///\brief Maximum matching algorithms in graph.

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

   ///\ingroup matching

   /// \ingroup matching

   ///\brief Edmonds' alternating forest maximum matching algorithm.

   /// \brief Edmonds' alternating forest maximum matching algorithm.

   ///This class provides Edmonds' alternating forest matching

   /// This class provides Edmonds' alternating forest matching

   ///algorithm. The starting matching (if any) can be passed to the

   /// algorithm. The starting matching (if any) can be passed to the

   ///algorithm using some of init functions.

   /// algorithm using some of init functions.

   ///The dual side of a matching is a map of the nodes to

   /// The dual side of a matching is a map of the nodes to

   ///MaxMatching::DecompType, having values \c D, \c A and \c C

   /// MaxMatching::Status, having values \c EVEN/D, \c ODD/A and \c

   ///showing the Gallai-Edmonds decomposition of the digraph. The nodes

   /// MATCHED/C showing the Gallai-Edmonds decomposition of the

   ///in \c D induce a digraph with factor-critical components, the nodes

   /// graph. The nodes in \c EVEN/D induce a graph with

   ///in \c A form the barrier, and the nodes in \c C induce a digraph

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

   ///having a perfect matching. This decomposition can be attained by

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

   ///calling \c decomposition() after running the algorithm.

   /// perfect matching. The number of the fractor critical components

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

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

   /// tight. This decomposition can be attained by calling \c

   /// decomposition() after running the algorithm.

   ///\param Digraph The graph type the algorithm runs on.

   /// \param _Graph The graph type the algorithm runs on.

   template <typename Graph>

   template <typename _Graph>

   public:

   protected:

     typedef _Graph Graph;

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

     MatchingMap;

     ///\brief Indicates the Gallai-Edmonds decomposition of the graph.

     ///

     ///Indicates the Gallai-Edmonds decomposition of the graph, which

     ///shows an upper bound on the size of a maximum matching. The

     ///nodes with Status \c EVEN/D induce a graph with factor-critical

     ///components, the nodes in \c ODD/A form the canonical barrier,

     ///and the nodes in \c MATCHED/C induce a graph having a perfect

     ///matching.

     enum Status {

       EVEN = 1, D = 1, MATCHED = 0, C = 0, ODD = -1, A = -1, UNMATCHED = -2

     };

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

   private:

     typedef typename Graph::template NodeMap<int> UFECrossRef;

     typedef UnionFindEnum<IntNodeMap> BlossomSet;

     typedef UnionFindEnum<UFECrossRef> UFE;

     typedef ExtendFindEnum<IntNodeMap> TreeSet;

     typedef std::vector<Node> NV;

     typedef RangeMap<Node> NodeIntMap;

     typedef MatchingMap EarMap;

     typedef typename Graph::template NodeMap<int> EFECrossRef;

     typedef std::vector<Node> NodeQueue;

     typedef ExtendFindEnum<EFECrossRef> EFE;

     const Graph& _graph;

     MatchingMap* _matching;

     StatusMap* _status;

     EarMap* _ear;

     IntNodeMap* _blossom_set_index;

     BlossomSet* _blossom_set;

     NodeIntMap* _blossom_rep;

     IntNodeMap* _tree_set_index;

     TreeSet* _tree_set;

     NodeQueue _node_queue;

     int _process, _postpone, _last;

     int _node_num;

   private:

     void createStructures() {

       _node_num = countNodes(_graph);

       if (!_matching) {

         _matching = new MatchingMap(_graph);

       }

       if (!_status) {

         _status = new StatusMap(_graph);

       }

       if (!_ear) {

         _ear = new EarMap(_graph);

       }

       if (!_blossom_set) {

         _blossom_set_index = new IntNodeMap(_graph);

         _blossom_set = new BlossomSet(*_blossom_set_index);

       }

       if (!_blossom_rep) {

         _blossom_rep = new NodeIntMap(_node_num);

       }

       if (!_tree_set) {

         _tree_set_index = new IntNodeMap(_graph);

         _tree_set = new TreeSet(*_tree_set_index);

       }

       _node_queue.resize(_node_num);

     }

     void destroyStructures() {

       if (_matching) {

         delete _matching;

       }

       if (_status) {

         delete _status;

       }

       if (_ear) {

         delete _ear;

       }

       if (_blossom_set) {

         delete _blossom_set;

         delete _blossom_set_index;

       }

       if (_blossom_rep) {

         delete _blossom_rep;

       }

       if (_tree_set) {

         delete _tree_set_index;

         delete _tree_set;

       }

     }

     void processDense(const Node& n) {

       _process = _postpone = _last = 0;

       _node_queue[_last++] = n;

       while (_process != _last) {

         Node u = _node_queue[_process++];

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

           Node v = _graph.target(a);

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

             extendOnArc(a);

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

             augmentOnArc(a);

             return;

           }

         }

       }

       while (_postpone != _last) {

         Node u = _node_queue[_postpone++];

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

           Node v = _graph.target(a);

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

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

               shrinkOnEdge(a);

             }

           }

           while (_process != _last) {

             Node w = _node_queue[_process++];

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

               Node x = _graph.target(b);

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

                 extendOnArc(b);

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

                 augmentOnArc(b);

                 return;

               }

             }

           }

         }

       }

     }

     void processSparse(const Node& n) {

       _process = _last = 0;

       _node_queue[_last++] = n;

       while (_process != _last) {

         Node u = _node_queue[_process++];

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

           Node v = _graph.target(a);

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

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

               shrinkOnEdge(a);

             }

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

             extendOnArc(a);

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

             augmentOnArc(a);

             return;

           }

         }

       }

     }

     void shrinkOnEdge(const Edge& e) {

       Node nca = INVALID;

       {

         std::set<Node> left_set, right_set;

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

         left_set.insert(left);

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

         right_set.insert(right);

         while (true) {

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

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

           left = (*_blossom_rep)[_blossom_set->

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

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

             nca = left;

             break;

           }

           left_set.insert(left);

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

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

           right = (*_blossom_rep)[_blossom_set->

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

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

             nca = right;

             break;

           }

           right_set.insert(right);

         }

         if (nca == INVALID) {

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

             nca = right;

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

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

               nca =(*_blossom_rep)[_blossom_set->

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

             }

           } else {

             nca = left;

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

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

               nca = (*_blossom_rep)[_blossom_set->

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

             }

           }

         }

       }

       {

         Node node = _graph.u(e);

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

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

         while (base != nca) {

           _ear->set(node, arc);

           Node n = node;

           while (n != base) {

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

             Arc a = (*_ear)[n];

             n = _graph.target(a);

             _ear->set(n, _graph.oppositeArc(a));

           }

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

           _tree_set->erase(base);

           _tree_set->erase(node);

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

           _status->set(node, EVEN);

           _node_queue[_last++] = node;

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

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

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

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

         }

       }

       _blossom_rep->set(_blossom_set->find(nca), nca);

       {

         Node node = _graph.v(e);

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

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

         while (base != nca) {

           _ear->set(node, arc);

           Node n = node;

           while (n != base) {

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

             Arc a = (*_ear)[n];

             n = _graph.target(a);

             _ear->set(n, _graph.oppositeArc(a));

           }

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

           _tree_set->erase(base);

           _tree_set->erase(node);

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

           _status->set(node, EVEN);

           _node_queue[_last++] = node;

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

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

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

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

         }

       }

       _blossom_rep->set(_blossom_set->find(nca), nca);

     }

     void extendOnArc(const Arc& a) {

       Node base = _graph.source(a);

       Node odd = _graph.target(a);

       _ear->set(odd, _graph.oppositeArc(a));

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

       _blossom_rep->set(_blossom_set->insert(even), even);

       _status->set(odd, ODD);

       _status->set(even, EVEN);

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

       _tree_set->insert(odd, tree);

       _tree_set->insert(even, tree);

       _node_queue[_last++] = even;

     }

     void augmentOnArc(const Arc& a) {

       Node even = _graph.source(a);

       Node odd = _graph.target(a);

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

       _matching->set(odd, _graph.oppositeArc(a));

       _status->set(odd, MATCHED);

       Arc arc = (*_matching)[even];

       _matching->set(even, a);

       while (arc != INVALID) {

         odd = _graph.target(arc);

         arc = (*_ear)[odd];

         even = _graph.target(arc);

         _matching->set(odd, arc);

         arc = (*_matching)[even];

         _matching->set(even, _graph.oppositeArc((*_matching)[odd]));

       }

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

            it != INVALID; ++it) {

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

           _status->set(it, MATCHED);

         } else {

           int blossom = _blossom_set->find(it);

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

                jt != INVALID; ++jt) {

             _status->set(jt, MATCHED);

           }

           _blossom_set->eraseClass(blossom);

         }

       }

       _tree_set->eraseClass(tree);

     }

     ///\brief Indicates the Gallai-Edmonds decomposition of the digraph.

     /// \brief Constructor

     ///

     ///

     ///Indicates the Gallai-Edmonds decomposition of the digraph, which

     /// Constructor.

     ///shows an upper bound on the size of a maximum matching. The

     MaxMatching(const Graph& graph)

     ///nodes with DecompType \c D induce a digraph with factor-critical

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

     ///components, the nodes in \c A form the canonical barrier, and the

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

     ///nodes in \c C induce a digraph having a perfect matching.

         _tree_set_index(0), _tree_set(0) {}

     enum DecompType {

       D=0,

     ~MaxMatching() {

       A=1,

       destroyStructures();

       C=2

     }

     };

     /// \name Execution control

   protected:

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

     /// \c run() member function.

     static const int HEUR_density=2;

     /// \n

     const Graph& g;

     typename Graph::template NodeMap<Node> _mate;

     /// If you need more control on the execution, first you must call

     typename Graph::template NodeMap<DecompType> position;

     /// \ref init(), \ref greedyInit() or \ref matchingInit()

     /// functions, then you can start the algorithm with the \ref

   public:

     /// startParse() or startDense() functions.

     MaxMatching(const Graph& _g)

     ///@{

       : g(_g), _mate(_g), position(_g) {}

     /// \brief Sets the actual matching to the empty matching.

     ///\brief Sets the actual matching to the empty matching.

     ///

     ///

     /// Sets the actual matching to the empty matching.

     ///Sets the actual matching to the empty matching.

       for(NodeIt v(g); v!=INVALID; ++v) {

       createStructures();

         _mate.set(v,INVALID);

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

         position.set(v,C);

         _matching->set(n, INVALID);

         _status->set(n, UNMATCHED);

       for(NodeIt v(g); v!=INVALID; ++v) {

       createStructures();

         _mate.set(v,INVALID);

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

         position.set(v,C);

         _matching->set(n, INVALID);

       }

         _status->set(n, UNMATCHED);

       for(NodeIt v(g); v!=INVALID; ++v)

       }

         if ( _mate[v]==INVALID ) {

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

           for( IncEdgeIt e(g,v); e!=INVALID ; ++e ) {

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

             Node y=g.runningNode(e);

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

             if ( _mate[y]==INVALID && y!=v ) {

             Node v = _graph.target(a);

               _mate.set(v,y);

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

               _mate.set(y,v);

               _matching->set(n, a);

               _status->set(n, MATCHED);

               _matching->set(v, _graph.oppositeArc(a));

               _status->set(v, MATCHED);

     }

       }

     }

     ///\brief Initialize the matching from each nodes' mate.

     ///

     ///Initialize the matching from a \c Node valued \c Node map. This

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

     ///map must be \e symmetric, i.e. if \c map[u]==v then \c

     ///

     ///map[v]==u must hold, and \c uv will be an arc of the initial

     /// Initialize the matching from a \c bool valued \c Edge map. This

     ///matching.

     /// map must have the property that there are no two incident edges

     template <typename MateMap>

     /// with true value, ie. it contains a matching.

     void mateMapInit(MateMap& map) {

     /// \return %True if the map contains a matching.

       for(NodeIt v(g); v!=INVALID; ++v) {

         _mate.set(v,map[v]);

         position.set(v,C);

       }

     }

     ///\brief Initialize the matching from a node map with the

     ///incident matching arcs.

     ///

     ///Initialize the matching from an \c Edge valued \c Node map. \c

     ///map[v] must be an \c Edge incident to \c v. This map must have

     ///the property that if \c g.oppositeNode(u,map[u])==v then \c \c

     ///g.oppositeNode(v,map[v])==u holds, and now some arc joining \c

     ///u to \c v will be an arc of the matching.

     template<typename MatchingMap>

     void matchingMapInit(MatchingMap& map) {

       for(NodeIt v(g); v!=INVALID; ++v) {

         position.set(v,C);

         Edge e=map[v];

         if ( e!=INVALID )

           _mate.set(v,g.oppositeNode(v,e));

         else

           _mate.set(v,INVALID);

       }

     }

     ///\brief Initialize the matching from the map containing the

     ///undirected matching arcs.

     ///

     ///Initialize the matching from a \c bool valued \c Edge map. This

     ///map must have the property that there are no two incident arcs

     ///\c e, \c f with \c map[e]==map[f]==true. The arcs \c e with \c

     ///map[e]==true form the matching.

     void matchingInit(MatchingMap& map) {

     bool matchingInit(const MatchingMap& matching) {

       for(NodeIt v(g); v!=INVALID; ++v) {

       createStructures();

         _mate.set(v,INVALID);

         position.set(v,C);

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

       }

         _matching->set(n, INVALID);

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

         _status->set(n, UNMATCHED);

         if ( map[e] ) {

       }

           Node u=g.u(e);

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

           Node v=g.v(e);

         if (matching[e]) {

           _mate.set(u,v);

           _mate.set(v,u);

           Node u = _graph.u(e);

         }

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

       }

           _matching->set(u, _graph.direct(e, true));

     }

           _status->set(u, MATCHED);

           Node v = _graph.v(e);

     ///\brief Runs Edmonds' algorithm.

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

     ///

           _matching->set(v, _graph.direct(e, false));

     ///Runs Edmonds' algorithm for sparse digraphs (number of arcs <

           _status->set(v, MATCHED);

     ///2*number of nodes), and a heuristical Edmonds' algorithm with a

         }

     ///heuristic of postponing shrinks for dense digraphs.

       }

       return true;

     }

     /// \brief Starts Edmonds' algorithm

     ///

     /// If runs the original Edmonds' algorithm.

     void startSparse() {

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

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

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

           _tree_set->insert(n);

           _status->set(n, EVEN);

           processSparse(n);

         }

       }

     }

     /// \brief Starts Edmonds' algorithm.

     ///

     /// It runs Edmonds' algorithm with a heuristic of postponing

     /// shrinks, giving a faster algorithm for dense graphs.

     void startDense() {

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

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

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

           _tree_set->insert(n);

           _status->set(n, EVEN);

           processDense(n);

         }

       }

     }

     /// \brief Runs Edmonds' algorithm

     ///

     /// Runs Edmonds' algorithm for sparse graphs (<tt>m<2*n</tt>)

     /// or Edmonds' algorithm with a heuristic of

     /// postponing shrinks for dense graphs.

       if (countEdges(g) < HEUR_density * countNodes(g)) {

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

     /// @}

     ///\brief Starts Edmonds' algorithm.

     ///

     /// \name Primal solution

     ///If runs the original Edmonds' algorithm.

     /// Functions for get the primal solution, ie. the matching.

     void startSparse() {

     /// @{

       typename Graph::template NodeMap<Node> ear(g,INVALID);

       //undefined for the base nodes of the blossoms (i.e. for the

       //representative elements of UFE blossom) and for the nodes in C

       UFECrossRef blossom_base(g);

       UFE blossom(blossom_base);

       NV rep(countNodes(g));

       EFECrossRef tree_base(g);

       EFE tree(tree_base);

       //If these UFE's would be members of the class then also

       //blossom_base and tree_base should be a member.

       //We build only one tree and the other vertices uncovered by the

       //matching belong to C. (They can be considered as singleton

       //trees.) If this tree can be augmented or no more

       //grow/augmentation/shrink is possible then we return to this

       //"for" cycle.

       for(NodeIt v(g); v!=INVALID; ++v) {

         if (position[v]==C && _mate[v]==INVALID) {

           rep[blossom.insert(v)] = v;

           tree.insert(v);

           position.set(v,D);

           normShrink(v, ear, blossom, rep, tree);

         }

       }

     }

     ///\brief Starts Edmonds' algorithm.

     ///

     ///It runs Edmonds' algorithm with a heuristic of postponing

     ///shrinks, giving a faster algorithm for dense digraphs.

     void startDense() {

       typename Graph::template NodeMap<Node> ear(g,INVALID);

       //undefined for the base nodes of the blossoms (i.e. for the

       //representative elements of UFE blossom) and for the nodes in C

       UFECrossRef blossom_base(g);

       UFE blossom(blossom_base);

       NV rep(countNodes(g));

       EFECrossRef tree_base(g);

       EFE tree(tree_base);

       //If these UFE's would be members of the class then also

       //blossom_base and tree_base should be a member.

       //We build only one tree and the other vertices uncovered by the

       //matching belong to C. (They can be considered as singleton

       //trees.) If this tree can be augmented or no more

       //grow/augmentation/shrink is possible then we return to this

       //"for" cycle.

       for(NodeIt v(g); v!=INVALID; ++v) {

         if ( position[v]==C && _mate[v]==INVALID ) {

           rep[blossom.insert(v)] = v;

           tree.insert(v);

           position.set(v,D);

           lateShrink(v, ear, blossom, rep, tree);

         }

       }

     }

     ///run() it returns the size of a maximum matching in the digraph.

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

     int size() const {

     int matchingSize() const {

       int s=0;

       int size = 0;

       for(NodeIt v(g); v!=INVALID; ++v) {

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

         if ( _mate[v]!=INVALID ) {

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

           ++s;

           ++size;

         }

         }

       }

       }

       return s/2;

       return size / 2;

     }

     }

     /// \brief Returns true when the edge is in the matching.

     ///

     /// Returns true when the edge is in the matching.

     bool matching(const Edge& edge) const {

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

     }

     /// \brief Returns the matching edge incident to the given node.

     ///

     /// Returns the matching edge of a \c node in the actual matching or

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

     Arc matching(const Node& n) const {

       return (*_matching)[n];

     }

     ///Returns the mate of a \c node in the actual matching.

     ///Returns the mate of a \c node in the actual matching or

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

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

     Node mate(const Node& node) const {

     Node mate(const Node& n) const {

       return _mate[node];

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

     }

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

     }

     ///\brief Returns the matching arc incident to the given node.

     ///

     /// @}

     ///Returns the matching arc of a \c node in the actual matching.

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

     /// \name Dual solution

     Edge matchingArc(const Node& node) const {

     /// Functions for get the dual solution, ie. the decomposition.

       if (_mate[node] == INVALID) return INVALID;

       Node n = node < _mate[node] ? node : _mate[node];

     /// @{

       for (IncEdgeIt e(g, n); e != INVALID; ++e) {

         if (g.oppositeNode(n, e) == _mate[n]) {

           return e;

         }

       }

       return INVALID;

     }

     DecompType decomposition(const Node& n) {

     Status decomposition(const Node& n) const {

       return position[n] == A;

       return (*_status)[n];

     bool barrier(const Node& n) {

     bool barrier(const Node& n) const {

       return position[n] == A;

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

     }

     }

     ///\brief Gives back the matching in a \c Node of mates.

     /// @}

     ///

     ///Writes the stored matching to a \c Node valued \c Node map. The

     ///resulting map will be \e symmetric, i.e. if \c map[u]==v then \c

     ///map[v]==u will hold, and now \c uv is an arc of the matching.

     template <typename MateMap>

     void mateMap(MateMap& map) const {

       for(NodeIt v(g); v!=INVALID; ++v) {

         map.set(v,_mate[v]);

       }

     }

     ///\brief Gives back the matching in an \c Edge valued \c Node

     ///map.

     ///

     ///Writes the stored matching to an \c Edge valued \c Node

     ///map. \c map[v] will be an \c Edge incident to \c v. This

     ///map will have the property that if \c g.oppositeNode(u,map[u])

     ///== v then \c map[u]==map[v] holds, and now this arc is an arc

     ///of the matching.

     template <typename MatchingMap>

     void matchingMap(MatchingMap& map)  const {

       typename Graph::template NodeMap<bool> todo(g,true);

       for(NodeIt v(g); v!=INVALID; ++v) {

         if (_mate[v]!=INVALID && v < _mate[v]) {

           Node u=_mate[v];

           for(IncEdgeIt e(g,v); e!=INVALID; ++e) {

             if ( g.runningNode(e) == u ) {

               map.set(u,e);

               map.set(v,e);

               todo.set(u,false);

               todo.set(v,false);

               break;

             }

           }

         }

       }

     }

     ///\brief Gives back the matching in a \c bool valued \c Edge

     ///map.

     ///

     ///Writes the matching stored to a \c bool valued \c Arc

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

     ///incident arcs \c e, \c f with \c map[e]==map[f]==true. The

     ///arcs \c e with \c map[e]==true form the matching.

     template<typename MatchingMap>

     void matching(MatchingMap& map) const {

       for(EdgeIt e(g); e!=INVALID; ++e) map.set(e,false);

       typename Graph::template NodeMap<bool> todo(g,true);

       for(NodeIt v(g); v!=INVALID; ++v) {

         if ( todo[v] && _mate[v]!=INVALID ) {

           Node u=_mate[v];

           for(IncEdgeIt e(g,v); e!=INVALID; ++e) {

             if ( g.runningNode(e) == u ) {

               map.set(e,true);

               todo.set(u,false);

               todo.set(v,false);

               break;

             }

           }

         }

       }

     }

     ///\brief Returns the canonical decomposition of the digraph after running

     ///the algorithm.

     ///

     ///After calling any run methods of the class, it writes the

     ///Gallai-Edmonds canonical decomposition of the digraph. \c map

     ///must be a node map of \ref DecompType 's.

     template <typename DecompositionMap>

     void decomposition(DecompositionMap& map) const {

       for(NodeIt v(g); v!=INVALID; ++v) map.set(v,position[v]);

     }

     ///\brief Returns a barrier on the nodes.

     ///

     ///After calling any run methods of the class, it writes a

     ///canonical barrier on the nodes. The odd component number of the

     ///remaining digraph minus the barrier size is a lower bound for the

     ///uncovered nodes in the digraph. The \c map must be a node map of

     ///bools.

     template <typename BarrierMap>

     void barrier(BarrierMap& barrier) {

       for(NodeIt v(g); v!=INVALID; ++v) barrier.set(v,position[v] == A);

     }

   private:

     void lateShrink(Node v, typename Graph::template NodeMap<Node>& ear,

                     UFE& blossom, NV& rep, EFE& tree) {

       //We have one tree which we grow, and also shrink but only if it

       //cannot be postponed. If we augment then we return to the "for"

       //cycle of runEdmonds().

       std::queue<Node> Q;   //queue of the totally unscanned nodes

       Q.push(v);

       std::queue<Node> R;

       //queue of the nodes which must be scanned for a possible shrink

       while ( !Q.empty() ) {

         Node x=Q.front();

         Q.pop();

         for( IncEdgeIt e(g,x); e!= INVALID; ++e ) {

           Node y=g.runningNode(e);

           //growOrAugment grows if y is covered by the matching and

           //augments if not. In this latter case it returns 1.

           if (position[y]==C &&

               growOrAugment(y, x, ear, blossom, rep, tree, Q)) return;

         }

         R.push(x);

       }

       while ( !R.empty() ) {

         Node x=R.front();

         R.pop();

         for( IncEdgeIt e(g,x); e!=INVALID ; ++e ) {

           Node y=g.runningNode(e);

           if ( position[y] == D && blossom.find(x) != blossom.find(y) )

             //Recall that we have only one tree.

             shrink( x, y, ear, blossom, rep, tree, Q);

           while ( !Q.empty() ) {

             Node z=Q.front();

             Q.pop();

             for( IncEdgeIt f(g,z); f!= INVALID; ++f ) {

               Node w=g.runningNode(f);

               //growOrAugment grows if y is covered by the matching and

               //augments if not. In this latter case it returns 1.

               if (position[w]==C &&

                   growOrAugment(w, z, ear, blossom, rep, tree, Q)) return;

             }

             R.push(z);

           }

         } //for e

       } // while ( !R.empty() )

     }

     void normShrink(Node v, typename Graph::template NodeMap<Node>& ear,

                     UFE& blossom, NV& rep, EFE& tree) {

       //We have one tree, which we grow and shrink. If we augment then we

       //return to the "for" cycle of runEdmonds().

       std::queue<Node> Q;   //queue of the unscanned nodes

       Q.push(v);

       while ( !Q.empty() ) {

         Node x=Q.front();

         Q.pop();

         for( IncEdgeIt e(g,x); e!=INVALID; ++e ) {

           Node y=g.runningNode(e);

           switch ( position[y] ) {

           case D:          //x and y must be in the same tree

             if ( blossom.find(x) != blossom.find(y))

               //x and y are in the same tree

               shrink(x, y, ear, blossom, rep, tree, Q);

             break;

           case C:

             //growOrAugment grows if y is covered by the matching and

             //augments if not. In this latter case it returns 1.

             if (growOrAugment(y, x, ear, blossom, rep, tree, Q)) return;

             break;

           default: break;

           }

         }

       }

     }

     void shrink(Node x,Node y, typename Graph::template NodeMap<Node>& ear,

                 UFE& blossom, NV& rep, EFE& tree,std::queue<Node>& Q) {

       //x and y are the two adjacent vertices in two blossoms.

       typename Graph::template NodeMap<bool> path(g,false);

       Node b=rep[blossom.find(x)];

       path.set(b,true);

       b=_mate[b];

       while ( b!=INVALID ) {

         b=rep[blossom.find(ear[b])];

         path.set(b,true);

         b=_mate[b];

       } //we go until the root through bases of blossoms and odd vertices

       Node top=y;

       Node middle=rep[blossom.find(top)];

       Node bottom=x;

       while ( !path[middle] )

         shrinkStep(top, middle, bottom, ear, blossom, rep, tree, Q);

       //Until we arrive to a node on the path, we update blossom, tree

       //and the positions of the odd nodes.

       Node base=middle;

       top=x;

       middle=rep[blossom.find(top)];

       bottom=y;

       Node blossom_base=rep[blossom.find(base)];

       while ( middle!=blossom_base )

         shrinkStep(top, middle, bottom, ear, blossom, rep, tree, Q);

       //Until we arrive to a node on the path, we update blossom, tree

       //and the positions of the odd nodes.

       rep[blossom.find(base)] = base;

     }

     void shrinkStep(Node& top, Node& middle, Node& bottom,

                     typename Graph::template NodeMap<Node>& ear,

                     UFE& blossom, NV& rep, EFE& tree, std::queue<Node>& Q) {

       //We traverse a blossom and update everything.

       ear.set(top,bottom);

       Node t=top;

       while ( t!=middle ) {

         Node u=_mate[t];

         t=ear[u];

         ear.set(t,u);

       }

       bottom=_mate[middle];

       position.set(bottom,D);

       Q.push(bottom);

       top=ear[bottom];

       Node oldmiddle=middle;

       middle=rep[blossom.find(top)];

       tree.erase(bottom);

       tree.erase(oldmiddle);

       blossom.insert(bottom);

       blossom.join(bottom, oldmiddle);

       blossom.join(top, oldmiddle);

     }

     bool growOrAugment(Node& y, Node& x, typename Graph::template

                        NodeMap<Node>& ear, UFE& blossom, NV& rep, EFE& tree,

                        std::queue<Node>& Q) {

       //x is in a blossom in the tree, y is outside. If y is covered by

       //the matching we grow, otherwise we augment. In this case we

       //return 1.

       if ( _mate[y]!=INVALID ) {       //grow

         ear.set(y,x);

         Node w=_mate[y];

         rep[blossom.insert(w)] = w;

         position.set(y,A);

         position.set(w,D);

         int t = tree.find(rep[blossom.find(x)]);

         tree.insert(y,t);

         tree.insert(w,t);

         Q.push(w);

       } else {                      //augment

         augment(x, ear, blossom, rep, tree);

         _mate.set(x,y);

         _mate.set(y,x);

         return true;

       }

       return false;

     }

     void augment(Node x, typename Graph::template NodeMap<Node>& ear,

                  UFE& blossom, NV& rep, EFE& tree) {

       Node v=_mate[x];

       while ( v!=INVALID ) {

         Node u=ear[v];

         _mate.set(v,u);

         Node tmp=v;

         v=_mate[u];

         _mate.set(u,tmp);

       }

       int y = tree.find(rep[blossom.find(x)]);

       for (typename EFE::ItemIt tit(tree, y); tit != INVALID; ++tit) {

         if ( position[tit] == D ) {

           int b = blossom.find(tit);

           for (typename UFE::ItemIt bit(blossom, b); bit != INVALID; ++bit) {

             position.set(bit, C);

           }

           blossom.eraseClass(b);

         } else position.set(tit, C);

       }

       tree.eraseClass(y);

     }

   /// arcs in the digraph with maximum overall weight and no two of

   /// edges in the graph with maximum overall weight and no two of

   /// them shares their endpoints. The problem can be formulated with

   /// them shares their ends. The problem can be formulated with the

   /// the next linear program:

   /// following linear program.

   ///\f[ \sum_{e \in \gamma(B)}x_e \le \frac{\vert B \vert - 1}{2} \quad \forall B\in\mathcal{O}\f]

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

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

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

   /// 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 arcs with both endpoints in

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

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

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

   /// the nodes.

   /// subsets of the nodes.

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

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

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

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

   /// \f[\min \sum_{u \in V}y_u + \sum_{B \in \mathcal{O}}\frac{\vert B \vert - 1}{2}z_B\f]

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

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

     typedef HeapUnionFind<Value, NodeIntMap> BlossomSet;

     typedef HeapUnionFind<Value, IntNodeMap> BlossomSet;

     NodeIntMap *_blossom_index;

     IntNodeMap *_blossom_index;

     NodeIntMap *_node_index;

     IntNodeMap *_node_index;

     ArcIntMap *_node_heap_index;

     IntArcMap *_node_heap_index;

       NodeData(ArcIntMap& node_heap_index)

       NodeData(IntArcMap& node_heap_index)

       BinHeap<Value, ArcIntMap> heap;

       BinHeap<Value, IntArcMap> heap;

     NodeIntMap *_delta1_index;

     IntNodeMap *_delta1_index;

     BinHeap<Value, NodeIntMap> *_delta1;

     BinHeap<Value, IntNodeMap> *_delta1;

     EdgeIntMap *_delta3_index;

     IntEdgeMap *_delta3_index;

     BinHeap<Value, EdgeIntMap> *_delta3;

     BinHeap<Value, IntEdgeMap> *_delta3;

         _blossom_index = new NodeIntMap(_graph);

         _blossom_index = new IntNodeMap(_graph);

         _node_index = new NodeIntMap(_graph);

         _node_index = new IntNodeMap(_graph);

         _node_heap_index = new ArcIntMap(_graph);

         _node_heap_index = new IntArcMap(_graph);

         _delta1_index = new NodeIntMap(_graph);

         _delta1_index = new IntNodeMap(_graph);

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

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

         _delta3_index = new EdgeIntMap(_graph);

         _delta3_index = new IntEdgeMap(_graph);

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

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

     void augmentOnArc(const Edge& arc) {

     void augmentOnEdge(const Edge& edge) {

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

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

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

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

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

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

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

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