/* -*- 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