/* -*- 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_MATCHING_H

#define LEMON_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>

#include <lemon/fractional_matching.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

  /// for finding a maximum cardinality matching in a general undirected graph.

  /// 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

  /// tight. This decomposition can be obtained using \ref status() or

  /// \ref statusMap() after running the algorithm.

  ///

  /// \tparam GR The undirected graph type the algorithm runs on.

  template <typename GR>

  class MaxMatching {

  public:

    /// The graph type of the algorithm

    typedef GR Graph;

    /// The type of the matching map

    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.

    };

    /// The type of the status map

    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;

          }

        }

      }

      while (_postpone != _last) {

        Node u = _node_queue[_postpone++];

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

          Node v = _graph.target(a);

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

            if (_blossom_set->find(u) != _blossom_set->find(v)) {

              shrinkOnEdge(a);

            }

          }

          while (_process != _last) {

            Node w = _node_queue[_process++];

            for (OutArcIt b(_graph, w); b != INVALID; ++b) {

              Node x = _graph.target(b);

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

                extendOnArc(b);

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

                augmentOnArc(b);

                return;

              }

            }

          }

        }

      }

    }

    void processSparse(const Node& n) {

      _process = _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] == EVEN) {

            if (_blossom_set->find(u) != _blossom_set->find(v)) {

              shrinkOnEdge(a);

            }

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

            extendOnArc(a);

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

            augmentOnArc(a);

            return;

          }

        }

      }

    }

    void shrinkOnEdge(const Edge& e) {

      Node nca = INVALID;

      {

        std::set<Node> left_set, right_set;

        Node left = (*_blossom_rep)[_blossom_set->find(_graph.u(e))];

        left_set.insert(left);

        Node right = (*_blossom_rep)[_blossom_set->find(_graph.v(e))];

        right_set.insert(right);

        while (true) {

          if ((*_matching)[left] == INVALID) break;

          left = _graph.target((*_matching)[left]);

          left = (*_blossom_rep)[_blossom_set->

                                 find(_graph.target((*_ear)[left]))];

          if (right_set.find(left) != right_set.end()) {

            nca = left;

            break;

          }

          left_set.insert(left);

          if ((*_matching)[right] == INVALID) break;

          right = _graph.target((*_matching)[right]);

          right = (*_blossom_rep)[_blossom_set->

                                  find(_graph.target((*_ear)[right]))];

          if (left_set.find(right) != left_set.end()) {

            nca = right;

            break;

          }

          right_set.insert(right);

        }

        if (nca == INVALID) {

          if ((*_matching)[left] == INVALID) {

            nca = right;

            while (left_set.find(nca) == left_set.end()) {

              nca = _graph.target((*_matching)[nca]);

              nca =(*_blossom_rep)[_blossom_set->

                                   find(_graph.target((*_ear)[nca]))];

            }

          } else {

            nca = left;

            while (right_set.find(nca) == right_set.end()) {

              nca = _graph.target((*_matching)[nca]);

              nca = (*_blossom_rep)[_blossom_set->

                                   find(_graph.target((*_ear)[nca]))];

            }

          }

        }

      }

      {

        Node node = _graph.u(e);

        Arc arc = _graph.direct(e, true);

        Node base = (*_blossom_rep)[_blossom_set->find(node)];

        while (base != nca) {

          (*_ear)[node] = arc;

          Node n = node;

          while (n != base) {

            n = _graph.target((*_matching)[n]);

            Arc a = (*_ear)[n];

            n = _graph.target(a);

            (*_ear)[n] = _graph.oppositeArc(a);

          }

          node = _graph.target((*_matching)[base]);

          _tree_set->erase(base);

          _tree_set->erase(node);

          _blossom_set->insert(node, _blossom_set->find(base));

          (*_status)[node] = EVEN;

          _node_queue[_last++] = node;

          arc = _graph.oppositeArc((*_ear)[node]);

          node = _graph.target((*_ear)[node]);

          base = (*_blossom_rep)[_blossom_set->find(node)];

          _blossom_set->join(_graph.target(arc), base);

        }

      }

      (*_blossom_rep)[_blossom_set->find(nca)] = nca;

      {

        Node node = _graph.v(e);

        Arc arc = _graph.direct(e, false);

        Node base = (*_blossom_rep)[_blossom_set->find(node)];

        while (base != nca) {

          (*_ear)[node] = arc;

          Node n = node;

          while (n != base) {

            n = _graph.target((*_matching)[n]);

            Arc a = (*_ear)[n];

            n = _graph.target(a);

            (*_ear)[n] = _graph.oppositeArc(a);

          }

          node = _graph.target((*_matching)[base]);

          _tree_set->erase(base);

          _tree_set->erase(node);

          _blossom_set->insert(node, _blossom_set->find(base));

          (*_status)[node] = EVEN;

          _node_queue[_last++] = node;

          arc = _graph.oppositeArc((*_ear)[node]);

          node = _graph.target((*_ear)[node]);

          base = (*_blossom_rep)[_blossom_set->find(node)];

          _blossom_set->join(_graph.target(arc), base);

        }

      }

      (*_blossom_rep)[_blossom_set->find(nca)] = nca;

    }

    void extendOnArc(const Arc& a) {

      Node base = _graph.source(a);

      Node odd = _graph.target(a);

      (*_ear)[odd] = _graph.oppositeArc(a);

      Node even = _graph.target((*_matching)[odd]);

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

      (*_status)[odd] = ODD;

      (*_status)[even] = EVEN;

      int tree = _tree_set->find((*_blossom_rep)[_blossom_set->find(base)]);

      _tree_set->insert(odd, tree);

      _tree_set->insert(even, tree);

      _node_queue[_last++] = even;

    }

    void augmentOnArc(const Arc& a) {

      Node even = _graph.source(a);

      Node odd = _graph.target(a);

      int tree = _tree_set->find((*_blossom_rep)[_blossom_set->find(even)]);

      (*_matching)[odd] = _graph.oppositeArc(a);

      (*_status)[odd] = MATCHED;

      Arc arc = (*_matching)[even];

      (*_matching)[even] = a;

      while (arc != INVALID) {

        odd = _graph.target(arc);

        arc = (*_ear)[odd];

        even = _graph.target(arc);

        (*_matching)[odd] = arc;

        arc = (*_matching)[even];

        (*_matching)[even] = _graph.oppositeArc((*_matching)[odd]);

      }

      for (typename TreeSet::ItemIt it(*_tree_set, tree);

           it != INVALID; ++it) {

        if ((*_status)[it] == ODD) {

          (*_status)[it] = MATCHED;

        } else {

          int blossom = _blossom_set->find(it);

          for (typename BlossomSet::ItemIt jt(*_blossom_set, blossom);

               jt != INVALID; ++jt) {

            (*_status)[jt] = MATCHED;

          }

          _blossom_set->eraseClass(blossom);

        }

      }

      _tree_set->eraseClass(tree);

    }

  public:

    /// \brief Constructor

    ///

    /// Constructor.

    MaxMatching(const Graph& graph)

      : _graph(graph), _matching(0), _status(0), _ear(0),

        _blossom_set_index(0), _blossom_set(0), _blossom_rep(0),

        _tree_set_index(0), _tree_set(0) {}

    ~MaxMatching() {

      destroyStructures();

    }

    /// \name Execution Control

    /// The simplest way to execute the algorithm is to use the

    /// \c run() member function.\n

    /// If you need better control on the execution, you have to call

    /// one of the functions \ref init(), \ref greedyInit() or

    /// \ref matchingInit() first, then you can start the algorithm with

    /// \ref startSparse() or \ref startDense().

    ///@{

    /// \brief Set the initial matching to the empty matching.

    ///

    /// This function sets the initial matching to the empty matching.

    void init() {

      createStructures();

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

        (*_matching)[n] = INVALID;

        (*_status)[n] = UNMATCHED;

      }

    }

    /// \brief Find an initial matching in a greedy way.

    ///

    /// This function finds an initial matching in a greedy way.

    void greedyInit() {

      createStructures();

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

        (*_matching)[n] = INVALID;

        (*_status)[n] = UNMATCHED;

      }

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

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

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

            Node v = _graph.target(a);

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

              (*_matching)[n] = a;

              (*_status)[n] = MATCHED;

              (*_matching)[v] = _graph.oppositeArc(a);

              (*_status)[v] = MATCHED;

              break;

            }

          }

        }

      }

    }

    /// \brief Initialize the matching from a map.

    ///

    /// This function initializes the matching from a \c bool valued edge

    /// map. This map should have the property that there are no two incident

    /// edges with \c true value, i.e. it really contains a matching.

    /// \return \c true if the map contains a matching.

    template <typename MatchingMap>

    bool matchingInit(const MatchingMap& matching) {

      createStructures();

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

        (*_matching)[n] = INVALID;

        (*_status)[n] = UNMATCHED;

      }

      for(EdgeIt e(_graph); e!=INVALID; ++e) {

        if (matching[e]) {

          Node u = _graph.u(e);

          if ((*_matching)[u] != INVALID) return false;

          (*_matching)[u] = _graph.direct(e, true);

          (*_status)[u] = MATCHED;

          Node v = _graph.v(e);

          if ((*_matching)[v] != INVALID) return false;

          (*_matching)[v] = _graph.direct(e, false);

          (*_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 a const reference to the matching map.

    ///

    /// This function returns a const reference to a node map that stores

    /// the matching arc (or edge) incident to each node.

    const MatchingMap& matchingMap() const {

      return *_matching;

    }

    /// \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.

    Status status(const Node& n) const {

      return (*_status)[n];

    }

    /// \brief Return a const reference to the status map, which stores

    /// the Edmonds-Gallai decomposition.

    ///

    /// This function returns a const reference to a node map that stores the

    /// \ref Status "status" of each node in the Edmonds-Gallai decomposition.

    const StatusMap& statusMap() const {

      return *_status;

    }

    /// \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 GR The undirected graph type the algorithm runs on.

  /// \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;

    /// The type of the matching map

    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

    };

    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;

    int _unmatched;

    typedef MaxWeightedFractionalMatching<Graph, WeightMap> FractionalMatching;

    FractionalMatching *_fractional;

    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);

      }

      if (!_blossom_set) {

        _blossom_index = new IntNodeMap(_graph);

        _blossom_set = new BlossomSet(*_blossom_index);

        _blossom_data = new RangeMap<BlossomData>(_blossom_num);

      } else if (_blossom_data->size() != _blossom_num) {

        delete _blossom_data;

        _blossom_data = new RangeMap<BlossomData>(_blossom_num);

      }

      if (!_node_index) {

        _node_index = new IntNodeMap(_graph);

        _node_heap_index = new IntArcMap(_graph);

        _node_data = new RangeMap<NodeData>(_node_num,

                                            NodeData(*_node_heap_index));

      } else {

        delete _node_data;

        _node_data = new RangeMap<NodeData>(_node_num,

                                            NodeData(*_node_heap_index));

      }

      if (!_tree_set) {

        _tree_set_index = new IntIntMap(_blossom_num);

        _tree_set = new TreeSet(*_tree_set_index);

      } else {

        _tree_set_index->resize(_blossom_num);

      }

      if (!_delta1) {

        _delta1_index = new IntNodeMap(_graph);

        _delta1 = new BinHeap<Value, IntNodeMap>(*_delta1_index);

      }

      if (!_delta2) {

        _delta2_index = new IntIntMap(_blossom_num);

        _delta2 = new BinHeap<Value, IntIntMap>(*_delta2_index);

      } else {

        _delta2_index->resize(_blossom_num);

      }

      if (!_delta3) {

        _delta3_index = new IntEdgeMap(_graph);

        _delta3 = new BinHeap<Value, IntEdgeMap>(*_delta3_index);

      }

      if (!_delta4) {

        _delta4_index = new IntIntMap(_blossom_num);

        _delta4 = new BinHeap<Value, IntIntMap>(*_delta4_index);

      } else {

        _delta4_index->resize(_blossom_num);

      }

    }

    void destroyStructures() {

      if (_matching) {

        delete _matching;

      }

      if (_node_potential) {

        delete _node_potential;

      }

      if (_blossom_set) {

        delete _blossom_index;

        delete _blossom_set;

        delete _blossom_data;

      }

      if (_node_index) {

        delete _node_index;

        delete _node_heap_index;

        delete _node_data;

      }

      if (_tree_set) {

        delete _tree_set_index;

        delete _tree_set;

      }

      if (_delta1) {

        delete _delta1_index;

        delete _delta1;

      }

      if (_delta2) {

        delete _delta2_index;

        delete _delta2;

      }

      if (_delta3) {

        delete _delta3_index;

        delete _delta3;

      }

      if (_delta4) {

        delete _delta4_index;

        delete _delta4;

      }

    }

    void matchedToEven(int blossom, int tree) {

      if (_delta2->state(blossom) == _delta2->IN_HEAP) {

        _delta2->erase(blossom);

      }

      if (!_blossom_set->trivial(blossom)) {

        (*_blossom_data)[blossom].pot -=

          2 * (_delta_sum - (*_blossom_data)[blossom].offset);

      }

      for (typename BlossomSet::ItemIt n(*_blossom_set, blossom);

           n != INVALID; ++n) {

        _blossom_set->increase(n, std::numeric_limits<Value>::max());

        int ni = (*_node_index)[n];

        (*_node_data)[ni].heap.clear();

        (*_node_data)[ni].heap_index.clear();

        (*_node_data)[ni].pot += _delta_sum - (*_blossom_data)[blossom].offset;

        _delta1->push(n, (*_node_data)[ni].pot);

        for (InArcIt e(_graph, n); e != INVALID; ++e) {

          Node v = _graph.source(e);

          int vb = _blossom_set->find(v);

          int vi = (*_node_index)[v];

          Value rw = (*_node_data)[ni].pot + (*_node_data)[vi].pot -

            dualScale * _weight[e];

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

            if (_delta3->state(e) != _delta3->IN_HEAP && blossom != vb) {

              _delta3->push(e, rw / 2);

            }

          } else {

            typename std::map<int, Arc>::iterator it =

              (*_node_data)[vi].heap_index.find(tree);

            if (it != (*_node_data)[vi].heap_index.end()) {

              if ((*_node_data)[vi].heap[it->second] > rw) {

                (*_node_data)[vi].heap.replace(it->second, e);

                (*_node_data)[vi].heap.decrease(e, rw);

                it->second = e;

              }

            } else {

              (*_node_data)[vi].heap.push(e, rw);

              (*_node_data)[vi].heap_index.insert(std::make_pair(tree, e));

            }

            if ((*_blossom_set)[v] > (*_node_data)[vi].heap.prio()) {

              _blossom_set->decrease(v, (*_node_data)[vi].heap.prio());

              if ((*_blossom_data)[vb].status == MATCHED) {

                if (_delta2->state(vb) != _delta2->IN_HEAP) {

                  _delta2->push(vb, _blossom_set->classPrio(vb) -

                               (*_blossom_data)[vb].offset);

                } else if ((*_delta2)[vb] > _blossom_set->classPrio(vb) -

                           (*_blossom_data)[vb].offset) {

                  _delta2->decrease(vb, _blossom_set->classPrio(vb) -

                                   (*_blossom_data)[vb].offset);

                }

              }

            }

          }

        }

      }

      (*_blossom_data)[blossom].offset = 0;

    }

    void matchedToOdd(int blossom) {

      if (_delta2->state(blossom) == _delta2->IN_HEAP) {

        _delta2->erase(blossom);

      }

      (*_blossom_data)[blossom].offset += _delta_sum;

      if (!_blossom_set->trivial(blossom)) {

        _delta4->push(blossom, (*_blossom_data)[blossom].pot / 2 +

                      (*_blossom_data)[blossom].offset);

      }

    }

    void evenToMatched(int blossom, int tree) {

      if (!_blossom_set->trivial(blossom)) {