Changeset 2548:a3ba22ebccc6 in lemon-0.x

Timestamp:

12/28/07 12:00:51 (16 years ago)

Author:

Balazs Dezso

Branch:

default

Phase:

public

Convert:

svn:c9d7d8f5-90d6-0310-b91f-818b3a526b0e/lemon/trunk@3425

Message:

Edmond's Blossom shrinking algroithm:
MaxWeightedMatching?
MaxWeightedPerfectMatching?

Files:

• doc/groups.dox (modified) (1 diff)
• lemon/bin_heap.h (modified) (2 diffs)
• lemon/max_matching.h (modified) (3 diffs)
• lemon/unionfind.h (modified) (1 diff)

doc/groups.dox

@@ -319 +319 @@
 for find matchings in graphs and bipartite graphs.
 
-This group provides some algorithm objects and function
-to calculate matchings in graphs and bipartite graphs.
+This group provides some algorithm objects and function to calculate
+matchings in graphs and bipartite graphs. The general matching problem is
+finding a subset of the edges which does not shares common endpoints.
+ 
+There are several different algorithms for calculate matchings in
+graphs.  The matching problems in bipartite graphs are generally
+easier than in general graphs. The goal of the matching optimization
+can be the finding maximum cardinality, maximum weight or minimum cost
+matching. The search can be constrained to find perfect or
+maximum cardinality matching.
+
+Lemon contains the next algorithms:
+- \ref lemon::MaxBipartiteMatching "MaxBipartiteMatching" Hopcroft-Karp 
+  augmenting path algorithm for calculate maximum cardinality matching in 
+  bipartite graphs
+- \ref lemon::PrBipartiteMatching "PrBipartiteMatching" Push-Relabel 
+  algorithm for calculate maximum cardinality matching in bipartite graphs 
+- \ref lemon::MaxWeightedBipartiteMatching "MaxWeightedBipartiteMatching" 
+  Successive shortest path algorithm for calculate maximum weighted matching 
+  and maximum weighted bipartite matching in bipartite graph
+- \ref lemon::MinCostMaxBipartiteMatching "MinCostMaxBipartiteMatching" 
+  Successive shortest path algorithm for calculate minimum cost maximum 
+  matching in bipartite graph
+- \ref lemon::MaxMatching "MaxMatching" Edmond's blossom shrinking algorithm
+  for calculate maximum cardinality matching in general graph
+- \ref lemon::MaxWeightedMatching "MaxWeightedMatching" Edmond's blossom
+  shrinking algorithm for calculate maximum weighted matching in general
+  graph
+- \ref lemon::MaxWeightedPerfectMatching "MaxWeightedPerfectMatching"
+  Edmond's blossom shrinking algorithm for calculate maximum weighted
+  perfect matching in general graph
 
 \image html bipartite_matching.png

lemon/bin_heap.h

@@ -53 +53 @@
 
   public:
+    ///\e
     typedef _ItemIntMap ItemIntMap;
+    ///\e
     typedef _Prio Prio;
+    ///\e
     typedef typename ItemIntMap::Key Item;
+    ///\e
     typedef std::pair<Item,Prio> Pair;
+    ///\e
     typedef _Compare Compare;
 
@@ -322 +327 @@
     }
 
+    /// \brief Replaces an item in the heap.
+    ///
+    /// The \c i item is replaced with \c j item. The \c i item should
+    /// be in the heap, while the \c j should be out of the heap. The
+    /// \c i item will out of the heap and \c j will be in the heap
+    /// with the same prioriority as prevoiusly the \c i item.
+    void replace(const Item& i, const Item& j) {
+      int idx = iim[i];
+      iim.set(i, iim[j]);
+      iim.set(j, idx);
+      data[idx].first = j;
+    }
+
   }; // class BinHeap
   

lemon/max_matching.h

@@ -20 +20 @@
 #define LEMON_MAX_MATCHING_H
 
+#include <vector>
 #include <queue>
+#include <set>
+
 #include <lemon/bits/invalid.h>
 #include <lemon/unionfind.h>
 #include <lemon/graph_utils.h>
+#include <lemon/bin_heap.h>
 
 ///\ingroup matching
@@ -32 +36 @@
 
   ///\ingroup matching
-
+  ///
   ///\brief Edmonds' alternating forest maximum matching algorithm.
   ///
@@ -619 +623 @@
 
   };
+
+  /// \ingroup matching
+  ///
+  /// \brief Weighted matching in general undirected 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 undirected
+  /// edges in the graph with maximum overall weight and no two of
+  /// them shares their endpoints. The problem can be formulated with
+  /// the next 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 endpoints 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 next:
+  /// \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 \c run() or the \c init() and
+  /// then the \c start() member functions. After it the matching can
+  /// be asked with \c matching() or mate() functions. The dual
+  /// solution can be get with \c nodeValue(), \c blossomNum() and \c
+  /// blossomValue() members and \ref MaxWeightedMatching::BlossomIt
+  /// "BlossomIt" nested class which is able to iterate on the nodes
+  /// of a blossom. If the value type is integral then the dual
+  /// solution is multiplied by \ref MaxWeightedMatching::dualScale "4".
+  ///
+  /// \author Balazs Dezso
+  template <typename _UGraph, 
+            typename _WeightMap = typename _UGraph::template UEdgeMap<int> >
+  class MaxWeightedMatching {
+  public:
+
+    typedef _UGraph UGraph;
+    typedef _WeightMap WeightMap;
+    typedef typename WeightMap::Value Value;
+
+    /// \brief Scaling factor for dual solution
+    ///
+    /// Scaling factor for dual solution, it is equal to 4 or 1
+    /// according to the value type.
+    static const int dualScale = 
+      std::numeric_limits<Value>::is_integer ? 4 : 1;
+
+    typedef typename UGraph::template NodeMap<typename UGraph::Edge> 
+    MatchingMap;
+    
+  private:
+
+    UGRAPH_TYPEDEFS(typename UGraph);
+
+    typedef typename UGraph::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 UGraph& _ugraph;
+    const WeightMap& _weight;
+
+    MatchingMap* _matching;
+
+    NodePotential* _node_potential;
+
+    BlossomPotential _blossom_potential;
+    BlossomNodeList _blossom_node_list;
+
+    int _node_num;
+    int _blossom_num;
+
+    typedef typename UGraph::template NodeMap<int> NodeIntMap;
+    typedef typename UGraph::template EdgeMap<int> EdgeIntMap;
+    typedef typename UGraph::template UEdgeMap<int> UEdgeIntMap;
+    typedef IntegerMap<int> IntIntMap;
+
+    enum Status {
+      EVEN = -1, MATCHED = 0, ODD = 1, UNMATCHED = -2
+    };
+
+    typedef HeapUnionFind<Value, NodeIntMap> BlossomSet;
+    struct BlossomData {
+      int tree;
+      Status status;
+      Edge pred, next;
+      Value pot, offset;
+      Node base;
+    };
+
+    NodeIntMap *_blossom_index;
+    BlossomSet *_blossom_set;
+    IntegerMap<BlossomData>* _blossom_data;
+
+    NodeIntMap *_node_index;
+    EdgeIntMap *_node_heap_index;
+
+    struct NodeData {
+      
+      NodeData(EdgeIntMap& node_heap_index) 
+        : heap(node_heap_index) {}
+      
+      int blossom;
+      Value pot;
+      BinHeap<Value, EdgeIntMap> heap;
+      std::map<int, Edge> heap_index;
+      
+      int tree;
+    };
+
+    IntegerMap<NodeData>* _node_data;
+
+    typedef ExtendFindEnum<IntIntMap> TreeSet;
+
+    IntIntMap *_tree_set_index;
+    TreeSet *_tree_set;
+
+    NodeIntMap *_delta1_index;
+    BinHeap<Value, NodeIntMap> *_delta1;
+
+    IntIntMap *_delta2_index;
+    BinHeap<Value, IntIntMap> *_delta2;
+    
+    UEdgeIntMap *_delta3_index;
+    BinHeap<Value, UEdgeIntMap> *_delta3;
+
+    IntIntMap *_delta4_index;
+    BinHeap<Value, IntIntMap> *_delta4;
+
+    Value _delta_sum;
+
+    void createStructures() {
+      _node_num = countNodes(_ugraph); 
+      _blossom_num = _node_num * 3 / 2;
+
+      if (!_matching) {
+        _matching = new MatchingMap(_ugraph);
+      }
+      if (!_node_potential) {
+        _node_potential = new NodePotential(_ugraph);
+      }
+      if (!_blossom_set) {
+        _blossom_index = new NodeIntMap(_ugraph);
+        _blossom_set = new BlossomSet(*_blossom_index);
+        _blossom_data = new IntegerMap<BlossomData>(_blossom_num);
+      }
+
+      if (!_node_index) {
+        _node_index = new NodeIntMap(_ugraph);
+        _node_heap_index = new EdgeIntMap(_ugraph);
+        _node_data = new IntegerMap<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 NodeIntMap(_ugraph);
+        _delta1 = new BinHeap<Value, NodeIntMap>(*_delta1_index);
+      }
+      if (!_delta2) {
+        _delta2_index = new IntIntMap(_blossom_num);
+        _delta2 = new BinHeap<Value, IntIntMap>(*_delta2_index);
+      }
+      if (!_delta3) {
+        _delta3_index = new UEdgeIntMap(_ugraph);
+        _delta3 = new BinHeap<Value, UEdgeIntMap>(*_delta3_index);
+      }
+      if (!_delta4) {
+        _delta4_index = new IntIntMap(_blossom_num);
+        _delta4 = new BinHeap<Value, IntIntMap>(*_delta4_index);
+      }
+    }
+
+    void destroyStructures() {
+      _node_num = countNodes(_ugraph); 
+      _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 (InEdgeIt e(_ugraph, n); e != INVALID; ++e) {
+          Node v = _ugraph.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, Edge>::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 (InEdgeIt e(_ugraph, n); e != INVALID; ++e) {
+          Node v = _ugraph.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) {
+
+              Edge r = _ugraph.oppositeEdge(e);
+
+              typename std::map<int, Edge>::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 if ((*_blossom_data)[vb].status == UNMATCHED) {
+            if (_delta3->state(e) == _delta3->IN_HEAP) {
+              _delta3->erase(e);
+            }
+          } else {
+          
+            typename std::map<int, Edge>::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 (InEdgeIt e(_ugraph, n); e != INVALID; ++e) {
+          Node v = _ugraph.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, Edge>::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 matchedToUnmatched(int blossom) {
+      if (_delta2->state(blossom) == _delta2->IN_HEAP) {
+        _delta2->erase(blossom);
+      }
+
+      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();
+
+        for (OutEdgeIt e(_ugraph, n); e != INVALID; ++e) {
+          Node v = _ugraph.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) {
+            if (_delta3->state(e) != _delta3->IN_HEAP) {
+              _delta3->push(e, rw);
+            }
+          }
+        }
+      }
+    }
+
+    void unmatchedToMatched(int blossom) {
+      for (typename BlossomSet::ItemIt n(*_blossom_set, blossom);
+           n != INVALID; ++n) {
+        int ni = (*_node_index)[n];
+
+        for (InEdgeIt e(_ugraph, n); e != INVALID; ++e) {
+          Node v = _ugraph.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);
+            
+            Edge r = _ugraph.oppositeEdge(e);
+            
+            typename std::map<int, Edge>::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 if ((*_blossom_data)[vb].status == UNMATCHED) {
+            if (_delta3->state(e) == _delta3->IN_HEAP) {
+              _delta3->erase(e);
+            }
+          }
+        }
+      }
+    }
+
+    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(_ugraph.target((*_blossom_data)[even].pred));
+        (*_blossom_data)[odd].status = MATCHED;
+        oddToMatched(odd);
+        (*_blossom_data)[odd].next = (*_blossom_data)[odd].pred;
+      
+        even = _blossom_set->find(_ugraph.target((*_blossom_data)[odd].pred));
+        (*_blossom_data)[even].status = MATCHED;
+        evenToMatched(even, tree);
+        (*_blossom_data)[even].next = 
+          _ugraph.oppositeEdge((*_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].status = UNMATCHED;
+      (*_blossom_data)[blossom].base = node;
+      matchedToUnmatched(blossom);
+    }
+
+
+    void augmentOnEdge(const UEdge& edge) {
+      
+      int left = _blossom_set->find(_ugraph.source(edge));
+      int right = _blossom_set->find(_ugraph.target(edge));
+
+      if ((*_blossom_data)[left].status == EVEN) {
+        int left_tree = _tree_set->find(left);
+        alternatePath(left, left_tree);
+        destroyTree(left_tree);
+      } else {
+        (*_blossom_data)[left].status = MATCHED;
+        unmatchedToMatched(left);
+      }
+
+      if ((*_blossom_data)[right].status == EVEN) {
+        int right_tree = _tree_set->find(right);
+        alternatePath(right, right_tree);
+        destroyTree(right_tree);
+      } else {
+        (*_blossom_data)[right].status = MATCHED;
+        unmatchedToMatched(right);
+      }
+
+      (*_blossom_data)[left].next = _ugraph.direct(edge, true);
+      (*_blossom_data)[right].next = _ugraph.direct(edge, false);
+    }
+
+    void extendOnEdge(const Edge& edge) {
+      int base = _blossom_set->find(_ugraph.target(edge));
+      int tree = _tree_set->find(base);
+      
+      int odd = _blossom_set->find(_ugraph.source(edge));
+      _tree_set->insert(odd, tree);
+      (*_blossom_data)[odd].status = ODD;
+      matchedToOdd(odd);
+      (*_blossom_data)[odd].pred = edge;
+
+      int even = _blossom_set->find(_ugraph.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 UEdge& uedge, int tree) {
+      int nca = -1;
+      std::vector<int> left_path, right_path;
+
+      {
+        std::set<int> left_set, right_set;
+        int left = _blossom_set->find(_ugraph.source(uedge));
+        left_path.push_back(left);
+        left_set.insert(left);
+
+        int right = _blossom_set->find(_ugraph.target(uedge));
+        right_path.push_back(right);
+        right_set.insert(right);
+
+        while (true) {
+
+          if ((*_blossom_data)[left].pred == INVALID) break;
+
+          left = 
+            _blossom_set->find(_ugraph.target((*_blossom_data)[left].pred));
+          left_path.push_back(left);
+          left = 
+            _blossom_set->find(_ugraph.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(_ugraph.target((*_blossom_data)[right].pred));
+          right_path.push_back(right);
+          right = 
+            _blossom_set->find(_ugraph.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(_ugraph.target((*_blossom_data)[nca].pred));
+              right_path.push_back(nca);
+              nca = 
+                _blossom_set->find(_ugraph.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(_ugraph.target((*_blossom_data)[nca].pred));
+              left_path.push_back(nca);
+              nca = 
+                _blossom_set->find(_ugraph.target((*_blossom_data)[nca].pred));
+              left_path.push_back(nca);
+            }
+          }
+        }
+      }
+
+      std::vector<int> subblossoms;
+      Edge prev;
+
+      prev = _ugraph.direct(uedge, 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 = _ugraph.oppositeEdge((*_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) {
+      Edge next = (*_blossom_data)[blossom].next; 
+      Edge 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(_ugraph.source(pred));
+      int d = _blossom_set->find(_ugraph.source(next));
+      
+      int ib, id;
+      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 = 
+            _ugraph.oppositeEdge((*_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 = 
+                           _ugraph.oppositeEdge((*_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 = 
+            _ugraph.oppositeEdge((*_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 =
+            _ugraph.oppositeEdge((*_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 = 
+            _ugraph.oppositeEdge((*_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 Edge& matching) {
+      if (_blossom_set->trivial(blossom)) {
+        int bi = (*_node_index)[base];
+        Value pot = (*_node_data)[bi].pot;
+
+        _matching->set(base, matching);
+        _blossom_node_list.push_back(base);
+        _node_potential->set(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()];
+          
+          Edge m = (*_blossom_data)[tb].next;
+          extractBlossom(sb, _ugraph.target(m), _ugraph.oppositeEdge(m));
+          extractBlossom(tb, _ugraph.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;
+          }
+
+          Edge matching = (*_blossom_data)[blossoms[i]].next;
+          Node base = _ugraph.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 UGraph& ugraph, const WeightMap& weight)
+      : _ugraph(ugraph), _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 member
+    /// \c run() member function.
+
+    ///@{
+
+    /// \brief Initialize the algorithm
+    ///
+    /// Initialize the algorithm
+    void init() {
+      createStructures();
+
+      for (EdgeIt e(_ugraph); e != INVALID; ++e) {
+        _node_heap_index->set(e, BinHeap<Value, EdgeIntMap>::PRE_HEAP);
+      }
+      for (NodeIt n(_ugraph); n != INVALID; ++n) {
+        _delta1_index->set(n, _delta1->PRE_HEAP);
+      }
+      for (UEdgeIt e(_ugraph); e != INVALID; ++e) {
+        _delta3_index->set(e, _delta3->PRE_HEAP);
+      }
+      for (int i = 0; i < _blossom_num; ++i) {
+        _delta2_index->set(i, _delta2->PRE_HEAP);
+        _delta4_index->set(i, _delta4->PRE_HEAP);
+      }
+
+      int index = 0;
+      for (NodeIt n(_ugraph); n != INVALID; ++n) {
+        Value max = 0;
+        for (OutEdgeIt e(_ugraph, n); e != INVALID; ++e) {
+          if (_ugraph.target(e) == n) continue;
+          if ((dualScale * _weight[e]) / 2 > max) {
+            max = (dualScale * _weight[e]) / 2;
+          }
+        }
+        _node_index->set(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 (UEdgeIt e(_ugraph); e != INVALID; ++e) {
+        int si = (*_node_index)[_ugraph.source(e)];
+        int ti = (*_node_index)[_ugraph.target(e)];
+        if (_ugraph.source(e) != _ugraph.target(e)) {
+          _delta3->push(e, ((*_node_data)[si].pot + (*_node_data)[ti].pot - 
+                            dualScale * _weight[e]) / 2);
+        }
+      }
+    }
+
+    /// \brief Starts the algorithm
+    ///
+    /// Starts the algorithm
+    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);
+            Edge e = (*_node_data)[(*_node_index)[n]].heap.top();
+            extendOnEdge(e);
+          }
+          break;
+        case D3:
+          {
+            UEdge e = _delta3->top();
+            
+            int left_blossom = _blossom_set->find(_ugraph.source(e));
+            int right_blossom = _blossom_set->find(_ugraph.target(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 Runs %MaxWeightedMatching algorithm.
+    ///
+    /// This method runs the %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 for get the primal solution, ie. the matching.
+    
+    /// @{
+
+    /// \brief Returns the matching value.
+    ///
+    /// Returns the matching value.
+    Value matchingValue() const {
+      Value sum = 0;
+      for (NodeIt n(_ugraph); n != INVALID; ++n) {
+        if ((*_matching)[n] != INVALID) {
+          sum += _weight[(*_matching)[n]];
+        }
+      }
+      return sum /= 2;
+    }
+
+    /// \brief Returns true when the edge is in the matching.
+    ///
+    /// Returns true when the edge is in the matching.
+    bool matching(const UEdge& edge) const {
+      return (*_matching)[_ugraph.source(edge)] == _ugraph.direct(edge, true);
+    }
+
+    /// \brief Returns the incident matching edge.
+    ///
+    /// Returns the incident matching edge from given node. If the
+    /// node is not matched then it gives back \c INVALID.
+    Edge matching(const Node& node) const {
+      return (*_matching)[node];
+    }
+
+    /// \brief Returns the mate of the node.
+    ///
+    /// Returns the adjancent node in a mathcing edge. If the node is
+    /// not matched then it gives back \c INVALID.
+    Node mate(const Node& node) const {
+      return (*_matching)[node] != INVALID ?
+        _ugraph.target((*_matching)[node]) : INVALID;
+    }
+
+    /// @}
+
+    /// \name Dual solution
+    /// Functions for get the dual solution.
+    
+    /// @{
+
+    /// \brief Returns the value of the dual solution.
+    ///
+    /// Returns the value of the dual solution. It should be equal to
+    /// the primal value scaled by \ref dualScale "dual scale".
+    Value dualValue() const {
+      Value sum = 0;
+      for (NodeIt n(_ugraph); n != INVALID; ++n) {
+        sum += nodeValue(n);
+      }
+      for (int i = 0; i < blossomNum(); ++i) {
+        sum += blossomValue(i) * (blossomSize(i) / 2);
+      }
+      return sum;
+    }
+
+    /// \brief Returns the value of the node.
+    ///
+    /// Returns the the value of the node.
+    Value nodeValue(const Node& n) const {
+      return (*_node_potential)[n];
+    }
+
+    /// \brief Returns the number of the blossoms in the basis.
+    ///
+    /// Returns the number of the blossoms in the basis.
+    /// \see BlossomIt
+    int blossomNum() const {
+      return _blossom_potential.size();
+    }
+
+
+    /// \brief Returns the number of the nodes in the blossom.
+    ///
+    /// Returns the number of the nodes in the blossom.
+    int blossomSize(int k) const {
+      return _blossom_potential[k].end - _blossom_potential[k].begin;
+    }
+
+    /// \brief Returns the value of the blossom.
+    ///
+    /// Returns the the value of the blossom.
+    /// \see BlossomIt
+    Value blossomValue(int k) const {
+      return _blossom_potential[k].value;
+    }
+
+    /// \brief Lemon iterator for get the items of the blossom.
+    ///
+    /// Lemon iterator for get the nodes of the blossom. This class
+    /// provides a common style lemon iterator which gives back a
+    /// subset of the nodes.
+    class BlossomIt {
+    public:
+
+      /// \brief Constructor.
+      ///
+      /// Constructor for get the nodes of the variable. 
+      BlossomIt(const MaxWeightedMatching& algorithm, int variable) 
+        : _algorithm(&algorithm)
+      {
+        _index = _algorithm->_blossom_potential[variable].begin;
+        _last = _algorithm->_blossom_potential[variable].end;
+      }
+
+      /// \brief Invalid constructor.
+      ///
+      /// Invalid constructor.
+      BlossomIt(Invalid) : _index(-1) {}
+
+      /// \brief Conversion to node.
+      ///
+      /// Conversion to node.
+      operator Node() const { 
+        return _algorithm ? _algorithm->_blossom_node_list[_index] : INVALID;
+      }
+
+      /// \brief Increment operator.
+      ///
+      /// Increment operator.
+      BlossomIt& operator++() {
+        ++_index;
+        if (_index == _last) {
+          _index = -1;
+        }
+        return *this; 
+      }
+
+      bool operator==(const BlossomIt& it) const { 
+        return _index == it._index; 
+      }
+      bool operator!=(const BlossomIt& it) const { 
+        return _index != it._index;
+      }
+
+    private:
+      const MaxWeightedMatching* _algorithm;
+      int _last;
+      int _index;
+    };
+
+    /// @}
+    
+  };
+
+  /// \ingroup matching
+  ///
+  /// \brief Weighted perfect matching in general undirected graphs
+  ///
+  /// This class provides an efficient implementation of Edmond's
+  /// maximum weighted perfecr 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 undirected
+  /// edges in the graph with maximum overall weight and no two of
+  /// them shares their endpoints and covers all nodes. The problem
+  /// can be formulated with the next 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 endpoints 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 next:
+  /// \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 \c run() or the \c init() and
+  /// then the \c start() member functions. After it the matching can
+  /// be asked with \c matching() or mate() functions. The dual
+  /// solution can be get with \c nodeValue(), \c blossomNum() and \c
+  /// blossomValue() members and \ref MaxWeightedMatching::BlossomIt
+  /// "BlossomIt" nested class which is able to iterate on the nodes
+  /// of a blossom. If the value type is integral then the dual
+  /// solution is multiplied by \ref MaxWeightedMatching::dualScale "4".
+  ///
+  /// \author Balazs Dezso
+  template <typename _UGraph, 
+            typename _WeightMap = typename _UGraph::template UEdgeMap<int> >
+  class MaxWeightedPerfectMatching {
+  public:
+
+    typedef _UGraph UGraph;
+    typedef _WeightMap WeightMap;
+    typedef typename WeightMap::Value Value;
+
+    /// \brief Scaling factor for dual solution
+    ///
+    /// Scaling factor for dual solution, it is equal to 4 or 1
+    /// according to the value type.
+    static const int dualScale = 
+      std::numeric_limits<Value>::is_integer ? 4 : 1;
+
+    typedef typename UGraph::template NodeMap<typename UGraph::Edge> 
+    MatchingMap;
+    
+  private:
+
+    UGRAPH_TYPEDEFS(typename UGraph);
+
+    typedef typename UGraph::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 UGraph& _ugraph;
+    const WeightMap& _weight;
+
+    MatchingMap* _matching;
+
+    NodePotential* _node_potential;
+
+    BlossomPotential _blossom_potential;
+    BlossomNodeList _blossom_node_list;
+
+    int _node_num;
+    int _blossom_num;
+
+    typedef typename UGraph::template NodeMap<int> NodeIntMap;
+    typedef typename UGraph::template EdgeMap<int> EdgeIntMap;
+    typedef typename UGraph::template UEdgeMap<int> UEdgeIntMap;
+    typedef IntegerMap<int> IntIntMap;
+
+    enum Status {
+      EVEN = -1, MATCHED = 0, ODD = 1
+    };
+
+    typedef HeapUnionFind<Value, NodeIntMap> BlossomSet;
+    struct BlossomData {
+      int tree;
+      Status status;
+      Edge pred, next;
+      Value pot, offset;
+    };
+
+    NodeIntMap *_blossom_index;
+    BlossomSet *_blossom_set;
+    IntegerMap<BlossomData>* _blossom_data;
+
+    NodeIntMap *_node_index;
+    EdgeIntMap *_node_heap_index;
+
+    struct NodeData {
+      
+      NodeData(EdgeIntMap& node_heap_index) 
+        : heap(node_heap_index) {}
+      
+      int blossom;
+      Value pot;
+      BinHeap<Value, EdgeIntMap> heap;
+      std::map<int, Edge> heap_index;
+      
+      int tree;
+    };
+
+    IntegerMap<NodeData>* _node_data;
+
+    typedef ExtendFindEnum<IntIntMap> TreeSet;
+
+    IntIntMap *_tree_set_index;
+    TreeSet *_tree_set;
+
+    IntIntMap *_delta2_index;
+    BinHeap<Value, IntIntMap> *_delta2;
+    
+    UEdgeIntMap *_delta3_index;
+    BinHeap<Value, UEdgeIntMap> *_delta3;
+
+    IntIntMap *_delta4_index;
+    BinHeap<Value, IntIntMap> *_delta4;
+
+    Value _delta_sum;
+
+    void createStructures() {
+      _node_num = countNodes(_ugraph); 
+      _blossom_num = _node_num * 3 / 2;
+
+      if (!_matching) {
+        _matching = new MatchingMap(_ugraph);
+      }
+      if (!_node_potential) {
+        _node_potential = new NodePotential(_ugraph);
+      }
+      if (!_blossom_set) {
+        _blossom_index = new NodeIntMap(_ugraph);
+        _blossom_set = new BlossomSet(*_blossom_index);
+        _blossom_data = new IntegerMap<BlossomData>(_blossom_num);
+      }
+
+      if (!_node_index) {
+        _node_index = new NodeIntMap(_ugraph);
+        _node_heap_index = new EdgeIntMap(_ugraph);
+        _node_data = new IntegerMap<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 UEdgeIntMap(_ugraph);
+        _delta3 = new BinHeap<Value, UEdgeIntMap>(*_delta3_index);
+      }
+      if (!_delta4) {
+        _delta4_index = new IntIntMap(_blossom_num);
+        _delta4 = new BinHeap<Value, IntIntMap>(*_delta4_index);
+      }
+    }
+
+    void destroyStructures() {
+      _node_num = countNodes(_ugraph); 
+      _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 (_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 (InEdgeIt e(_ugraph, n); e != INVALID; ++e) {
+          Node v = _ugraph.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, Edge>::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 (InEdgeIt e(_ugraph, n); e != INVALID; ++e) {
+          Node v = _ugraph.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) {
+
+              Edge r = _ugraph.oppositeEdge(e);
+
+              typename std::map<int, Edge>::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, Edge>::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 (InEdgeIt e(_ugraph, n); e != INVALID; ++e) {
+          Node v = _ugraph.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, Edge>::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(_ugraph.target((*_blossom_data)[even].pred));
+        (*_blossom_data)[odd].status = MATCHED;
+        oddToMatched(odd);
+        (*_blossom_data)[odd].next = (*_blossom_data)[odd].pred;
+      
+        even = _blossom_set->find(_ugraph.target((*_blossom_data)[odd].pred));
+        (*_blossom_data)[even].status = MATCHED;
+        evenToMatched(even, tree);
+        (*_blossom_data)[even].next = 
+          _ugraph.oppositeEdge((*_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 augmentOnEdge(const UEdge& edge) {
+      
+      int left = _blossom_set->find(_ugraph.source(edge));
+      int right = _blossom_set->find(_ugraph.target(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 = _ugraph.direct(edge, true);
+      (*_blossom_data)[right].next = _ugraph.direct(edge, false);
+    }
+
+    void extendOnEdge(const Edge& edge) {
+      int base = _blossom_set->find(_ugraph.target(edge));
+      int tree = _tree_set->find(base);
+      
+      int odd = _blossom_set->find(_ugraph.source(edge));
+      _tree_set->insert(odd, tree);
+      (*_blossom_data)[odd].status = ODD;
+      matchedToOdd(odd);
+      (*_blossom_data)[odd].pred = edge;
+
+      int even = _blossom_set->find(_ugraph.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 UEdge& uedge, int tree) {
+      int nca = -1;
+      std::vector<int> left_path, right_path;
+
+      {
+        std::set<int> left_set, right_set;
+        int left = _blossom_set->find(_ugraph.source(uedge));
+        left_path.push_back(left);
+        left_set.insert(left);
+
+        int right = _blossom_set->find(_ugraph.target(uedge));
+        right_path.push_back(right);
+        right_set.insert(right);
+
+        while (true) {
+
+          if ((*_blossom_data)[left].pred == INVALID) break;
+
+          left = 
+            _blossom_set->find(_ugraph.target((*_blossom_data)[left].pred));
+          left_path.push_back(left);
+          left = 
+            _blossom_set->find(_ugraph.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(_ugraph.target((*_blossom_data)[right].pred));
+          right_path.push_back(right);
+          right = 
+            _blossom_set->find(_ugraph.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(_ugraph.target((*_blossom_data)[nca].pred));
+              right_path.push_back(nca);
+              nca = 
+                _blossom_set->find(_ugraph.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(_ugraph.target((*_blossom_data)[nca].pred));
+              left_path.push_back(nca);
+              nca = 
+                _blossom_set->find(_ugraph.target((*_blossom_data)[nca].pred));
+              left_path.push_back(nca);
+            }
+          }
+        }
+      }
+
+      std::vector<int> subblossoms;
+      Edge prev;
+
+      prev = _ugraph.direct(uedge, 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 = _ugraph.oppositeEdge((*_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) {
+      Edge next = (*_blossom_data)[blossom].next; 
+      Edge 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(_ugraph.source(pred));
+      int d = _blossom_set->find(_ugraph.source(next));
+      
+      int ib, id;
+      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 = 
+            _ugraph.oppositeEdge((*_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 = 
+                           _ugraph.oppositeEdge((*_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 = 
+            _ugraph.oppositeEdge((*_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 =
+            _ugraph.oppositeEdge((*_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 = 
+            _ugraph.oppositeEdge((*_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 Edge& matching) {
+      if (_blossom_set->trivial(blossom)) {
+        int bi = (*_node_index)[base];
+        Value pot = (*_node_data)[bi].pot;
+
+        _matching->set(base, matching);
+        _blossom_node_list.push_back(base);
+        _node_potential->set(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()];
+          
+          Edge m = (*_blossom_data)[tb].next;
+          extractBlossom(sb, _ugraph.target(m), _ugraph.oppositeEdge(m));
+          extractBlossom(tb, _ugraph.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) {
+
+        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;
+        }
+        
+        Edge matching = (*_blossom_data)[blossoms[i]].next;
+        Node base = _ugraph.source(matching);
+        extractBlossom(blossoms[i], base, matching);
+      }
+    }
+    
+  public:
+
+    /// \brief Constructor
+    ///
+    /// Constructor.
+    MaxWeightedPerfectMatching(const UGraph& ugraph, const WeightMap& weight)
+      : _ugraph(ugraph), _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),
+
+        _delta2_index(0), _delta2(0),
+        _delta3_index(0), _delta3(0), 
+        _delta4_index(0), _delta4(0),
+
+        _delta_sum() {}
+
+    ~MaxWeightedPerfectMatching() {
+      destroyStructures();
+    }
+
+    /// \name Execution control 
+    /// The simplest way to execute the algorithm is to use the member
+    /// \c run() member function.
+
+    ///@{
+
+    /// \brief Initialize the algorithm
+    ///
+    /// Initialize the algorithm
+    void init() {
+      createStructures();
+
+      for (EdgeIt e(_ugraph); e != INVALID; ++e) {
+        _node_heap_index->set(e, BinHeap<Value, EdgeIntMap>::PRE_HEAP);
+      }
+      for (UEdgeIt e(_ugraph); e != INVALID; ++e) {
+        _delta3_index->set(e, _delta3->PRE_HEAP);
+      }
+      for (int i = 0; i < _blossom_num; ++i) {
+        _delta2_index->set(i, _delta2->PRE_HEAP);
+        _delta4_index->set(i, _delta4->PRE_HEAP);
+      }
+
+      int index = 0;
+      for (NodeIt n(_ugraph); n != INVALID; ++n) {
+        Value max = std::numeric_limits<Value>::min();
+        for (OutEdgeIt e(_ugraph, n); e != INVALID; ++e) {
+          if (_ugraph.target(e) == n) continue;
+          if ((dualScale * _weight[e]) / 2 > max) {
+            max = (dualScale * _weight[e]) / 2;
+          }
+        }
+        _node_index->set(n, index);
+        (*_node_data)[index].pot = 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 (UEdgeIt e(_ugraph); e != INVALID; ++e) {
+        int si = (*_node_index)[_ugraph.source(e)];
+        int ti = (*_node_index)[_ugraph.target(e)];
+        if (_ugraph.source(e) != _ugraph.target(e)) {
+          _delta3->push(e, ((*_node_data)[si].pot + (*_node_data)[ti].pot - 
+                            dualScale * _weight[e]) / 2);
+        }
+      }
+    }
+
+    /// \brief Starts the algorithm
+    ///
+    /// Starts the algorithm
+    bool start() {
+      enum OpType {
+        D2, D3, D4
+      };
+
+      int unmatched = _node_num;
+      while (unmatched > 0) {
+        Value d2 = !_delta2->empty() ? 
+          _delta2->prio() : std::numeric_limits<Value>::max();
+
+        Value d3 = !_delta3->empty() ? 
+          _delta3->prio() : std::numeric_limits<Value>::max();
+
+        Value d4 = !_delta4->empty() ? 
+          _delta4->prio() : std::numeric_limits<Value>::max();
+
+        _delta_sum = d2; OpType ot = D2;
+        if (d3 < _delta_sum) { _delta_sum = d3; ot = D3; }
+        if (d4 < _delta_sum) { _delta_sum = d4; ot = D4; }
+
+        if (_delta_sum == std::numeric_limits<Value>::max()) {
+          return false;
+        }
+        
+        switch (ot) {
+        case D2:
+          {
+            int blossom = _delta2->top();
+            Node n = _blossom_set->classTop(blossom);
+            Edge e = (*_node_data)[(*_node_index)[n]].heap.top();
+            extendOnEdge(e);
+          }
+          break;
+        case D3:
+          {
+            UEdge e = _delta3->top();
+            
+            int left_blossom = _blossom_set->find(_ugraph.source(e));
+            int right_blossom = _blossom_set->find(_ugraph.target(e));
+          
+            if (left_blossom == right_blossom) {
+              _delta3->pop();
+            } else {
+              int left_tree = _tree_set->find(left_blossom);
+              int right_tree = _tree_set->find(right_blossom);
+            
+              if (left_tree == right_tree) {
+                shrinkOnEdge(e, left_tree);
+              } else {
+                augmentOnEdge(e);
+                unmatched -= 2;
+              }
+            }
+          } break;
+        case D4:
+          splitBlossom(_delta4->top());
+          break;
+        }
+      }
+      extractMatching();
+      return true;
+    }
+
+    /// \brief Runs %MaxWeightedPerfectMatching algorithm.
+    ///
+    /// This method runs the %MaxWeightedPerfectMatching algorithm.
+    ///
+    /// \note mwm.run() is just a shortcut of the following code.
+    /// \code
+    ///   mwm.init();
+    ///   mwm.start();
+    /// \endcode
+    bool run() {
+      init();
+      return start();
+    }
+
+    /// @}
+
+    /// \name Primal solution
+    /// Functions for get the primal solution, ie. the matching.
+    
+    /// @{
+
+    /// \brief Returns the matching value.
+    ///
+    /// Returns the matching value.
+    Value matchingValue() const {
+      Value sum = 0;
+      for (NodeIt n(_ugraph); n != INVALID; ++n) {
+        if ((*_matching)[n] != INVALID) {
+          sum += _weight[(*_matching)[n]];
+        }
+      }
+      return sum /= 2;
+    }
+
+    /// \brief Returns true when the edge is in the matching.
+    ///
+    /// Returns true when the edge is in the matching.
+    bool matching(const UEdge& edge) const {
+      return (*_matching)[_ugraph.source(edge)] == _ugraph.direct(edge, true);
+    }
+
+    /// \brief Returns the incident matching edge.
+    ///
+    /// Returns the incident matching edge from given node.
+    Edge matching(const Node& node) const {
+      return (*_matching)[node];
+    }
+
+    /// \brief Returns the mate of the node.
+    ///
+    /// Returns the adjancent node in a mathcing edge.
+    Node mate(const Node& node) const {
+      return _ugraph.target((*_matching)[node]);
+    }
+
+    /// @}
+
+    /// \name Dual solution
+    /// Functions for get the dual solution.
+    
+    /// @{
+
+    /// \brief Returns the value of the dual solution.
+    ///
+    /// Returns the value of the dual solution. It should be equal to
+    /// the primal value scaled by \ref dualScale "dual scale".
+    Value dualValue() const {
+      Value sum = 0;
+      for (NodeIt n(_ugraph); n != INVALID; ++n) {
+        sum += nodeValue(n);
+      }
+      for (int i = 0; i < blossomNum(); ++i) {
+        sum += blossomValue(i) * (blossomSize(i) / 2);
+      }
+      return sum;
+    }
+
+    /// \brief Returns the value of the node.
+    ///
+    /// Returns the the value of the node.
+    Value nodeValue(const Node& n) const {
+      return (*_node_potential)[n];
+    }
+
+    /// \brief Returns the number of the blossoms in the basis.
+    ///
+    /// Returns the number of the blossoms in the basis.
+    /// \see BlossomIt
+    int blossomNum() const {
+      return _blossom_potential.size();
+    }
+
+
+    /// \brief Returns the number of the nodes in the blossom.
+    ///
+    /// Returns the number of the nodes in the blossom.
+    int blossomSize(int k) const {
+      return _blossom_potential[k].end - _blossom_potential[k].begin;
+    }
+
+    /// \brief Returns the value of the blossom.
+    ///
+    /// Returns the the value of the blossom.
+    /// \see BlossomIt
+    Value blossomValue(int k) const {
+      return _blossom_potential[k].value;
+    }
+
+    /// \brief Lemon iterator for get the items of the blossom.
+    ///
+    /// Lemon iterator for get the nodes of the blossom. This class
+    /// provides a common style lemon iterator which gives back a
+    /// subset of the nodes.
+    class BlossomIt {
+    public:
+
+      /// \brief Constructor.
+      ///
+      /// Constructor for get the nodes of the variable. 
+      BlossomIt(const MaxWeightedPerfectMatching& algorithm, int variable) 
+        : _algorithm(&algorithm)
+      {
+        _index = _algorithm->_blossom_potential[variable].begin;
+        _last = _algorithm->_blossom_potential[variable].end;
+      }
+
+      /// \brief Invalid constructor.
+      ///
+      /// Invalid constructor.
+      BlossomIt(Invalid) : _index(-1) {}
+
+      /// \brief Conversion to node.
+      ///
+      /// Conversion to node.
+      operator Node() const { 
+        return _algorithm ? _algorithm->_blossom_node_list[_index] : INVALID;
+      }
+
+      /// \brief Increment operator.
+      ///
+      /// Increment operator.
+      BlossomIt& operator++() {
+        ++_index;
+        if (_index == _last) {
+          _index = -1;
+        }
+        return *this; 
+      }
+
+      bool operator==(const BlossomIt& it) const { 
+        return _index == it._index; 
+      }
+      bool operator!=(const BlossomIt& it) const { 
+        return _index != it._index;
+      }
+
+    private:
+      const MaxWeightedPerfectMatching* _algorithm;
+      int _last;
+      int _index;
+    };
+
+    /// @}
+    
+  };
+
   
 } //END OF NAMESPACE LEMON

lemon/unionfind.h

@@ -925 +925 @@
   };
 
+  /// \ingroup auxdat
+  ///
+  /// \brief A \e Union-Find data structure implementation which
+  /// is able to store a priority for each item and retrieve the minimum of
+  /// each class.
+  ///
+  /// A \e Union-Find data structure implementation which is able to
+  /// store a priority for each item and retrieve the minimum of each
+  /// class. In addition, it supports the joining and splitting the
+  /// components. If you don't need this feature then you makes
+  /// better to use the \ref UnionFind class which is more efficient.
+  ///
+  /// The union-find data strcuture based on a (2, 16)-tree with a
+  /// tournament minimum selection on the internal nodes. The insert
+  /// operation takes O(1), the find, set, decrease and increase takes
+  /// O(log(n)), where n is the number of nodes in the current
+  /// component.  The complexity of join and split is O(log(n)*k),
+  /// where n is the sum of the number of the nodes and k is the
+  /// number of joined components or the number of the components
+  /// after the split.
+  ///
+  /// \pre You need to add all the elements by the \ref insert()
+  /// method.
+  ///
+  template <typename _Value, typename _ItemIntMap, 
+            typename _Comp = std::less<_Value> >
+  class HeapUnionFind {
+  public:
+    
+    typedef _Value Value;
+    typedef typename _ItemIntMap::Key Item;
+
+    typedef _ItemIntMap ItemIntMap;
+
+    typedef _Comp Comp;
+
+  private:
+
+    static const int cmax = 3;
+
+    ItemIntMap& index;
+
+    struct ClassNode {
+      int parent;
+      int depth;
+
+      int left, right;
+      int next, prev;
+    };
+
+    int first_class;
+    int first_free_class;
+    std::vector<ClassNode> classes;
+
+    int newClass() {
+      if (first_free_class < 0) {
+        int id = classes.size();
+        classes.push_back(ClassNode());
+        return id;
+      } else {
+        int id = first_free_class;
+        first_free_class = classes[id].next;
+        return id;
+      }
+    }
+
+    void deleteClass(int id) {
+      classes[id].next = first_free_class;
+      first_free_class = id;
+    }
+
+    struct ItemNode {
+      int parent;
+      Item item;
+      Value prio;
+      int next, prev;
+      int left, right;
+      int size;
+    };
+
+    int first_free_node;
+    std::vector<ItemNode> nodes;
+
+    int newNode() {
+      if (first_free_node < 0) {
+        int id = nodes.size();
+        nodes.push_back(ItemNode());
+        return id;
+      } else {
+        int id = first_free_node;
+        first_free_node = nodes[id].next;
+        return id;
+      }
+    }
+
+    void deleteNode(int id) {
+      nodes[id].next = first_free_node;
+      first_free_node = id;
+    }
+
+    Comp comp;
+
+    int findClass(int id) const {
+      int kd = id;
+      while (kd >= 0) {
+        kd = nodes[kd].parent;
+      }
+      return ~kd;
+    }
+
+    int leftNode(int id) const {
+      int kd = ~(classes[id].parent);
+      for (int i = 0; i < classes[id].depth; ++i) {
+        kd = nodes[kd].left;
+      }
+      return kd;
+    }
+
+    int nextNode(int id) const {
+      int depth = 0;
+      while (id >= 0 && nodes[id].next == -1) {
+        id = nodes[id].parent;
+        ++depth;
+      }
+      if (id < 0) {
+        return -1;
+      }
+      id = nodes[id].next;
+      while (depth--) {
+        id = nodes[id].left;      
+      }
+      return id;
+    }
+
+
+    void setPrio(int id) {
+      int jd = nodes[id].left;
+      nodes[id].prio = nodes[jd].prio;
+      nodes[id].item = nodes[jd].item;
+      jd = nodes[jd].next;
+      while (jd != -1) {
+        if (comp(nodes[jd].prio, nodes[id].prio)) {
+          nodes[id].prio = nodes[jd].prio;
+          nodes[id].item = nodes[jd].item;
+        }
+        jd = nodes[jd].next;
+      }
+    }
+
+    void push(int id, int jd) {
+      nodes[id].size = 1;
+      nodes[id].left = nodes[id].right = jd;
+      nodes[jd].next = nodes[jd].prev = -1;
+      nodes[jd].parent = id;
+    }
+
+    void pushAfter(int id, int jd) {
+      int kd = nodes[id].parent;
+      if (nodes[id].next != -1) {
+        nodes[nodes[id].next].prev = jd;
+        if (kd >= 0) {
+          nodes[kd].size += 1;
+        }
+      } else {
+        if (kd >= 0) {
+          nodes[kd].right = jd;
+          nodes[kd].size += 1;
+        }
+      }
+      nodes[jd].next = nodes[id].next;
+      nodes[jd].prev = id;
+      nodes[id].next = jd;
+      nodes[jd].parent = kd;
+    }
+
+    void pushRight(int id, int jd) {
+      nodes[id].size += 1;
+      nodes[jd].prev = nodes[id].right;
+      nodes[jd].next = -1;
+      nodes[nodes[id].right].next = jd;
+      nodes[id].right = jd;
+      nodes[jd].parent = id;
+    }
+
+    void popRight(int id) {
+      nodes[id].size -= 1;
+      int jd = nodes[id].right;
+      nodes[nodes[jd].prev].next = -1;
+      nodes[id].right = nodes[jd].prev;
+    }
+
+    void splice(int id, int jd) {
+      nodes[id].size += nodes[jd].size;
+      nodes[nodes[id].right].next = nodes[jd].left;
+      nodes[nodes[jd].left].prev = nodes[id].right;
+      int kd = nodes[jd].left;
+      while (kd != -1) {
+        nodes[kd].parent = id;
+        kd = nodes[kd].next;
+      }
+    }
+
+    void split(int id, int jd) {
+      int kd = nodes[id].parent;
+      nodes[kd].right = nodes[id].prev;
+      nodes[nodes[id].prev].next = -1;
+      
+      nodes[jd].left = id;
+      nodes[id].prev = -1;
+      int num = 0;
+      while (id != -1) {
+        nodes[id].parent = jd;
+        nodes[jd].right = id;
+        id = nodes[id].next;
+        ++num;
+      }      
+      nodes[kd].size -= num;
+      nodes[jd].size = num;
+    }
+
+    void pushLeft(int id, int jd) {
+      nodes[id].size += 1;
+      nodes[jd].next = nodes[id].left;
+      nodes[jd].prev = -1;
+      nodes[nodes[id].left].prev = jd;
+      nodes[id].left = jd;
+      nodes[jd].parent = id;
+    }
+
+    void popLeft(int id) {
+      nodes[id].size -= 1;
+      int jd = nodes[id].left;
+      nodes[nodes[jd].next].prev = -1;
+      nodes[id].left = nodes[jd].next;
+    }
+
+    void repairLeft(int id) {
+      int jd = ~(classes[id].parent);
+      while (nodes[jd].left != -1) {
+        int kd = nodes[jd].left;
+        if (nodes[jd].size == 1) {
+          if (nodes[jd].parent < 0) {
+            classes[id].parent = ~kd;
+            classes[id].depth -= 1;
+            nodes[kd].parent = ~id;
+            deleteNode(jd);
+            jd = kd;
+          } else {
+            int pd = nodes[jd].parent;
+            if (nodes[nodes[jd].next].size < cmax) {
+              pushLeft(nodes[jd].next, nodes[jd].left);
+              if (less(nodes[jd].left, nodes[jd].next)) {
+                nodes[nodes[jd].next].prio = nodes[nodes[jd].left].prio;
+                nodes[nodes[jd].next].item = nodes[nodes[jd].left].item;
+              } 
+              popLeft(pd);
+              deleteNode(jd);
+              jd = pd;
+            } else {
+              int ld = nodes[nodes[jd].next].left;
+              popLeft(nodes[jd].next);
+              pushRight(jd, ld);
+              if (less(ld, nodes[jd].left)) {
+                nodes[jd].item = nodes[ld].item;
+                nodes[jd].prio = nodes[jd].prio;
+              }
+              if (nodes[nodes[jd].next].item == nodes[ld].item) {
+                setPrio(nodes[jd].next);
+              }
+              jd = nodes[jd].left;
+            }
+          }
+        } else {
+          jd = nodes[jd].left;
+        }
+      }
+    }    
+
+    void repairRight(int id) {
+      int jd = ~(classes[id].parent);
+      while (nodes[jd].right != -1) {
+        int kd = nodes[jd].right;
+        if (nodes[jd].size == 1) {
+          if (nodes[jd].parent < 0) {
+            classes[id].parent = ~kd;
+            classes[id].depth -= 1;
+            nodes[kd].parent = ~id;
+            deleteNode(jd);
+            jd = kd;
+          } else {
+            int pd = nodes[jd].parent;
+            if (nodes[nodes[jd].prev].size < cmax) {
+              pushRight(nodes[jd].prev, nodes[jd].right);
+              if (less(nodes[jd].right, nodes[jd].prev)) {
+                nodes[nodes[jd].prev].prio = nodes[nodes[jd].right].prio;
+                nodes[nodes[jd].prev].item = nodes[nodes[jd].right].item;
+              } 
+              popRight(pd);
+              deleteNode(jd);
+              jd = pd;
+            } else {
+              int ld = nodes[nodes[jd].prev].right;
+              popRight(nodes[jd].prev);
+              pushLeft(jd, ld);
+              if (less(ld, nodes[jd].right)) {
+                nodes[jd].item = nodes[ld].item;
+                nodes[jd].prio = nodes[jd].prio;
+              }
+              if (nodes[nodes[jd].prev].item == nodes[ld].item) {
+                setPrio(nodes[jd].prev);
+              }
+              jd = nodes[jd].right;
+            }
+          }
+        } else {
+          jd = nodes[jd].right;
+        }
+      }
+    }
+
+
+    bool less(int id, int jd) const {
+      return comp(nodes[id].prio, nodes[jd].prio);
+    }
+
+    bool equal(int id, int jd) const {
+      return !less(id, jd) && !less(jd, id);
+    }
+
+
+  public:
+
+    /// \brief Returns true when the given class is alive.
+    ///
+    /// Returns true when the given class is alive, ie. the class is
+    /// not nested into other class.
+    bool alive(int cls) const {
+      return classes[cls].parent < 0;
+    }
+
+    /// \brief Returns true when the given class is trivial.
+    ///
+    /// Returns true when the given class is trivial, ie. the class
+    /// contains just one item directly.
+    bool trivial(int cls) const {
+      return classes[cls].left == -1;
+    }
+
+    /// \brief Constructs the union-find.
+    ///
+    /// Constructs the union-find.  
+    /// \brief _index The index map of the union-find. The data
+    /// structure uses internally for store references.
+    HeapUnionFind(ItemIntMap& _index) 
+      : index(_index), first_class(-1), 
+        first_free_class(-1), first_free_node(-1) {}
+
+    /// \brief Insert a new node into a new component.
+    ///
+    /// Insert a new node into a new component.
+    /// \param item The item of the new node.
+    /// \param prio The priority of the new node.
+    /// \return The class id of the one-item-heap.
+    int insert(const Item& item, const Value& prio) {
+      int id = newNode();
+      nodes[id].item = item;
+      nodes[id].prio = prio;
+      nodes[id].size = 0;
+
+      nodes[id].prev = -1;
+      nodes[id].next = -1;
+
+      nodes[id].left = -1;
+      nodes[id].right = -1;
+
+      nodes[id].item = item;
+      index[item] = id;
+      
+      int class_id = newClass();
+      classes[class_id].parent = ~id;
+      classes[class_id].depth = 0;
+
+      classes[class_id].left = -1;
+      classes[class_id].right = -1;
+      
+      if (first_class != -1) {
+        classes[first_class].prev = class_id;
+      }
+      classes[class_id].next = first_class;
+      classes[class_id].prev = -1;
+      first_class = class_id;
+
+      nodes[id].parent = ~class_id;
+      
+      return class_id;
+    }
+
+    /// \brief The class of the item.
+    ///
+    /// \return The alive class id of the item, which is not nested into
+    /// other classes.
+    ///
+    /// The time complexity is O(log(n)).
+    int find(const Item& item) const {
+      return findClass(index[item]);
+    }
+    
+    /// \brief Joins the classes.
+    ///
+    /// The current function joins the given classes. The parameter is
+    /// an STL range which should be contains valid class ids. The
+    /// time complexity is O(log(n)*k) where n is the overall number
+    /// of the joined nodes and k is the number of classes.
+    /// \return The class of the joined classes.
+    /// \pre The range should contain at least two class ids.
+    template <typename Iterator>
+    int join(Iterator begin, Iterator end) {
+      std::vector<int> cs;
+      for (Iterator it = begin; it != end; ++it) {
+        cs.push_back(*it);
+      }
+
+      int class_id = newClass();
+      { // creation union-find
+
+        if (first_class != -1) {
+          classes[first_class].prev = class_id;
+        }
+        classes[class_id].next = first_class;
+        classes[class_id].prev = -1;
+        first_class = class_id;
+
+        classes[class_id].depth = classes[cs[0]].depth;
+        classes[class_id].parent = classes[cs[0]].parent;
+        nodes[~(classes[class_id].parent)].parent = ~class_id;
+
+        int l = cs[0];
+
+        classes[class_id].left = l;
+        classes[class_id].right = l;
+
+        if (classes[l].next != -1) {
+          classes[classes[l].next].prev = classes[l].prev;
+        }
+        classes[classes[l].prev].next = classes[l].next;
+        
+        classes[l].prev = -1;
+        classes[l].next = -1;
+
+        classes[l].depth = leftNode(l);
+        classes[l].parent = class_id;
+        
+      }
+
+      { // merging of heap
+        int l = class_id;
+        for (int ci = 1; ci < int(cs.size()); ++ci) {
+          int r = cs[ci];
+          int rln = leftNode(r);
+          if (classes[l].depth > classes[r].depth) {
+            int id = ~(classes[l].parent);
+            for (int i = classes[r].depth + 1; i < classes[l].depth; ++i) {
+              id = nodes[id].right;
+            }
+            while (id >= 0 && nodes[id].size == cmax) {
+              int new_id = newNode();
+              int right_id = nodes[id].right;
+
+              popRight(id);
+              if (nodes[id].item == nodes[right_id].item) {
+                setPrio(id);
+              }
+              push(new_id, right_id);
+              pushRight(new_id, ~(classes[r].parent));
+              setPrio(new_id);
+
+              id = nodes[id].parent;
+              classes[r].parent = ~new_id;
+            }
+            if (id < 0) {
+              int new_parent = newNode();
+              nodes[new_parent].next = -1;
+              nodes[new_parent].prev = -1;
+              nodes[new_parent].parent = ~l;
+
+              push(new_parent, ~(classes[l].parent));
+              pushRight(new_parent, ~(classes[r].parent));
+              setPrio(new_parent);
+
+              classes[l].parent = ~new_parent;
+              classes[l].depth += 1;
+            } else {
+              pushRight(id, ~(classes[r].parent));
+              while (id >= 0 && less(~(classes[r].parent), id)) {
+                nodes[id].prio = nodes[~(classes[r].parent)].prio;
+                nodes[id].item = nodes[~(classes[r].parent)].item;
+                id = nodes[id].parent;
+              }
+            }
+          } else if (classes[r].depth > classes[l].depth) {
+            int id = ~(classes[r].parent);
+            for (int i = classes[l].depth + 1; i < classes[r].depth; ++i) {
+              id = nodes[id].left;
+            }
+            while (id >= 0 && nodes[id].size == cmax) {
+              int new_id = newNode();
+              int left_id = nodes[id].left;
+
+              popLeft(id);
+              if (nodes[id].prio == nodes[left_id].prio) {
+                setPrio(id);
+              }
+              push(new_id, left_id);
+              pushLeft(new_id, ~(classes[l].parent));
+              setPrio(new_id);
+
+              id = nodes[id].parent;
+              classes[l].parent = ~new_id;
+
+            }
+            if (id < 0) {
+              int new_parent = newNode();
+              nodes[new_parent].next = -1;
+              nodes[new_parent].prev = -1;
+              nodes[new_parent].parent = ~l;
+
+              push(new_parent, ~(classes[r].parent));
+              pushLeft(new_parent, ~(classes[l].parent));
+              setPrio(new_parent);
+              
+              classes[r].parent = ~new_parent;
+              classes[r].depth += 1;
+            } else {
+              pushLeft(id, ~(classes[l].parent));
+              while (id >= 0 && less(~(classes[l].parent), id)) {
+                nodes[id].prio = nodes[~(classes[l].parent)].prio;
+                nodes[id].item = nodes[~(classes[l].parent)].item;
+                id = nodes[id].parent;
+              }
+            }
+            nodes[~(classes[r].parent)].parent = ~l;
+            classes[l].parent = classes[r].parent;
+            classes[l].depth = classes[r].depth;
+          } else {
+            if (classes[l].depth != 0 && 
+                nodes[~(classes[l].parent)].size + 
+                nodes[~(classes[r].parent)].size <= cmax) {
+              splice(~(classes[l].parent), ~(classes[r].parent));
+              deleteNode(~(classes[r].parent));
+              if (less(~(classes[r].parent), ~(classes[l].parent))) {
+                nodes[~(classes[l].parent)].prio = 
+                  nodes[~(classes[r].parent)].prio;
+                nodes[~(classes[l].parent)].item = 
+                  nodes[~(classes[r].parent)].item;
+              }
+            } else {
+              int new_parent = newNode();
+              nodes[new_parent].next = nodes[new_parent].prev = -1;
+              push(new_parent, ~(classes[l].parent));
+              pushRight(new_parent, ~(classes[r].parent));
+              setPrio(new_parent);
+            
+              classes[l].parent = ~new_parent;
+              classes[l].depth += 1;
+              nodes[new_parent].parent = ~l;
+            }
+          }
+          if (classes[r].next != -1) {
+            classes[classes[r].next].prev = classes[r].prev;
+          }
+          classes[classes[r].prev].next = classes[r].next;
+
+          classes[r].prev = classes[l].right;
+          classes[classes[l].right].next = r;
+          classes[l].right = r;
+          classes[r].parent = l;
+
+          classes[r].next = -1;
+          classes[r].depth = rln;
+        }
+      }
+      return class_id;
+    }
+
+    /// \brief Split the class to subclasses.
+    ///
+    /// The current function splits the given class. The join, which
+    /// made the current class, stored a reference to the
+    /// subclasses. The \c splitClass() member restores the classes
+    /// and creates the heaps. The parameter is an STL output iterator
+    /// which will be filled with the subclass ids. The time
+    /// complexity is O(log(n)*k) where n is the overall number of
+    /// nodes in the splitted classes and k is the number of the
+    /// classes.
+    template <typename Iterator>
+    void split(int cls, Iterator out) {
+      std::vector<int> cs;
+      { // splitting union-find
+        int id = cls;
+        int l = classes[id].left;
+
+        classes[l].parent = classes[id].parent;
+        classes[l].depth = classes[id].depth;
+
+        nodes[~(classes[l].parent)].parent = ~l;
+
+        *out++ = l;
+
+        while (l != -1) {
+          cs.push_back(l);
+          l = classes[l].next;
+        }
+
+        classes[classes[id].right].next = first_class;
+        classes[first_class].prev = classes[id].right;
+        first_class = classes[id].left;
+        
+        if (classes[id].next != -1) {
+          classes[classes[id].next].prev = classes[id].prev;
+        }
+        classes[classes[id].prev].next = classes[id].next;
+        
+        deleteClass(id);
+      }
+
+      {
+        for (int i = 1; i < int(cs.size()); ++i) {
+          int l = classes[cs[i]].depth;
+          while (nodes[nodes[l].parent].left == l) {
+            l = nodes[l].parent;
+          }
+          int r = l;    
+          while (nodes[l].parent >= 0) {
+            l = nodes[l].parent;
+            int new_node = newNode();
+
+            nodes[new_node].prev = -1;
+            nodes[new_node].next = -1;
+
+            split(r, new_node);
+            pushAfter(l, new_node);
+            setPrio(l);
+            setPrio(new_node);
+            r = new_node;
+          }
+          classes[cs[i]].parent = ~r;
+          classes[cs[i]].depth = classes[~(nodes[l].parent)].depth;
+          nodes[r].parent = ~cs[i];
+
+          nodes[l].next = -1;
+          nodes[r].prev = -1;
+
+          repairRight(~(nodes[l].parent));
+          repairLeft(cs[i]);
+          
+          *out++ = cs[i];
+        }
+      }
+    }
+
+    /// \brief Gives back the priority of the current item.
+    ///
+    /// \return Gives back the priority of the current item.
+    const Value& operator[](const Item& item) const {
+      return nodes[index[item]].prio;
+    }
+
+    /// \brief Sets the priority of the current item.
+    ///
+    /// Sets the priority of the current item.
+    void set(const Item& item, const Value& prio) {
+      if (comp(prio, nodes[index[item]].prio)) {
+        decrease(item, prio);
+      } else if (!comp(prio, nodes[index[item]].prio)) {
+        increase(item, prio);
+      }
+    }
+      
+    /// \brief Increase the priority of the current item.
+    ///
+    /// Increase the priority of the current item.
+    void increase(const Item& item, const Value& prio) {
+      int id = index[item];
+      int kd = nodes[id].parent;
+      nodes[id].prio = prio;
+      while (kd >= 0 && nodes[kd].item == item) {
+        setPrio(kd);
+        kd = nodes[kd].parent;
+      }
+    }
+
+    /// \brief Increase the priority of the current item.
+    ///
+    /// Increase the priority of the current item.
+    void decrease(const Item& item, const Value& prio) {
+      int id = index[item];
+      int kd = nodes[id].parent;
+      nodes[id].prio = prio;
+      while (kd >= 0 && less(id, kd)) {
+        nodes[kd].prio = prio;
+        nodes[kd].item = item;
+        kd = nodes[kd].parent;
+      }
+    }
+    
+    /// \brief Gives back the minimum priority of the class.
+    ///
+    /// \return Gives back the minimum priority of the class.
+    const Value& classPrio(int cls) const {
+      return nodes[~(classes[cls].parent)].prio;
+    }
+
+    /// \brief Gives back the minimum priority item of the class.
+    ///
+    /// \return Gives back the minimum priority item of the class.
+    const Item& classTop(int cls) const {
+      return nodes[~(classes[cls].parent)].item;
+    }
+
+    /// \brief Gives back a representant item of the class.
+    /// 
+    /// The representant is indpendent from the priorities of the
+    /// items. 
+    /// \return Gives back a representant item of the class.
+    const Item& classRep(int id) const {
+      int parent = classes[id].parent;
+      return nodes[parent >= 0 ? classes[id].depth : leftNode(id)].item;
+    }
+
+    /// \brief Lemon style iterator for the items of a class.
+    ///
+    /// ClassIt is a lemon style iterator for the components. It iterates
+    /// on the items of a class. By example if you want to iterate on
+    /// each items of each classes then you may write the next code.
+    ///\code
+    /// for (ClassIt cit(huf); cit != INVALID; ++cit) {
+    ///   std::cout << "Class: ";
+    ///   for (ItemIt iit(huf, cit); iit != INVALID; ++iit) {
+    ///     std::cout << toString(iit) << ' ' << std::endl;
+    ///   }
+    ///   std::cout << std::endl;
+    /// }
+    ///\endcode
+    class ItemIt {
+    private:
+
+      const HeapUnionFind* _huf;
+      int _id, _lid;
+      
+    public:
+
+      /// \brief Default constructor 
+      ///
+      /// Default constructor 
+      ItemIt() {}
+
+      ItemIt(const HeapUnionFind& huf, int cls) : _huf(&huf) {
+        int id = cls;
+        int parent = _huf->classes[id].parent;
+        if (parent >= 0) {
+          _id = _huf->classes[id].depth;
+          if (_huf->classes[id].next != -1) {
+            _lid = _huf->classes[_huf->classes[id].next].depth;
+          } else {
+            _lid = -1;
+          }
+        } else {
+          _id = _huf->leftNode(id);
+          _lid = -1;
+        } 
+      }
+      
+      /// \brief Increment operator
+      ///
+      /// It steps to the next item in the class.
+      ItemIt& operator++() {
+        _id = _huf->nextNode(_id);
+        return *this;
+      }
+
+      /// \brief Conversion operator
+      ///
+      /// It converts the iterator to the current item.
+      operator const Item&() const {
+        return _huf->nodes[_id].item;
+      }
+      
+      /// \brief Equality operator
+      ///
+      /// Equality operator
+      bool operator==(const ItemIt& i) { 
+        return i._id == _id;
+      }
+
+      /// \brief Inequality operator
+      ///
+      /// Inequality operator
+      bool operator!=(const ItemIt& i) { 
+        return i._id != _id;
+      }
+
+      /// \brief Equality operator
+      ///
+      /// Equality operator
+      bool operator==(Invalid) { 
+        return _id == _lid;
+      }
+
+      /// \brief Inequality operator
+      ///
+      /// Inequality operator
+      bool operator!=(Invalid) { 
+        return _id != _lid;
+      }
+      
+    };
+
+    /// \brief Class iterator
+    ///
+    /// The iterator stores 
+    class ClassIt {
+    private:
+
+      const HeapUnionFind* _huf;
+      int _id;
+
+    public:
+
+      ClassIt(const HeapUnionFind& huf) 
+        : _huf(&huf), _id(huf.first_class) {}
+
+      ClassIt(const HeapUnionFind& huf, int cls) 
+        : _huf(&huf), _id(huf.classes[cls].left) {}
+
+      ClassIt(Invalid) : _huf(0), _id(-1) {}
+      
+      const ClassIt& operator++() {
+        _id = _huf->classes[_id].next;
+        return *this;
+      }
+
+      /// \brief Equality operator
+      ///
+      /// Equality operator
+      bool operator==(const ClassIt& i) { 
+        return i._id == _id;
+      }
+
+      /// \brief Inequality operator
+      ///
+      /// Inequality operator
+      bool operator!=(const ClassIt& i) { 
+        return i._id != _id;
+      }      
+      
+      operator int() const {
+        return _id;
+      }
+            
+    };
+
+  };
+
   //! @}