RhodeCode

    /// 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, UNMATCHED = -2

    };

    typedef HeapUnionFind<Value, IntNodeMap> BlossomSet;

    struct BlossomData {

      int tree;

      Status status;

      Arc pred, next;

      Value pot, offset;

      Node base;

    };

    IntNodeMap *_blossom_index;

    BlossomSet *_blossom_set;

    RangeMap<BlossomData>* _blossom_data;

    IntNodeMap *_node_index;

    IntArcMap *_node_heap_index;

    struct NodeData {

      NodeData(IntArcMap& node_heap_index)

        : heap(node_heap_index) {}

      int blossom;

      Value pot;

      BinHeap<Value, IntArcMap> heap;

      std::map<int, Arc> heap_index;

      int tree;

    };

    RangeMap<NodeData>* _node_data;

    typedef ExtendFindEnum<IntIntMap> TreeSet;

    IntIntMap *_tree_set_index;

    TreeSet *_tree_set;

    IntNodeMap *_delta1_index;

    BinHeap<Value, IntNodeMap> *_delta1;

    IntIntMap *_delta2_index;

    BinHeap<Value, IntIntMap> *_delta2;

    IntEdgeMap *_delta3_index;

    BinHeap<Value, IntEdgeMap> *_delta3;

    IntIntMap *_delta4_index;

    BinHeap<Value, IntIntMap> *_delta4;

    Value _delta_sum;

    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);

      }

      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));

      }

      if (!_tree_set) {

        _tree_set_index = new IntIntMap(_blossom_num);

        _tree_set = new TreeSet(*_tree_set_index);

      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);

      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);

      }

    }

    void destroyStructures() {

      _node_num = countNodes(_graph);

      _blossom_num = _node_num * 3 / 2;

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

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

              _delta3->push(e, rw);

            }

          } 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) -

      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]].status == MATCHED) {

          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() {}

    ~MaxWeightedMatching() {

      destroyStructures();

    }

    /// \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();

      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;

      }

      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].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 Start the algorithm

    ///

    /// This function starts the algorithm.

    ///

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

    void start() {

      enum OpType {

        D1, D2, D3, D4

      };

      int unmatched = _node_num;

      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 = d1; OpType ot = D1;

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

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

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

        switch (ot) {

        case D1:

          {

            Node n = _delta1->top();

            unmatchNode(n);

            --unmatched;

          }

          break;

        case D2:

          {

            int blossom = _delta2->top();

            Node n = _blossom_set->classTop(blossom);

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

            extendOnArc(e);

          }

          break;

        case D3:

          {

            Edge e = _delta3->top();

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

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

            if (left_blossom == right_blossom) {

              _delta3->pop();

            } else {

              int left_tree;

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

                left_tree = _tree_set->find(left_blossom);

              } else {

                left_tree = -1;

                ++unmatched;

              }

              int right_tree;

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

                right_tree = _tree_set->find(right_blossom);

              } else {

                right_tree = -1;

                ++unmatched;

              }

              if (left_tree == right_tree) {

                shrinkOnEdge(e, left_tree);

              } else {

                augmentOnEdge(e);

                unmatched -= 2;

              }

            }

          } break;

        case D4:

          splitBlossom(_delta4->top());

          break;

        }

      }

      extractMatching();

    }

    /// \brief Run the algorithm.

    ///

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

    ///

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

    /// \code

    ///   mwm.init();

    ///   mwm.start();

    /// \endcode

    void run() {

      init();

      start();

    }

    /// @}

    /// \name Primal Solution

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

    /// matching.\n

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

    /// using them.

    /// @{

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

    ///

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

    ///

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

    Value 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;

        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;

    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);

      }

      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));

      }

      if (!_tree_set) {

        _tree_set_index = new IntIntMap(_blossom_num);

        _tree_set = new TreeSet(*_tree_set_index);

      if (!_delta2) {

        _delta2_index = new IntIntMap(_blossom_num);

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

      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);

      }

    }

    void destroyStructures() {

      _node_num = countNodes(_graph);

      _blossom_num = _node_num * 3 / 2;

      if (_matching) {

        delete _matching;

      }

      if (_node_potential) {