diff -r 6a17a722b50e -r e8349c6f12ca lemon/network_simplex.h
--- /dev/null	Thu Jan 01 00:00:00 1970 +0000
+++ b/lemon/network_simplex.h	Tue Feb 24 09:46:02 2009 +0100
@@ -0,0 +1,1191 @@
+/* -*- mode: C++; indent-tabs-mode: nil; -*-
+ *
+ * This file is a part of LEMON, a generic C++ optimization library.
+ *
+ * Copyright (C) 2003-2009
+ * Egervary Jeno Kombinatorikus Optimalizalasi Kutatocsoport
+ * (Egervary Research Group on Combinatorial Optimization, EGRES).
+ *
+ * Permission to use, modify and distribute this software is granted
+ * provided that this copyright notice appears in all copies. For
+ * precise terms see the accompanying LICENSE file.
+ *
+ * This software is provided "AS IS" with no warranty of any kind,
+ * express or implied, and with no claim as to its suitability for any
+ * purpose.
+ *
+ */
+
+#ifndef LEMON_NETWORK_SIMPLEX_H
+#define LEMON_NETWORK_SIMPLEX_H
+
+/// \ingroup min_cost_flow
+///
+/// \file
+/// \brief Network simplex algorithm for finding a minimum cost flow.
+
+#include <vector>
+#include <limits>
+#include <algorithm>
+
+#include <lemon/math.h>
+
+namespace lemon {
+
+  /// \addtogroup min_cost_flow
+  /// @{
+
+  /// \brief Implementation of the primal network simplex algorithm
+  /// for finding a \ref min_cost_flow "minimum cost flow".
+  ///
+  /// \ref NetworkSimplex implements the primal network simplex algorithm
+  /// for finding a \ref min_cost_flow "minimum cost flow".
+  ///
+  /// \tparam Digraph The digraph type the algorithm runs on.
+  /// \tparam LowerMap The type of the lower bound map.
+  /// \tparam CapacityMap The type of the capacity (upper bound) map.
+  /// \tparam CostMap The type of the cost (length) map.
+  /// \tparam SupplyMap The type of the supply map.
+  ///
+  /// \warning
+  /// - Arc capacities and costs should be \e non-negative \e integers.
+  /// - Supply values should be \e signed \e integers.
+  /// - The value types of the maps should be convertible to each other.
+  /// - \c CostMap::Value must be signed type.
+  ///
+  /// \note \ref NetworkSimplex provides five different pivot rule
+  /// implementations that significantly affect the efficiency of the
+  /// algorithm.
+  /// By default "Block Search" pivot rule is used, which proved to be
+  /// by far the most efficient according to our benchmark tests.
+  /// However another pivot rule can be selected using \ref run()
+  /// function with the proper parameter.
+#ifdef DOXYGEN
+  template < typename Digraph,
+             typename LowerMap,
+             typename CapacityMap,
+             typename CostMap,
+             typename SupplyMap >
+
+#else
+  template < typename Digraph,
+             typename LowerMap = typename Digraph::template ArcMap<int>,
+             typename CapacityMap = typename Digraph::template ArcMap<int>,
+             typename CostMap = typename Digraph::template ArcMap<int>,
+             typename SupplyMap = typename Digraph::template NodeMap<int> >
+#endif
+  class NetworkSimplex
+  {
+    TEMPLATE_DIGRAPH_TYPEDEFS(Digraph);
+
+    typedef typename CapacityMap::Value Capacity;
+    typedef typename CostMap::Value Cost;
+    typedef typename SupplyMap::Value Supply;
+
+    typedef std::vector<Arc> ArcVector;
+    typedef std::vector<Node> NodeVector;
+    typedef std::vector<int> IntVector;
+    typedef std::vector<bool> BoolVector;
+    typedef std::vector<Capacity> CapacityVector;
+    typedef std::vector<Cost> CostVector;
+    typedef std::vector<Supply> SupplyVector;
+
+  public:
+
+    /// The type of the flow map
+    typedef typename Digraph::template ArcMap<Capacity> FlowMap;
+    /// The type of the potential map
+    typedef typename Digraph::template NodeMap<Cost> PotentialMap;
+
+  public:
+
+    /// Enum type for selecting the pivot rule used by \ref run()
+    enum PivotRuleEnum {
+      FIRST_ELIGIBLE_PIVOT,
+      BEST_ELIGIBLE_PIVOT,
+      BLOCK_SEARCH_PIVOT,
+      CANDIDATE_LIST_PIVOT,
+      ALTERING_LIST_PIVOT
+    };
+
+  private:
+
+    // State constants for arcs
+    enum ArcStateEnum {
+      STATE_UPPER = -1,
+      STATE_TREE  =  0,
+      STATE_LOWER =  1
+    };
+
+  private:
+
+    // References for the original data
+    const Digraph &_orig_graph;
+    const LowerMap *_orig_lower;
+    const CapacityMap &_orig_cap;
+    const CostMap &_orig_cost;
+    const SupplyMap *_orig_supply;
+    Node _orig_source;
+    Node _orig_target;
+    Capacity _orig_flow_value;
+
+    // Result maps
+    FlowMap *_flow_result;
+    PotentialMap *_potential_result;
+    bool _local_flow;
+    bool _local_potential;
+
+    // Data structures for storing the graph
+    ArcVector _arc;
+    NodeVector _node;
+    IntNodeMap _node_id;
+    IntVector _source;
+    IntVector _target;
+
+    // The number of nodes and arcs in the original graph
+    int _node_num;
+    int _arc_num;
+
+    // Node and arc maps
+    CapacityVector _cap;
+    CostVector _cost;
+    CostVector _supply;
+    CapacityVector _flow;
+    CostVector _pi;
+
+    // Node and arc maps for the spanning tree structure
+    IntVector _depth;
+    IntVector _parent;
+    IntVector _pred;
+    IntVector _thread;
+    BoolVector _forward;
+    IntVector _state;
+
+    // The root node
+    int _root;
+
+    // The entering arc in the current pivot iteration
+    int _in_arc;
+
+    // Temporary data used in the current pivot iteration
+    int join, u_in, v_in, u_out, v_out;
+    int right, first, second, last;
+    int stem, par_stem, new_stem;
+    Capacity delta;
+
+  private:
+
+    /// \brief Implementation of the "First Eligible" pivot rule for the
+    /// \ref NetworkSimplex "network simplex" algorithm.
+    ///
+    /// This class implements the "First Eligible" pivot rule
+    /// for the \ref NetworkSimplex "network simplex" algorithm.
+    ///
+    /// For more information see \ref NetworkSimplex::run().
+    class FirstEligiblePivotRule
+    {
+    private:
+
+      // References to the NetworkSimplex class
+      const ArcVector &_arc;
+      const IntVector  &_source;
+      const IntVector  &_target;
+      const CostVector &_cost;
+      const IntVector  &_state;
+      const CostVector &_pi;
+      int &_in_arc;
+      int _arc_num;
+
+      // Pivot rule data
+      int _next_arc;
+
+    public:
+
+      /// Constructor
+      FirstEligiblePivotRule(NetworkSimplex &ns) :
+        _arc(ns._arc), _source(ns._source), _target(ns._target),
+        _cost(ns._cost), _state(ns._state), _pi(ns._pi),
+        _in_arc(ns._in_arc), _arc_num(ns._arc_num), _next_arc(0)
+      {}
+
+      /// Find next entering arc
+      bool findEnteringArc() {
+        Cost c;
+        for (int e = _next_arc; e < _arc_num; ++e) {
+          c = _state[e] * (_cost[e] + _pi[_source[e]] - _pi[_target[e]]);
+          if (c < 0) {
+            _in_arc = e;
+            _next_arc = e + 1;
+            return true;
+          }
+        }
+        for (int e = 0; e < _next_arc; ++e) {
+          c = _state[e] * (_cost[e] + _pi[_source[e]] - _pi[_target[e]]);
+          if (c < 0) {
+            _in_arc = e;
+            _next_arc = e + 1;
+            return true;
+          }
+        }
+        return false;
+      }
+
+    }; //class FirstEligiblePivotRule
+
+
+    /// \brief Implementation of the "Best Eligible" pivot rule for the
+    /// \ref NetworkSimplex "network simplex" algorithm.
+    ///
+    /// This class implements the "Best Eligible" pivot rule
+    /// for the \ref NetworkSimplex "network simplex" algorithm.
+    ///
+    /// For more information see \ref NetworkSimplex::run().
+    class BestEligiblePivotRule
+    {
+    private:
+
+      // References to the NetworkSimplex class
+      const ArcVector &_arc;
+      const IntVector  &_source;
+      const IntVector  &_target;
+      const CostVector &_cost;
+      const IntVector  &_state;
+      const CostVector &_pi;
+      int &_in_arc;
+      int _arc_num;
+
+    public:
+
+      /// Constructor
+      BestEligiblePivotRule(NetworkSimplex &ns) :
+        _arc(ns._arc), _source(ns._source), _target(ns._target),
+        _cost(ns._cost), _state(ns._state), _pi(ns._pi),
+        _in_arc(ns._in_arc), _arc_num(ns._arc_num)
+      {}
+
+      /// Find next entering arc
+      bool findEnteringArc() {
+        Cost c, min = 0;
+        for (int e = 0; e < _arc_num; ++e) {
+          c = _state[e] * (_cost[e] + _pi[_source[e]] - _pi[_target[e]]);
+          if (c < min) {
+            min = c;
+            _in_arc = e;
+          }
+        }
+        return min < 0;
+      }
+
+    }; //class BestEligiblePivotRule
+
+
+    /// \brief Implementation of the "Block Search" pivot rule for the
+    /// \ref NetworkSimplex "network simplex" algorithm.
+    ///
+    /// This class implements the "Block Search" pivot rule
+    /// for the \ref NetworkSimplex "network simplex" algorithm.
+    ///
+    /// For more information see \ref NetworkSimplex::run().
+    class BlockSearchPivotRule
+    {
+    private:
+
+      // References to the NetworkSimplex class
+      const ArcVector &_arc;
+      const IntVector  &_source;
+      const IntVector  &_target;
+      const CostVector &_cost;
+      const IntVector  &_state;
+      const CostVector &_pi;
+      int &_in_arc;
+      int _arc_num;
+
+      // Pivot rule data
+      int _block_size;
+      int _next_arc;
+
+    public:
+
+      /// Constructor
+      BlockSearchPivotRule(NetworkSimplex &ns) :
+        _arc(ns._arc), _source(ns._source), _target(ns._target),
+        _cost(ns._cost), _state(ns._state), _pi(ns._pi),
+        _in_arc(ns._in_arc), _arc_num(ns._arc_num), _next_arc(0)
+      {
+        // The main parameters of the pivot rule
+        const double BLOCK_SIZE_FACTOR = 2.0;
+        const int MIN_BLOCK_SIZE = 10;
+
+        _block_size = std::max( int(BLOCK_SIZE_FACTOR * sqrt(_arc_num)),
+                                MIN_BLOCK_SIZE );
+      }
+
+      /// Find next entering arc
+      bool findEnteringArc() {
+        Cost c, min = 0;
+        int cnt = _block_size;
+        int e, min_arc = _next_arc;
+        for (e = _next_arc; e < _arc_num; ++e) {
+          c = _state[e] * (_cost[e] + _pi[_source[e]] - _pi[_target[e]]);
+          if (c < min) {
+            min = c;
+            min_arc = e;
+          }
+          if (--cnt == 0) {
+            if (min < 0) break;
+            cnt = _block_size;
+          }
+        }
+        if (min == 0 || cnt > 0) {
+          for (e = 0; e < _next_arc; ++e) {
+            c = _state[e] * (_cost[e] + _pi[_source[e]] - _pi[_target[e]]);
+            if (c < min) {
+              min = c;
+              min_arc = e;
+            }
+            if (--cnt == 0) {
+              if (min < 0) break;
+              cnt = _block_size;
+            }
+          }
+        }
+        if (min >= 0) return false;
+        _in_arc = min_arc;
+        _next_arc = e;
+        return true;
+      }
+
+    }; //class BlockSearchPivotRule
+
+
+    /// \brief Implementation of the "Candidate List" pivot rule for the
+    /// \ref NetworkSimplex "network simplex" algorithm.
+    ///
+    /// This class implements the "Candidate List" pivot rule
+    /// for the \ref NetworkSimplex "network simplex" algorithm.
+    ///
+    /// For more information see \ref NetworkSimplex::run().
+    class CandidateListPivotRule
+    {
+    private:
+
+      // References to the NetworkSimplex class
+      const ArcVector &_arc;
+      const IntVector  &_source;
+      const IntVector  &_target;
+      const CostVector &_cost;
+      const IntVector  &_state;
+      const CostVector &_pi;
+      int &_in_arc;
+      int _arc_num;
+
+      // Pivot rule data
+      IntVector _candidates;
+      int _list_length, _minor_limit;
+      int _curr_length, _minor_count;
+      int _next_arc;
+
+    public:
+
+      /// Constructor
+      CandidateListPivotRule(NetworkSimplex &ns) :
+        _arc(ns._arc), _source(ns._source), _target(ns._target),
+        _cost(ns._cost), _state(ns._state), _pi(ns._pi),
+        _in_arc(ns._in_arc), _arc_num(ns._arc_num), _next_arc(0)
+      {
+        // The main parameters of the pivot rule
+        const double LIST_LENGTH_FACTOR = 1.0;
+        const int MIN_LIST_LENGTH = 10;
+        const double MINOR_LIMIT_FACTOR = 0.1;
+        const int MIN_MINOR_LIMIT = 3;
+
+        _list_length = std::max( int(LIST_LENGTH_FACTOR * sqrt(_arc_num)),
+                                 MIN_LIST_LENGTH );
+        _minor_limit = std::max( int(MINOR_LIMIT_FACTOR * _list_length),
+                                 MIN_MINOR_LIMIT );
+        _curr_length = _minor_count = 0;
+        _candidates.resize(_list_length);
+      }
+
+      /// Find next entering arc
+      bool findEnteringArc() {
+        Cost min, c;
+        int e, min_arc = _next_arc;
+        if (_curr_length > 0 && _minor_count < _minor_limit) {
+          // Minor iteration: select the best eligible arc from the
+          // current candidate list
+          ++_minor_count;
+          min = 0;
+          for (int i = 0; i < _curr_length; ++i) {
+            e = _candidates[i];
+            c = _state[e] * (_cost[e] + _pi[_source[e]] - _pi[_target[e]]);
+            if (c < min) {
+              min = c;
+              min_arc = e;
+            }
+            if (c >= 0) {
+              _candidates[i--] = _candidates[--_curr_length];
+            }
+          }
+          if (min < 0) {
+            _in_arc = min_arc;
+            return true;
+          }
+        }
+
+        // Major iteration: build a new candidate list
+        min = 0;
+        _curr_length = 0;
+        for (e = _next_arc; e < _arc_num; ++e) {
+          c = _state[e] * (_cost[e] + _pi[_source[e]] - _pi[_target[e]]);
+          if (c < 0) {
+            _candidates[_curr_length++] = e;
+            if (c < min) {
+              min = c;
+              min_arc = e;
+            }
+            if (_curr_length == _list_length) break;
+          }
+        }
+        if (_curr_length < _list_length) {
+          for (e = 0; e < _next_arc; ++e) {
+            c = _state[e] * (_cost[e] + _pi[_source[e]] - _pi[_target[e]]);
+            if (c < 0) {
+              _candidates[_curr_length++] = e;
+              if (c < min) {
+                min = c;
+                min_arc = e;
+              }
+              if (_curr_length == _list_length) break;
+            }
+          }
+        }
+        if (_curr_length == 0) return false;
+        _minor_count = 1;
+        _in_arc = min_arc;
+        _next_arc = e;
+        return true;
+      }
+
+    }; //class CandidateListPivotRule
+
+
+    /// \brief Implementation of the "Altering Candidate List" pivot rule
+    /// for the \ref NetworkSimplex "network simplex" algorithm.
+    ///
+    /// This class implements the "Altering Candidate List" pivot rule
+    /// for the \ref NetworkSimplex "network simplex" algorithm.
+    ///
+    /// For more information see \ref NetworkSimplex::run().
+    class AlteringListPivotRule
+    {
+    private:
+
+      // References to the NetworkSimplex class
+      const ArcVector &_arc;
+      const IntVector  &_source;
+      const IntVector  &_target;
+      const CostVector &_cost;
+      const IntVector  &_state;
+      const CostVector &_pi;
+      int &_in_arc;
+      int _arc_num;
+
+      // Pivot rule data
+      int _block_size, _head_length, _curr_length;
+      int _next_arc;
+      IntVector _candidates;
+      CostVector _cand_cost;
+
+      // Functor class to compare arcs during sort of the candidate list
+      class SortFunc
+      {
+      private:
+        const CostVector &_map;
+      public:
+        SortFunc(const CostVector &map) : _map(map) {}
+        bool operator()(int left, int right) {
+          return _map[left] > _map[right];
+        }
+      };
+
+      SortFunc _sort_func;
+
+    public:
+
+      /// Constructor
+      AlteringListPivotRule(NetworkSimplex &ns) :
+        _arc(ns._arc), _source(ns._source), _target(ns._target),
+        _cost(ns._cost), _state(ns._state), _pi(ns._pi),
+        _in_arc(ns._in_arc), _arc_num(ns._arc_num),
+        _next_arc(0), _cand_cost(ns._arc_num), _sort_func(_cand_cost)
+      {
+        // The main parameters of the pivot rule
+        const double BLOCK_SIZE_FACTOR = 1.5;
+        const int MIN_BLOCK_SIZE = 10;
+        const double HEAD_LENGTH_FACTOR = 0.1;
+        const int MIN_HEAD_LENGTH = 3;
+
+        _block_size = std::max( int(BLOCK_SIZE_FACTOR * sqrt(_arc_num)),
+                                MIN_BLOCK_SIZE );
+        _head_length = std::max( int(HEAD_LENGTH_FACTOR * _block_size),
+                                 MIN_HEAD_LENGTH );
+        _candidates.resize(_head_length + _block_size);
+        _curr_length = 0;
+      }
+
+      /// Find next entering arc
+      bool findEnteringArc() {
+        // Check the current candidate list
+        int e;
+        for (int i = 0; i < _curr_length; ++i) {
+          e = _candidates[i];
+          _cand_cost[e] = _state[e] *
+            (_cost[e] + _pi[_source[e]] - _pi[_target[e]]);
+          if (_cand_cost[e] >= 0) {
+            _candidates[i--] = _candidates[--_curr_length];
+          }
+        }
+
+        // Extend the list
+        int cnt = _block_size;
+        int last_edge = 0;
+        int limit = _head_length;
+
+        for (int e = _next_arc; e < _arc_num; ++e) {
+          _cand_cost[e] = _state[e] *
+            (_cost[e] + _pi[_source[e]] - _pi[_target[e]]);
+          if (_cand_cost[e] < 0) {
+            _candidates[_curr_length++] = e;
+            last_edge = e;
+          }
+          if (--cnt == 0) {
+            if (_curr_length > limit) break;
+            limit = 0;
+            cnt = _block_size;
+          }
+        }
+        if (_curr_length <= limit) {
+          for (int e = 0; e < _next_arc; ++e) {
+            _cand_cost[e] = _state[e] *
+              (_cost[e] + _pi[_source[e]] - _pi[_target[e]]);
+            if (_cand_cost[e] < 0) {
+              _candidates[_curr_length++] = e;
+              last_edge = e;
+            }
+            if (--cnt == 0) {
+              if (_curr_length > limit) break;
+              limit = 0;
+              cnt = _block_size;
+            }
+          }
+        }
+        if (_curr_length == 0) return false;
+        _next_arc = last_edge + 1;
+
+        // Make heap of the candidate list (approximating a partial sort)
+        make_heap( _candidates.begin(), _candidates.begin() + _curr_length,
+                   _sort_func );
+
+        // Pop the first element of the heap
+        _in_arc = _candidates[0];
+        pop_heap( _candidates.begin(), _candidates.begin() + _curr_length,
+                  _sort_func );
+        _curr_length = std::min(_head_length, _curr_length - 1);
+        return true;
+      }
+
+    }; //class AlteringListPivotRule
+
+  public:
+
+    /// \brief General constructor (with lower bounds).
+    ///
+    /// General constructor (with lower bounds).
+    ///
+    /// \param digraph The digraph the algorithm runs on.
+    /// \param lower The lower bounds of the arcs.
+    /// \param capacity The capacities (upper bounds) of the arcs.
+    /// \param cost The cost (length) values of the arcs.
+    /// \param supply The supply values of the nodes (signed).
+    NetworkSimplex( const Digraph &digraph,
+                    const LowerMap &lower,
+                    const CapacityMap &capacity,
+                    const CostMap &cost,
+                    const SupplyMap &supply ) :
+      _orig_graph(digraph), _orig_lower(&lower), _orig_cap(capacity),
+      _orig_cost(cost), _orig_supply(&supply),
+      _flow_result(NULL), _potential_result(NULL),
+      _local_flow(false), _local_potential(false),
+      _node_id(digraph)
+    {}
+
+    /// \brief General constructor (without lower bounds).
+    ///
+    /// General constructor (without lower bounds).
+    ///
+    /// \param digraph The digraph the algorithm runs on.
+    /// \param capacity The capacities (upper bounds) of the arcs.
+    /// \param cost The cost (length) values of the arcs.
+    /// \param supply The supply values of the nodes (signed).
+    NetworkSimplex( const Digraph &digraph,
+                    const CapacityMap &capacity,
+                    const CostMap &cost,
+                    const SupplyMap &supply ) :
+      _orig_graph(digraph), _orig_lower(NULL), _orig_cap(capacity),
+      _orig_cost(cost), _orig_supply(&supply),
+      _flow_result(NULL), _potential_result(NULL),
+      _local_flow(false), _local_potential(false),
+      _node_id(digraph)
+    {}
+
+    /// \brief Simple constructor (with lower bounds).
+    ///
+    /// Simple constructor (with lower bounds).
+    ///
+    /// \param digraph The digraph the algorithm runs on.
+    /// \param lower The lower bounds of the arcs.
+    /// \param capacity The capacities (upper bounds) of the arcs.
+    /// \param cost The cost (length) values of the arcs.
+    /// \param s The source node.
+    /// \param t The target node.
+    /// \param flow_value The required amount of flow from node \c s
+    /// to node \c t (i.e. the supply of \c s and the demand of \c t).
+    NetworkSimplex( const Digraph &digraph,
+                    const LowerMap &lower,
+                    const CapacityMap &capacity,
+                    const CostMap &cost,
+                    Node s, Node t,
+                    Capacity flow_value ) :
+      _orig_graph(digraph), _orig_lower(&lower), _orig_cap(capacity),
+      _orig_cost(cost), _orig_supply(NULL),
+      _orig_source(s), _orig_target(t), _orig_flow_value(flow_value),
+      _flow_result(NULL), _potential_result(NULL),
+      _local_flow(false), _local_potential(false),
+      _node_id(digraph)
+    {}
+
+    /// \brief Simple constructor (without lower bounds).
+    ///
+    /// Simple constructor (without lower bounds).
+    ///
+    /// \param digraph The digraph the algorithm runs on.
+    /// \param capacity The capacities (upper bounds) of the arcs.
+    /// \param cost The cost (length) values of the arcs.
+    /// \param s The source node.
+    /// \param t The target node.
+    /// \param flow_value The required amount of flow from node \c s
+    /// to node \c t (i.e. the supply of \c s and the demand of \c t).
+    NetworkSimplex( const Digraph &digraph,
+                    const CapacityMap &capacity,
+                    const CostMap &cost,
+                    Node s, Node t,
+                    Capacity flow_value ) :
+      _orig_graph(digraph), _orig_lower(NULL), _orig_cap(capacity),
+      _orig_cost(cost), _orig_supply(NULL),
+      _orig_source(s), _orig_target(t), _orig_flow_value(flow_value),
+      _flow_result(NULL), _potential_result(NULL),
+      _local_flow(false), _local_potential(false),
+      _node_id(digraph)
+    {}
+
+    /// Destructor.
+    ~NetworkSimplex() {
+      if (_local_flow) delete _flow_result;
+      if (_local_potential) delete _potential_result;
+    }
+
+    /// \brief Set the flow map.
+    ///
+    /// This function sets the flow map.
+    ///
+    /// \return <tt>(*this)</tt>
+    NetworkSimplex& flowMap(FlowMap &map) {
+      if (_local_flow) {
+        delete _flow_result;
+        _local_flow = false;
+      }
+      _flow_result = &map;
+      return *this;
+    }
+
+    /// \brief Set the potential map.
+    ///
+    /// This function sets the potential map.
+    ///
+    /// \return <tt>(*this)</tt>
+    NetworkSimplex& potentialMap(PotentialMap &map) {
+      if (_local_potential) {
+        delete _potential_result;
+        _local_potential = false;
+      }
+      _potential_result = &map;
+      return *this;
+    }
+
+    /// \name Execution control
+    /// The algorithm can be executed using the
+    /// \ref lemon::NetworkSimplex::run() "run()" function.
+    /// @{
+
+    /// \brief Run the algorithm.
+    ///
+    /// This function runs the algorithm.
+    ///
+    /// \param pivot_rule The pivot rule that is used during the
+    /// algorithm.
+    ///
+    /// The available pivot rules:
+    ///
+    /// - FIRST_ELIGIBLE_PIVOT The next eligible arc is selected in
+    /// a wraparound fashion in every iteration
+    /// (\ref FirstEligiblePivotRule).
+    ///
+    /// - BEST_ELIGIBLE_PIVOT The best eligible arc is selected in
+    /// every iteration (\ref BestEligiblePivotRule).
+    ///
+    /// - BLOCK_SEARCH_PIVOT A specified number of arcs are examined in
+    /// every iteration in a wraparound fashion and the best eligible
+    /// arc is selected from this block (\ref BlockSearchPivotRule).
+    ///
+    /// - CANDIDATE_LIST_PIVOT In a major iteration a candidate list is
+    /// built from eligible arcs in a wraparound fashion and in the
+    /// following minor iterations the best eligible arc is selected
+    /// from this list (\ref CandidateListPivotRule).
+    ///
+    /// - ALTERING_LIST_PIVOT It is a modified version of the
+    /// "Candidate List" pivot rule. It keeps only the several best
+    /// eligible arcs from the former candidate list and extends this
+    /// list in every iteration (\ref AlteringListPivotRule).
+    ///
+    /// According to our comprehensive benchmark tests the "Block Search"
+    /// pivot rule proved to be the fastest and the most robust on
+    /// various test inputs. Thus it is the default option.
+    ///
+    /// \return \c true if a feasible flow can be found.
+    bool run(PivotRuleEnum pivot_rule = BLOCK_SEARCH_PIVOT) {
+      return init() && start(pivot_rule);
+    }
+
+    /// @}
+
+    /// \name Query Functions
+    /// The results of the algorithm can be obtained using these
+    /// functions.\n
+    /// \ref lemon::NetworkSimplex::run() "run()" must be called before
+    /// using them.
+    /// @{
+
+    /// \brief Return a const reference to the flow map.
+    ///
+    /// This function returns a const reference to an arc map storing
+    /// the found flow.
+    ///
+    /// \pre \ref run() must be called before using this function.
+    const FlowMap& flowMap() const {
+      return *_flow_result;
+    }
+
+    /// \brief Return a const reference to the potential map
+    /// (the dual solution).
+    ///
+    /// This function returns a const reference to a node map storing
+    /// the found potentials (the dual solution).
+    ///
+    /// \pre \ref run() must be called before using this function.
+    const PotentialMap& potentialMap() const {
+      return *_potential_result;
+    }
+
+    /// \brief Return the flow on the given arc.
+    ///
+    /// This function returns the flow on the given arc.
+    ///
+    /// \pre \ref run() must be called before using this function.
+    Capacity flow(const Arc& arc) const {
+      return (*_flow_result)[arc];
+    }
+
+    /// \brief Return the potential of the given node.
+    ///
+    /// This function returns the potential of the given node.
+    ///
+    /// \pre \ref run() must be called before using this function.
+    Cost potential(const Node& node) const {
+      return (*_potential_result)[node];
+    }
+
+    /// \brief Return the total cost of the found flow.
+    ///
+    /// This function returns the total cost of the found flow.
+    /// The complexity of the function is \f$ O(e) \f$.
+    ///
+    /// \pre \ref run() must be called before using this function.
+    Cost totalCost() const {
+      Cost c = 0;
+      for (ArcIt e(_orig_graph); e != INVALID; ++e)
+        c += (*_flow_result)[e] * _orig_cost[e];
+      return c;
+    }
+
+    /// @}
+
+  private:
+
+    // Initialize internal data structures
+    bool init() {
+      // Initialize result maps
+      if (!_flow_result) {
+        _flow_result = new FlowMap(_orig_graph);
+        _local_flow = true;
+      }
+      if (!_potential_result) {
+        _potential_result = new PotentialMap(_orig_graph);
+        _local_potential = true;
+      }
+
+      // Initialize vectors
+      _node_num = countNodes(_orig_graph);
+      _arc_num = countArcs(_orig_graph);
+      int all_node_num = _node_num + 1;
+      int all_edge_num = _arc_num + _node_num;
+
+      _arc.resize(_arc_num);
+      _node.reserve(_node_num);
+      _source.resize(all_edge_num);
+      _target.resize(all_edge_num);
+
+      _cap.resize(all_edge_num);
+      _cost.resize(all_edge_num);
+      _supply.resize(all_node_num);
+      _flow.resize(all_edge_num, 0);
+      _pi.resize(all_node_num, 0);
+
+      _depth.resize(all_node_num);
+      _parent.resize(all_node_num);
+      _pred.resize(all_node_num);
+      _thread.resize(all_node_num);
+      _forward.resize(all_node_num);
+      _state.resize(all_edge_num, STATE_LOWER);
+
+      // Initialize node related data
+      bool valid_supply = true;
+      if (_orig_supply) {
+        Supply sum = 0;
+        int i = 0;
+        for (NodeIt n(_orig_graph); n != INVALID; ++n, ++i) {
+          _node.push_back(n);
+          _node_id[n] = i;
+          _supply[i] = (*_orig_supply)[n];
+          sum += _supply[i];
+        }
+        valid_supply = (sum == 0);
+      } else {
+        int i = 0;
+        for (NodeIt n(_orig_graph); n != INVALID; ++n, ++i) {
+          _node.push_back(n);
+          _node_id[n] = i;
+          _supply[i] = 0;
+        }
+        _supply[_node_id[_orig_source]] =  _orig_flow_value;
+        _supply[_node_id[_orig_target]] = -_orig_flow_value;
+      }
+      if (!valid_supply) return false;
+
+      // Set data for the artificial root node
+      _root = _node_num;
+      _depth[_root] = 0;
+      _parent[_root] = -1;
+      _pred[_root] = -1;
+      _thread[_root] = 0;
+      _supply[_root] = 0;
+      _pi[_root] = 0;
+
+      // Store the arcs in a mixed order
+      int k = std::max(int(sqrt(_arc_num)), 10);
+      int i = 0;
+      for (ArcIt e(_orig_graph); e != INVALID; ++e) {
+        _arc[i] = e;
+        if ((i += k) >= _arc_num) i = (i % k) + 1;
+      }
+
+      // Initialize arc maps
+      for (int i = 0; i != _arc_num; ++i) {
+        Arc e = _arc[i];
+        _source[i] = _node_id[_orig_graph.source(e)];
+        _target[i] = _node_id[_orig_graph.target(e)];
+        _cost[i] = _orig_cost[e];
+        _cap[i] = _orig_cap[e];
+      }
+
+      // Remove non-zero lower bounds
+      if (_orig_lower) {
+        for (int i = 0; i != _arc_num; ++i) {
+          Capacity c = (*_orig_lower)[_arc[i]];
+          if (c != 0) {
+            _cap[i] -= c;
+            _supply[_source[i]] -= c;
+            _supply[_target[i]] += c;
+          }
+        }
+      }
+
+      // Add artificial arcs and initialize the spanning tree data structure
+      Cost max_cost = std::numeric_limits<Cost>::max() / 4;
+      Capacity max_cap = std::numeric_limits<Capacity>::max();
+      for (int u = 0, e = _arc_num; u != _node_num; ++u, ++e) {
+        _thread[u] = u + 1;
+        _depth[u] = 1;
+        _parent[u] = _root;
+        _pred[u] = e;
+        if (_supply[u] >= 0) {
+          _flow[e] = _supply[u];
+          _forward[u] = true;
+          _pi[u] = -max_cost;
+        } else {
+          _flow[e] = -_supply[u];
+          _forward[u] = false;
+          _pi[u] = max_cost;
+        }
+        _cost[e] = max_cost;
+        _cap[e] = max_cap;
+        _state[e] = STATE_TREE;
+      }
+
+      return true;
+    }
+
+    // Find the join node
+    void findJoinNode() {
+      int u = _source[_in_arc];
+      int v = _target[_in_arc];
+      while (_depth[u] > _depth[v]) u = _parent[u];
+      while (_depth[v] > _depth[u]) v = _parent[v];
+      while (u != v) {
+        u = _parent[u];
+        v = _parent[v];
+      }
+      join = u;
+    }
+
+    // Find the leaving arc of the cycle and returns true if the
+    // leaving arc is not the same as the entering arc
+    bool findLeavingArc() {
+      // Initialize first and second nodes according to the direction
+      // of the cycle
+      if (_state[_in_arc] == STATE_LOWER) {
+        first  = _source[_in_arc];
+        second = _target[_in_arc];
+      } else {
+        first  = _target[_in_arc];
+        second = _source[_in_arc];
+      }
+      delta = _cap[_in_arc];
+      int result = 0;
+      Capacity d;
+      int e;
+
+      // Search the cycle along the path form the first node to the root
+      for (int u = first; u != join; u = _parent[u]) {
+        e = _pred[u];
+        d = _forward[u] ? _flow[e] : _cap[e] - _flow[e];
+        if (d < delta) {
+          delta = d;
+          u_out = u;
+          result = 1;
+        }
+      }
+      // Search the cycle along the path form the second node to the root
+      for (int u = second; u != join; u = _parent[u]) {
+        e = _pred[u];
+        d = _forward[u] ? _cap[e] - _flow[e] : _flow[e];
+        if (d <= delta) {
+          delta = d;
+          u_out = u;
+          result = 2;
+        }
+      }
+
+      if (result == 1) {
+        u_in = first;
+        v_in = second;
+      } else {
+        u_in = second;
+        v_in = first;
+      }
+      return result != 0;
+    }
+
+    // Change _flow and _state vectors
+    void changeFlow(bool change) {
+      // Augment along the cycle
+      if (delta > 0) {
+        Capacity val = _state[_in_arc] * delta;
+        _flow[_in_arc] += val;
+        for (int u = _source[_in_arc]; u != join; u = _parent[u]) {
+          _flow[_pred[u]] += _forward[u] ? -val : val;
+        }
+        for (int u = _target[_in_arc]; u != join; u = _parent[u]) {
+          _flow[_pred[u]] += _forward[u] ? val : -val;
+        }
+      }
+      // Update the state of the entering and leaving arcs
+      if (change) {
+        _state[_in_arc] = STATE_TREE;
+        _state[_pred[u_out]] =
+          (_flow[_pred[u_out]] == 0) ? STATE_LOWER : STATE_UPPER;
+      } else {
+        _state[_in_arc] = -_state[_in_arc];
+      }
+    }
+
+    // Update _thread and _parent vectors
+    void updateThreadParent() {
+      int u;
+      v_out = _parent[u_out];
+
+      // Handle the case when join and v_out coincide
+      bool par_first = false;
+      if (join == v_out) {
+        for (u = join; u != u_in && u != v_in; u = _thread[u]) ;
+        if (u == v_in) {
+          par_first = true;
+          while (_thread[u] != u_out) u = _thread[u];
+          first = u;
+        }
+      }
+
+      // Find the last successor of u_in (u) and the node after it (right)
+      // according to the thread index
+      for (u = u_in; _depth[_thread[u]] > _depth[u_in]; u = _thread[u]) ;
+      right = _thread[u];
+      if (_thread[v_in] == u_out) {
+        for (last = u; _depth[last] > _depth[u_out]; last = _thread[last]) ;
+        if (last == u_out) last = _thread[last];
+      }
+      else last = _thread[v_in];
+
+      // Update stem nodes
+      _thread[v_in] = stem = u_in;
+      par_stem = v_in;
+      while (stem != u_out) {
+        _thread[u] = new_stem = _parent[stem];
+
+        // Find the node just before the stem node (u) according to
+        // the original thread index
+        for (u = new_stem; _thread[u] != stem; u = _thread[u]) ;
+        _thread[u] = right;
+
+        // Change the parent node of stem and shift stem and par_stem nodes
+        _parent[stem] = par_stem;
+        par_stem = stem;
+        stem = new_stem;
+
+        // Find the last successor of stem (u) and the node after it (right)
+        // according to the thread index
+        for (u = stem; _depth[_thread[u]] > _depth[stem]; u = _thread[u]) ;
+        right = _thread[u];
+      }
+      _parent[u_out] = par_stem;
+      _thread[u] = last;
+
+      if (join == v_out && par_first) {
+        if (first != v_in) _thread[first] = right;
+      } else {
+        for (u = v_out; _thread[u] != u_out; u = _thread[u]) ;
+        _thread[u] = right;
+      }
+    }
+
+    // Update _pred and _forward vectors
+    void updatePredArc() {
+      int u = u_out, v;
+      while (u != u_in) {
+        v = _parent[u];
+        _pred[u] = _pred[v];
+        _forward[u] = !_forward[v];
+        u = v;
+      }
+      _pred[u_in] = _in_arc;
+      _forward[u_in] = (u_in == _source[_in_arc]);
+    }
+
+    // Update _depth and _potential vectors
+    void updateDepthPotential() {
+      _depth[u_in] = _depth[v_in] + 1;
+      Cost sigma = _forward[u_in] ?
+        _pi[v_in] - _pi[u_in] - _cost[_pred[u_in]] :
+        _pi[v_in] - _pi[u_in] + _cost[_pred[u_in]];
+      _pi[u_in] += sigma;
+      for(int u = _thread[u_in]; _parent[u] != -1; u = _thread[u]) {
+        _depth[u] = _depth[_parent[u]] + 1;
+        if (_depth[u] <= _depth[u_in]) break;
+        _pi[u] += sigma;
+      }
+    }
+
+    // Execute the algorithm
+    bool start(PivotRuleEnum pivot_rule) {
+      // Select the pivot rule implementation
+      switch (pivot_rule) {
+        case FIRST_ELIGIBLE_PIVOT:
+          return start<FirstEligiblePivotRule>();
+        case BEST_ELIGIBLE_PIVOT:
+          return start<BestEligiblePivotRule>();
+        case BLOCK_SEARCH_PIVOT:
+          return start<BlockSearchPivotRule>();
+        case CANDIDATE_LIST_PIVOT:
+          return start<CandidateListPivotRule>();
+        case ALTERING_LIST_PIVOT:
+          return start<AlteringListPivotRule>();
+      }
+      return false;
+    }
+
+    template<class PivotRuleImplementation>
+    bool start() {
+      PivotRuleImplementation pivot(*this);
+
+      // Execute the network simplex algorithm
+      while (pivot.findEnteringArc()) {
+        findJoinNode();
+        bool change = findLeavingArc();
+        changeFlow(change);
+        if (change) {
+          updateThreadParent();
+          updatePredArc();
+          updateDepthPotential();
+        }
+      }
+
+      // Check if the flow amount equals zero on all the artificial arcs
+      for (int e = _arc_num; e != _arc_num + _node_num; ++e) {
+        if (_flow[e] > 0) return false;
+      }
+
+      // Copy flow values to _flow_result
+      if (_orig_lower) {
+        for (int i = 0; i != _arc_num; ++i) {
+          Arc e = _arc[i];
+          (*_flow_result)[e] = (*_orig_lower)[e] + _flow[i];
+        }
+      } else {
+        for (int i = 0; i != _arc_num; ++i) {
+          (*_flow_result)[_arc[i]] = _flow[i];
+        }
+      }
+      // Copy potential values to _potential_result
+      for (int i = 0; i != _node_num; ++i) {
+        (*_potential_result)[_node[i]] = _pi[i];
+      }
+
+      return true;
+    }
+
+  }; //class NetworkSimplex
+
+  ///@}
+
+} //namespace lemon
+
+#endif //LEMON_NETWORK_SIMPLEX_H