RhodeCode

#include <limits>

#include <algorithm>

#include <lemon/core.h>

#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".

  /// 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.

  ///

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

  ///

  ///

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

  /// implementations. For more information see \ref PivotRule.

  class NetworkSimplex

  {

  public:

    /// The type of the flow map

    /// The type of the potential map

  public:

    /// \brief Enum type for selecting the pivot rule.

    ///

    /// Enum type 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 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;

    // 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;

    // Parameters of the problem

    bool _pstsup;

    Node _psource, _ptarget;

    // Result maps

    FlowMap *_flow_map;

    PotentialMap *_potential_map;

    bool _local_flow;

    bool _local_potential;

    // Data structures for storing the digraph

    IntNodeMap _node_id;

    ArcVector _arc_ref;

    IntVector _source;

    IntVector _target;

    // Node and arc data

    // 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;

  private:

    // Implementation of the First Eligible pivot rule

    class FirstEligiblePivotRule

    {

    private:

      // References to the NetworkSimplex class

      const IntVector  &_source;

      const IntVector  &_target;

      const IntVector  &_state;

      int &_in_arc;

      int _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), _arc_num(ns._arc_num), _next_arc(0)

      {}

      // Find next entering arc

      bool findEnteringArc() {

        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

    // Implementation of the Best Eligible pivot rule

    class BestEligiblePivotRule

    {

    private:

      // References to the NetworkSimplex class

      const IntVector  &_source;

      const IntVector  &_target;

      const IntVector  &_state;

      int &_in_arc;

      int _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), _arc_num(ns._arc_num)

      {}

      // Find next entering arc

      bool findEnteringArc() {

        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

    // Implementation of the Block Search pivot rule

    class BlockSearchPivotRule

    {

    private:

      // References to the NetworkSimplex class

      const IntVector  &_source;

      const IntVector  &_target;

      const IntVector  &_state;

      int &_in_arc;

      int _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), _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() {

        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

    // Implementation of the Candidate List pivot rule

    class CandidateListPivotRule

    {

    private:

      // References to the NetworkSimplex class

      const IntVector  &_source;

      const IntVector  &_target;

      const IntVector  &_state;

      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) :

        _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() {

        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_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

    // Implementation of the Altering Candidate List pivot rule

    class AlteringListPivotRule

    {

    private:

      // References to the NetworkSimplex class

      const IntVector  &_source;

      const IntVector  &_target;

      const IntVector  &_state;

      int &_in_arc;

      int _arc_num;

      // Pivot rule data

      int _block_size, _head_length, _curr_length;

      int _next_arc;

      IntVector _candidates;

      // Functor class to compare arcs during sort of the candidate list

      class SortFunc

      {

      private:

      public:

        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), _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 );

        _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 Constructor.

    ///

    /// Constructor.

    ///

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

    NetworkSimplex(const GR& graph) :

      _graph(graph),

      _plower(NULL), _pupper(NULL), _pcost(NULL),

      _psupply(NULL), _pstsup(false),

      _flow_map(NULL), _potential_map(NULL),

      _local_flow(false), _local_potential(false),

      _node_id(graph)

    {

    }

    /// Destructor.

    ~NetworkSimplex() {

      if (_local_flow) delete _flow_map;

      if (_local_potential) delete _potential_map;

    }

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

    ///

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

    /// If neither this function nor \ref boundMaps() is 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.

    /// of the algorithm.

    ///

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

    template <typename LOWER>

    NetworkSimplex& lowerMap(const LOWER& map) {

      delete _plower;

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

        (*_plower)[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 none of the functions \ref upperMap(), \ref capacityMap()

    /// and \ref boundMaps() is used before calling \ref run(),

    /// the upper bounds (capacities) will be set to

    ///

    /// \param map An arc map storing the upper bounds.

    /// of the algorithm.

    ///

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

    template<typename UPPER>

    NetworkSimplex& upperMap(const UPPER& map) {

      delete _pupper;

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

        (*_pupper)[a] = map[a];

      }

      return *this;

    }

    /// \brief Set the upper bounds (capacities) on the arcs.

    ///

    /// This function sets the upper bounds (capacities) on the arcs.

    /// It is just an alias for \ref upperMap().

    ///

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

    template<typename CAP>

    NetworkSimplex& capacityMap(const CAP& map) {

      return upperMap(map);

    }

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

    ///

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

    /// If neither this function nor \ref lowerMap() is used before

    /// calling \ref run(), the lower bounds will be set to zero

    /// on all arcs.

    /// If none of the functions \ref upperMap(), \ref capacityMap()

    /// and \ref boundMaps() is used before calling \ref run(),

    /// the upper bounds (capacities) will be set to

    ///

    /// \param lower An arc map storing the lower bounds.

    /// \param upper An arc map storing the upper bounds.

    ///

    /// The \c Value type of the maps must be convertible to the

    ///

    /// \note This function is just a shortcut of calling \ref lowerMap()

    /// and \ref upperMap() separately.

    ///

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

    template <typename LOWER, typename UPPER>

    NetworkSimplex& boundMaps(const LOWER& lower, const UPPER& upper) {

      return lowerMap(lower).upperMap(upper);

    }

    /// \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.

    /// of the algorithm.

    ///

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

    template<typename COST>

    NetworkSimplex& costMap(const COST& map) {

      delete _pcost;

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

        (*_pcost)[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.

    /// of the algorithm.

    ///

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

    template<typename SUP>

    NetworkSimplex& supplyMap(const SUP& map) {

      delete _psupply;

      _pstsup = false;

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

        (*_psupply)[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.)

    ///

    /// \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>

      delete _psupply;

      _psupply = NULL;

      _pstsup = true;

      _psource = s;

      _ptarget = t;

      _pstflow = k;

      return *this;

    }

    /// \brief Set the flow map.

    ///

    /// This function sets the flow map.

    /// If it is not used before calling \ref run(), an instance will

    /// be allocated automatically. The destructor deallocates this

    /// automatically allocated map, of course.

    ///

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

    NetworkSimplex& flowMap(FlowMap& map) {

      if (_local_flow) {

        delete _flow_map;

        _local_flow = false;

      }

      _flow_map = ↦

      return *this;

      delete _plower;

      delete _pupper;

      delete _pcost;

      delete _psupply;

      _plower = NULL;

      _pupper = NULL;

      _pcost = NULL;

      _psupply = NULL;

      _pstsup = false;

      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.

    ///

    /// \note The return type of the function can be specified as a

    /// template parameter. For example,

    /// \code

    ///   ns.totalCost<double>();

    /// \endcode

    /// 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 Num>

    Num totalCost() const {

      Num c = 0;

      if (_pcost) {

        for (ArcIt e(_graph); e != INVALID; ++e)

          c += (*_flow_map)[e] * (*_pcost)[e];

      } else {

        for (ArcIt e(_graph); e != INVALID; ++e)

          c += (*_flow_map)[e];

      }

      return c;

    }

#ifndef DOXYGEN

    }

#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.

      return (*_flow_map)[a];

    }

    /// \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_map;

    }

    /// \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.

      return (*_potential_map)[n];

    }

    /// \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, which form the dual solution of the

    /// \ref min_cost_flow "minimum cost flow" problem.

    ///

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

    const PotentialMap& potentialMap() const {

      return *_potential_map;

    }

    /// @}

  private:

    // Initialize internal data structures

    bool init() {

      // Initialize result maps

      if (!_flow_map) {

        _flow_map = new FlowMap(_graph);

      _cap.resize(all_arc_num);

      _cost.resize(all_arc_num);

      _supply.resize(all_node_num);

      _flow.resize(all_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(all_arc_num);

      // Initialize node related data

      bool valid_supply = true;

      if (!_pstsup && !_psupply) {

        _pstsup = true;

        _psource = _ptarget = NodeIt(_graph);

        _pstflow = 0;

      }

      if (_psupply) {

        int i = 0;

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

          _node_id[n] = i;

          _supply[i] = (*_psupply)[n];

          sum += _supply[i];

        }

        valid_supply = (sum == 0);

      } else {

        int i = 0;

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

          _node_id[n] = i;

          _supply[i] = 0;

        }

        _supply[_node_id[_psource]] =  _pstflow;

        _supply[_node_id[_ptarget]]   = -_pstflow;

      }

      if (!valid_supply) return false;

      // 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] = all_node_num;

      _last_succ[_root] = _root - 1;

      _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(_graph); e != INVALID; ++e) {

        _arc_ref[i] = e;

        if ((i += k) >= _arc_num) i = (i % k) + 1;

      }

      // Initialize arc maps

      if (_pupper && _pcost) {

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

          Arc e = _arc_ref[i];

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

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

          _cap[i] = (*_pupper)[e];

          _cost[i] = (*_pcost)[e];

          _flow[i] = 0;

          _state[i] = STATE_LOWER;

        }

      } else {

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

          Arc e = _arc_ref[i];

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

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

          _flow[i] = 0;

          _state[i] = STATE_LOWER;

        }

        if (_pupper) {

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

            _cap[i] = (*_pupper)[_arc_ref[i]];

        } else {

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

        }

        if (_pcost) {

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

            _cost[i] = (*_pcost)[_arc_ref[i]];

        } else {

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

            _cost[i] = 1;

        }

      }

      // Remove non-zero lower bounds

      if (_plower) {

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

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

        _thread[u] = u + 1;

        _rev_thread[u + 1] = u;

        _succ_num[u] = 1;

        _last_succ[u] = u;

        _parent[u] = _root;

        _pred[u] = e;

        _cost[e] = max_cost;

        _cap[e] = max_cap;

        _state[e] = STATE_TREE;

        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;

        }

      }

      return true;

    }

      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;

      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) {

        _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];

      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() {

        _pi[v_in] - _pi[u_in] - _cost[_pred[u_in]] :

        _pi[v_in] - _pi[u_in] + _cost[_pred[u_in]];

      if (_succ_num[u_in] > _node_num / 2) {

        // Update in the upper subtree (which contains the root)