source: lemon/lemon/network_simplex.h @ 775:e2bdd1a988f3

/* -*- 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_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".
  /// This algorithm is a specialized version of the linear programming
  /// simplex method directly for the minimum cost flow problem.
  /// It is one of the most efficient solution methods.
  ///
  /// In general this class is the fastest implementation available
  /// in LEMON for the minimum cost flow problem.
  /// Moreover it 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 value type used for flow amounts, capacity bounds
  /// and supply values in the algorithm. By default it is \c int.
  /// \tparam C The value type used for costs and potentials in the
  /// algorithm. By default it is the same as \c V.
  ///
  /// \warning Both value types must be signed and all input data 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 pivot rule
    /// implementations that significantly affect the running time
    /// of the algorithm.
    /// By default \ref BLOCK_SEARCH "Block Search" is used, which
    /// proved to be the most efficient and the most robust on various
    /// test inputs according to our benchmark tests.
    /// However another pivot rule can be selected using the \ref run()
    /// function with the proper parameter.
    enum PivotRule {

      /// The First Eligible pivot rule.
      /// The next eligible arc is selected in a wraparound fashion
      /// in every iteration.
      FIRST_ELIGIBLE,

      /// The Best Eligible pivot rule.
      /// The best eligible arc is selected in every iteration.
      BEST_ELIGIBLE,

      /// The Block Search pivot rule.
      /// A specified number of arcs are examined in every iteration
      /// in a wraparound fashion and the best eligible arc is selected
      /// from this block.
      BLOCK_SEARCH,

      /// The Candidate List pivot rule.
      /// In a major iteration a candidate list is built from eligible arcs
      /// in a wraparound fashion and in the following minor iterations
      /// the best eligible arc is selected from this list.
      CANDIDATE_LIST,

      /// The Altering Candidate List pivot rule.
      /// It is a modified version of the Candidate List method.
      /// It keeps only the several best eligible arcs from the former
      /// candidate list and extends this list in every iteration.
      ALTERING_LIST
    };
    
  private:

    TEMPLATE_DIGRAPH_TYPEDEFS(GR);

    typedef std::vector<Arc> ArcVector;
    typedef std::vector<Node> NodeVector;
    typedef std::vector<int> IntVector;
    typedef std::vector<bool> BoolVector;
    typedef std::vector<Value> ValueVector;
    typedef std::vector<Cost> CostVector;

    // State constants for arcs
    enum ArcStateEnum {
      STATE_UPPER = -1,
      STATE_TREE  =  0,
      STATE_LOWER =  1
    };

  private:

    // Data related to the underlying digraph
    const GR &_graph;
    int _node_num;
    int _arc_num;
    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;

    // 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;
    IntVector _dirty_revs;
    BoolVector _forward;
    IntVector _state;
    int _root;

    // Temporary data used in the current pivot iteration
    int in_arc, join, u_in, v_in, u_out, v_out;
    int first, second, right, last;
    int stem, par_stem, new_stem;
    Value delta;

  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 IntVector  &_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 IntVector  &_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 IntVector  &_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 = 0.5;
        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 IntVector  &_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 IntVector  &_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 = 0.1;
        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;
        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 limit = _head_length;

        for (e = _next_arc; e < _search_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;
          }
          if (--cnt == 0) {
            if (_curr_length > limit) goto search_end;
            limit = 0;
            cnt = _block_size;
          }
        }
        for (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;
          }
          if (--cnt == 0) {
            if (_curr_length > limit) goto search_end;
            limit = 0;
            cnt = _block_size;
          }
        }
        if (_curr_length == 0) return false;
        
      search_end:

        // 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];
        _next_arc = e;
        pop_heap( _candidates.begin(), _candidates.begin() + _curr_length,
                  _sort_func );
        _curr_length = std::min(_head_length, _curr_length - 1);
        return true;
      }

    }; //class AlteringListPivotRule

  public:

    /// \brief Constructor.
    ///
    /// The constructor of the class.
    ///
    /// \param graph The digraph the algorithm runs on.
    /// \param arc_mixing Indicate if the arcs have to be stored in a
    /// mixed order in the internal data structure. 
    /// In special cases, it could lead to better overall performance,
    /// but it is usually slower. Therefore it is disabled by default.
    NetworkSimplex(const GR& graph, bool arc_mixing = false) :
      _graph(graph), _node_id(graph), _arc_id(graph),
      INF(std::numeric_limits<Value>::has_infinity ?
          std::numeric_limits<Value>::infinity() :
          std::numeric_limits<Value>::max())
    {
      // Check the value 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");
        
      // 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);
      _forward.resize(all_node_num);
      _thread.resize(all_node_num);
      _rev_thread.resize(all_node_num);
      _succ_num.resize(all_node_num);
      _last_succ.resize(all_node_num);
      _state.resize(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
        int k = std::max(int(std::sqrt(double(_arc_num))), 10);
        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 += k) >= _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)];
        }
      }
      
      // Initialize maps
      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;
    }

    /// \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 on each arc).
    ///
    /// \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.
    /// (It makes sense only if non-zero lower bounds are given.)
    ///
    /// \param map A node map storing the supply values.
    /// Its \c Value type must be convertible to the \c Value type
    /// of the algorithm.
    ///
    /// \return <tt>(*this)</tt>
    template<typename 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.
    /// (It makes sense only if non-zero lower bounds are given.)
    ///
    /// Using this function has the same effect as using \ref supplyMap()
    /// with such 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 parameters
    /// that have been given are kept for the next call, unless
    /// \ref reset() is called, thus only the modified parameters
    /// have to be set again. See \ref reset() for examples.
    /// However the underlying digraph must not be modified after this
    /// class have been constructed, since it copies and extends the graph.
    ///
    /// \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
    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 run() calls. If this function is not
    /// used, all the parameters given before are kept for the next
    /// \ref run() call.
    /// However the underlying digraph must not be modified after this
    /// class have been constructed, since it copies and extends the graph.
    ///
    /// 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 (reset() 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 reset()
    ///   // (the lower bounds will be set to zero on all arcs)
    ///   ns.reset();
    ///   ns.upperMap(capacity).costMap(cost)
    ///     .supplyMap(sup).run();
    /// \endcode
    ///
    /// \return <tt>(*this)</tt>
    NetworkSimplex& reset() {
      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;
    }

    /// @}

    /// \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 Return the flow map (the primal solution).
    ///
    /// 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 Return the potential map (the dual solution).
    ///
    /// 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] < INF ? _upper[i] - c : INF;
          } else {
            _cap[i] = _upper[i] < INF + 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 = std::numeric_limits<Cost>::min();
        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) {
            _forward[u] = true;
            _pi[u] = 0;
            _source[e] = u;
            _target[e] = _root;
            _flow[e] = _supply[u];
            _cost[e] = 0;
          } else {
            _forward[u] = false;
            _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) {
            _forward[u] = true;
            _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 {
            _forward[u] = false;
            _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) {
            _forward[u] = false;
            _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 {
            _forward[u] = true;
            _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
      if (_state[in_arc] == STATE_LOWER) {
        first  = _source[in_arc];
        second = _target[in_arc];
      } else {
        first  = _target[in_arc];
        second = _source[in_arc];
      }
      delta = _cap[in_arc];
      int result = 0;
      Value d;
      int e;

      // Search the cycle along the path form the first node to the root
      for (int u = first; u != join; u = _parent[u]) {
        e = _pred[u];
        d = _forward[u] ?
          _flow[e] : (_cap[e] == INF ? INF : _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] == INF ? INF : _cap[e] - _flow[e]) : _flow[e];
        if (d <= delta) {
          delta = d;
          u_out = u;
          result = 2;
        }
      }

      if (result == 1) {
        u_in = first;
        v_in = second;
      } else {
        u_in = second;
        v_in = first;
      }
      return result != 0;
    }

    // Change _flow and _state vectors
    void changeFlow(bool change) {
      // Augment along the cycle
      if (delta > 0) {
        Value val = _state[in_arc] * delta;
        _flow[in_arc] += val;
        for (int u = _source[in_arc]; u != join; u = _parent[u]) {
          _flow[_pred[u]] += _forward[u] ? -val : val;
        }
        for (int u = _target[in_arc]; u != join; u = _parent[u]) {
          _flow[_pred[u]] += _forward[u] ? val : -val;
        }
      }
      // Update the state of the entering and leaving arcs
      if (change) {
        _state[in_arc] = STATE_TREE;
        _state[_pred[u_out]] =
          (_flow[_pred[u_out]] == 0) ? STATE_LOWER : STATE_UPPER;
      } else {
        _state[in_arc] = -_state[in_arc];
      }
    }

    // Update the tree structure
    void updateTreeStructure() {
      int u, w;
      int old_rev_thread = _rev_thread[u_out];
      int old_succ_num = _succ_num[u_out];
      int old_last_succ = _last_succ[u_out];
      v_out = _parent[u_out];

      u = _last_succ[u_in];  // the last successor of u_in
      right = _thread[u];    // the node after it

      // Handle the case when old_rev_thread equals to v_in
      // (it also means that join and v_out coincide)
      if (old_rev_thread == v_in) {
        last = _thread[_last_succ[u_out]];
      } else {
        last = _thread[v_in];
      }

      // Update _thread and _parent along the stem nodes (i.e. the nodes
      // between u_in and u_out, whose parent have to be changed)
      _thread[v_in] = stem = u_in;
      _dirty_revs.clear();
      _dirty_revs.push_back(v_in);
      par_stem = v_in;
      while (stem != u_out) {
        // Insert the next stem node into the thread list
        new_stem = _parent[stem];
        _thread[u] = new_stem;
        _dirty_revs.push_back(u);

        // Remove the subtree of stem from the thread list
        w = _rev_thread[stem];
        _thread[w] = right;
        _rev_thread[right] = w;

        // Change the parent node and shift stem nodes
        _parent[stem] = par_stem;
        par_stem = stem;
        stem = new_stem;

        // Update u and right
        u = _last_succ[stem] == _last_succ[par_stem] ?
          _rev_thread[par_stem] : _last_succ[stem];
        right = _thread[u];
      }
      _parent[u_out] = par_stem;
      _thread[u] = last;
      _rev_thread[last] = u;
      _last_succ[u_out] = u;

      // Remove the subtree of u_out from the thread list except for
      // the case when old_rev_thread equals to v_in
      // (it also means that join and v_out coincide)
      if (old_rev_thread != v_in) {
        _thread[old_rev_thread] = right;
        _rev_thread[right] = old_rev_thread;
      }

      // Update _rev_thread using the new _thread values
      for (int i = 0; i < int(_dirty_revs.size()); ++i) {
        u = _dirty_revs[i];
        _rev_thread[_thread[u]] = u;
      }

      // Update _pred, _forward, _last_succ and _succ_num for the
      // stem nodes from u_out to u_in
      int tmp_sc = 0, tmp_ls = _last_succ[u_out];
      u = u_out;
      while (u != u_in) {
        w = _parent[u];
        _pred[u] = _pred[w];
        _forward[u] = !_forward[w];
        tmp_sc += _succ_num[u] - _succ_num[w];
        _succ_num[u] = tmp_sc;
        _last_succ[w] = tmp_ls;
        u = w;
      }
      _pred[u_in] = in_arc;
      _forward[u_in] = (u_in == _source[in_arc]);
      _succ_num[u_in] = old_succ_num;

      // Set limits for updating _last_succ form v_in and v_out
      // towards the root
      int up_limit_in = -1;
      int up_limit_out = -1;
      if (_last_succ[join] == v_in) {
        up_limit_out = join;
      } else {
        up_limit_in = join;
      }

      // Update _last_succ from v_in towards the root
      for (u = v_in; u != up_limit_in && _last_succ[u] == v_in;
           u = _parent[u]) {
        _last_succ[u] = _last_succ[u_out];
      }
      // Update _last_succ from v_out towards the root
      if (join != old_rev_thread && v_in != old_rev_thread) {
        for (u = v_out; u != up_limit_out && _last_succ[u] == old_last_succ;
             u = _parent[u]) {
          _last_succ[u] = old_rev_thread;
        }
      } else {
        for (u = v_out; u != up_limit_out && _last_succ[u] == old_last_succ;
             u = _parent[u]) {
          _last_succ[u] = _last_succ[u_out];
        }
      }

      // Update _succ_num from v_in to join
      for (u = v_in; u != join; u = _parent[u]) {
        _succ_num[u] += old_succ_num;
      }
      // Update _succ_num from v_out to join
      for (u = v_out; u != join; u = _parent[u]) {
        _succ_num[u] -= old_succ_num;
      }
    }

    // Update potentials
    void updatePotential() {
      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]];
      // Update potentials in the subtree, which has been moved
      int end = _thread[_last_succ[u_in]];
      for (int u = u_in; u != end; u = _thread[u]) {
        _pi[u] += sigma;
      }
    }

    // Execute the algorithm
    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);

      // Execute the Network Simplex algorithm
      while (pivot.findEnteringArc()) {
        findJoinNode();
        bool change = findLeavingArc();
        if (delta >= INF) 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>::min();
          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