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

  *

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

  *

  * Copyright (C) 2003-2010

  * 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_algs

 ///

 /// \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_algs

   /// @{

   /// \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"

   /// \ref amo93networkflows, \ref dantzig63linearprog,

   /// \ref kellyoneill91netsimplex.

   /// This algorithm is a highly efficient specialized version of the

   /// linear programming simplex method directly for the minimum cost

   /// flow problem.

   ///

   /// In general, \ref NetworkSimplex and \ref CostScaling are the fastest

   /// implementations available in LEMON for solving this problem.

   /// (For more information, see \ref min_cost_flow_algs "the module page".)

   /// Furthermore, this class supports both directions of the supply/demand

   /// inequality constraints. For more information, see \ref SupplyType.

   ///

   /// Most of the parameters of the problem (except for the digraph)

   /// can be given using separate functions, and the algorithm can be

   /// executed using the \ref run() function. If some parameters are not

   /// specified, then default values will be used.

   ///

   /// \tparam GR The digraph type the algorithm runs on.

   /// \tparam V The number type used for flow amounts, capacity bounds

   /// and supply values in the algorithm. By default, it is \c int.

   /// \tparam C The number type used for costs and potentials in the

   /// algorithm. By default, it is the same as \c V.

   ///

   /// \warning Both \c V and \c C must be signed number types.

   /// \warning All input data (capacities, supply values, and costs) must

   /// be integer.

   ///

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

   /// implementations, from which the most efficient one is used

   /// by default. For more information, see \ref PivotRule.

   template <typename GR, typename V = int, typename C = V>

   class NetworkSimplex

   {

   public:

     /// The type of the flow amounts, capacity bounds and supply values

     typedef V Value;

     /// The type of the arc costs

     typedef C Cost;

   public:

     /// \brief Problem type constants for the \c run() function.

     ///

     /// Enum type containing the problem type constants that can be

     /// returned by the \ref run() function of the algorithm.

     enum ProblemType {

       /// The problem has no feasible solution (flow).

       INFEASIBLE,

       /// The problem has optimal solution (i.e. it is feasible and

       /// bounded), and the algorithm has found optimal flow and node

       /// potentials (primal and dual solutions).

       OPTIMAL,

       /// The objective function of the problem is unbounded, i.e.

       /// there is a directed cycle having negative total cost and

       /// infinite upper bound.

       UNBOUNDED

     };

     /// \brief Constants for selecting the type of the supply constraints.

     ///

     /// Enum type containing constants for selecting the supply type,

     /// i.e. the direction of the inequalities in the supply/demand

     /// constraints of the \ref min_cost_flow "minimum cost flow problem".

     ///

     /// The default supply type is \c GEQ, the \c LEQ type can be

     /// selected using \ref supplyType().

     /// The equality form is a special case of both supply types.

     enum SupplyType {

       /// This option means that there are <em>"greater or equal"</em>

       /// supply/demand constraints in the definition of the problem.

       GEQ,

       /// This option means that there are <em>"less or equal"</em>

       /// supply/demand constraints in the definition of the problem.

       LEQ

     };

     /// \brief Constants for selecting the pivot rule.

     ///

     /// Enum type containing constants for selecting the pivot rule for

     /// the \ref run() function.

     ///

     /// \ref NetworkSimplex provides five different implementations for

     /// the pivot strategy that significantly affects the running time

     /// of the algorithm.

     /// According to experimental tests conducted on various problem

     /// instances, \ref BLOCK_SEARCH "Block Search" and

     /// \ref ALTERING_LIST "Altering Candidate List" rules turned out

     /// to be the most efficient.

     /// Since \ref BLOCK_SEARCH "Block Search" is a simpler strategy that

     /// seemed to be slightly more robust, it is used by default.

     /// However, another pivot rule can easily be selected using the

     /// \ref run() function with the proper parameter.

     enum PivotRule {

       /// The \e First \e Eligible pivot rule.

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

       /// in every iteration.

       FIRST_ELIGIBLE,

       /// The \e Best \e Eligible pivot rule.

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

       BEST_ELIGIBLE,

       /// The \e Block \e 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 \e Candidate \e 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 \e Altering \e Candidate \e List pivot rule.

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

       /// It keeps only a few of the best eligible arcs from the former

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

       ALTERING_LIST

     };

   private:

     TEMPLATE_DIGRAPH_TYPEDEFS(GR);

     typedef std::vector<int> IntVector;

     typedef std::vector<Value> ValueVector;

     typedef std::vector<Cost> CostVector;

     typedef std::vector<signed char> CharVector;

     // Note: vector<signed char> is used instead of vector<ArcState> and

     // vector<ArcDirection> for efficiency reasons

     // State constants for arcs

     enum ArcState {

       STATE_UPPER = -1,

       STATE_TREE  =  0,

       STATE_LOWER =  1

     };

     // Direction constants for tree arcs

     enum ArcDirection {

       DIR_DOWN = -1,

       DIR_UP   =  1

     };

   private:

     // Data related to the underlying digraph

     const GR &_graph;

     int _node_num;

     int _arc_num;

     int _all_arc_num;

     int _search_arc_num;

     // Parameters of the problem

     bool _have_lower;

     SupplyType _stype;

     Value _sum_supply;

     // Data structures for storing the digraph

     IntNodeMap _node_id;

     IntArcMap _arc_id;

     IntVector _source;

     IntVector _target;

     bool _arc_mixing;

     // Node and arc data

     ValueVector _lower;

     ValueVector _upper;

     ValueVector _cap;

     CostVector _cost;

     ValueVector _supply;

     ValueVector _flow;

     CostVector _pi;

     // Data for storing the spanning tree structure

     IntVector _parent;

     IntVector _pred;

     IntVector _thread;

     IntVector _rev_thread;

     IntVector _succ_num;

     IntVector _last_succ;

     CharVector _pred_dir;

     CharVector _state;

     IntVector _dirty_revs;

     int _root;

     // Temporary data used in the current pivot iteration

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

     Value delta;

     const Value MAX;

   public:

     /// \brief Constant for infinite upper bounds (capacities).

     ///

     /// Constant for infinite upper bounds (capacities).

     /// It is \c std::numeric_limits<Value>::infinity() if available,

     /// \c std::numeric_limits<Value>::max() otherwise.

     const Value INF;

   private:

     // Implementation of the First Eligible pivot rule

     class FirstEligiblePivotRule

     {

     private:

       // References to the NetworkSimplex class

       const IntVector  &_source;

       const IntVector  &_target;

       const CostVector &_cost;

       const CharVector &_state;

       const CostVector &_pi;

       int &_in_arc;

       int _search_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), _search_arc_num(ns._search_arc_num),

         _next_arc(0)

       {}

       // Find next entering arc

       bool findEnteringArc() {

         Cost c;

         for (int e = _next_arc; e != _search_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 CostVector &_cost;

       const CharVector &_state;

       const CostVector &_pi;

       int &_in_arc;

       int _search_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), _search_arc_num(ns._search_arc_num)

       {}

       // Find next entering arc

       bool findEnteringArc() {

         Cost c, min = 0;

         for (int e = 0; e != _search_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 CostVector &_cost;

       const CharVector &_state;

       const CostVector &_pi;

       int &_in_arc;

       int _search_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), _search_arc_num(ns._search_arc_num),

         _next_arc(0)

       {

         // The main parameters of the pivot rule

         const double BLOCK_SIZE_FACTOR = 1.0;

         const int MIN_BLOCK_SIZE = 10;

         _block_size = std::max( int(BLOCK_SIZE_FACTOR *

                                     std::sqrt(double(_search_arc_num))),

                                 MIN_BLOCK_SIZE );

       }

       // Find next entering arc

       bool findEnteringArc() {

         Cost c, min = 0;

         int cnt = _block_size;

         int e;

         for (e = _next_arc; e != _search_arc_num; ++e) {

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

           if (c < min) {

             min = c;

             _in_arc = e;

           }

           if (--cnt == 0) {

             if (min < 0) goto search_end;

             cnt = _block_size;

           }

         }

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

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

           if (c < min) {

             min = c;

             _in_arc = e;

           }

           if (--cnt == 0) {

             if (min < 0) goto search_end;

             cnt = _block_size;

           }

         }

         if (min >= 0) return false;

       search_end:

         _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 CostVector &_cost;

       const CharVector &_state;

       const CostVector &_pi;

       int &_in_arc;

       int _search_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), _search_arc_num(ns._search_arc_num),

         _next_arc(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 *

                                      std::sqrt(double(_search_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;

         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;

               _in_arc = 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_arc; e != _search_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;

               _in_arc = e;

             }

             if (_curr_length == _list_length) goto search_end;

           }

         }

         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;

               _in_arc = e;

             }

             if (_curr_length == _list_length) goto search_end;

           }

         }

         if (_curr_length == 0) return false;

       search_end:

         _minor_count = 1;

         _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 CostVector &_cost;

       const CharVector &_state;

       const CostVector &_pi;

       int &_in_arc;

       int _search_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), _search_arc_num(ns._search_arc_num),

         _next_arc(0), _cand_cost(ns._search_arc_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 = 0.01;

         const int MIN_HEAD_LENGTH = 3;

         _block_size = std::max( int(BLOCK_SIZE_FACTOR *

                                     std::sqrt(double(_search_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;

         Cost c;

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

           e = _candidates[i];

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

           if (c < 0) {

             _cand_cost[e] = c;

           } else {

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

           }

         }

         // Extend the list

         int cnt = _block_size;

         int limit = _head_length;

         for (e = _next_arc; e != _search_arc_num; ++e) {

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

           if (c < 0) {

             _cand_cost[e] = c;

             _candidates[_curr_length++] = e;

           }

           if (--cnt == 0) {

             if (_curr_length > limit) goto search_end;

             limit = 0;

             cnt = _block_size;

           }

         }

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

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

           if (c < 0) {

             _cand_cost[e] = c;

             _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:

         // Perform partial sort operation on the candidate list

         int new_length = std::min(_head_length + 1, _curr_length);

         std::partial_sort(_candidates.begin(), _candidates.begin() + new_length,

                           _candidates.begin() + _curr_length, _sort_func);

         // Select the entering arc and remove it from the list

         _in_arc = _candidates[0];

         _next_arc = e;

         _candidates[0] = _candidates[new_length - 1];

         _curr_length = new_length - 1;

         return true;

       }

     }; //class AlteringListPivotRule

   public:

     /// \brief Constructor.

     ///

     /// The constructor of the class.

     ///

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

     /// \param arc_mixing Indicate if the arcs will be stored in a

     /// mixed order in the internal data structure.

     /// In general, it leads to similar performance as using the original

     /// arc order, but it makes the algorithm more robust and in special

     /// cases, even significantly faster. Therefore, it is enabled by default.

     NetworkSimplex(const GR& graph, bool arc_mixing = true) :

       _graph(graph), _node_id(graph), _arc_id(graph),

       _arc_mixing(arc_mixing),

       MAX(std::numeric_limits<Value>::max()),

       INF(std::numeric_limits<Value>::has_infinity ?

           std::numeric_limits<Value>::infinity() : MAX)

     {

       // Check the number types

       LEMON_ASSERT(std::numeric_limits<Value>::is_signed,

         "The flow type of NetworkSimplex must be signed");

       LEMON_ASSERT(std::numeric_limits<Cost>::is_signed,

         "The cost type of NetworkSimplex must be signed");

       // Reset data structures

       reset();

     }

     /// \name Parameters

     /// The parameters of the algorithm can be specified using these

     /// functions.

     /// @{

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

     ///

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

     /// If it is not 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 LowerMap>

     NetworkSimplex& lowerMap(const LowerMap& map) {

       _have_lower = true;

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

         _lower[_arc_id[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 it is not used before calling \ref run(), the upper bounds

     /// will be set to \ref INF on all arcs (i.e. the flow value will be

     /// unbounded from above).

     ///

     /// \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 UpperMap>

     NetworkSimplex& upperMap(const UpperMap& map) {

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

         _upper[_arc_id[a]] = map[a];

       }

       return *this;

     }

     /// \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 Cost type

     /// of the algorithm.

     ///

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

     template<typename CostMap>

     NetworkSimplex& costMap(const CostMap& map) {

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

         _cost[_arc_id[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.

     ///

     /// \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>

     ///

     /// \sa supplyType()

     template<typename SupplyMap>

     NetworkSimplex& supplyMap(const SupplyMap& map) {

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

         _supply[_node_id[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.

     ///

     /// Using this function has the same effect as using \ref supplyMap()

     /// with a map in which \c k is assigned to \c s, \c -k is

     /// assigned to \c t and all other nodes have zero supply value.

     ///

     /// \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) {

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

         _supply[i] = 0;

       }

       _supply[_node_id[s]] =  k;

       _supply[_node_id[t]] = -k;

       return *this;

     }

     /// \brief Set the type of the supply constraints.

     ///

     /// This function sets the type of the supply/demand constraints.

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

     /// type will be used.

     ///

     /// For more information, see \ref SupplyType.

     ///

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

     NetworkSimplex& supplyType(SupplyType supply_type) {

       _stype = supply_type;

       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 functions \ref lowerMap(),

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

     /// \ref supplyType().

     /// For example,

     /// \code

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

     ///   ns.lowerMap(lower).upperMap(upper).costMap(cost)

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

     /// \endcode

     ///

     /// This function can be called more than once. All the given parameters

     /// are kept for the next call, unless \ref resetParams() or \ref reset()

     /// is used, thus only the modified parameters have to be set again.

     /// If the underlying digraph was also modified after the construction

     /// of the class (or the last \ref reset() call), then the \ref reset()

     /// function must be called.

     ///

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

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

     ///

     /// \return \c INFEASIBLE if no feasible flow exists,

     /// \n \c OPTIMAL if the problem has optimal solution

     /// (i.e. it is feasible and bounded), and the algorithm has found

     /// optimal flow and node potentials (primal and dual solutions),

     /// \n \c UNBOUNDED if the objective function of the problem is

     /// unbounded, i.e. there is a directed cycle having negative total

     /// cost and infinite upper bound.

     ///

     /// \see ProblemType, PivotRule

     /// \see resetParams(), reset()

     ProblemType run(PivotRule pivot_rule = BLOCK_SEARCH) {

       if (!init()) return INFEASIBLE;

       return start(pivot_rule);

     }

     /// \brief Reset all the parameters that have been given before.

     ///

     /// This function resets all the paramaters that have been given

     /// before using functions \ref lowerMap(), \ref upperMap(),

     /// \ref costMap(), \ref supplyMap(), \ref stSupply(), \ref supplyType().

     ///

     /// It is useful for multiple \ref run() calls. Basically, all the given

     /// parameters are kept for the next \ref run() call, unless

     /// \ref resetParams() or \ref reset() is used.

     /// If the underlying digraph was also modified after the construction

     /// of the class or the last \ref reset() call, then the \ref reset()

     /// function must be used, otherwise \ref resetParams() is sufficient.

     ///

     /// For example,

     /// \code

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

     ///

     ///   // First run

     ///   ns.lowerMap(lower).upperMap(upper).costMap(cost)

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

     ///

     ///   // Run again with modified cost map (resetParams() is not called,

     ///   // so only the cost map have to be set again)

     ///   cost[e] += 100;

     ///   ns.costMap(cost).run();

     ///

     ///   // Run again from scratch using resetParams()

     ///   // (the lower bounds will be set to zero on all arcs)

     ///   ns.resetParams();

     ///   ns.upperMap(capacity).costMap(cost)

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

     /// \endcode

     ///

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

     ///

     /// \see reset(), run()

     NetworkSimplex& resetParams() {

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

         _supply[i] = 0;

       }

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

         _lower[i] = 0;

         _upper[i] = INF;

         _cost[i] = 1;

       }

       _have_lower = false;

       _stype = GEQ;

       return *this;

     }

     /// \brief Reset the internal data structures and all the parameters

     /// that have been given before.

     ///

     /// This function resets the internal data structures and all the

     /// paramaters that have been given before using functions \ref lowerMap(),

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

     /// \ref supplyType().

     ///

     /// It is useful for multiple \ref run() calls. Basically, all the given

     /// parameters are kept for the next \ref run() call, unless

     /// \ref resetParams() or \ref reset() is used.

     /// If the underlying digraph was also modified after the construction

     /// of the class or the last \ref reset() call, then the \ref reset()

     /// function must be used, otherwise \ref resetParams() is sufficient.

     ///

     /// See \ref resetParams() for examples.

     ///

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

     ///

     /// \see resetParams(), run()

     NetworkSimplex& reset() {

       // Resize vectors

       _node_num = countNodes(_graph);

       _arc_num = countArcs(_graph);

       int all_node_num = _node_num + 1;

       int max_arc_num = _arc_num + 2 * _node_num;

       _source.resize(max_arc_num);

       _target.resize(max_arc_num);

       _lower.resize(_arc_num);

       _upper.resize(_arc_num);

       _cap.resize(max_arc_num);

       _cost.resize(max_arc_num);

       _supply.resize(all_node_num);

       _flow.resize(max_arc_num);

       _pi.resize(all_node_num);

       _parent.resize(all_node_num);

       _pred.resize(all_node_num);

       _pred_dir.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(max_arc_num);

       // Copy the graph

       int i = 0;

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

         _node_id[n] = i;

       }

       if (_arc_mixing) {

         // Store the arcs in a mixed order

         const int skip = std::max(_arc_num / _node_num, 3);

         int i = 0, j = 0;

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

           _arc_id[a] = i;

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

           _target[i] = _node_id[_graph.target(a)];

           if ((i += skip) >= _arc_num) i = ++j;

         }

       } else {

         // Store the arcs in the original order

         int i = 0;

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

           _arc_id[a] = i;

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

           _target[i] = _node_id[_graph.target(a)];

         }

       }

       // Reset parameters

       resetParams();

       return *this;

     }

     /// @}

     /// \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.

     /// Its complexity is O(e).

     ///

     /// \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 Cost

     /// 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 Number>

     Number totalCost() const {

       Number c = 0;

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

         int i = _arc_id[a];

         c += Number(_flow[i]) * Number(_cost[i]);

       }

       return c;

     }

 #ifndef DOXYGEN

     Cost totalCost() const {

       return totalCost<Cost>();

     }

 #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[_arc_id[a]];

     }

     /// \brief Copy the flow values (the primal solution) into the

     /// given map.

     ///

     /// This function copies the flow value on each arc into the given

     /// map. The \c Value type of the algorithm must be convertible to

     /// the \c Value type of the map.

     ///

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

     template <typename FlowMap>

     void flowMap(FlowMap &map) const {

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

         map.set(a, _flow[_arc_id[a]]);

       }

     }

     /// \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.

     Cost potential(const Node& n) const {

       return _pi[_node_id[n]];

     }

     /// \brief Copy the potential values (the dual solution) into the

     /// given map.

     ///

     /// This function copies the potential (dual value) of each node

     /// into the given map.

     /// The \c Cost type of the algorithm must be convertible to the

     /// \c Value type of the map.

     ///

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

     template <typename PotentialMap>

     void potentialMap(PotentialMap &map) const {

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

         map.set(n, _pi[_node_id[n]]);

       }

     }

     /// @}

   private:

     // Initialize internal data structures

     bool init() {

       if (_node_num == 0) return false;

       // Check the sum of supply values

       _sum_supply = 0;

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

         _sum_supply += _supply[i];

       }

       if ( !((_stype == GEQ && _sum_supply <= 0) ||

              (_stype == LEQ && _sum_supply >= 0)) ) return false;

       // Remove non-zero lower bounds

       if (_have_lower) {

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

           Value c = _lower[i];

           if (c >= 0) {

             _cap[i] = _upper[i] < MAX ? _upper[i] - c : INF;

           } else {

             _cap[i] = _upper[i] < MAX + c ? _upper[i] - c : INF;

           }

           _supply[_source[i]] -= c;

           _supply[_target[i]] += c;

         }

       } else {

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

           _cap[i] = _upper[i];

         }

       }

       // Initialize artifical cost

       Cost ART_COST;

       if (std::numeric_limits<Cost>::is_exact) {

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

       } else {

         ART_COST = 0;

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

           if (_cost[i] > ART_COST) ART_COST = _cost[i];

         }

         ART_COST = (ART_COST + 1) * _node_num;

       }

       // Initialize arc maps

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

         _flow[i] = 0;

         _state[i] = STATE_LOWER;

       }

       // 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] = _node_num + 1;

       _last_succ[_root] = _root - 1;

       _supply[_root] = -_sum_supply;

       _pi[_root] = 0;

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

       if (_sum_supply == 0) {

         // EQ supply constraints

         _search_arc_num = _arc_num;

         _all_arc_num = _arc_num + _node_num;

         for (int u = 0, e = _arc_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] = INF;

           _state[e] = STATE_TREE;

           if (_supply[u] >= 0) {

             _pred_dir[u] = DIR_UP;

             _pi[u] = 0;

             _source[e] = u;

             _target[e] = _root;

             _flow[e] = _supply[u];

             _cost[e] = 0;

           } else {

             _pred_dir[u] = DIR_DOWN;

             _pi[u] = ART_COST;

             _source[e] = _root;

             _target[e] = u;

             _flow[e] = -_supply[u];

             _cost[e] = ART_COST;

           }

         }

       }

       else if (_sum_supply > 0) {

         // LEQ supply constraints

         _search_arc_num = _arc_num + _node_num;

         int f = _arc_num + _node_num;

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

           _parent[u] = _root;

           _thread[u] = u + 1;

           _rev_thread[u + 1] = u;

           _succ_num[u] = 1;

           _last_succ[u] = u;

           if (_supply[u] >= 0) {

             _pred_dir[u] = DIR_UP;

             _pi[u] = 0;

             _pred[u] = e;

             _source[e] = u;

             _target[e] = _root;

             _cap[e] = INF;

             _flow[e] = _supply[u];

             _cost[e] = 0;

             _state[e] = STATE_TREE;

           } else {

             _pred_dir[u] = DIR_DOWN;

             _pi[u] = ART_COST;

             _pred[u] = f;

             _source[f] = _root;

             _target[f] = u;

             _cap[f] = INF;

             _flow[f] = -_supply[u];

             _cost[f] = ART_COST;

             _state[f] = STATE_TREE;

             _source[e] = u;

             _target[e] = _root;

             _cap[e] = INF;

             _flow[e] = 0;

             _cost[e] = 0;

             _state[e] = STATE_LOWER;

             ++f;

           }

         }

         _all_arc_num = f;

       }

       else {

         // GEQ supply constraints

         _search_arc_num = _arc_num + _node_num;

         int f = _arc_num + _node_num;

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

           _parent[u] = _root;

           _thread[u] = u + 1;

           _rev_thread[u + 1] = u;

           _succ_num[u] = 1;

           _last_succ[u] = u;

           if (_supply[u] <= 0) {

             _pred_dir[u] = DIR_DOWN;

             _pi[u] = 0;

             _pred[u] = e;

             _source[e] = _root;

             _target[e] = u;

             _cap[e] = INF;

             _flow[e] = -_supply[u];

             _cost[e] = 0;

             _state[e] = STATE_TREE;

           } else {

             _pred_dir[u] = DIR_UP;

             _pi[u] = -ART_COST;

             _pred[u] = f;

             _source[f] = u;

             _target[f] = _root;

             _cap[f] = INF;

             _flow[f] = _supply[u];

             _state[f] = STATE_TREE;

             _cost[f] = ART_COST;

             _source[e] = _root;

             _target[e] = u;

             _cap[e] = INF;

             _flow[e] = 0;

             _cost[e] = 0;

             _state[e] = STATE_LOWER;

             ++f;

           }

         }

         _all_arc_num = f;

       }

       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

       int first, second;

       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 c, d;

       int e;

       // Search the cycle form the first node to the join node

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

         e = _pred[u];

         d = _flow[e];

         if (_pred_dir[u] == DIR_DOWN) {

           c = _cap[e];

           d = c >= MAX ? INF : c - d;

         }

         if (d < delta) {

           delta = d;

           u_out = u;

           result = 1;

         }

       }

       // Search the cycle form the second node to the join node

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

         e = _pred[u];

         d = _flow[e];

         if (_pred_dir[u] == DIR_UP) {

           c = _cap[e];

           d = c >= MAX ? INF : c - d;

         }

         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]] -= _pred_dir[u] * val;

         }

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

           _flow[_pred[u]] += _pred_dir[u] * 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 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];

       // Check if u_in and u_out coincide

       if (u_in == u_out) {

         // Update _parent, _pred, _pred_dir

         _parent[u_in] = v_in;

         _pred[u_in] = in_arc;

         _pred_dir[u_in] = u_in == _source[in_arc] ? DIR_UP : DIR_DOWN;

         // Update _thread and _rev_thread

         if (_thread[v_in] != u_out) {

           int after = _thread[old_last_succ];

           _thread[old_rev_thread] = after;

           _rev_thread[after] = old_rev_thread;

           after = _thread[v_in];

           _thread[v_in] = u_out;

           _rev_thread[u_out] = v_in;

           _thread[old_last_succ] = after;

           _rev_thread[after] = old_last_succ;

         }

       } else {

         // Handle the case when old_rev_thread equals to v_in

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

         int thread_continue = old_rev_thread == v_in ?

           _thread[old_last_succ] : _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)

         int stem = u_in;              // the current stem node

         int par_stem = v_in;          // the new parent of stem

         int next_stem;                // the next stem node

         int last = _last_succ[u_in];  // the last successor of stem

         int before, after = _thread[last];

         _thread[v_in] = u_in;

         _dirty_revs.clear();

         _dirty_revs.push_back(v_in);

         while (stem != u_out) {

           // Insert the next stem node into the thread list

           next_stem = _parent[stem];

           _thread[last] = next_stem;

           _dirty_revs.push_back(last);

           // Remove the subtree of stem from the thread list

           before = _rev_thread[stem];

           _thread[before] = after;

           _rev_thread[after] = before;

           // Change the parent node and shift stem nodes

           _parent[stem] = par_stem;

           par_stem = stem;

           stem = next_stem;

           // Update last and after

           last = _last_succ[stem] == _last_succ[par_stem] ?

             _rev_thread[par_stem] : _last_succ[stem];

           after = _thread[last];

         }

         _parent[u_out] = par_stem;

         _thread[last] = thread_continue;

         _rev_thread[thread_continue] = last;

         _last_succ[u_out] = last;

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

         // the case when old_rev_thread equals to v_in

         if (old_rev_thread != v_in) {

           _thread[old_rev_thread] = after;

           _rev_thread[after] = old_rev_thread;

         }

         // Update _rev_thread using the new _thread values

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

           int u = _dirty_revs[i];

           _rev_thread[_thread[u]] = u;

         }

         // Update _pred, _pred_dir, _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];

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

           _pred[u] = _pred[p];

           _pred_dir[u] = -_pred_dir[p];

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

           _succ_num[u] = tmp_sc;

           _last_succ[p] = tmp_ls;

         }

         _pred[u_in] = in_arc;

         _pred_dir[u_in] = u_in == _source[in_arc] ? DIR_UP : DIR_DOWN;

         _succ_num[u_in] = old_succ_num;

       }

       // Update _last_succ from v_in towards the root

       int up_limit_out = _last_succ[join] == v_in ? join : -1;

       int last_succ_out = _last_succ[u_out];

       for (int u = v_in; u != -1 && _last_succ[u] == v_in; u = _parent[u]) {

         _last_succ[u] = last_succ_out;

       }

       // Update _last_succ from v_out towards the root

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

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

              u = _parent[u]) {

           _last_succ[u] = old_rev_thread;

         }

       }

       else if (last_succ_out != old_last_succ) {

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

              u = _parent[u]) {

           _last_succ[u] = last_succ_out;

         }

       }

       // Update _succ_num from v_in to join

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

         _succ_num[u] += old_succ_num;

       }

       // Update _succ_num from v_out to join

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

         _succ_num[u] -= old_succ_num;

       }

     }

     // Update potentials in the subtree that has been moved

     void updatePotential() {

       Cost sigma = _pi[v_in] - _pi[u_in] -

                    _pred_dir[u_in] * _cost[in_arc];

       int end = _thread[_last_succ[u_in]];

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

         _pi[u] += sigma;

       }

     }

     // Heuristic initial pivots

     bool initialPivots() {

       Value curr, total = 0;

       std::vector<Node> supply_nodes, demand_nodes;

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

         curr = _supply[_node_id[u]];

         if (curr > 0) {

           total += curr;

           supply_nodes.push_back(u);

         }

         else if (curr < 0) {

           demand_nodes.push_back(u);

         }

       }

       if (_sum_supply > 0) total -= _sum_supply;

       if (total <= 0) return true;

       IntVector arc_vector;

       if (_sum_supply >= 0) {

         if (supply_nodes.size() == 1 && demand_nodes.size() == 1) {

           // Perform a reverse graph search from the sink to the source

           typename GR::template NodeMap<bool> reached(_graph, false);

           Node s = supply_nodes[0], t = demand_nodes[0];

           std::vector<Node> stack;

           reached[t] = true;

           stack.push_back(t);

           while (!stack.empty()) {

             Node u, v = stack.back();

             stack.pop_back();

             if (v == s) break;

             for (InArcIt a(_graph, v); a != INVALID; ++a) {

               if (reached[u = _graph.source(a)]) continue;

               int j = _arc_id[a];

               if (_cap[j] >= total) {

                 arc_vector.push_back(j);

                 reached[u] = true;

                 stack.push_back(u);

               }

             }

           }

         } else {

           // Find the min. cost incoming arc for each demand node

           for (int i = 0; i != int(demand_nodes.size()); ++i) {

             Node v = demand_nodes[i];

             Cost c, min_cost = std::numeric_limits<Cost>::max();

             Arc min_arc = INVALID;

             for (InArcIt a(_graph, v); a != INVALID; ++a) {

               c = _cost[_arc_id[a]];

               if (c < min_cost) {

                 min_cost = c;

                 min_arc = a;

               }

             }

             if (min_arc != INVALID) {

               arc_vector.push_back(_arc_id[min_arc]);

             }

           }

         }

       } else {

         // Find the min. cost outgoing arc for each supply node

         for (int i = 0; i != int(supply_nodes.size()); ++i) {

           Node u = supply_nodes[i];

           Cost c, min_cost = std::numeric_limits<Cost>::max();

           Arc min_arc = INVALID;

           for (OutArcIt a(_graph, u); a != INVALID; ++a) {

             c = _cost[_arc_id[a]];

             if (c < min_cost) {

               min_cost = c;

               min_arc = a;

             }

           }

           if (min_arc != INVALID) {

             arc_vector.push_back(_arc_id[min_arc]);

           }

         }

       }

       // Perform heuristic initial pivots

       for (int i = 0; i != int(arc_vector.size()); ++i) {

         in_arc = arc_vector[i];

         if (_state[in_arc] * (_cost[in_arc] + _pi[_source[in_arc]] -

             _pi[_target[in_arc]]) >= 0) continue;

         findJoinNode();

         bool change = findLeavingArc();

         if (delta >= MAX) return false;

         changeFlow(change);

         if (change) {

           updateTreeStructure();

           updatePotential();

         }

       }

       return true;

     }

     // Execute the algorithm

     ProblemType 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 INFEASIBLE; // avoid warning

     }

     template <typename PivotRuleImpl>

     ProblemType start() {

       PivotRuleImpl pivot(*this);

       // Perform heuristic initial pivots

       if (!initialPivots()) return UNBOUNDED;

       // Execute the Network Simplex algorithm

       while (pivot.findEnteringArc()) {

         findJoinNode();

         bool change = findLeavingArc();

         if (delta >= MAX) return UNBOUNDED;

         changeFlow(change);

         if (change) {

           updateTreeStructure();

           updatePotential();

         }

       }

       // Check feasibility

       for (int e = _search_arc_num; e != _all_arc_num; ++e) {

         if (_flow[e] != 0) return INFEASIBLE;

       }

       // Transform the solution and the supply map to the original form

       if (_have_lower) {

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

           Value c = _lower[i];

           if (c != 0) {

             _flow[i] += c;

             _supply[_source[i]] += c;

             _supply[_target[i]] -= c;

           }

         }

       }

       // Shift potentials to meet the requirements of the GEQ/LEQ type

       // optimality conditions

       if (_sum_supply == 0) {

         if (_stype == GEQ) {

           Cost max_pot = -std::numeric_limits<Cost>::max();

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

             if (_pi[i] > max_pot) max_pot = _pi[i];

           }

           if (max_pot > 0) {

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

               _pi[i] -= max_pot;

           }

         } else {

           Cost min_pot = std::numeric_limits<Cost>::max();

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

             if (_pi[i] < min_pot) min_pot = _pi[i];

           }

           if (min_pot < 0) {

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

               _pi[i] -= min_pot;

           }

         }

       }

       return OPTIMAL;

     }

   }; //class NetworkSimplex

   ///@}

 } //namespace lemon

 #endif //LEMON_NETWORK_SIMPLEX_H