/* -*- mode: C++; indent-tabs-mode: nil; -*-

  *

  * This file is a part of LEMON, a generic C++ optimization library.

  *

  * Copyright (C) 2003-2013

  * 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, _unmatched;

   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);

             --_unmatched;

             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);

                 --_unmatched;

                 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);

             --_unmatched;

             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;

       }

       _unmatched = _node_num;

     }

     /// \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;

       }

       _unmatched = _node_num;

       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;

               _unmatched -= 2;

               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;

       }

       _unmatched = _node_num;

       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;

           _unmatched -= 2;

         }

       }

       return true;

     }

     /// \brief Start Edmonds' algorithm

     ///

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

     /// \c decomposition parameter is set to false, then the Gallai-Edmonds

     /// decomposition is not computed.

     ///

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

     /// called before using this function.

     void startSparse(bool decomposition = true) {

       int _unmatched_limit = decomposition ? 0 : 1;

       for(NodeIt n(_graph); _unmatched > _unmatched_limit; ++n) {

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

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

           _tree_set->insert(n);

           (*_status)[n] = EVEN;

           --_unmatched;

           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.  If

     /// the \c decomposition parameter is set to false, then the Gallai-Edmonds

     /// decomposition is not computed.

     ///

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

     /// called before using this function.

     void startDense(bool decomposition = true) {

       int _unmatched_limit = decomposition ? 0 : 1;

       for(NodeIt n(_graph); _unmatched > _unmatched_limit; ++n) {

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

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

           _tree_set->insert(n);

           (*_status)[n] = EVEN;

           --_unmatched;

           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). If the \c decomposition parameter is set to

     /// false, then the Gallai-Edmonds decomposition is not computed. In some

     /// cases, this can speed up the algorithm significantly, especially when a

     /// maximum matching is computed in a dense graph with odd number of nodes.

     void run(bool decomposition = true) {

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

         greedyInit();

         startSparse(decomposition);

       } else {

         init();

         startDense(decomposition);

       }

     }

     /// @}

     /// \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)) {

         (*_blossom_data)[blossom].pot += 2 * _delta_sum;

       }

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

            n != INVALID; ++n) {

         int ni = (*_node_index)[n];

         (*_node_data)[ni].pot -= _delta_sum;

         _delta1->erase(n);

         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 (vb == blossom) {

             if (_delta3->state(e) == _delta3->IN_HEAP) {

               _delta3->erase(e);

             }

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

             if (_delta3->state(e) == _delta3->IN_HEAP) {

               _delta3->erase(e);

             }

             int vt = _tree_set->find(vb);

             if (vt != tree) {

               Arc r = _graph.oppositeArc(e);

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

                 (*_node_data)[ni].heap_index.find(vt);

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

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

                   (*_node_data)[ni].heap.replace(it->second, r);

                   (*_node_data)[ni].heap.decrease(r, rw);

                   it->second = r;

                 }

               } else {

                 (*_node_data)[ni].heap.push(r, rw);

                 (*_node_data)[ni].heap_index.insert(std::make_pair(vt, r));

               }

               if ((*_blossom_set)[n] > (*_node_data)[ni].heap.prio()) {

                 _blossom_set->decrease(n, (*_node_data)[ni].heap.prio());

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

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

                                (*_blossom_data)[blossom].offset);

                 } else if ((*_delta2)[blossom] >

                            _blossom_set->classPrio(blossom) -

                            (*_blossom_data)[blossom].offset){

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

                                    (*_blossom_data)[blossom].offset);

                 }

               }

             }

           } else {

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

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

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

               (*_node_data)[vi].heap.erase(it->second);

               (*_node_data)[vi].heap_index.erase(it);

               if ((*_node_data)[vi].heap.empty()) {

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

               } else if ((*_blossom_set)[v] < (*_node_data)[vi].heap.prio()) {

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

               }

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

                 if (_blossom_set->classPrio(vb) ==

                     std::numeric_limits<Value>::max()) {

                   _delta2->erase(vb);

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

                            (*_blossom_data)[vb].offset) {

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

                                    (*_blossom_data)[vb].offset);

                 }

               }

             }

           }

         }

       }

     }

     void oddToMatched(int blossom) {

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

       if (_blossom_set->classPrio(blossom) !=

           std::numeric_limits<Value>::max()) {

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

                       (*_blossom_data)[blossom].offset);

       }

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

         _delta4->erase(blossom);

       }

     }

     void oddToEven(int blossom, int tree) {

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

         _delta4->erase(blossom);

         (*_blossom_data)[blossom].pot -=

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

       }

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

            n != INVALID; ++n) {

         int ni = (*_node_index)[n];

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

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

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

         (*_node_data)[ni].pot +=

           2 * _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 alternatePath(int even, int tree) {

       int odd;

       evenToMatched(even, tree);

       (*_blossom_data)[even].status = MATCHED;

       while ((*_blossom_data)[even].pred != INVALID) {

         odd = _blossom_set->find(_graph.target((*_blossom_data)[even].pred));

         (*_blossom_data)[odd].status = MATCHED;

         oddToMatched(odd);

         (*_blossom_data)[odd].next = (*_blossom_data)[odd].pred;

         even = _blossom_set->find(_graph.target((*_blossom_data)[odd].pred));

         (*_blossom_data)[even].status = MATCHED;

         evenToMatched(even, tree);

         (*_blossom_data)[even].next =

           _graph.oppositeArc((*_blossom_data)[odd].pred);

       }

     }

     void destroyTree(int tree) {

       for (TreeSet::ItemIt b(*_tree_set, tree); b != INVALID; ++b) {

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

           (*_blossom_data)[b].status = MATCHED;

           evenToMatched(b, tree);

         } else if ((*_blossom_data)[b].status == ODD) {

           (*_blossom_data)[b].status = MATCHED;

           oddToMatched(b);

         }

       }

       _tree_set->eraseClass(tree);

     }

     void unmatchNode(const Node& node) {

       int blossom = _blossom_set->find(node);

       int tree = _tree_set->find(blossom);

       alternatePath(blossom, tree);

       destroyTree(tree);

       (*_blossom_data)[blossom].base = node;

       (*_blossom_data)[blossom].next = INVALID;

     }

     void augmentOnEdge(const Edge& edge) {

       int left = _blossom_set->find(_graph.u(edge));

       int right = _blossom_set->find(_graph.v(edge));

       int left_tree = _tree_set->find(left);

       alternatePath(left, left_tree);

       destroyTree(left_tree);

       int right_tree = _tree_set->find(right);

       alternatePath(right, right_tree);

       destroyTree(right_tree);

       (*_blossom_data)[left].next = _graph.direct(edge, true);

       (*_blossom_data)[right].next = _graph.direct(edge, false);

     }

     void augmentOnArc(const Arc& arc) {

       int left = _blossom_set->find(_graph.source(arc));

       int right = _blossom_set->find(_graph.target(arc));

       (*_blossom_data)[left].status = MATCHED;

       int right_tree = _tree_set->find(right);

       alternatePath(right, right_tree);

       destroyTree(right_tree);

       (*_blossom_data)[left].next = arc;

       (*_blossom_data)[right].next = _graph.oppositeArc(arc);

     }

     void extendOnArc(const Arc& arc) {

       int base = _blossom_set->find(_graph.target(arc));

       int tree = _tree_set->find(base);

       int odd = _blossom_set->find(_graph.source(arc));

       _tree_set->insert(odd, tree);

       (*_blossom_data)[odd].status = ODD;

       matchedToOdd(odd);

       (*_blossom_data)[odd].pred = arc;

       int even = _blossom_set->find(_graph.target((*_blossom_data)[odd].next));

       (*_blossom_data)[even].pred = (*_blossom_data)[even].next;

       _tree_set->insert(even, tree);

       (*_blossom_data)[even].status = EVEN;

       matchedToEven(even, tree);

     }

     void shrinkOnEdge(const Edge& edge, int tree) {

       int nca = -1;

       std::vector<int> left_path, right_path;

       {

         std::set<int> left_set, right_set;

         int left = _blossom_set->find(_graph.u(edge));

         left_path.push_back(left);

         left_set.insert(left);

         int right = _blossom_set->find(_graph.v(edge));

         right_path.push_back(right);

         right_set.insert(right);

         while (true) {

           if ((*_blossom_data)[left].pred == INVALID) break;

           left =

             _blossom_set->find(_graph.target((*_blossom_data)[left].pred));

           left_path.push_back(left);

           left =

             _blossom_set->find(_graph.target((*_blossom_data)[left].pred));

           left_path.push_back(left);

           left_set.insert(left);

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

             nca = left;

             break;

           }

           if ((*_blossom_data)[right].pred == INVALID) break;

           right =

             _blossom_set->find(_graph.target((*_blossom_data)[right].pred));

           right_path.push_back(right);

           right =

             _blossom_set->find(_graph.target((*_blossom_data)[right].pred));

           right_path.push_back(right);

           right_set.insert(right);

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

             nca = right;

             break;

           }

         }

         if (nca == -1) {

           if ((*_blossom_data)[left].pred == INVALID) {

             nca = right;

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

               nca =

                 _blossom_set->find(_graph.target((*_blossom_data)[nca].pred));

               right_path.push_back(nca);

               nca =

                 _blossom_set->find(_graph.target((*_blossom_data)[nca].pred));

               right_path.push_back(nca);

             }

           } else {

             nca = left;

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

               nca =

                 _blossom_set->find(_graph.target((*_blossom_data)[nca].pred));

               left_path.push_back(nca);

               nca =

                 _blossom_set->find(_graph.target((*_blossom_data)[nca].pred));

               left_path.push_back(nca);

             }

           }

         }

       }

       std::vector<int> subblossoms;

       Arc prev;

       prev = _graph.direct(edge, true);

       for (int i = 0; left_path[i] != nca; i += 2) {

         subblossoms.push_back(left_path[i]);

         (*_blossom_data)[left_path[i]].next = prev;

         _tree_set->erase(left_path[i]);

         subblossoms.push_back(left_path[i + 1]);

         (*_blossom_data)[left_path[i + 1]].status = EVEN;

         oddToEven(left_path[i + 1], tree);

         _tree_set->erase(left_path[i + 1]);

         prev = _graph.oppositeArc((*_blossom_data)[left_path[i + 1]].pred);

       }

       int k = 0;

       while (right_path[k] != nca) ++k;

       subblossoms.push_back(nca);

       (*_blossom_data)[nca].next = prev;

       for (int i = k - 2; i >= 0; i -= 2) {

         subblossoms.push_back(right_path[i + 1]);

         (*_blossom_data)[right_path[i + 1]].status = EVEN;

         oddToEven(right_path[i + 1], tree);

         _tree_set->erase(right_path[i + 1]);

         (*_blossom_data)[right_path[i + 1]].next =

           (*_blossom_data)[right_path[i + 1]].pred;

         subblossoms.push_back(right_path[i]);

         _tree_set->erase(right_path[i]);

       }

       int surface =

         _blossom_set->join(subblossoms.begin(), subblossoms.end());

       for (int i = 0; i < int(subblossoms.size()); ++i) {

         if (!_blossom_set->trivial(subblossoms[i])) {

           (*_blossom_data)[subblossoms[i]].pot += 2 * _delta_sum;

         }

         (*_blossom_data)[subblossoms[i]].status = MATCHED;

       }

       (*_blossom_data)[surface].pot = -2 * _delta_sum;

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

       (*_blossom_data)[surface].status = EVEN;

       (*_blossom_data)[surface].pred = (*_blossom_data)[nca].pred;

       (*_blossom_data)[surface].next = (*_blossom_data)[nca].pred;

       _tree_set->insert(surface, tree);

       _tree_set->erase(nca);

     }

     void splitBlossom(int blossom) {

       Arc next = (*_blossom_data)[blossom].next;

       Arc pred = (*_blossom_data)[blossom].pred;

       int tree = _tree_set->find(blossom);

       (*_blossom_data)[blossom].status = MATCHED;

       oddToMatched(blossom);

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

         _delta2->erase(blossom);

       }

       std::vector<int> subblossoms;

       _blossom_set->split(blossom, std::back_inserter(subblossoms));

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

       int b = _blossom_set->find(_graph.source(pred));

       int d = _blossom_set->find(_graph.source(next));

       int ib = -1, id = -1;

       for (int i = 0; i < int(subblossoms.size()); ++i) {

         if (subblossoms[i] == b) ib = i;

         if (subblossoms[i] == d) id = i;

         (*_blossom_data)[subblossoms[i]].offset = offset;

         if (!_blossom_set->trivial(subblossoms[i])) {

           (*_blossom_data)[subblossoms[i]].pot -= 2 * offset;

         }

         if (_blossom_set->classPrio(subblossoms[i]) !=

             std::numeric_limits<Value>::max()) {

           _delta2->push(subblossoms[i],

                         _blossom_set->classPrio(subblossoms[i]) -

                         (*_blossom_data)[subblossoms[i]].offset);

         }

       }

       if (id > ib ? ((id - ib) % 2 == 0) : ((ib - id) % 2 == 1)) {

         for (int i = (id + 1) % subblossoms.size();

              i != ib; i = (i + 2) % subblossoms.size()) {

           int sb = subblossoms[i];

           int tb = subblossoms[(i + 1) % subblossoms.size()];

           (*_blossom_data)[sb].next =

             _graph.oppositeArc((*_blossom_data)[tb].next);

         }

         for (int i = ib; i != id; i = (i + 2) % subblossoms.size()) {

           int sb = subblossoms[i];

           int tb = subblossoms[(i + 1) % subblossoms.size()];

           int ub = subblossoms[(i + 2) % subblossoms.size()];

           (*_blossom_data)[sb].status = ODD;

           matchedToOdd(sb);

           _tree_set->insert(sb, tree);

           (*_blossom_data)[sb].pred = pred;

           (*_blossom_data)[sb].next =

             _graph.oppositeArc((*_blossom_data)[tb].next);

           pred = (*_blossom_data)[ub].next;

           (*_blossom_data)[tb].status = EVEN;

           matchedToEven(tb, tree);

           _tree_set->insert(tb, tree);

           (*_blossom_data)[tb].pred = (*_blossom_data)[tb].next;

         }

         (*_blossom_data)[subblossoms[id]].status = ODD;

         matchedToOdd(subblossoms[id]);

         _tree_set->insert(subblossoms[id], tree);

         (*_blossom_data)[subblossoms[id]].next = next;

         (*_blossom_data)[subblossoms[id]].pred = pred;

       } else {

         for (int i = (ib + 1) % subblossoms.size();

              i != id; i = (i + 2) % subblossoms.size()) {

           int sb = subblossoms[i];

           int tb = subblossoms[(i + 1) % subblossoms.size()];

           (*_blossom_data)[sb].next =

             _graph.oppositeArc((*_blossom_data)[tb].next);

         }

         for (int i = id; i != ib; i = (i + 2) % subblossoms.size()) {

           int sb = subblossoms[i];

           int tb = subblossoms[(i + 1) % subblossoms.size()];

           int ub = subblossoms[(i + 2) % subblossoms.size()];

           (*_blossom_data)[sb].status = ODD;

           matchedToOdd(sb);

           _tree_set->insert(sb, tree);

           (*_blossom_data)[sb].next = next;

           (*_blossom_data)[sb].pred =

             _graph.oppositeArc((*_blossom_data)[tb].next);

           (*_blossom_data)[tb].status = EVEN;

           matchedToEven(tb, tree);

           _tree_set->insert(tb, tree);

           (*_blossom_data)[tb].pred =

             (*_blossom_data)[tb].next =

             _graph.oppositeArc((*_blossom_data)[ub].next);

           next = (*_blossom_data)[ub].next;

         }

         (*_blossom_data)[subblossoms[ib]].status = ODD;

         matchedToOdd(subblossoms[ib]);

         _tree_set->insert(subblossoms[ib], tree);

         (*_blossom_data)[subblossoms[ib]].next = next;

         (*_blossom_data)[subblossoms[ib]].pred = pred;

       }

       _tree_set->erase(blossom);

     }

     void extractBlossom(int blossom, const Node& base, const Arc& matching) {

       if (_blossom_set->trivial(blossom)) {

         int bi = (*_node_index)[base];

         Value pot = (*_node_data)[bi].pot;

         (*_matching)[base] = matching;

         _blossom_node_list.push_back(base);

         (*_node_potential)[base] = pot;

       } else {

         Value pot = (*_blossom_data)[blossom].pot;

         int bn = _blossom_node_list.size();

         std::vector<int> subblossoms;

         _blossom_set->split(blossom, std::back_inserter(subblossoms));

         int b = _blossom_set->find(base);

         int ib = -1;

         for (int i = 0; i < int(subblossoms.size()); ++i) {

           if (subblossoms[i] == b) { ib = i; break; }

         }

         for (int i = 1; i < int(subblossoms.size()); i += 2) {

           int sb = subblossoms[(ib + i) % subblossoms.size()];

           int tb = subblossoms[(ib + i + 1) % subblossoms.size()];

           Arc m = (*_blossom_data)[tb].next;

           extractBlossom(sb, _graph.target(m), _graph.oppositeArc(m));

           extractBlossom(tb, _graph.source(m), m);

         }

         extractBlossom(subblossoms[ib], base, matching);

         int en = _blossom_node_list.size();

         _blossom_potential.push_back(BlossomVariable(bn, en, pot));

       }

     }

     void extractMatching() {

       std::vector<int> blossoms;

       for (typename BlossomSet::ClassIt c(*_blossom_set); c != INVALID; ++c) {

         blossoms.push_back(c);

       }

       for (int i = 0; i < int(blossoms.size()); ++i) {

         if ((*_blossom_data)[blossoms[i]].next != INVALID) {

           Value offset = (*_blossom_data)[blossoms[i]].offset;

           (*_blossom_data)[blossoms[i]].pot += 2 * offset;

           for (typename BlossomSet::ItemIt n(*_blossom_set, blossoms[i]);

                n != INVALID; ++n) {

             (*_node_data)[(*_node_index)[n]].pot -= offset;

           }

           Arc matching = (*_blossom_data)[blossoms[i]].next;

           Node base = _graph.source(matching);

           extractBlossom(blossoms[i], base, matching);

         } else {

           Node base = (*_blossom_data)[blossoms[i]].base;

           extractBlossom(blossoms[i], base, INVALID);

         }

       }

     }

   public:

     /// \brief Constructor

     ///

     /// Constructor.

     MaxWeightedMatching(const Graph& graph, const WeightMap& weight)

       : _graph(graph), _weight(weight), _matching(0),

         _node_potential(0), _blossom_potential(), _blossom_node_list(),

         _node_num(0), _blossom_num(0),

         _blossom_index(0), _blossom_set(0), _blossom_data(0),

         _node_index(0), _node_heap_index(0), _node_data(0),

         _tree_set_index(0), _tree_set(0),

         _delta1_index(0), _delta1(0),

         _delta2_index(0), _delta2(0),

         _delta3_index(0), _delta3(0),

         _delta4_index(0), _delta4(0),

         _delta_sum(), _unmatched(0),

         _fractional(0)

     {}

     ~MaxWeightedMatching() {

       destroyStructures();

       if (_fractional) {

         delete _fractional;

       }

     }

     /// \name Execution Control

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

     /// \ref run() member function.

     ///@{

     /// \brief Initialize the algorithm

     ///

     /// This function initializes the algorithm.

     void init() {

       createStructures();

       _blossom_node_list.clear();

       _blossom_potential.clear();

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

         (*_node_heap_index)[e] = BinHeap<Value, IntArcMap>::PRE_HEAP;

       }

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

         (*_delta1_index)[n] = _delta1->PRE_HEAP;

       }

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

         (*_delta3_index)[e] = _delta3->PRE_HEAP;

       }

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

         (*_delta2_index)[i] = _delta2->PRE_HEAP;

         (*_delta4_index)[i] = _delta4->PRE_HEAP;

       }

       _unmatched = _node_num;

       _delta1->clear();

       _delta2->clear();

       _delta3->clear();

       _delta4->clear();

       _blossom_set->clear();

       _tree_set->clear();

       int index = 0;

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

         Value max = 0;

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

           if (_graph.target(e) == n) continue;

           if ((dualScale * _weight[e]) / 2 > max) {

             max = (dualScale * _weight[e]) / 2;

           }

         }

         (*_node_index)[n] = index;

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

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

         (*_node_data)[index].pot = max;

         _delta1->push(n, max);

         int blossom =

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

         _tree_set->insert(blossom);

         (*_blossom_data)[blossom].status = EVEN;

         (*_blossom_data)[blossom].pred = INVALID;

         (*_blossom_data)[blossom].next = INVALID;

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

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

         ++index;

       }

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

         int si = (*_node_index)[_graph.u(e)];

         int ti = (*_node_index)[_graph.v(e)];

         if (_graph.u(e) != _graph.v(e)) {

           _delta3->push(e, ((*_node_data)[si].pot + (*_node_data)[ti].pot -

                             dualScale * _weight[e]) / 2);

         }

       }

     }

     /// \brief Initialize the algorithm with fractional matching

     ///

     /// This function initializes the algorithm with a fractional

     /// matching. This initialization is also called jumpstart heuristic.

     void fractionalInit() {

       createStructures();

       _blossom_node_list.clear();

       _blossom_potential.clear();

       if (_fractional == 0) {

         _fractional = new FractionalMatching(_graph, _weight, false);

       }

       _fractional->run();

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

         (*_node_heap_index)[e] = BinHeap<Value, IntArcMap>::PRE_HEAP;

       }

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

         (*_delta1_index)[n] = _delta1->PRE_HEAP;

       }

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

         (*_delta3_index)[e] = _delta3->PRE_HEAP;

       }

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

         (*_delta2_index)[i] = _delta2->PRE_HEAP;

         (*_delta4_index)[i] = _delta4->PRE_HEAP;

       }

       _unmatched = 0;

       _delta1->clear();

       _delta2->clear();

       _delta3->clear();

       _delta4->clear();

       _blossom_set->clear();

       _tree_set->clear();

       int index = 0;

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

         Value pot = _fractional->nodeValue(n);

         (*_node_index)[n] = index;

         (*_node_data)[index].pot = pot;

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

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

         int blossom =

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

         (*_blossom_data)[blossom].status = MATCHED;

         (*_blossom_data)[blossom].pred = INVALID;

         (*_blossom_data)[blossom].next = _fractional->matching(n);

         if (_fractional->matching(n) == INVALID) {

           (*_blossom_data)[blossom].base = n;

         }

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

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

         ++index;

       }

       typename Graph::template NodeMap<bool> processed(_graph, false);

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

         if (processed[n]) continue;

         processed[n] = true;

         if (_fractional->matching(n) == INVALID) continue;

         int num = 1;

         Node v = _graph.target(_fractional->matching(n));

         while (n != v) {

           processed[v] = true;

           v = _graph.target(_fractional->matching(v));

           ++num;

         }

         if (num % 2 == 1) {

           std::vector<int> subblossoms(num);

           subblossoms[--num] = _blossom_set->find(n);

           _delta1->push(n, _fractional->nodeValue(n));

           v = _graph.target(_fractional->matching(n));

           while (n != v) {

             subblossoms[--num] = _blossom_set->find(v);

             _delta1->push(v, _fractional->nodeValue(v));

             v = _graph.target(_fractional->matching(v));

           }

           int surface =

             _blossom_set->join(subblossoms.begin(), subblossoms.end());

           (*_blossom_data)[surface].status = EVEN;

           (*_blossom_data)[surface].pred = INVALID;

           (*_blossom_data)[surface].next = INVALID;

           (*_blossom_data)[surface].pot = 0;

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

           _tree_set->insert(surface);

           ++_unmatched;

         }

       }

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

         int si = (*_node_index)[_graph.u(e)];

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

         int ti = (*_node_index)[_graph.v(e)];

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

         if ((*_blossom_data)[sb].status == EVEN &&

             (*_blossom_data)[tb].status == EVEN && sb != tb) {

           _delta3->push(e, ((*_node_data)[si].pot + (*_node_data)[ti].pot -

                             dualScale * _weight[e]) / 2);

         }

       }

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

         int nb = _blossom_set->find(n);

         if ((*_blossom_data)[nb].status != MATCHED) continue;

         int ni = (*_node_index)[n];

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

           Node v = _graph.target(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) {

             int vt = _tree_set->find(vb);

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

               (*_node_data)[ni].heap_index.find(vt);

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

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

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

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

                 it->second = e;

               }

             } else {

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

               (*_node_data)[ni].heap_index.insert(std::make_pair(vt, e));

             }

           }

         }

         if (!(*_node_data)[ni].heap.empty()) {

           _blossom_set->decrease(n, (*_node_data)[ni].heap.prio());

           _delta2->push(nb, _blossom_set->classPrio(nb));

         }

       }

     }

     /// \brief Start the algorithm

     ///

     /// This function starts the algorithm.

     ///

     /// \pre \ref init() or \ref fractionalInit() must be called

     /// before using this function.

     void start() {

       enum OpType {

         D1, D2, D3, D4

       };

       while (_unmatched > 0) {

         Value d1 = !_delta1->empty() ?

           _delta1->prio() : std::numeric_limits<Value>::max();

         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 = d3; OpType ot = D3;

         if (d1 < _delta_sum) { _delta_sum = d1; ot = D1; }

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

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

             if ((*_blossom_data)[blossom].next == INVALID) {

               augmentOnArc(a);

               --_unmatched;

             } else {

               extendOnArc(a);

             }

           }

           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();

     }

     /// \brief Run the algorithm.

     ///

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

     ///

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

     /// \code

     ///   mwm.fractionalInit();

     ///   mwm.start();

     /// \endcode

     void run() {

       fractionalInit();

       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 matchingWeight() const {

       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 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 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 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 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;

     /// The type of the matching map

     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;

     int _unmatched;

     typedef MaxWeightedPerfectFractionalMatching<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 if (_node_data->size() != _node_num) {

         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 (!_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 (_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;

         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)) {

         (*_blossom_data)[blossom].pot += 2 * _delta_sum;

       }

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

            n != INVALID; ++n) {

         int ni = (*_node_index)[n];

         (*_node_data)[ni].pot -= _delta_sum;

         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 (vb == blossom) {

             if (_delta3->state(e) == _delta3->IN_HEAP) {

               _delta3->erase(e);

             }

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

             if (_delta3->state(e) == _delta3->IN_HEAP) {

               _delta3->erase(e);

             }

             int vt = _tree_set->find(vb);

             if (vt != tree) {

               Arc r = _graph.oppositeArc(e);

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

                 (*_node_data)[ni].heap_index.find(vt);

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

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

                   (*_node_data)[ni].heap.replace(it->second, r);

                   (*_node_data)[ni].heap.decrease(r, rw);

                   it->second = r;

                 }

               } else {

                 (*_node_data)[ni].heap.push(r, rw);

                 (*_node_data)[ni].heap_index.insert(std::make_pair(vt, r));

               }

               if ((*_blossom_set)[n] > (*_node_data)[ni].heap.prio()) {

                 _blossom_set->decrease(n, (*_node_data)[ni].heap.prio());

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

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

                                (*_blossom_data)[blossom].offset);

                 } else if ((*_delta2)[blossom] >

                            _blossom_set->classPrio(blossom) -

                            (*_blossom_data)[blossom].offset){

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

                                    (*_blossom_data)[blossom].offset);

                 }

               }

             }

           } else {

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

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

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

               (*_node_data)[vi].heap.erase(it->second);

               (*_node_data)[vi].heap_index.erase(it);

               if ((*_node_data)[vi].heap.empty()) {

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

               } else if ((*_blossom_set)[v] < (*_node_data)[vi].heap.prio()) {

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

               }

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

                 if (_blossom_set->classPrio(vb) ==

                     std::numeric_limits<Value>::max()) {

                   _delta2->erase(vb);

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

                            (*_blossom_data)[vb].offset) {

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

                                    (*_blossom_data)[vb].offset);

                 }

               }

             }

           }

         }

       }

     }

     void oddToMatched(int blossom) {

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

       if (_blossom_set->classPrio(blossom) !=

           std::numeric_limits<Value>::max()) {

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

                        (*_blossom_data)[blossom].offset);

       }

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

         _delta4->erase(blossom);

       }

     }

     void oddToEven(int blossom, int tree) {

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

         _delta4->erase(blossom);

         (*_blossom_data)[blossom].pot -=

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

       }

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

            n != INVALID; ++n) {

         int ni = (*_node_index)[n];

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

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

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

         (*_node_data)[ni].pot +=

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

         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 alternatePath(int even, int tree) {

       int odd;

       evenToMatched(even, tree);

       (*_blossom_data)[even].status = MATCHED;

       while ((*_blossom_data)[even].pred != INVALID) {

         odd = _blossom_set->find(_graph.target((*_blossom_data)[even].pred));