/* -*- 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/core.h>

 #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 &_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_map;

     PotentialMap *_potential_map;

     bool _local_flow;

     bool _local_potential;

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

     int _node_num;

     int _arc_num;

     // Data structures for storing the graph

     IntNodeMap _node_id;

     ArcVector _arc_ref;

     IntVector _source;

     IntVector _target;

     // Node and arc maps

     CapacityVector _cap;

     CostVector _cost;

     CostVector _supply;

     CapacityVector _flow;

     CostVector _pi;

     // Data for storing the spanning tree structure

     IntVector _depth;

     IntVector _parent;

     IntVector _pred;

     IntVector _thread;

     BoolVector _forward;

     IntVector _state;

     int _root;

     // Temporary data used in the current pivot iteration

     int in_arc, join, u_in, v_in, u_out, v_out;

     int first, second, right, 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 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) :

         _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 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) :

         _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 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) :

         _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 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) :

         _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 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) :

         _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_arc = 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_arc = 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_arc = e;

             }

             if (--cnt == 0) {

               if (_curr_length > limit) break;

               limit = 0;

               cnt = _block_size;

             }

           }

         }

         if (_curr_length == 0) return false;

         _next_arc = last_arc + 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 graph 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 &graph,

                     const LowerMap &lower,

                     const CapacityMap &capacity,

                     const CostMap &cost,

                     const SupplyMap &supply ) :

       _graph(graph), _orig_lower(&lower), _orig_cap(capacity),

       _orig_cost(cost), _orig_supply(&supply),

       _flow_map(NULL), _potential_map(NULL),

       _local_flow(false), _local_potential(false),

       _node_id(graph)

     {}

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

     ///

     /// General constructor (without lower bounds).

     ///

     /// \param graph 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 &graph,

                     const CapacityMap &capacity,

                     const CostMap &cost,

                     const SupplyMap &supply ) :

       _graph(graph), _orig_lower(NULL), _orig_cap(capacity),

       _orig_cost(cost), _orig_supply(&supply),

       _flow_map(NULL), _potential_map(NULL),

       _local_flow(false), _local_potential(false),

       _node_id(graph)

     {}

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

     ///

     /// Simple constructor (with lower bounds).

     ///

     /// \param graph 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 &graph,

                     const LowerMap &lower,

                     const CapacityMap &capacity,

                     const CostMap &cost,

                     Node s, Node t,

                     Capacity flow_value ) :

       _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_map(NULL), _potential_map(NULL),

       _local_flow(false), _local_potential(false),

       _node_id(graph)

     {}

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

     ///

     /// Simple constructor (without lower bounds).

     ///

     /// \param graph 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 &graph,

                     const CapacityMap &capacity,

                     const CostMap &cost,

                     Node s, Node t,

                     Capacity flow_value ) :

       _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_map(NULL), _potential_map(NULL),

       _local_flow(false), _local_potential(false),

       _node_id(graph)

     {}

     /// Destructor.

     ~NetworkSimplex() {

       if (_local_flow) delete _flow_map;

       if (_local_potential) delete _potential_map;

     }

     /// \brief Set the flow map.

     ///

     /// This function sets the flow map.

     ///

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

     NetworkSimplex& flowMap(FlowMap &map) {

       if (_local_flow) {

         delete _flow_map;

         _local_flow = false;

       }

       _flow_map = ↦

       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_map;

         _local_potential = false;

       }

       _potential_map = ↦

       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_map;

     }

     /// \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_map;

     }

     /// \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_map)[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_map)[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(_graph); e != INVALID; ++e)

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

       return c;

     }

     /// @}

   private:

     // Initialize internal data structures

     bool init() {

       // Initialize result maps

       if (!_flow_map) {

         _flow_map = new FlowMap(_graph);

         _local_flow = true;

       }

       if (!_potential_map) {

         _potential_map = new PotentialMap(_graph);

         _local_potential = true;

       }

       // Initialize vectors

       _node_num = countNodes(_graph);

       _arc_num = countArcs(_graph);

       int all_node_num = _node_num + 1;

       int all_arc_num = _arc_num + _node_num;

       _arc_ref.resize(_arc_num);

       _source.resize(all_arc_num);

       _target.resize(all_arc_num);

       _cap.resize(all_arc_num);

       _cost.resize(all_arc_num);

       _supply.resize(all_node_num);

       _flow.resize(all_arc_num, 0);

       _pi.resize(all_node_num, 0);

       _depth.resize(all_node_num);

       _parent.resize(all_node_num);

       _pred.resize(all_node_num);

       _forward.resize(all_node_num);

       _thread.resize(all_node_num);

       _state.resize(all_arc_num, STATE_LOWER);

       // Initialize node related data

       bool valid_supply = true;

       if (_orig_supply) {

         Supply sum = 0;

         int i = 0;

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

           _node_id[n] = i;

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

           sum += _supply[i];

         }

         valid_supply = (sum == 0);

       } else {

         int i = 0;

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

           _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(_graph); e != INVALID; ++e) {

         _arc_ref[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_ref[i];

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

         _target[i] = _node_id[_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_ref[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;

           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_arc] * delta;

         _flow[in_arc] += val;

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

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

         }

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

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

         }

       }

       // Update the state of the entering and leaving arcs

       if (change) {

         _state[in_arc] = STATE_TREE;

         _state[_pred[u_out]] =

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

       } else {

         _state[in_arc] = -_state[in_arc];

       }

     }

     // Update _thread and _parent vectors

     void updateThreadParent() {

       int u;

       v_out = _parent[u_out];

       // Handle the case when join and v_out coincide

       bool par_first = false;

       if (join == v_out) {

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

         if (u == v_in) {

           par_first = true;

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

           first = u;

         }

       }

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

       // according to the thread index

       for (u = u_in; _depth[_thread[u]] > _depth[u_in]; u = _thread[u]) ;

       right = _thread[u];

       if (_thread[v_in] == u_out) {

         for (last = u; _depth[last] > _depth[u_out]; last = _thread[last]) ;

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

       }

       else last = _thread[v_in];

       // Update stem nodes

       _thread[v_in] = stem = u_in;

       par_stem = v_in;

       while (stem != u_out) {

         _thread[u] = new_stem = _parent[stem];

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

         // the original thread index

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

         _thread[u] = right;

         // Change the parent node of stem and shift stem and par_stem nodes

         _parent[stem] = par_stem;

         par_stem = stem;

         stem = new_stem;

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

         // according to the thread index

         for (u = stem; _depth[_thread[u]] > _depth[stem]; u = _thread[u]) ;

         right = _thread[u];

       }

       _parent[u_out] = par_stem;

       _thread[u] = last;

       if (join == v_out && par_first) {

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

       } else {

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

         _thread[u] = right;

       }

     }

     // Update _pred and _forward vectors

     void updatePredArc() {

       int u = u_out, v;

       while (u != u_in) {

         v = _parent[u];

         _pred[u] = _pred[v];

         _forward[u] = !_forward[v];

         u = v;

       }

       _pred[u_in] = in_arc;

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

     }

     // Update _depth and _potential vectors

     void updateDepthPotential() {

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

       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]];

       _pi[u_in] += sigma;

       for(int u = _thread[u_in]; _parent[u] != -1; u = _thread[u]) {

         _depth[u] = _depth[_parent[u]] + 1;

         if (_depth[u] <= _depth[u_in]) break;

         _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.findEnteringArc()) {

         findJoinNode();

         bool change = findLeavingArc();

         changeFlow(change);

         if (change) {

           updateThreadParent();

           updatePredArc();

           updateDepthPotential();

         }

       }

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

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

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

       }

       // Copy flow values to _flow_map

       if (_orig_lower) {

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

           Arc e = _arc_ref[i];

           _flow_map->set(e, (*_orig_lower)[e] + _flow[i]);

         }

       } else {

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

           _flow_map->set(_arc_ref[i], _flow[i]);

         }

       }

       // Copy potential values to _potential_map

       for (NodeIt n(_graph); n != INVALID; ++n) {

         _potential_map->set(n, _pi[_node_id[n]]);

       }

       return true;

     }

   }; //class NetworkSimplex

   ///@}

 } //namespace lemon

 #endif //LEMON_NETWORK_SIMPLEX_H