/* -*- 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_utils.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

   {

     GRAPH_TYPEDEFS(typename Graph);

     typedef typename CapacityMap::Value Capacity;

     typedef typename CostMap::Value Cost;

     typedef typename SupplyMap::Value Supply;

     typedef std::vector<Edge> EdgeVector;

     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;

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

   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 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 EdgeVector &_edge;

       const IntVector  &_source;

       const IntVector  &_target;

       const CostVector &_cost;

       const IntVector  &_state;

       const CostVector &_pi;

       int &_in_edge;

       int _edge_num;

       // Pivot rule data

       int _next_edge;

     public:

       /// Constructor

       FirstEligiblePivotRule(NetworkSimplex &ns) :

         _edge(ns._edge), _source(ns._source), _target(ns._target),

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

         _in_edge(ns._in_edge), _edge_num(ns._edge_num), _next_edge(0)

       {}

       /// Find next entering edge

       bool findEnteringEdge() {

         Cost c;

         for (int e = _next_edge; e < _edge_num; ++e) {

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

           if (c < 0) {

             _in_edge = e;

             _next_edge = e + 1;

             return true;

           }

         }

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

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

           if (c < 0) {

             _in_edge = e;

             _next_edge = 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 EdgeVector &_edge;

       const IntVector  &_source;

       const IntVector  &_target;

       const CostVector &_cost;

       const IntVector  &_state;

       const CostVector &_pi;

       int &_in_edge;

       int _edge_num;

     public:

       /// Constructor

       BestEligiblePivotRule(NetworkSimplex &ns) :

         _edge(ns._edge), _source(ns._source), _target(ns._target),

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

         _in_edge(ns._in_edge), _edge_num(ns._edge_num)

       {}

       /// Find next entering edge

       bool findEnteringEdge() {

         Cost c, min = 0;

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

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

           if (c < min) {

             min = c;

             _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

       const EdgeVector &_edge;

       const IntVector  &_source;

       const IntVector  &_target;

       const CostVector &_cost;

       const IntVector  &_state;

       const CostVector &_pi;

       int &_in_edge;

       int _edge_num;

       // Pivot rule data

       int _block_size;

       int _next_edge;

     public:

       /// Constructor

       BlockSearchPivotRule(NetworkSimplex &ns) :

         _edge(ns._edge), _source(ns._source), _target(ns._target),

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

         _in_edge(ns._in_edge), _edge_num(ns._edge_num), _next_edge(0)

       {

         // The main parameters of the pivot rule

         const double BLOCK_SIZE_FACTOR = 0.5;

         const int MIN_BLOCK_SIZE = 10;

         _block_size = std::max( int(BLOCK_SIZE_FACTOR * sqrt(_edge_num)),

                                 MIN_BLOCK_SIZE );

       }

       /// Find next entering edge

       bool findEnteringEdge() {

         Cost c, min = 0;

         int cnt = _block_size;

         int e;

         for (e = _next_edge; e < _edge_num; ++e) {

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

           if (c < min) {

             min = c;

             _in_edge = e;

           }

           if (--cnt == 0) {

             if (min < 0) goto search_end;

             cnt = _block_size;

           }

         }

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

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

           if (c < min) {

             min = c;

             _in_edge = e;

           }

           if (--cnt == 0) {

             if (min < 0) goto search_end;

             cnt = _block_size;

           }

         }

         if (min >= 0) return false;

       search_end:

         _next_edge = 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 EdgeVector &_edge;

       const IntVector  &_source;

       const IntVector  &_target;

       const CostVector &_cost;

       const IntVector  &_state;

       const CostVector &_pi;

       int &_in_edge;

       int _edge_num;

       // Pivot rule data

       IntVector _candidates;

       int _list_length, _minor_limit;

       int _curr_length, _minor_count;

       int _next_edge;

     public:

       /// Constructor

       CandidateListPivotRule(NetworkSimplex &ns) :

         _edge(ns._edge), _source(ns._source), _target(ns._target),

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

         _in_edge(ns._in_edge), _edge_num(ns._edge_num), _next_edge(0)

       {

         // The main parameters of the pivot rule

         const double LIST_LENGTH_FACTOR = 0.25;

         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(_edge_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 edge

       bool findEnteringEdge() {

         Cost min, c;

         int e;

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

           // Minor iteration: select the best eligible edge 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;

               _in_edge = e;

             }

             else if (c >= 0) {

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

             }

           }

           if (min < 0) return true;

         }

         // Major iteration: build a new candidate list

         min = 0;

         _curr_length = 0;

         for (e = _next_edge; e < _edge_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;

               _in_edge = e;

             }

             if (_curr_length == _list_length) goto search_end;

           }

         }

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

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

           if (c < 0) {

             _candidates[_curr_length++] = e;

             if (c < min) {

               min = c;

               _in_edge = e;

             }

             if (_curr_length == _list_length) goto search_end;

           }

         }

         if (_curr_length == 0) return false;

       search_end:        

         _minor_count = 1;

         _next_edge = 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 EdgeVector &_edge;

       const IntVector  &_source;

       const IntVector  &_target;

       const CostVector &_cost;

       const IntVector  &_state;

       const CostVector &_pi;

       int &_in_edge;

       int _edge_num;

       int _block_size, _head_length, _curr_length;

       int _next_edge;

       IntVector _candidates;

       CostVector _cand_cost;

       // Functor class to compare edges 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) :

         _edge(ns._edge), _source(ns._source), _target(ns._target),

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

         _in_edge(ns._in_edge), _edge_num(ns._edge_num),

          _next_edge(0), _cand_cost(ns._edge_num),_sort_func(_cand_cost)

       {

         // The main parameters of the pivot rule

         const double BLOCK_SIZE_FACTOR = 1.0;

         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(_edge_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 edge

       bool findEnteringEdge() {

         // 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 limit = _head_length;

         for (e = _next_edge; e < _edge_num; ++e) {

           _cand_cost[e] = _state[e] *

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

           if (_cand_cost[e] < 0) {

             _candidates[_curr_length++] = e;

           }

           if (--cnt == 0) {

             if (_curr_length > limit) goto search_end;

             limit = 0;

             cnt = _block_size;

           }

         }

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

           _cand_cost[e] = _state[e] *

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

           if (_cand_cost[e] < 0) {

             _candidates[_curr_length++] = e;

           }

           if (--cnt == 0) {

             if (_curr_length > limit) goto search_end;

             limit = 0;

             cnt = _block_size;

           }

         }

         if (_curr_length == 0) return false;

       search_end:

         // 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_edge = _candidates[0];

         _next_edge = e;

         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 original graph

     const Graph &_orig_graph;

     // The original lower bound map

     const LowerMap *_orig_lower;

     // The original capacity map

     const CapacityMap &_orig_cap;

     // The original cost map

     const CostMap &_orig_cost;

     // The original supply map

     const SupplyMap *_orig_supply;

     // The source node (if no supply map was given)

     Node _orig_source;

     // The target node (if no supply map was given)

     Node _orig_target;

     // The flow value (if no supply map was given)

     Capacity _orig_flow_value;

     // The flow result map

     FlowMap *_flow_result;

     // The potential result map

     PotentialMap *_potential_result;

     // Indicate if the flow result map is local

     bool _local_flow;

     // Indicate if the potential result map is local

     bool _local_potential;

     // The edge references

     EdgeVector _edge;

     // The node references

     NodeVector _node;

     // The node ids

     IntNodeMap _node_id;

     // The source nodes of the edges

     IntVector _source;

     // The target nodess of the edges

     IntVector _target;

     // The (modified) capacity vector

     CapacityVector _cap;

     // The cost vector

     CostVector _cost;

     // The (modified) supply vector

     CostVector _supply;

     // The current flow vector

     CapacityVector _flow;

     // The current potential vector

     CostVector _pi;

     // The number of nodes in the original graph

     int _node_num;

     // The number of edges in the original graph

     int _edge_num;

     // The parent vector of the spanning tree structure

     IntVector _parent;

     // The pred_edge vector of the spanning tree structure

     IntVector _pred;

     // The thread vector of the spanning tree structure

     IntVector _thread;

     IntVector _rev_thread;

     IntVector _succ_num;

     IntVector _last_succ;

     IntVector _dirty_revs;

     // The forward vector of the spanning tree structure

     BoolVector _forward;

     // The state vector

     IntVector _state;

     // The root node

     int _root;

     // The entering edge of the current pivot iteration

     int _in_edge;

     // Temporary nodes 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;

     // 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 ) :

       _orig_graph(graph), _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(graph)

     {}

     /// \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 ) :

       _orig_graph(graph), _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(graph)

     {}

     /// \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,

                     Node s, Node t,

                     Capacity flow_value ) :

       _orig_graph(graph), _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(graph)

     {}

     /// \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,

                     Node s, Node t,

                     Capacity flow_value ) :

       _orig_graph(graph), _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(graph)

     {}

     /// 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 (EdgeIt 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);

       _edge_num = countEdges(_orig_graph);

       int all_node_num = _node_num + 1;

       int all_edge_num = _edge_num + _node_num;

       _edge.resize(_edge_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);

       _parent.resize(all_node_num);

       _pred.resize(all_node_num);

       _forward.resize(all_node_num);

       _thread.resize(all_node_num);

       _rev_thread.resize(all_node_num);

       _succ_num.resize(all_node_num);

       _last_succ.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;

       _parent[_root] = -1;

       _pred[_root] = -1;

       _thread[_root] = 0;

       _rev_thread[0] = _root;

       _succ_num[_root] = all_node_num;

       _last_succ[_root] = _root - 1;

       _supply[_root] = 0;

       _pi[_root] = 0;

       // Store the edges

       int i = 0;

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

         _edge[i++] = e;

       }

       // Initialize edge maps

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

         Edge e = _edge[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 != _edge_num; ++i) {

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

           if (c != 0) {

             _cap[i] -= c;

             _supply[_source[i]] -= c;

             _supply[_target[i]] += c;

           }

         }

       }

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

       Cost max_cost = std::numeric_limits<Cost>::max() / 2 + 1;

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

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

         _parent[u] = _root;

         _pred[u] = e;

         _thread[u] = u + 1;

         _rev_thread[u + 1] = u;

         _succ_num[u] = 1;

         _last_succ[u] = u;

         _cap[e] = max_cap;

         _state[e] = STATE_TREE;

         if (_supply[u] >= 0) {

           _forward[u] = true;

           _pi[u] = 0;

           _source[e] = u;

           _target[e] = _root;

           _flow[e] = _supply[u];

           _cost[e] = 0;

         }

         else {

           _forward[u] = false;

           _pi[u] = max_cost;

           _source[e] = _root;

           _target[e] = u;

           _flow[e] = -_supply[u];

           _cost[e] = max_cost;

         }

       }

       return true;

     }

     // Find the join node

     void findJoinNode() {

       int u = _source[_in_edge];

       int v = _target[_in_edge];

       while (u != v) {

         if (_succ_num[u] < _succ_num[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  = _source[_in_edge];

         second = _target[_in_edge];

       } else {

         first  = _target[_in_edge];

         second = _source[_in_edge];

       }

       delta = _cap[_in_edge];

       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;

           result = 1;

         }

       }

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

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

         e = _pred[u];

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

         if (d <= delta) {

           delta = d;

           u_out = u;

           result = 2;

         }

       }

       if (result == 1) {

         u_in = first;

         v_in = second;

       } else {

         u_in = second;

         v_in = first;

       }

       return result != 0;

     }

     // Change _flow and _state vectors

     void changeFlow(bool change) {

       // Augment along the cycle

       if (delta > 0) {

         Capacity val = _state[_in_edge] * delta;

         _flow[_in_edge] += val;

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

           _flow[_pred[u]] += _forward[u] ? -val : val;

         }

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

           _flow[_pred[u]] += _forward[u] ? val : -val;

         }

       }

       // Update the state of the entering and leaving edges

       if (change) {

         _state[_in_edge] = STATE_TREE;

         _state[_pred[u_out]] =

           (_flow[_pred[u_out]] == 0) ? STATE_LOWER : STATE_UPPER;

       } else {

         _state[_in_edge] = -_state[_in_edge];

       }

     }

     // Update the tree structure

     void updateTreeStructure() {

       int u, w;

       int old_rev_thread = _rev_thread[u_out];

       int old_succ_num = _succ_num[u_out];

       int old_last_succ = _last_succ[u_out];

       v_out = _parent[u_out];

       u = _last_succ[u_in];  // the last successor of u_in

       right = _thread[u];    // the node after it

       // Handle the case when old_rev_thread equals to v_in 

       // (it also means that join and v_out coincide)

       if (old_rev_thread == v_in) {

         last = _thread[_last_succ[u_out]];

       } else {

         last = _thread[v_in];

       }

       // Update _thread and _parent along the stem nodes (i.e. the nodes 

       // between u_in and u_out, whose parent have to be changed)

       _thread[v_in] = stem = u_in;

       _dirty_revs.clear();

       _dirty_revs.push_back(v_in);

       par_stem = v_in;

       while (stem != u_out) {

         // Insert the next stem node into the thread list

         new_stem = _parent[stem];

         _thread[u] = new_stem;

         _dirty_revs.push_back(u);

         // Remove the subtree of stem from the thread list

         w = _rev_thread[stem];

         _thread[w] = right;

         _rev_thread[right] = w;

         // Change the parent node and shift stem nodes

         _parent[stem] = par_stem;        

         par_stem = stem;

         stem = new_stem;

         // Update u and right nodes

         u = _last_succ[stem] == _last_succ[par_stem] ?

           _rev_thread[par_stem] : _last_succ[stem];

         right = _thread[u];

       }

       _parent[u_out] = par_stem;

       _last_succ[u_out] = u;

       _thread[u] = last;

       _rev_thread[last] = u;

       // Remove the subtree of u_out from the thread list except for 

       // the case when old_rev_thread equals to v_in 

       // (it also means that join and v_out coincide)

       if (old_rev_thread != v_in) {

         _thread[old_rev_thread] = right;

         _rev_thread[right] = old_rev_thread;

       }

       // Update _rev_thread using the new _thread values

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

         u = _dirty_revs[i];

         _rev_thread[_thread[u]] = u;

       }

       // Update _last_succ for the stem nodes from u_out to u_in

       for (u = u_out; u != u_in; u = _parent[u]) {

         _last_succ[_parent[u]] = _last_succ[u];      

       }

       // Set limits for updating _last_succ form v_in and v_out

       // towards the root

       int up_limit_in = -1;

       int up_limit_out = -1;

       if (_last_succ[join] == v_in) {

         up_limit_out = join;

       } else {

         up_limit_in = join;

       }

       // Update _last_succ from v_in towards the root

       for (u = v_in; u != up_limit_in && _last_succ[u] == v_in; 

            u = _parent[u]) {

         _last_succ[u] = _last_succ[u_out];

       }

       // Update _last_succ from v_out towards the root

       if (join != old_rev_thread && v_in != old_rev_thread) {

         for (u = v_out; u != up_limit_out && _last_succ[u] == old_last_succ; 

              u = _parent[u]) {

           _last_succ[u] = old_rev_thread;

         }

       } else {

         for (u = v_out; u != up_limit_out && _last_succ[u] == old_last_succ;

              u = _parent[u]) {

           _last_succ[u] = _last_succ[u_out];

         }

       }

       // Update _pred and _forward for the stem nodes from u_out to u_in

       u = u_out;

       while (u != u_in) {

         w = _parent[u];

         _pred[u] = _pred[w];

         _forward[u] = !_forward[w];

         u = w;

       }

       _pred[u_in] = _in_edge;

       _forward[u_in] = (u_in == _source[_in_edge]);

       // Update _succ_num from v_in to join

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

         _succ_num[u] += old_succ_num;

       }

       // Update _succ_num from v_out to join

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

         _succ_num[u] -= old_succ_num;

       }

       // Update _succ_num for the stem nodes from u_out to u_in

       int tmp = 0;

       u = u_out;

       while (u != u_in) {

         w = _parent[u];

         tmp = _succ_num[u] - _succ_num[w] + tmp;

         _succ_num[u] = tmp;

         u = w;

       }

       _succ_num[u_in] = old_succ_num;

     }

     // Update potentials

     void updatePotential() {

       Cost sigma = _forward[u_in] ?

         _pi[v_in] - _pi[u_in] - _cost[_pred[u_in]] :

         _pi[v_in] - _pi[u_in] + _cost[_pred[u_in]];

       // Update in the lower subtree (which has been moved)

       int end = _thread[_last_succ[u_in]];

       for (int u = u_in; u != end; u = _thread[u]) {

         _pi[u] += sigma;

       }

     }

     // Execute the algorithm

     bool start(PivotRuleEnum pivot_rule) {

       // Select the pivot rule implementation

       switch (pivot_rule) {

         case FIRST_ELIGIBLE_PIVOT:

           return start<FirstEligiblePivotRule>();

         case BEST_ELIGIBLE_PIVOT:

           return start<BestEligiblePivotRule>();

         case BLOCK_SEARCH_PIVOT:

           return start<BlockSearchPivotRule>();

         case CANDIDATE_LIST_PIVOT:

           return start<CandidateListPivotRule>();

         case ALTERING_LIST_PIVOT:

           return start<AlteringListPivotRule>();

       }

       return false;

     }

     template<class PivotRuleImplementation>

     bool start() {

       PivotRuleImplementation pivot(*this);

       // Execute the network simplex algorithm

       while (pivot.findEnteringEdge()) {

         findJoinNode();

         bool change = findLeavingEdge();

         changeFlow(change);

         if (change) {

           updateTreeStructure();

           updatePotential();

         }

       }

       // Check if the flow amount equals zero on all the artificial edges

       for (int e = _edge_num; e != _edge_num + _node_num; ++e) {

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

       }

       // Copy flow values to _flow_result

       if (_orig_lower) {

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

           Edge e = _edge[i];

           (*_flow_result)[e] = (*_orig_lower)[e] + _flow[i];

         }

       } else {

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

           (*_flow_result)[_edge[i]] = _flow[i];

         }

       }

       // Copy potential values to _potential_result

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

         (*_potential_result)[_node[i]] = _pi[i];

       }

       return true;

     }

   }; //class NetworkSimplex

   ///@}

 } //namespace lemon

 #endif //LEMON_NETWORK_SIMPLEX_H