RhodeCode

/* -*- 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_MAX_MATCHING_H

#define LEMON_MAX_MATCHING_H

#include <vector>

#include <queue>

#include <set>

#include <limits>

#include <lemon/core.h>

#include <lemon/unionfind.h>

#include <lemon/bin_heap.h>

#include <lemon/maps.h>

///\ingroup matching

///\file

///\brief Maximum matching algorithms in general graphs.

namespace lemon {

  /// \ingroup matching

  ///

  /// \brief Maximum cardinality matching in general graphs

  ///

  /// This class implements Edmonds' alternating forest matching algorithm

  /// It can be started from an arbitrary initial matching 

  /// (the default is the empty one).

  ///

  /// The dual solution of the problem is a map of the nodes to

  /// \ref MaxMatching::Status "Status", having values \c EVEN (or \c D),

  /// \c ODD (or \c A) and \c MATCHED (or \c C) defining the Gallai-Edmonds

  /// decomposition of the graph. The nodes in \c EVEN/D induce a subgraph

  /// with factor-critical components, the nodes in \c ODD/A form the

  /// canonical barrier, and the nodes in \c MATCHED/C induce a graph having

  /// a perfect matching. The number of the factor-critical components

  /// minus the number of barrier nodes is a lower bound on the

  /// unmatched nodes, and the matching is optimal if and only if this bound is

  ///

  template <typename GR>

  class MaxMatching {

  public:

    /// The graph type of the algorithm

    typedef GR Graph;

    typedef typename Graph::template NodeMap<typename Graph::Arc>

    MatchingMap;

    ///\brief Status constants for Gallai-Edmonds decomposition.

    ///

    ///These constants are used for indicating the Gallai-Edmonds 

    ///decomposition of a graph. The nodes with status \c EVEN (or \c D)

    ///induce a subgraph with factor-critical components, the nodes with

    ///status \c ODD (or \c A) form the canonical barrier, and the nodes

    ///with status \c MATCHED (or \c C) induce a subgraph having a 

    ///perfect matching.

    enum Status {

      EVEN = 1,       ///< = 1. (\c D is an alias for \c EVEN.)

      D = 1,

      MATCHED = 0,    ///< = 0. (\c C is an alias for \c MATCHED.)

      C = 0,

      ODD = -1,       ///< = -1. (\c A is an alias for \c ODD.)

      A = -1,

      UNMATCHED = -2  ///< = -2.

    };

    typedef typename Graph::template NodeMap<Status> StatusMap;

  private:

    TEMPLATE_GRAPH_TYPEDEFS(Graph);

    typedef UnionFindEnum<IntNodeMap> BlossomSet;

    typedef ExtendFindEnum<IntNodeMap> TreeSet;

    typedef RangeMap<Node> NodeIntMap;

    typedef MatchingMap EarMap;

    typedef std::vector<Node> NodeQueue;

    const Graph& _graph;

    MatchingMap* _matching;

    StatusMap* _status;

    EarMap* _ear;

    IntNodeMap* _blossom_set_index;

    BlossomSet* _blossom_set;

    NodeIntMap* _blossom_rep;

    IntNodeMap* _tree_set_index;

    TreeSet* _tree_set;

    NodeQueue _node_queue;

    int _process, _postpone, _last;

    int _node_num;

  private:

    void createStructures() {

      _node_num = countNodes(_graph);

      if (!_matching) {

        _matching = new MatchingMap(_graph);

      }

      if (!_status) {

        _status = new StatusMap(_graph);

      }

      if (!_ear) {

        _ear = new EarMap(_graph);

      }

      if (!_blossom_set) {

        _blossom_set_index = new IntNodeMap(_graph);

        _blossom_set = new BlossomSet(*_blossom_set_index);

      }

      if (!_blossom_rep) {

        _blossom_rep = new NodeIntMap(_node_num);

      }

      if (!_tree_set) {

        _tree_set_index = new IntNodeMap(_graph);

        _tree_set = new TreeSet(*_tree_set_index);

      }

      _node_queue.resize(_node_num);

    }

    void destroyStructures() {

      if (_matching) {

        delete _matching;

      }

      if (_status) {

        delete _status;

      }

      if (_ear) {

        delete _ear;

      }

      if (_blossom_set) {

        delete _blossom_set;

        delete _blossom_set_index;

      }

      if (_blossom_rep) {

        delete _blossom_rep;

      }

      if (_tree_set) {

        delete _tree_set_index;

        delete _tree_set;

      }

    }

    void processDense(const Node& n) {

      _process = _postpone = _last = 0;

      _node_queue[_last++] = n;

      while (_process != _last) {

        Node u = _node_queue[_process++];

        for (OutArcIt a(_graph, u); a != INVALID; ++a) {

          Node v = _graph.target(a);

          if ((*_status)[v] == MATCHED) {

            extendOnArc(a);

          } else if ((*_status)[v] == UNMATCHED) {

            augmentOnArc(a);

            return;

          }

        }

      }

          (*_status)[v] = MATCHED;

        }

      }

      return true;

    }

    /// \brief Start Edmonds' algorithm

    ///

    /// This function runs the original Edmonds' algorithm.

    ///

    /// \pre \ref Init(), \ref greedyInit() or \ref matchingInit() must be

    /// called before using this function.

    void startSparse() {

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

        if ((*_status)[n] == UNMATCHED) {

          (*_blossom_rep)[_blossom_set->insert(n)] = n;

          _tree_set->insert(n);

          (*_status)[n] = EVEN;

          processSparse(n);

        }

      }

    }

    /// \brief Start Edmonds' algorithm with a heuristic improvement 

    /// for dense graphs

    ///

    /// This function runs Edmonds' algorithm with a heuristic of postponing

    /// shrinks, therefore resulting in a faster algorithm for dense graphs.

    ///

    /// \pre \ref Init(), \ref greedyInit() or \ref matchingInit() must be

    /// called before using this function.

    void startDense() {

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

        if ((*_status)[n] == UNMATCHED) {

          (*_blossom_rep)[_blossom_set->insert(n)] = n;

          _tree_set->insert(n);

          (*_status)[n] = EVEN;

          processDense(n);

        }

      }

    }

    /// \brief Run Edmonds' algorithm

    ///

    /// This function runs Edmonds' algorithm. An additional heuristic of 

    /// postponing shrinks is used for relatively dense graphs 

    /// (for which <tt>m>=2*n</tt> holds).

    void run() {

      if (countEdges(_graph) < 2 * countNodes(_graph)) {

        greedyInit();

        startSparse();

      } else {

        init();

        startDense();

      }

    }

    /// @}

    /// \name Primal Solution

    /// Functions to get the primal solution, i.e. the maximum matching.

    /// @{

    /// \brief Return the size (cardinality) of the matching.

    ///

    /// This function returns the size (cardinality) of the current matching. 

    /// After run() it returns the size of the maximum matching in the graph.

    int matchingSize() const {

      int size = 0;

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

        if ((*_matching)[n] != INVALID) {

          ++size;

        }

      }

      return size / 2;

    }

    /// \brief Return \c true if the given edge is in the matching.

    ///

    /// This function returns \c true if the given edge is in the current 

    /// matching.

    bool matching(const Edge& edge) const {

      return edge == (*_matching)[_graph.u(edge)];

    }

    /// \brief Return the matching arc (or edge) incident to the given node.

    ///

    /// This function returns the matching arc (or edge) incident to the

    /// given node in the current matching or \c INVALID if the node is 

    /// not covered by the matching.

    Arc matching(const Node& n) const {

      return (*_matching)[n];

    }

    /// \brief Return the mate of the given node.

    ///

    /// This function returns the mate of the given node in the current 

    /// matching or \c INVALID if the node is not covered by the matching.

    Node mate(const Node& n) const {

      return (*_matching)[n] != INVALID ?

        _graph.target((*_matching)[n]) : INVALID;

    }

    /// @}

    /// \name Dual Solution

    /// Functions to get the dual solution, i.e. the Gallai-Edmonds 

    /// decomposition.

    /// @{

    /// \brief Return the status of the given node in the Edmonds-Gallai

    /// decomposition.

    ///

    /// This function returns the \ref Status "status" of the given node

    /// in the Edmonds-Gallai decomposition.

      return (*_status)[n];

    }

    /// \brief Return \c true if the given node is in the barrier.

    ///

    /// This function returns \c true if the given node is in the barrier.

    bool barrier(const Node& n) const {

      return (*_status)[n] == ODD;

    }

    /// @}

  };

  /// \ingroup matching

  ///

  /// \brief Weighted matching in general graphs

  ///

  /// This class provides an efficient implementation of Edmond's

  /// maximum weighted matching algorithm. The implementation is based

  /// on extensive use of priority queues and provides

  /// \f$O(nm\log n)\f$ time complexity.

  ///

  /// The maximum weighted matching problem is to find a subset of the 

  /// edges in an undirected graph with maximum overall weight for which 

  /// each node has at most one incident edge.

  /// It can be formulated with the following linear program.

  /// \f[ \sum_{e \in \delta(u)}x_e \le 1 \quad \forall u\in V\f]

  /** \f[ \sum_{e \in \gamma(B)}x_e \le \frac{\vert B \vert - 1}{2}

      \quad \forall B\in\mathcal{O}\f] */

  /// \f[x_e \ge 0\quad \forall e\in E\f]

  /// \f[\max \sum_{e\in E}x_ew_e\f]

  /// where \f$\delta(X)\f$ is the set of edges incident to a node in

  /// \f$X\f$, \f$\gamma(X)\f$ is the set of edges with both ends in

  /// \f$X\f$ and \f$\mathcal{O}\f$ is the set of odd cardinality

  /// subsets of the nodes.

  ///

  /// The algorithm calculates an optimal matching and a proof of the

  /// optimality. The solution of the dual problem can be used to check

  /// the result of the algorithm. The dual linear problem is the

  /// following.

  /** \f[ y_u + y_v + \sum_{B \in \mathcal{O}, uv \in \gamma(B)}

      z_B \ge w_{uv} \quad \forall uv\in E\f] */

  /// \f[y_u \ge 0 \quad \forall u \in V\f]

  /// \f[z_B \ge 0 \quad \forall B \in \mathcal{O}\f]

  /** \f[\min \sum_{u \in V}y_u + \sum_{B \in \mathcal{O}}

      \frac{\vert B \vert - 1}{2}z_B\f] */

  ///

  /// The algorithm can be executed with the run() function. 

  /// After it the matching (the primal solution) and the dual solution

  /// can be obtained using the query functions and the 

  /// \ref MaxWeightedMatching::BlossomIt "BlossomIt" nested class, 

  /// which is able to iterate on the nodes of a blossom. 

  /// If the value type is integer, then the dual solution is multiplied

  /// by \ref MaxWeightedMatching::dualScale "4".

  ///

  /// \tparam WM The type edge weight map. The default type is 

  /// \ref concepts::Graph::EdgeMap "GR::EdgeMap<int>".

#ifdef DOXYGEN

  template <typename GR, typename WM>

#else

  template <typename GR,

            typename WM = typename GR::template EdgeMap<int> >

#endif

  class MaxWeightedMatching {

  public:

    /// The graph type of the algorithm

    typedef GR Graph;

    /// The type of the edge weight map

    typedef WM WeightMap;

    /// The value type of the edge weights

    typedef typename WeightMap::Value Value;

    typedef typename Graph::template NodeMap<typename Graph::Arc>

    MatchingMap;

    /// \brief Scaling factor for dual solution

    ///

    /// Scaling factor for dual solution. It is equal to 4 or 1

    /// according to the value type.

    static const int dualScale =

      std::numeric_limits<Value>::is_integer ? 4 : 1;

  private:

    TEMPLATE_GRAPH_TYPEDEFS(Graph);

    typedef typename Graph::template NodeMap<Value> NodePotential;

    typedef std::vector<Node> BlossomNodeList;

    struct BlossomVariable {

      int begin, end;

      Value value;

      BlossomVariable(int _begin, int _end, Value _value)

        : begin(_begin), end(_end), value(_value) {}

    };

    typedef std::vector<BlossomVariable> BlossomPotential;

    const Graph& _graph;

    const WeightMap& _weight;

    MatchingMap* _matching;

    NodePotential* _node_potential;

    BlossomPotential _blossom_potential;

    BlossomNodeList _blossom_node_list;

    int _node_num;

    int _blossom_num;

    typedef RangeMap<int> IntIntMap;

    enum Status {

      EVEN = -1, MATCHED = 0, ODD = 1, UNMATCHED = -2

    };

    typedef HeapUnionFind<Value, IntNodeMap> BlossomSet;

    struct BlossomData {

      int tree;

      Status status;

      Arc pred, next;

      Value pot, offset;

      Node base;

    };

    IntNodeMap *_blossom_index;

    BlossomSet *_blossom_set;

    RangeMap<BlossomData>* _blossom_data;

    IntNodeMap *_node_index;

    IntArcMap *_node_heap_index;

    struct NodeData {

      NodeData(IntArcMap& node_heap_index)

        : heap(node_heap_index) {}

      int blossom;

      Value pot;

      BinHeap<Value, IntArcMap> heap;

      std::map<int, Arc> heap_index;

      int tree;

    };

    RangeMap<NodeData>* _node_data;

    typedef ExtendFindEnum<IntIntMap> TreeSet;

    IntIntMap *_tree_set_index;

    TreeSet *_tree_set;

    IntNodeMap *_delta1_index;

    BinHeap<Value, IntNodeMap> *_delta1;

    IntIntMap *_delta2_index;

    BinHeap<Value, IntIntMap> *_delta2;

    IntEdgeMap *_delta3_index;

    BinHeap<Value, IntEdgeMap> *_delta3;

    IntIntMap *_delta4_index;

    BinHeap<Value, IntIntMap> *_delta4;

    Value _delta_sum;

        Value d4 = !_delta4->empty() ?

          _delta4->prio() : std::numeric_limits<Value>::max();

        _delta_sum = d1; OpType ot = D1;

        if (d2 < _delta_sum) { _delta_sum = d2; ot = D2; }

        if (d3 < _delta_sum) { _delta_sum = d3; ot = D3; }

        if (d4 < _delta_sum) { _delta_sum = d4; ot = D4; }

        switch (ot) {

        case D1:

          {

            Node n = _delta1->top();

            unmatchNode(n);

            --unmatched;

          }

          break;

        case D2:

          {

            int blossom = _delta2->top();

            Node n = _blossom_set->classTop(blossom);

            Arc e = (*_node_data)[(*_node_index)[n]].heap.top();

            extendOnArc(e);

          }

          break;

        case D3:

          {

            Edge e = _delta3->top();

            int left_blossom = _blossom_set->find(_graph.u(e));

            int right_blossom = _blossom_set->find(_graph.v(e));

            if (left_blossom == right_blossom) {

              _delta3->pop();

            } else {

              int left_tree;

              if ((*_blossom_data)[left_blossom].status == EVEN) {

                left_tree = _tree_set->find(left_blossom);

              } else {

                left_tree = -1;

                ++unmatched;

              }

              int right_tree;

              if ((*_blossom_data)[right_blossom].status == EVEN) {

                right_tree = _tree_set->find(right_blossom);

              } else {

                right_tree = -1;

                ++unmatched;

              }

              if (left_tree == right_tree) {

                shrinkOnEdge(e, left_tree);

              } else {

                augmentOnEdge(e);

                unmatched -= 2;

              }

            }

          } break;

        case D4:

          splitBlossom(_delta4->top());

          break;

        }

      }

      extractMatching();

    }

    /// \brief Run the algorithm.

    ///

    /// This method runs the \c %MaxWeightedMatching algorithm.

    ///

    /// \note mwm.run() is just a shortcut of the following code.

    /// \code

    ///   mwm.init();

    ///   mwm.start();

    /// \endcode

    void run() {

      init();

      start();

    }

    /// @}

    /// \name Primal Solution

    /// Functions to get the primal solution, i.e. the maximum weighted 

    /// matching.\n

    /// Either \ref run() or \ref start() function should be called before

    /// using them.

    /// @{

    /// \brief Return the weight of the matching.

    ///

    /// This function returns the weight of the found matching.

    ///

    /// \pre Either run() or start() must be called before using this function.

      Value sum = 0;

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

        if ((*_matching)[n] != INVALID) {

          sum += _weight[(*_matching)[n]];

        }

      }

      return sum /= 2;

    }

    /// \brief Return the size (cardinality) of the matching.

    ///

    /// This function returns the size (cardinality) of the found matching.

    ///

    /// \pre Either run() or start() must be called before using this function.

    int matchingSize() const {

      int num = 0;

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

        if ((*_matching)[n] != INVALID) {

          ++num;

        }

      }

      return num /= 2;

    }

    /// \brief Return \c true if the given edge is in the matching.

    ///

    /// This function returns \c true if the given edge is in the found 

    /// matching.

    ///

    /// \pre Either run() or start() must be called before using this function.

    bool matching(const Edge& edge) const {

      return edge == (*_matching)[_graph.u(edge)];

    }

    /// \brief Return the matching arc (or edge) incident to the given node.

    ///

    /// This function returns the matching arc (or edge) incident to the

    /// given node in the found matching or \c INVALID if the node is 

    /// not covered by the matching.

    ///

    /// \pre Either run() or start() must be called before using this function.

    Arc matching(const Node& node) const {

      return (*_matching)[node];

    }

    /// \brief Return the mate of the given node.

    ///

    /// This function returns the mate of the given node in the found 

    /// matching or \c INVALID if the node is not covered by the matching.

    ///

    /// \pre Either run() or start() must be called before using this function.

    Node mate(const Node& node) const {

      return (*_matching)[node] != INVALID ?

        _graph.target((*_matching)[node]) : INVALID;

    }

    /// @}

    /// \name Dual Solution

    /// Functions to get the dual solution.\n

    /// Either \ref run() or \ref start() function should be called before

    /// using them.

    /// @{

    /// \brief Return the value of the dual solution.

    ///

    /// This function returns the value of the dual solution. 

    /// It should be equal to the primal value scaled by \ref dualScale 

    /// "dual scale".

    ///

    /// \pre Either run() or start() must be called before using this function.

    Value dualValue() const {

      Value sum = 0;

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

        sum += nodeValue(n);

      }

      for (int i = 0; i < blossomNum(); ++i) {

        sum += blossomValue(i) * (blossomSize(i) / 2);

      }

      return sum;

    }

    /// \brief Return the dual value (potential) of the given node.

    ///

    /// This function returns the dual value (potential) of the given node.

    ///

    /// \pre Either run() or start() must be called before using this function.

    Value nodeValue(const Node& n) const {

      return (*_node_potential)[n];

    }

    /// \brief Return the number of the blossoms in the basis.

    ///

    /// This function returns the number of the blossoms in the basis.

    ///

    /// \pre Either run() or start() must be called before using this function.

    /// \see BlossomIt

    int blossomNum() const {

      return _blossom_potential.size();

    }

    /// \brief Return the number of the nodes in the given blossom.

    ///

    /// This function returns the number of the nodes in the given blossom.

    ///

    /// \pre Either run() or start() must be called before using this function.

    /// \see BlossomIt

    int blossomSize(int k) const {

      return _blossom_potential[k].end - _blossom_potential[k].begin;

    }

    /// \brief Return the dual value (ptential) of the given blossom.

    ///

    /// This function returns the dual value (ptential) of the given blossom.

    ///

    /// \pre Either run() or start() must be called before using this function.

    Value blossomValue(int k) const {

      return _blossom_potential[k].value;

    }

    /// \brief Iterator for obtaining the nodes of a blossom.

    ///

    /// This class provides an iterator for obtaining the nodes of the 

    /// given blossom. It lists a subset of the nodes.

    /// Before using this iterator, you must allocate a 

    /// MaxWeightedMatching class and execute it.

    class BlossomIt {

    public:

      /// \brief Constructor.

      ///

      /// Constructor to get the nodes of the given variable.

      ///

      /// \pre Either \ref MaxWeightedMatching::run() "algorithm.run()" or 

      /// \ref MaxWeightedMatching::start() "algorithm.start()" must be 

      /// called before initializing this iterator.

      BlossomIt(const MaxWeightedMatching& algorithm, int variable)

        : _algorithm(&algorithm)

      {

        _index = _algorithm->_blossom_potential[variable].begin;

        _last = _algorithm->_blossom_potential[variable].end;

      }

      /// \brief Conversion to \c Node.

      ///

      /// Conversion to \c Node.

      operator Node() const {

        return _algorithm->_blossom_node_list[_index];

      }

      /// \brief Increment operator.

      ///

      /// Increment operator.

      BlossomIt& operator++() {

        ++_index;

        return *this;

      }

      /// \brief Validity checking

      ///

      /// Checks whether the iterator is invalid.

      bool operator==(Invalid) const { return _index == _last; }

      /// \brief Validity checking

      ///

      /// Checks whether the iterator is valid.

      bool operator!=(Invalid) const { return _index != _last; }

    private:

      const MaxWeightedMatching* _algorithm;

      int _last;

      int _index;

    };

    /// @}

  };

  /// \ingroup matching

  ///

  /// \brief Weighted perfect matching in general graphs

  ///

  /// This class provides an efficient implementation of Edmond's

  /// maximum weighted perfect matching algorithm. The implementation

  /// is based on extensive use of priority queues and provides

  /// \f$O(nm\log n)\f$ time complexity.

  ///

  /// The maximum weighted perfect matching problem is to find a subset of 

  /// the edges in an undirected graph with maximum overall weight for which 

  /// each node has exactly one incident edge.

  /// It can be formulated with the following linear program.

  /// \f[ \sum_{e \in \delta(u)}x_e = 1 \quad \forall u\in V\f]

  /** \f[ \sum_{e \in \gamma(B)}x_e \le \frac{\vert B \vert - 1}{2}

      \quad \forall B\in\mathcal{O}\f] */

  /// \f[x_e \ge 0\quad \forall e\in E\f]

  /// \f[\max \sum_{e\in E}x_ew_e\f]

  /// where \f$\delta(X)\f$ is the set of edges incident to a node in

  /// \f$X\f$, \f$\gamma(X)\f$ is the set of edges with both ends in

  /// \f$X\f$ and \f$\mathcal{O}\f$ is the set of odd cardinality

  /// subsets of the nodes.

  ///

  /// The algorithm calculates an optimal matching and a proof of the

  /// optimality. The solution of the dual problem can be used to check

  /// the result of the algorithm. The dual linear problem is the

  /// following.

  /** \f[ y_u + y_v + \sum_{B \in \mathcal{O}, uv \in \gamma(B)}z_B \ge

      w_{uv} \quad \forall uv\in E\f] */

  /// \f[z_B \ge 0 \quad \forall B \in \mathcal{O}\f]

  /** \f[\min \sum_{u \in V}y_u + \sum_{B \in \mathcal{O}}

      \frac{\vert B \vert - 1}{2}z_B\f] */

  ///

  /// The algorithm can be executed with the run() function. 

  /// After it the matching (the primal solution) and the dual solution

  /// can be obtained using the query functions and the 

  /// \ref MaxWeightedPerfectMatching::BlossomIt "BlossomIt" nested class, 

  /// which is able to iterate on the nodes of a blossom. 

  /// If the value type is integer, then the dual solution is multiplied

  /// by \ref MaxWeightedMatching::dualScale "4".

  ///

  /// \tparam WM The type edge weight map. The default type is 

  /// \ref concepts::Graph::EdgeMap "GR::EdgeMap<int>".

#ifdef DOXYGEN

  template <typename GR, typename WM>

#else

  template <typename GR,

            typename WM = typename GR::template EdgeMap<int> >

#endif

  class MaxWeightedPerfectMatching {

  public:

    /// The graph type of the algorithm

    typedef GR Graph;

    /// The type of the edge weight map

    typedef WM WeightMap;

    /// The value type of the edge weights

    typedef typename WeightMap::Value Value;

    /// \brief Scaling factor for dual solution

    ///

    /// Scaling factor for dual solution, it is equal to 4 or 1

    /// according to the value type.

    static const int dualScale =

      std::numeric_limits<Value>::is_integer ? 4 : 1;

    typedef typename Graph::template NodeMap<typename Graph::Arc>

    MatchingMap;

  private:

    TEMPLATE_GRAPH_TYPEDEFS(Graph);

    typedef typename Graph::template NodeMap<Value> NodePotential;

    typedef std::vector<Node> BlossomNodeList;

    struct BlossomVariable {

      int begin, end;

      Value value;

      BlossomVariable(int _begin, int _end, Value _value)

        : begin(_begin), end(_end), value(_value) {}

    };

    typedef std::vector<BlossomVariable> BlossomPotential;

    const Graph& _graph;

    const WeightMap& _weight;

    MatchingMap* _matching;

    NodePotential* _node_potential;

    BlossomPotential _blossom_potential;

    BlossomNodeList _blossom_node_list;

    int _node_num;

    int _blossom_num;

    typedef RangeMap<int> IntIntMap;

    enum Status {

      EVEN = -1, MATCHED = 0, ODD = 1

    };

    typedef HeapUnionFind<Value, IntNodeMap> BlossomSet;

    struct BlossomData {

      int tree;

      Status status;

      Arc pred, next;

      Value pot, offset;

    };

    IntNodeMap *_blossom_index;

    BlossomSet *_blossom_set;

    RangeMap<BlossomData>* _blossom_data;

    IntNodeMap *_node_index;

    IntArcMap *_node_heap_index;

    struct NodeData {

      NodeData(IntArcMap& node_heap_index)

        : heap(node_heap_index) {}

      int blossom;

      Value pot;

      BinHeap<Value, IntArcMap> heap;

      std::map<int, Arc> heap_index;

      int tree;

    };

    RangeMap<NodeData>* _node_data;

    typedef ExtendFindEnum<IntIntMap> TreeSet;

    IntIntMap *_tree_set_index;

    TreeSet *_tree_set;

    IntIntMap *_delta2_index;

    BinHeap<Value, IntIntMap> *_delta2;

    IntEdgeMap *_delta3_index;

    BinHeap<Value, IntEdgeMap> *_delta3;

    IntIntMap *_delta4_index;

    BinHeap<Value, IntIntMap> *_delta4;

    Value _delta_sum;

    void createStructures() {

      _node_num = countNodes(_graph);

      _blossom_num = _node_num * 3 / 2;

      if (!_matching) {

        _matching = new MatchingMap(_graph);

      }

      if (!_node_potential) {

        _node_potential = new NodePotential(_graph);

      }

    ///

    /// This function starts the algorithm.

    ///

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

    bool start() {

      enum OpType {

        D2, D3, D4

      };

      int unmatched = _node_num;

      while (unmatched > 0) {

        Value d2 = !_delta2->empty() ?

          _delta2->prio() : std::numeric_limits<Value>::max();

        Value d3 = !_delta3->empty() ?

          _delta3->prio() : std::numeric_limits<Value>::max();

        Value d4 = !_delta4->empty() ?

          _delta4->prio() : std::numeric_limits<Value>::max();

        _delta_sum = d2; OpType ot = D2;

        if (d3 < _delta_sum) { _delta_sum = d3; ot = D3; }

        if (d4 < _delta_sum) { _delta_sum = d4; ot = D4; }

        if (_delta_sum == std::numeric_limits<Value>::max()) {

          return false;

        }

        switch (ot) {

        case D2:

          {

            int blossom = _delta2->top();

            Node n = _blossom_set->classTop(blossom);

            Arc e = (*_node_data)[(*_node_index)[n]].heap.top();

            extendOnArc(e);

          }

          break;

        case D3:

          {

            Edge e = _delta3->top();

            int left_blossom = _blossom_set->find(_graph.u(e));

            int right_blossom = _blossom_set->find(_graph.v(e));

            if (left_blossom == right_blossom) {

              _delta3->pop();

            } else {

              int left_tree = _tree_set->find(left_blossom);

              int right_tree = _tree_set->find(right_blossom);

              if (left_tree == right_tree) {

                shrinkOnEdge(e, left_tree);

              } else {

                augmentOnEdge(e);

                unmatched -= 2;

              }

            }

          } break;

        case D4:

          splitBlossom(_delta4->top());

          break;

        }

      }

      extractMatching();

      return true;

    }

    /// \brief Run the algorithm.

    ///

    /// This method runs the \c %MaxWeightedPerfectMatching algorithm.

    ///

    /// \note mwpm.run() is just a shortcut of the following code.

    /// \code

    ///   mwpm.init();

    ///   mwpm.start();

    /// \endcode

    bool run() {

      init();

      return start();

    }

    /// @}

    /// \name Primal Solution

    /// Functions to get the primal solution, i.e. the maximum weighted 

    /// perfect matching.\n

    /// Either \ref run() or \ref start() function should be called before

    /// using them.

    /// @{

    /// \brief Return the weight of the matching.

    ///

    /// This function returns the weight of the found matching.

    ///

    /// \pre Either run() or start() must be called before using this function.

      Value sum = 0;

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

        if ((*_matching)[n] != INVALID) {

          sum += _weight[(*_matching)[n]];

        }

      }

      return sum /= 2;

    }

    /// \brief Return \c true if the given edge is in the matching.

    ///

    /// This function returns \c true if the given edge is in the found 

    /// matching.

    ///

    /// \pre Either run() or start() must be called before using this function.

    bool matching(const Edge& edge) const {

      return static_cast<const Edge&>((*_matching)[_graph.u(edge)]) == edge;

    }

    /// \brief Return the matching arc (or edge) incident to the given node.

    ///

    /// This function returns the matching arc (or edge) incident to the

    /// given node in the found matching or \c INVALID if the node is 

    /// not covered by the matching.

    ///

    /// \pre Either run() or start() must be called before using this function.

    Arc matching(const Node& node) const {

      return (*_matching)[node];

    }

    /// \brief Return the mate of the given node.

    ///

    /// This function returns the mate of the given node in the found 

    /// matching or \c INVALID if the node is not covered by the matching.

    ///

    /// \pre Either run() or start() must be called before using this function.

    Node mate(const Node& node) const {

      return _graph.target((*_matching)[node]);

    }

    /// @}

    /// \name Dual Solution

    /// Functions to get the dual solution.\n

    /// Either \ref run() or \ref start() function should be called before

    /// using them.

    /// @{

    /// \brief Return the value of the dual solution.

    ///

    /// This function returns the value of the dual solution. 

    /// It should be equal to the primal value scaled by \ref dualScale 

    /// "dual scale".

    ///

    /// \pre Either run() or start() must be called before using this function.

    Value dualValue() const {

      Value sum = 0;

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