/* -*- 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 GR The digraph type the algorithm runs on.

   /// \tparam V The value type used in the algorithm.

   /// By default it is \c int.

   ///

   /// \warning \c V must be a signed integer type.

   ///

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

   /// implementations. For more information see \ref PivotRule.

   template <typename GR, typename V = int>

   class NetworkSimplex

   {

   public:

     /// The value type of the algorithm

     typedef V Value;

     /// The type of the flow map

     typedef typename GR::template ArcMap<Value> FlowMap;

     /// The type of the potential map

     typedef typename GR::template NodeMap<Value> PotentialMap;

   public:

     /// \brief Enum type for selecting the pivot rule.

     ///

     /// Enum type for selecting the pivot rule for the \ref run()

     /// function.

     ///

     /// \ref NetworkSimplex provides five different pivot rule

     /// implementations that significantly affect the running time

     /// of the algorithm.

     /// By default \ref BLOCK_SEARCH "Block Search" is used, which

     /// proved to be the most efficient and the most robust on various

     /// test inputs according to our benchmark tests.

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

     /// function with the proper parameter.

     enum PivotRule {

       /// The First Eligible pivot rule.

       /// The next eligible arc is selected in a wraparound fashion

       /// in every iteration.

       FIRST_ELIGIBLE,

       /// The Best Eligible pivot rule.

       /// The best eligible arc is selected in every iteration.

       BEST_ELIGIBLE,

       /// The Block Search pivot rule.

       /// A specified number of arcs are examined in every iteration

       /// in a wraparound fashion and the best eligible arc is selected

       /// from this block.

       BLOCK_SEARCH,

       /// The Candidate List pivot rule.

       /// 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.

       CANDIDATE_LIST,

       /// The Altering Candidate List pivot rule.

       /// It is a modified version of the Candidate List method.

       /// It keeps only the several best eligible arcs from the former

       /// candidate list and extends this list in every iteration.

       ALTERING_LIST

     };

   private:

     TEMPLATE_DIGRAPH_TYPEDEFS(GR);

     typedef typename GR::template ArcMap<Value> ValueArcMap;

     typedef typename GR::template NodeMap<Value> ValueNodeMap;

     typedef std::vector<Arc> ArcVector;

     typedef std::vector<Node> NodeVector;

     typedef std::vector<int> IntVector;

     typedef std::vector<bool> BoolVector;

     typedef std::vector<Value> ValueVector;

     // State constants for arcs

     enum ArcStateEnum {

       STATE_UPPER = -1,

       STATE_TREE  =  0,

       STATE_LOWER =  1

     };

   private:

     // Data related to the underlying digraph

     const GR &_graph;

     int _node_num;

     int _arc_num;

     // Parameters of the problem

     ValueArcMap *_plower;

     ValueArcMap *_pupper;

     ValueArcMap *_pcost;

     ValueNodeMap *_psupply;

     bool _pstsup;

     Node _psource, _ptarget;

     Value _pstflow;

     // Result maps

     FlowMap *_flow_map;

     PotentialMap *_potential_map;

     bool _local_flow;

     bool _local_potential;

     // Data structures for storing the digraph

     IntNodeMap _node_id;

     ArcVector _arc_ref;

     IntVector _source;

     IntVector _target;

     // Node and arc data

     ValueVector _cap;

     ValueVector _cost;

     ValueVector _supply;

     ValueVector _flow;

     ValueVector _pi;

     // Data for storing the spanning tree structure

     IntVector _parent;

     IntVector _pred;

     IntVector _thread;

     IntVector _rev_thread;

     IntVector _succ_num;

     IntVector _last_succ;

     IntVector _dirty_revs;

     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;

     Value delta;

   private:

     // Implementation of the First Eligible pivot rule

     class FirstEligiblePivotRule

     {

     private:

       // References to the NetworkSimplex class

       const IntVector  &_source;

       const IntVector  &_target;

       const ValueVector &_cost;

       const IntVector  &_state;

       const ValueVector &_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() {

         Value 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

     // Implementation of the Best Eligible pivot rule

     class BestEligiblePivotRule

     {

     private:

       // References to the NetworkSimplex class

       const IntVector  &_source;

       const IntVector  &_target;

       const ValueVector &_cost;

       const IntVector  &_state;

       const ValueVector &_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() {

         Value 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

     // Implementation of the Block Search pivot rule

     class BlockSearchPivotRule

     {

     private:

       // References to the NetworkSimplex class

       const IntVector  &_source;

       const IntVector  &_target;

       const ValueVector &_cost;

       const IntVector  &_state;

       const ValueVector &_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() {

         Value 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

     // Implementation of the Candidate List pivot rule

     class CandidateListPivotRule

     {

     private:

       // References to the NetworkSimplex class

       const IntVector  &_source;

       const IntVector  &_target;

       const ValueVector &_cost;

       const IntVector  &_state;

       const ValueVector &_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() {

         Value 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

     // Implementation of the Altering Candidate List pivot rule

     class AlteringListPivotRule

     {

     private:

       // References to the NetworkSimplex class

       const IntVector  &_source;

       const IntVector  &_target;

       const ValueVector &_cost;

       const IntVector  &_state;

       const ValueVector &_pi;

       int &_in_arc;

       int _arc_num;

       // Pivot rule data

       int _block_size, _head_length, _curr_length;

       int _next_arc;

       IntVector _candidates;

       ValueVector _cand_cost;

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

       class SortFunc

       {

       private:

         const ValueVector &_map;

       public:

         SortFunc(const ValueVector &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 Constructor.

     ///

     /// Constructor.

     ///

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

     NetworkSimplex(const GR& graph) :

       _graph(graph),

       _plower(NULL), _pupper(NULL), _pcost(NULL),

       _psupply(NULL), _pstsup(false),

       _flow_map(NULL), _potential_map(NULL),

       _local_flow(false), _local_potential(false),

       _node_id(graph)

     {

       LEMON_ASSERT(std::numeric_limits<Value>::is_integer &&

                    std::numeric_limits<Value>::is_signed,

         "The value type of NetworkSimplex must be a signed integer");

     }

     /// Destructor.

     ~NetworkSimplex() {

       if (_local_flow) delete _flow_map;

       if (_local_potential) delete _potential_map;

     }

     /// \brief Set the lower bounds on the arcs.

     ///

     /// This function sets the lower bounds on the arcs.

     /// If neither this function nor \ref boundMaps() is used before

     /// calling \ref run(), the lower bounds will be set to zero

     /// on all arcs.

     ///

     /// \param map An arc map storing the lower bounds.

     /// Its \c Value type must be convertible to the \c Value type

     /// of the algorithm.

     ///

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

     template <typename LOWER>

     NetworkSimplex& lowerMap(const LOWER& map) {

       delete _plower;

       _plower = new ValueArcMap(_graph);

       for (ArcIt a(_graph); a != INVALID; ++a) {

         (*_plower)[a] = map[a];

       }

       return *this;

     }

     /// \brief Set the upper bounds (capacities) on the arcs.

     ///

     /// This function sets the upper bounds (capacities) on the arcs.

     /// If none of the functions \ref upperMap(), \ref capacityMap()

     /// and \ref boundMaps() is used before calling \ref run(),

     /// the upper bounds (capacities) will be set to

     /// \c std::numeric_limits<Value>::max() on all arcs.

     ///

     /// \param map An arc map storing the upper bounds.

     /// Its \c Value type must be convertible to the \c Value type

     /// of the algorithm.

     ///

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

     template<typename UPPER>

     NetworkSimplex& upperMap(const UPPER& map) {

       delete _pupper;

       _pupper = new ValueArcMap(_graph);

       for (ArcIt a(_graph); a != INVALID; ++a) {

         (*_pupper)[a] = map[a];

       }

       return *this;

     }

     /// \brief Set the upper bounds (capacities) on the arcs.

     ///

     /// This function sets the upper bounds (capacities) on the arcs.

     /// It is just an alias for \ref upperMap().

     ///

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

     template<typename CAP>

     NetworkSimplex& capacityMap(const CAP& map) {

       return upperMap(map);

     }

     /// \brief Set the lower and upper bounds on the arcs.

     ///

     /// This function sets the lower and upper bounds on the arcs.

     /// If neither this function nor \ref lowerMap() is used before

     /// calling \ref run(), the lower bounds will be set to zero

     /// on all arcs.

     /// If none of the functions \ref upperMap(), \ref capacityMap()

     /// and \ref boundMaps() is used before calling \ref run(),

     /// the upper bounds (capacities) will be set to

     /// \c std::numeric_limits<Value>::max() on all arcs.

     ///

     /// \param lower An arc map storing the lower bounds.

     /// \param upper An arc map storing the upper bounds.

     ///

     /// The \c Value type of the maps must be convertible to the

     /// \c Value type of the algorithm.

     ///

     /// \note This function is just a shortcut of calling \ref lowerMap()

     /// and \ref upperMap() separately.

     ///

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

     template <typename LOWER, typename UPPER>

     NetworkSimplex& boundMaps(const LOWER& lower, const UPPER& upper) {

       return lowerMap(lower).upperMap(upper);

     }

     /// \brief Set the costs of the arcs.

     ///

     /// This function sets the costs of the arcs.

     /// If it is not used before calling \ref run(), the costs

     /// will be set to \c 1 on all arcs.

     ///

     /// \param map An arc map storing the costs.

     /// Its \c Value type must be convertible to the \c Value type

     /// of the algorithm.

     ///

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

     template<typename COST>

     NetworkSimplex& costMap(const COST& map) {

       delete _pcost;

       _pcost = new ValueArcMap(_graph);

       for (ArcIt a(_graph); a != INVALID; ++a) {

         (*_pcost)[a] = map[a];

       }

       return *this;

     }

     /// \brief Set the supply values of the nodes.

     ///

     /// This function sets the supply values of the nodes.

     /// If neither this function nor \ref stSupply() is used before

     /// calling \ref run(), the supply of each node will be set to zero.

     /// (It makes sense only if non-zero lower bounds are given.)

     ///

     /// \param map A node map storing the supply values.

     /// Its \c Value type must be convertible to the \c Value type

     /// of the algorithm.

     ///

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

     template<typename SUP>

     NetworkSimplex& supplyMap(const SUP& map) {

       delete _psupply;

       _pstsup = false;

       _psupply = new ValueNodeMap(_graph);

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

         (*_psupply)[n] = map[n];

       }

       return *this;

     }

     /// \brief Set single source and target nodes and a supply value.

     ///

     /// This function sets a single source node and a single target node

     /// and the required flow value.

     /// If neither this function nor \ref supplyMap() is used before

     /// calling \ref run(), the supply of each node will be set to zero.

     /// (It makes sense only if non-zero lower bounds are given.)

     ///

     /// \param s The source node.

     /// \param t The target node.

     /// \param k 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).

     ///

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

     NetworkSimplex& stSupply(const Node& s, const Node& t, Value k) {

       delete _psupply;

       _psupply = NULL;

       _pstsup = true;

       _psource = s;

       _ptarget = t;

       _pstflow = k;

       return *this;

     }

     /// \brief Set the flow map.

     ///

     /// This function sets the flow map.

     /// If it is not used before calling \ref run(), an instance will

     /// be allocated automatically. The destructor deallocates this

     /// automatically allocated map, of course.

     ///

     /// \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, which is used for storing

     /// the dual solution.

     /// If it is not used before calling \ref run(), an instance will

     /// be allocated automatically. The destructor deallocates this

     /// automatically allocated map, of course.

     ///

     /// \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 \ref run().

     /// @{

     /// \brief Run the algorithm.

     ///

     /// This function runs the algorithm.

     /// The paramters can be specified using \ref lowerMap(),

     /// \ref upperMap(), \ref capacityMap(), \ref boundMaps(), 

     /// \ref costMap(), \ref supplyMap() and \ref stSupply()

     /// functions. For example,

     /// \code

     ///   NetworkSimplex<ListDigraph> ns(graph);

     ///   ns.boundMaps(lower, upper).costMap(cost)

     ///     .supplyMap(sup).run();

     /// \endcode

     ///

     /// \param pivot_rule The pivot rule that will be used during the

     /// algorithm. For more information see \ref PivotRule.

     ///

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

     bool run(PivotRule pivot_rule = BLOCK_SEARCH) {

       return init() && start(pivot_rule);

     }

     /// @}

     /// \name Query Functions

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

     /// functions.\n

     /// The \ref run() function must be called before using them.

     /// @{

     /// \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$.

     ///

     /// \note The return type of the function can be specified as a

     /// template parameter. For example,

     /// \code

     ///   ns.totalCost<double>();

     /// \endcode

     /// It is useful if the total cost cannot be stored in the \c Value

     /// type of the algorithm, which is the default return type of the

     /// function.

     ///

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

     template <typename Num>

     Num totalCost() const {

       Num c = 0;

       if (_pcost) {

         for (ArcIt e(_graph); e != INVALID; ++e)

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

       } else {

         for (ArcIt e(_graph); e != INVALID; ++e)

           c += (*_flow_map)[e];

       }

       return c;

     }

 #ifndef DOXYGEN

     Value totalCost() const {

       return totalCost<Value>();

     }

 #endif

     /// \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.

     Value flow(const Arc& a) const {

       return (*_flow_map)[a];

     }

     /// \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 the potential (dual value) of the given node.

     ///

     /// This function returns the potential (dual value) of the

     /// given node.

     ///

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

     Value potential(const Node& n) const {

       return (*_potential_map)[n];

     }

     /// \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, which form the dual solution of the

     /// \ref min_cost_flow "minimum cost flow" problem.

     ///

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

     const PotentialMap& potentialMap() const {

       return *_potential_map;

     }

     /// @}

   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;

       if (_node_num == 0) return false;

       _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);

       _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_arc_num, STATE_LOWER);

       // Initialize node related data

       bool valid_supply = true;

       if (!_pstsup && !_psupply) {

         _pstsup = true;

         _psource = _ptarget = NodeIt(_graph);

         _pstflow = 0;

       }

       if (_psupply) {

         Value sum = 0;

         int i = 0;

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

           _node_id[n] = i;

           _supply[i] = (*_psupply)[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[_psource]] =  _pstflow;

         _supply[_node_id[_ptarget]]   = -_pstflow;

       }

       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 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

       if (_pupper && _pcost) {

         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)];

           _cap[i] = (*_pupper)[e];

           _cost[i] = (*_pcost)[e];

         }

       } else {

         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)];

         }

         if (_pupper) {

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

             _cap[i] = (*_pupper)[_arc_ref[i]];

         } else {

           Value val = std::numeric_limits<Value>::max();

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

             _cap[i] = val;

         }

         if (_pcost) {

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

             _cost[i] = (*_pcost)[_arc_ref[i]];

         } else {

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

             _cost[i] = 1;

         }

       }

       // Remove non-zero lower bounds

       if (_plower) {

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

           Value c = (*_plower)[_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

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

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

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

         _thread[u] = u + 1;

         _rev_thread[u + 1] = u;

         _succ_num[u] = 1;

         _last_succ[u] = u;

         _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 (u != v) {

         if (_succ_num[u] < _succ_num[v]) {

           u = _parent[u];

         } else {

           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;

       Value 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) {

         Value 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 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

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

           _rev_thread[par_stem] : _last_succ[stem];

         right = _thread[u];

       }

       _parent[u_out] = par_stem;

       _thread[u] = last;

       _rev_thread[last] = u;

       _last_succ[u_out] = 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 _pred, _forward, _last_succ and _succ_num for the

       // stem nodes from u_out to u_in

       int tmp_sc = 0, tmp_ls = _last_succ[u_out];

       u = u_out;

       while (u != u_in) {

         w = _parent[u];

         _pred[u] = _pred[w];

         _forward[u] = !_forward[w];

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

         _succ_num[u] = tmp_sc;

         _last_succ[w] = tmp_ls;

         u = w;

       }

       _pred[u_in] = in_arc;

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

       _succ_num[u_in] = old_succ_num;

       // 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 _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 potentials

     void updatePotential() {

       Value sigma = _forward[u_in] ?

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

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

       if (_succ_num[u_in] > _node_num / 2) {

         // Update in the upper subtree (which contains the root)

         int before = _rev_thread[u_in];

         int after = _thread[_last_succ[u_in]];

         _thread[before] = after;

         _pi[_root] -= sigma;

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

           _pi[u] -= sigma;

         }

         _thread[before] = u_in;

       } else {

         // 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(PivotRule pivot_rule) {

       // Select the pivot rule implementation

       switch (pivot_rule) {

         case FIRST_ELIGIBLE:

           return start<FirstEligiblePivotRule>();

         case BEST_ELIGIBLE:

           return start<BestEligiblePivotRule>();

         case BLOCK_SEARCH:

           return start<BlockSearchPivotRule>();

         case CANDIDATE_LIST:

           return start<CandidateListPivotRule>();

         case ALTERING_LIST:

           return start<AlteringListPivotRule>();

       }

       return false;

     }

     template <typename PivotRuleImpl>

     bool start() {

       PivotRuleImpl pivot(*this);

       // Execute the Network Simplex algorithm

       while (pivot.findEnteringArc()) {

         findJoinNode();

         bool change = findLeavingArc();

         changeFlow(change);

         if (change) {

           updateTreeStructure();

           updatePotential();

         }

       }

       // 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 (_plower) {

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

           Arc e = _arc_ref[i];

           _flow_map->set(e, (*_plower)[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