structure

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

      Capacity max_cap = std::numeric_limits<Capacity>::max();

      for (int u = 0, e = _arc_num; u != _node_num; ++u, ++e) {

        _thread[u] = u + 1;

        _depth[u] = 1;

        _parent[u] = _root;

        _pred[u] = e;

        if (_supply[u] >= 0) {

          _flow[e] = _supply[u];

          _forward[u] = true;

          _pi[u] = -max_cost;

        } else {

          _flow[e] = -_supply[u];

          _forward[u] = false;

          _pi[u] = max_cost;

        }

        _cost[e] = max_cost;

        _cap[e] = max_cap;

        _state[e] = STATE_TREE;

      }

      return true;

    }

    // Find the join node

    void findJoinNode() {

      int u = _source[_in_arc];

      int v = _target[_in_arc];

      while (_depth[u] > _depth[v]) u = _parent[u];

      while (_depth[v] > _depth[u]) v = _parent[v];

      while (u != v) {

        u = _parent[u];

        v = _parent[v];

      }

      join = u;

    }

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

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

    bool findLeavingArc() {

      // Initialize first and second nodes according to the direction

      // of the cycle

      if (_state[_in_arc] == STATE_LOWER) {

        first  = _source[_in_arc];

        second = _target[_in_arc];

      } else {

        first  = _target[_in_arc];

        second = _source[_in_arc];

      }

      delta = _cap[_in_arc];

      int result = 0;

      Capacity d;

      int e;

      // Search the cycle along the path form the first node to the root

      for (int u = first; u != join; u = _parent[u]) {

        e = _pred[u];

        d = _forward[u] ? _flow[e] : _cap[e] - _flow[e];

        if (d < delta) {

          delta = d;

          u_out = u;