/* -*- C++ -*-

  *

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

  *

  * Copyright (C) 2003-2008

  * Egervary Jeno Kombinatorikus Optimalizalasi Kutatocsoport

  * (Egervary Research Group on Combinatorial Optimization, EGRES).

  *

  * Permission to use, modify and distribute this software is granted

  * provided that this copyright notice appears in all copies. For

  * precise terms see the accompanying LICENSE file.

  *

  * This software is provided "AS IS" with no warranty of any kind,

  * express or implied, and with no claim as to its suitability for any

  * purpose.

  *

  */

 #ifndef LEMON_NETWORK_SIMPLEX_H

 #define LEMON_NETWORK_SIMPLEX_H

 /// \ingroup min_cost_flow

 ///

 /// \file

 /// \brief Network simplex algorithm for finding a minimum cost flow.

 #include <vector>

 #include <limits>

 #include <lemon/graph_adaptor.h>

 #include <lemon/graph_utils.h>

 #include <lemon/smart_graph.h>

 #include <lemon/math.h>

 namespace lemon {

   /// \addtogroup min_cost_flow

   /// @{

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

   /// finding a minimum cost flow.

   ///

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

   /// finding a minimum cost flow.

   ///

   /// \tparam Graph The directed graph type the algorithm runs on.

   /// \tparam LowerMap The type of the lower bound map.

   /// \tparam CapacityMap The type of the capacity (upper bound) map.

   /// \tparam CostMap The type of the cost (length) map.

   /// \tparam SupplyMap The type of the supply map.

   ///

   /// \warning

   /// - Edge capacities and costs should be \e non-negative \e integers.

   /// - Supply values should be \e signed \e integers.

   /// - The value types of the maps should be convertible to each other.

   /// - \c CostMap::Value must be signed type.

   ///

   /// \note \ref NetworkSimplex provides six different pivot rule

   /// implementations that significantly affect the efficiency of the

   /// algorithm.

   /// By default a combined pivot rule is used, which is the fastest

   /// implementation according to our benchmark tests.

   /// Another pivot rule can be selected using \ref run() function

   /// with the proper parameter.

   ///

   /// \author Peter Kovacs

   template < typename Graph,

              typename LowerMap = typename Graph::template EdgeMap<int>,

              typename CapacityMap = typename Graph::template EdgeMap<int>,

              typename CostMap = typename Graph::template EdgeMap<int>,

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

   class NetworkSimplex

   {

     typedef typename CapacityMap::Value Capacity;

     typedef typename CostMap::Value Cost;

     typedef typename SupplyMap::Value Supply;

     typedef SmartGraph SGraph;

     GRAPH_TYPEDEFS(typename SGraph);

     typedef typename SGraph::template EdgeMap<Capacity> SCapacityMap;

     typedef typename SGraph::template EdgeMap<Cost> SCostMap;

     typedef typename SGraph::template NodeMap<Supply> SSupplyMap;

     typedef typename SGraph::template NodeMap<Cost> SPotentialMap;

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

     typedef typename SGraph::template NodeMap<bool> BoolNodeMap;

     typedef typename SGraph::template NodeMap<Node> NodeNodeMap;

     typedef typename SGraph::template NodeMap<Edge> EdgeNodeMap;

     typedef typename SGraph::template EdgeMap<int> IntEdgeMap;

     typedef typename Graph::template NodeMap<Node> NodeRefMap;

     typedef typename Graph::template EdgeMap<Edge> EdgeRefMap;

   public:

     /// The type of the flow map.

     typedef typename Graph::template EdgeMap<Capacity> FlowMap;

     /// The type of the potential map.

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

   public:

     /// Enum type to select the pivot rule used by \ref run().

     enum PivotRuleEnum {

       FIRST_ELIGIBLE_PIVOT,

       BEST_ELIGIBLE_PIVOT,

       BLOCK_SEARCH_PIVOT,

       LIMITED_SEARCH_PIVOT,

       CANDIDATE_LIST_PIVOT,

       COMBINED_PIVOT

     };

   private:

     /// \brief Map adaptor class for handling reduced edge costs.

     ///

     /// Map adaptor class for handling reduced edge costs.

     class ReducedCostMap : public MapBase<Edge, Cost>

     {

     private:

       const SGraph &_gr;

       const SCostMap &_cost_map;

       const SPotentialMap &_pot_map;

     public:

       ///\e

       ReducedCostMap( const SGraph &gr,

                       const SCostMap &cost_map,

                       const SPotentialMap &pot_map ) :

         _gr(gr), _cost_map(cost_map), _pot_map(pot_map) {}

       ///\e

       Cost operator[](const Edge &e) const {

         return _cost_map[e] + _pot_map[_gr.source(e)]

                            - _pot_map[_gr.target(e)];

       }

     }; //class ReducedCostMap

   private:

     /// \brief Implementation of the "First Eligible" pivot rule for the

     /// \ref NetworkSimplex "network simplex" algorithm.

     ///

     /// This class implements the "First Eligible" pivot rule

     /// for the \ref NetworkSimplex "network simplex" algorithm.

     class FirstEligiblePivotRule

     {

     private:

       NetworkSimplex &_ns;

       EdgeIt _next_edge;

     public:

       /// Constructor.

       FirstEligiblePivotRule(NetworkSimplex &ns) :

         _ns(ns), _next_edge(ns._graph) {}

       /// Finds the next entering edge.

       bool findEnteringEdge() {

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

           if (_ns._state[e] * _ns._red_cost[e] < 0) {

             _ns._in_edge = e;

             _next_edge = ++e;

             return true;

           }

         }

         for (EdgeIt e(_ns._graph); e != _next_edge; ++e) {

           if (_ns._state[e] * _ns._red_cost[e] < 0) {

             _ns._in_edge = e;

             _next_edge = ++e;

             return true;

           }

         }

         return false;

       }

     }; //class FirstEligiblePivotRule

     /// \brief Implementation of the "Best Eligible" pivot rule for the

     /// \ref NetworkSimplex "network simplex" algorithm.

     ///

     /// This class implements the "Best Eligible" pivot rule

     /// for the \ref NetworkSimplex "network simplex" algorithm.

     class BestEligiblePivotRule

     {

     private:

       NetworkSimplex &_ns;

     public:

       /// Constructor.

       BestEligiblePivotRule(NetworkSimplex &ns) : _ns(ns) {}

       /// Finds the next entering edge.

       bool findEnteringEdge() {

         Cost min = 0;

         for (EdgeIt e(_ns._graph); e != INVALID; ++e) {

           if (_ns._state[e] * _ns._red_cost[e] < min) {

             min = _ns._state[e] * _ns._red_cost[e];

             _ns._in_edge = e;

           }

         }

         return min < 0;

       }

     }; //class BestEligiblePivotRule

     /// \brief Implementation of the "Block Search" pivot rule for the

     /// \ref NetworkSimplex "network simplex" algorithm.

     ///

     /// This class implements the "Block Search" pivot rule

     /// for the \ref NetworkSimplex "network simplex" algorithm.

     class BlockSearchPivotRule

     {

     private:

       NetworkSimplex &_ns;

       EdgeIt _next_edge, _min_edge;

       int _block_size;

       static const int MIN_BLOCK_SIZE = 10;

     public:

       /// Constructor.

       BlockSearchPivotRule(NetworkSimplex &ns) :

         _ns(ns), _next_edge(ns._graph), _min_edge(ns._graph)

       {

         _block_size = 2 * int(sqrt(countEdges(_ns._graph)));

         if (_block_size < MIN_BLOCK_SIZE) _block_size = MIN_BLOCK_SIZE;

       }

       /// Finds the next entering edge.

       bool findEnteringEdge() {

         Cost curr, min = 0;

         int cnt = 0;

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

           if ((curr = _ns._state[e] * _ns._red_cost[e]) < min) {

             min = curr;

             _min_edge = e;

           }

           if (++cnt == _block_size) {

             if (min < 0) break;

             cnt = 0;

           }

         }

         if (min == 0) {

           for (EdgeIt e(_ns._graph); e != _next_edge; ++e) {

             if ((curr = _ns._state[e] * _ns._red_cost[e]) < min) {

               min = curr;

               _min_edge = e;

             }

             if (++cnt == _block_size) {

               if (min < 0) break;

               cnt = 0;

             }

           }

         }

         _ns._in_edge = _min_edge;

         _next_edge = ++_min_edge;

         return min < 0;

       }

     }; //class BlockSearchPivotRule

     /// \brief Implementation of the "Limited Search" pivot rule for the

     /// \ref NetworkSimplex "network simplex" algorithm.

     ///

     /// This class implements the "Limited Search" pivot rule

     /// for the \ref NetworkSimplex "network simplex" algorithm.

     class LimitedSearchPivotRule

     {

     private:

       NetworkSimplex &_ns;

       EdgeIt _next_edge, _min_edge;

       int _sample_size;

       static const int SAMPLE_SIZE_FACTOR = 15;

       static const int MIN_SAMPLE_SIZE = 10;

     public:

       /// Constructor.

       LimitedSearchPivotRule(NetworkSimplex &ns) :

         _ns(ns), _next_edge(ns._graph), _min_edge(ns._graph)

       {

         _sample_size = countEdges(_ns._graph) *

                        SAMPLE_SIZE_FACTOR / 10000;

         if (_sample_size < MIN_SAMPLE_SIZE)

           _sample_size = MIN_SAMPLE_SIZE;

       }

       /// Finds the next entering edge.

       bool findEnteringEdge() {

         Cost curr, min = 0;

         int cnt = 0;

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

           if ((curr = _ns._state[e] * _ns._red_cost[e]) < min) {

             min = curr;

             _min_edge = e;

           }

           if (curr < 0 && ++cnt == _sample_size) break;

         }

         if (min == 0) {

           for (EdgeIt e(_ns._graph); e != _next_edge; ++e) {

             if ((curr = _ns._state[e] * _ns._red_cost[e]) < min) {

               min = curr;

               _min_edge = e;

             }

             if (curr < 0 && ++cnt == _sample_size) break;

           }

         }

         _ns._in_edge = _min_edge;

         _next_edge = ++_min_edge;

         return min < 0;

       }

     }; //class LimitedSearchPivotRule

     /// \brief Implementation of the "Candidate List" pivot rule for the

     /// \ref NetworkSimplex "network simplex" algorithm.

     ///

     /// This class implements the "Candidate List" pivot rule

     /// for the \ref NetworkSimplex "network simplex" algorithm.

     class CandidateListPivotRule

     {

     private:

       NetworkSimplex &_ns;

       // The list of candidate edges.

       std::vector<Edge> _candidates;

       // The maximum length of the edge list.

       int _list_length;

       // The maximum number of minor iterations between two major

       // itarations.

       int _minor_limit;

       int _minor_count;

       EdgeIt _next_edge;

       static const int LIST_LENGTH_FACTOR = 20;

       static const int MINOR_LIMIT_FACTOR = 10;

       static const int MIN_LIST_LENGTH = 10;

       static const int MIN_MINOR_LIMIT = 2;

     public:

       /// Constructor.

       CandidateListPivotRule(NetworkSimplex &ns) :

         _ns(ns), _next_edge(ns._graph)

       {

         int edge_num = countEdges(_ns._graph);

         _minor_count = 0;

         _list_length = edge_num * LIST_LENGTH_FACTOR / 10000;

         if (_list_length < MIN_LIST_LENGTH)

           _list_length = MIN_LIST_LENGTH;

         _minor_limit = _list_length * MINOR_LIMIT_FACTOR / 100;

         if (_minor_limit < MIN_MINOR_LIMIT)

           _minor_limit = MIN_MINOR_LIMIT;

       }

       /// Finds the next entering edge.

       bool findEnteringEdge() {

         Cost min, curr;

         if (_minor_count < _minor_limit && _candidates.size() > 0) {

           // Minor iteration

           ++_minor_count;

           Edge e;

           min = 0;

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

             e = _candidates[i];

             if ((curr = _ns._state[e] * _ns._red_cost[e]) < min) {

               min = curr;

               _ns._in_edge = e;

             }

           }

           if (min < 0) return true;

         }

         // Major iteration

         _candidates.clear();

         EdgeIt e = _next_edge;

         min = 0;

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

           if ((curr = _ns._state[e] * _ns._red_cost[e]) < 0) {

             _candidates.push_back(e);

             if (curr < min) {

               min = curr;

               _ns._in_edge = e;

             }

             if (int(_candidates.size()) == _list_length) break;

           }

         }

         if (int(_candidates.size()) < _list_length) {

           for (e = EdgeIt(_ns._graph); e != _next_edge; ++e) {

             if ((curr = _ns._state[e] * _ns._red_cost[e]) < 0) {

               _candidates.push_back(e);

               if (curr < min) {

                 min = curr;

                 _ns._in_edge = e;

               }

               if (int(_candidates.size()) == _list_length) break;

             }

           }

         }

         if (_candidates.size() == 0) return false;

         _minor_count = 1;

         _next_edge = ++e;

         return true;

       }

     }; //class CandidateListPivotRule

   private:

     // State constants for edges

     enum EdgeStateEnum {

       STATE_UPPER = -1,

       STATE_TREE  =  0,

       STATE_LOWER =  1

     };

     // Constant for the combined pivot rule.

     static const int COMBINED_PIVOT_MAX_DEG = 5;

   private:

     // The directed graph the algorithm runs on

     SGraph _graph;

     // The original graph

     const Graph &_graph_ref;

     // The original lower bound map

     const LowerMap *_lower;

     // The capacity map

     SCapacityMap _capacity;

     // The cost map

     SCostMap _cost;

     // The supply map

     SSupplyMap _supply;

     bool _valid_supply;

     // Edge map of the current flow

     SCapacityMap _flow;

     // Node map of the current potentials

     SPotentialMap _potential;

     // The depth node map of the spanning tree structure

     IntNodeMap _depth;

     // The parent node map of the spanning tree structure

     NodeNodeMap _parent;

     // The pred_edge node map of the spanning tree structure

     EdgeNodeMap _pred_edge;

     // The thread node map of the spanning tree structure

     NodeNodeMap _thread;

     // The forward node map of the spanning tree structure

     BoolNodeMap _forward;

     // The state edge map

     IntEdgeMap _state;

     // The root node of the starting spanning tree

     Node _root;

     // The reduced cost map

     ReducedCostMap _red_cost;

     // Members for handling the original graph

     FlowMap *_flow_result;

     PotentialMap *_potential_result;

     bool _local_flow;

     bool _local_potential;

     NodeRefMap _node_ref;

     EdgeRefMap _edge_ref;

     // The entering edge of the current pivot iteration.

     Edge _in_edge;

     // Temporary nodes used in the current pivot iteration.

     Node join, u_in, v_in, u_out, v_out;

     Node right, first, second, last;

     Node stem, par_stem, new_stem;

     // The maximum augment amount along the found cycle in the current

     // pivot iteration.

     Capacity delta;

   public :

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

     ///

     /// General constructor (with lower bounds).

     ///

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

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

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

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

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

     NetworkSimplex( const Graph &graph,

                     const LowerMap &lower,

                     const CapacityMap &capacity,

                     const CostMap &cost,

                     const SupplyMap &supply ) :

       _graph(), _graph_ref(graph), _lower(&lower), _capacity(_graph),

       _cost(_graph), _supply(_graph), _flow(_graph),

       _potential(_graph), _depth(_graph), _parent(_graph),

       _pred_edge(_graph), _thread(_graph), _forward(_graph),

       _state(_graph), _red_cost(_graph, _cost, _potential),

       _flow_result(0), _potential_result(0),

       _local_flow(false), _local_potential(false),

       _node_ref(graph), _edge_ref(graph)

     {

       // Checking the sum of supply values

       Supply sum = 0;

       for (typename Graph::NodeIt n(_graph_ref); n != INVALID; ++n)

         sum += supply[n];

       if (!(_valid_supply = sum == 0)) return;

       // Copying _graph_ref to _graph

       _graph.reserveNode(countNodes(_graph_ref) + 1);

       _graph.reserveEdge(countEdges(_graph_ref) + countNodes(_graph_ref));

       copyGraph(_graph, _graph_ref)

         .edgeMap(_cost, cost)

         .nodeRef(_node_ref)

         .edgeRef(_edge_ref)

         .run();

       // Removing non-zero lower bounds

       for (typename Graph::EdgeIt e(_graph_ref); e != INVALID; ++e) {

         _capacity[_edge_ref[e]] = capacity[e] - lower[e];

       }

       for (typename Graph::NodeIt n(_graph_ref); n != INVALID; ++n) {

         Supply s = supply[n];

         for (typename Graph::InEdgeIt e(_graph_ref, n); e != INVALID; ++e)

           s += lower[e];

         for (typename Graph::OutEdgeIt e(_graph_ref, n); e != INVALID; ++e)

           s -= lower[e];

         _supply[_node_ref[n]] = s;

       }

     }

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

     ///

     /// General constructor (without lower bounds).

     ///

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

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

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

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

     NetworkSimplex( const Graph &graph,

                     const CapacityMap &capacity,

                     const CostMap &cost,

                     const SupplyMap &supply ) :

       _graph(), _graph_ref(graph), _lower(NULL), _capacity(_graph),

       _cost(_graph), _supply(_graph), _flow(_graph),

       _potential(_graph), _depth(_graph), _parent(_graph),

       _pred_edge(_graph), _thread(_graph), _forward(_graph),

       _state(_graph), _red_cost(_graph, _cost, _potential),

       _flow_result(0), _potential_result(0),

       _local_flow(false), _local_potential(false),

       _node_ref(graph), _edge_ref(graph)

     {

       // Checking the sum of supply values

       Supply sum = 0;

       for (typename Graph::NodeIt n(_graph_ref); n != INVALID; ++n)

         sum += supply[n];

       if (!(_valid_supply = sum == 0)) return;

       // Copying _graph_ref to graph

       copyGraph(_graph, _graph_ref)

         .edgeMap(_capacity, capacity)

         .edgeMap(_cost, cost)

         .nodeMap(_supply, supply)

         .nodeRef(_node_ref)

         .edgeRef(_edge_ref)

         .run();

     }

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

     ///

     /// Simple constructor (with lower bounds).

     ///

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

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

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

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

     /// \param s The source node.

     /// \param t The target node.

     /// \param flow_value The required amount of flow from node \c s

     /// to node \c t (i.e. the supply of \c s and the demand of \c t).

     NetworkSimplex( const Graph &graph,

                     const LowerMap &lower,

                     const CapacityMap &capacity,

                     const CostMap &cost,

                     typename Graph::Node s,

                     typename Graph::Node t,

                     typename SupplyMap::Value flow_value ) :

       _graph(), _graph_ref(graph), _lower(&lower), _capacity(_graph),

       _cost(_graph), _supply(_graph), _flow(_graph),

       _potential(_graph), _depth(_graph), _parent(_graph),

       _pred_edge(_graph), _thread(_graph), _forward(_graph),

       _state(_graph), _red_cost(_graph, _cost, _potential),

       _flow_result(0), _potential_result(0),

       _local_flow(false), _local_potential(false),

       _node_ref(graph), _edge_ref(graph)

     {

       // Copying _graph_ref to graph

       copyGraph(_graph, _graph_ref)

         .edgeMap(_cost, cost)

         .nodeRef(_node_ref)

         .edgeRef(_edge_ref)

         .run();

       // Removing non-zero lower bounds

       for (typename Graph::EdgeIt e(_graph_ref); e != INVALID; ++e) {

         _capacity[_edge_ref[e]] = capacity[e] - lower[e];

       }

       for (typename Graph::NodeIt n(_graph_ref); n != INVALID; ++n) {

         Supply sum = 0;

         if (n == s) sum =  flow_value;

         if (n == t) sum = -flow_value;

         for (typename Graph::InEdgeIt e(_graph_ref, n); e != INVALID; ++e)

           sum += lower[e];

         for (typename Graph::OutEdgeIt e(_graph_ref, n); e != INVALID; ++e)

           sum -= lower[e];

         _supply[_node_ref[n]] = sum;

       }

       _valid_supply = true;

     }

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

     ///

     /// Simple constructor (without lower bounds).

     ///

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

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

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

     /// \param s The source node.

     /// \param t The target node.

     /// \param flow_value The required amount of flow from node \c s

     /// to node \c t (i.e. the supply of \c s and the demand of \c t).

     NetworkSimplex( const Graph &graph,

                     const CapacityMap &capacity,

                     const CostMap &cost,

                     typename Graph::Node s,

                     typename Graph::Node t,

                     typename SupplyMap::Value flow_value ) :

       _graph(), _graph_ref(graph), _lower(NULL), _capacity(_graph),

       _cost(_graph), _supply(_graph, 0), _flow(_graph),

       _potential(_graph), _depth(_graph), _parent(_graph),

       _pred_edge(_graph), _thread(_graph), _forward(_graph),

       _state(_graph), _red_cost(_graph, _cost, _potential),

       _flow_result(0), _potential_result(0),

       _local_flow(false), _local_potential(false),

       _node_ref(graph), _edge_ref(graph)

     {

       // Copying _graph_ref to graph

       copyGraph(_graph, _graph_ref)

         .edgeMap(_capacity, capacity)

         .edgeMap(_cost, cost)

         .nodeRef(_node_ref)

         .edgeRef(_edge_ref)

         .run();

       _supply[_node_ref[s]] =  flow_value;

       _supply[_node_ref[t]] = -flow_value;

       _valid_supply = true;

     }

     /// Destructor.

     ~NetworkSimplex() {

       if (_local_flow) delete _flow_result;

       if (_local_potential) delete _potential_result;

     }

     /// \brief Sets the flow map.

     ///

     /// Sets the flow map.

     ///

     /// \return \c (*this)

     NetworkSimplex& flowMap(FlowMap &map) {

       if (_local_flow) {

         delete _flow_result;

         _local_flow = false;

       }

       _flow_result = ↦

       return *this;

     }

     /// \brief Sets the potential map.

     ///

     /// Sets the potential map.

     ///

     /// \return \c (*this)

     NetworkSimplex& potentialMap(PotentialMap &map) {

       if (_local_potential) {

         delete _potential_result;

         _local_potential = false;

       }

       _potential_result = ↦

       return *this;

     }

     /// \name Execution control

     /// The only way to execute the algorithm is to call the run()

     /// function.

     /// @{

     /// \brief Runs the algorithm.

     ///

     /// Runs the algorithm.

     ///

     /// \param pivot_rule The pivot rule that is used during the

     /// algorithm.

     ///

     /// The available pivot rules:

     ///

     /// - FIRST_ELIGIBLE_PIVOT The next eligible edge is selected in

     /// a wraparound fashion in every iteration

     /// (\ref FirstEligiblePivotRule).

     ///

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

     /// every iteration (\ref BestEligiblePivotRule).

     ///

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

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

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

     ///

     /// - LIMITED_SEARCH_PIVOT A specified number of eligible edges are

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

     /// one is selected from them (\ref LimitedSearchPivotRule).

     ///

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

     /// built from eligible edges and it is used for edge selection in

     /// the following minor iterations (\ref CandidateListPivotRule).

     ///

     /// - COMBINED_PIVOT This is a combined version of the two fastest

     /// pivot rules.

     /// For rather sparse graphs \ref LimitedSearchPivotRule

     /// "Limited Search" implementation is used, otherwise

     /// \ref BlockSearchPivotRule "Block Search" pivot rule is used.

     /// According to our benchmark tests this combined method is the

     /// most efficient.

     ///

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

     bool run(PivotRuleEnum pivot_rule = COMBINED_PIVOT) {

       return init() && start(pivot_rule);

     }

     /// @}

     /// \name Query Functions

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

     /// functions.

     /// \n run() must be called before using them.

     /// @{

     /// \brief Returns a const reference to the edge map storing the

     /// found flow.

     ///

     /// Returns a const reference to the edge map storing the found flow.

     ///

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

     const FlowMap& flowMap() const {

       return *_flow_result;

     }

     /// \brief Returns a const reference to the node map storing the

     /// found potentials (the dual solution).

     ///

     /// Returns a const reference to the node map storing the found

     /// potentials (the dual solution).

     ///

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

     const PotentialMap& potentialMap() const {

       return *_potential_result;

     }

     /// \brief Returns the flow on the given edge.

     ///

     /// Returns the flow on the given edge.

     ///

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

     Capacity flow(const typename Graph::Edge& edge) const {

       return (*_flow_result)[edge];

     }

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

     ///

     /// Returns the potential of the given node.

     ///

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

     Cost potential(const typename Graph::Node& node) const {

       return (*_potential_result)[node];

     }

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

     ///

     /// Returns the total cost of the found flow. The complexity of the

     /// function is \f$ O(e) \f$.

     ///

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

     Cost totalCost() const {

       Cost c = 0;

       for (typename Graph::EdgeIt e(_graph_ref); e != INVALID; ++e)

         c += (*_flow_result)[e] * _cost[_edge_ref[e]];

       return c;

     }

     /// @}

   private:

     /// \brief Extends the underlaying graph and initializes all the

     /// node and edge maps.

     bool init() {

       if (!_valid_supply) return false;

       // Initializing result maps

       if (!_flow_result) {

         _flow_result = new FlowMap(_graph_ref);

         _local_flow = true;

       }

       if (!_potential_result) {

         _potential_result = new PotentialMap(_graph_ref);

         _local_potential = true;

       }

       // Initializing state and flow maps

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

         _flow[e] = 0;

         _state[e] = STATE_LOWER;

       }

       // Adding an artificial root node to the graph

       _root = _graph.addNode();

       _parent[_root] = INVALID;

       _pred_edge[_root] = INVALID;

       _depth[_root] = 0;

       _supply[_root] = 0;

       _potential[_root] = 0;

       // Adding artificial edges to the graph and initializing the node

       // maps of the spanning tree data structure

       Node last = _root;

       Edge e;

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

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

         if (u == _root) continue;

         _thread[last] = u;

         last = u;

         _parent[u] = _root;

         _depth[u] = 1;

         if (_supply[u] >= 0) {

           e = _graph.addEdge(u, _root);

           _flow[e] = _supply[u];

           _forward[u] = true;

           _potential[u] = -max_cost;

         } else {

           e = _graph.addEdge(_root, u);

           _flow[e] = -_supply[u];

           _forward[u] = false;

           _potential[u] = max_cost;

         }

         _cost[e] = max_cost;

         _capacity[e] = std::numeric_limits<Capacity>::max();

         _state[e] = STATE_TREE;

         _pred_edge[u] = e;

       }

       _thread[last] = _root;

       return true;

     }

     /// Finds the join node.

     Node findJoinNode() {

       Node u = _graph.source(_in_edge);

       Node v = _graph.target(_in_edge);

       while (u != v) {

         if (_depth[u] == _depth[v]) {

           u = _parent[u];

           v = _parent[v];

         }

         else if (_depth[u] > _depth[v]) u = _parent[u];

         else v = _parent[v];

       }

       return u;

     }

     /// \brief Finds the leaving edge of the cycle. Returns \c true if

     /// the leaving edge is not the same as the entering edge.

     bool findLeavingEdge() {

       // Initializing first and second nodes according to the direction

       // of the cycle

       if (_state[_in_edge] == STATE_LOWER) {

         first  = _graph.source(_in_edge);

         second = _graph.target(_in_edge);

       } else {

         first  = _graph.target(_in_edge);

         second = _graph.source(_in_edge);

       }

       delta = _capacity[_in_edge];

       bool result = false;

       Capacity d;

       Edge e;

       // Searching the cycle along the path form the first node to the

       // root node

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

         e = _pred_edge[u];

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

         if (d < delta) {

           delta = d;

           u_out = u;

           u_in = first;

           v_in = second;

           result = true;

         }

       }

       // Searching the cycle along the path form the second node to the

       // root node

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

         e = _pred_edge[u];

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

         if (d <= delta) {

           delta = d;

           u_out = u;

           u_in = second;

           v_in = first;

           result = true;

         }

       }

       return result;

     }

     /// Changes \c flow and \c state edge maps.

     void changeFlows(bool change) {

       // Augmenting along the cycle

       if (delta > 0) {

         Capacity val = _state[_in_edge] * delta;

         _flow[_in_edge] += val;

         for (Node u = _graph.source(_in_edge); u != join; u = _parent[u]) {

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

         }

         for (Node u = _graph.target(_in_edge); u != join; u = _parent[u]) {

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

         }

       }

       // Updating the state of the entering and leaving edges

       if (change) {

         _state[_in_edge] = STATE_TREE;

         _state[_pred_edge[u_out]] =

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

       } else {

         _state[_in_edge] = -_state[_in_edge];

       }

     }

     /// Updates \c thread and \c parent node maps.

     void updateThreadParent() {

       Node u;

       v_out = _parent[u_out];

       // Handling the case when join and v_out coincide

       bool par_first = false;

       if (join == v_out) {

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

         if (u == v_in) {

           par_first = true;

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

           first = u;

         }

       }

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

       // (right) according to the thread index

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

       right = _thread[u];

       if (_thread[v_in] == u_out) {

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

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

       }

       else last = _thread[v_in];

       // Updating stem nodes

       _thread[v_in] = stem = u_in;

       par_stem = v_in;

       while (stem != u_out) {

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

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

         // the original thread index

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

         _thread[u] = right;

         // Changing the parent node of stem and shifting stem and

         // par_stem nodes

         _parent[stem] = par_stem;

         par_stem = stem;

         stem = new_stem;

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

         // (right) according to the thread index

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

         right = _thread[u];

       }

       _parent[u_out] = par_stem;

       _thread[u] = last;

       if (join == v_out && par_first) {

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

       } else {

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

         _thread[u] = right;

       }

     }

     /// Updates \c pred_edge and \c forward node maps.

     void updatePredEdge() {

       Node u = u_out, v;

       while (u != u_in) {

         v = _parent[u];

         _pred_edge[u] = _pred_edge[v];

         _forward[u] = !_forward[v];

         u = v;

       }

       _pred_edge[u_in] = _in_edge;

       _forward[u_in] = (u_in == _graph.source(_in_edge));

     }

     /// Updates \c depth and \c potential node maps.

     void updateDepthPotential() {

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

       _potential[u_in] = _forward[u_in] ?

         _potential[v_in] - _cost[_pred_edge[u_in]] :

         _potential[v_in] + _cost[_pred_edge[u_in]];

       Node u = _thread[u_in], v;

       while (true) {

         v = _parent[u];

         if (v == INVALID) break;

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

         _potential[u] = _forward[u] ?

           _potential[v] - _cost[_pred_edge[u]] :

           _potential[v] + _cost[_pred_edge[u]];

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

         u = _thread[u];

       }

     }

     /// Executes the algorithm.

     bool start(PivotRuleEnum pivot_rule) {

       switch (pivot_rule) {

         case FIRST_ELIGIBLE_PIVOT:

           return start<FirstEligiblePivotRule>();

         case BEST_ELIGIBLE_PIVOT:

           return start<BestEligiblePivotRule>();

         case BLOCK_SEARCH_PIVOT:

           return start<BlockSearchPivotRule>();

         case LIMITED_SEARCH_PIVOT:

           return start<LimitedSearchPivotRule>();

         case CANDIDATE_LIST_PIVOT:

           return start<CandidateListPivotRule>();

         case COMBINED_PIVOT:

           if ( countEdges(_graph) / countNodes(_graph) <=

                COMBINED_PIVOT_MAX_DEG )

             return start<LimitedSearchPivotRule>();

           else

             return start<BlockSearchPivotRule>();

       }

       return false;

     }

     template<class PivotRuleImplementation>

     bool start() {

       PivotRuleImplementation pivot(*this);

       // Executing the network simplex algorithm

       while (pivot.findEnteringEdge()) {

         join = findJoinNode();

         bool change = findLeavingEdge();

         changeFlows(change);

         if (change) {

           updateThreadParent();

           updatePredEdge();

           updateDepthPotential();

         }

       }

       // Checking if the flow amount equals zero on all the artificial

       // edges

       for (InEdgeIt e(_graph, _root); e != INVALID; ++e)

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

       for (OutEdgeIt e(_graph, _root); e != INVALID; ++e)

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

       // Copying flow values to _flow_result

       if (_lower) {

         for (typename Graph::EdgeIt e(_graph_ref); e != INVALID; ++e)

           (*_flow_result)[e] = (*_lower)[e] + _flow[_edge_ref[e]];

       } else {

         for (typename Graph::EdgeIt e(_graph_ref); e != INVALID; ++e)

           (*_flow_result)[e] = _flow[_edge_ref[e]];

       }

       // Copying potential values to _potential_result

       for (typename Graph::NodeIt n(_graph_ref); n != INVALID; ++n)

         (*_potential_result)[n] = _potential[_node_ref[n]];

       return true;

     }

   }; //class NetworkSimplex

   ///@}

 } //namespace lemon

 #endif //LEMON_NETWORK_SIMPLEX_H