/// \ingroup matching

   ///\ingroup matching

   ///Edmonds' alternating forest maximum matching algorithm.

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

   ///

   ///algorithm using read-in functions \ref readNMapNode, \ref

   ///algorithm using some of init functions.

   ///readNMapEdge or \ref readEMapBool depending on the container. The

   ///resulting maximum matching can be attained by write-out functions

   ///\ref writeNMapNode, \ref writeNMapEdge or \ref writeEMapBool

   ///depending on the preferred container. 

   ///MaxMatching::pos_enum, having values D, A and C showing the

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

   ///Gallai-Edmonds decomposition of the graph. The nodes in D induce

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

   ///a graph with factor-critical components, the nodes in A form the

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

   ///barrier, and the nodes in C induce a graph having a perfect

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

   ///matching. This decomposition can be attained by calling \ref

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

   ///writePos after running the algorithm. 

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

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

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

     ///

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

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

     enum pos_enum {

     enum DecompType {

     typename Graph::template NodeMap<pos_enum> position;

     typename Graph::template NodeMap<DecompType> position;

     MaxMatching(const Graph& _g) : g(_g), _mate(_g,INVALID), position(_g) {}

     MaxMatching(const Graph& _g) 

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

     ///Runs Edmonds' algorithm.

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

     ///Runs Edmonds' algorithm for sparse graphs (number of edges <

     ///

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

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

     ///heuristic of postponing shrinks for dense graphs. 

     ///

     void run() {

     void init() {

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

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

 	greedyMatching();

 	_mate.set(v,INVALID);      

 	runEdmonds(0);

 	position.set(v,C);

       } else runEdmonds(1);

       }

     }

     }

     ///\brief Finds a greedy matching for initial matching.

     ///Runs Edmonds' algorithm.

     ///

     ///For initial matchig it finds a maximal greedy matching.

     ///If heur=0 it runs Edmonds' algorithm. If heur=1 it runs

     void greedyInit() {

     ///Edmonds' algorithm with a heuristic of postponing shrinks,

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

     ///giving a faster algorithm for dense graphs.  

 	_mate.set(v,INVALID);      

     void runEdmonds( int heur = 1 ) {

 	position.set(v,C);

       }

       //each vertex is put to C

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

 	position.set(v,C);      

       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);

       UFECrossRef tree_base(g);

       UFE 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 ) {

 	  blossom.insert(v);

 	  tree.insert(v); 

 	  position.set(v,D);

 	  if ( heur == 1 ) lateShrink( v, ear, blossom, tree );

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

 	}

       }

     }

     ///Finds a greedy matching starting from the actual matching.

     ///Starting form the actual matching stored, it finds a maximal

     ///greedy matching.

     void greedyMatching() {

     ///Returns the size of the actual matching stored.

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

     ///

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

     ///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 edge of the initial

     ///matching.

     template <typename MateMap>

     void mateMapInit(MateMap& map) {

       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 edges.

     ///

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

     ///map[v] must be an \c UEdge 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 edge joining \c

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

     template<typename MatchingMap>

     void matchingMapInit(MatchingMap& map) {

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

 	position.set(v,C);

 	UEdge 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 edges.

     ///

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

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

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

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

     template <typename MatchingMap>

     void matchingInit(MatchingMap& map) {

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

 	_mate.set(v,INVALID);      

 	position.set(v,C);

       }

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

 	if ( map[e] ) {

 	  Node u=g.source(e);	  

 	  Node v=g.target(e);

 	  _mate.set(u,v);

 	  _mate.set(v,u);

 	} 

       } 

     }

     ///\brief Runs Edmonds' algorithm.

     ///

     ///Runs Edmonds' algorithm for sparse graphs (number of edges <

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

     ///heuristic of postponing shrinks for dense graphs. 

     void run() {

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

 	greedyInit();

 	startSparse();

       } else {

 	init();

 	startDense();

       }

     }

     ///\brief Starts Edmonds' algorithm.

     /// 

     ///If runs the original Edmonds' algorithm.  

     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 graphs.

     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);

 	}

       }

     }

     ///\brief Returns the size of the actual matching stored.

     ///

     ///Resets the actual matching to the empty matching.

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

     ///

     ///Resets the actual matching to the empty matching.  

     ///

     void resetMatching() {

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

 	_mate.set(v,INVALID);      

     }

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

     Node mate(Node& node) const {

     Node mate(const Node& node) const {

     ///Reads a matching from a \c Node valued \c Node map.

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

     ///

     ///Reads a matching from a \c Node valued \c Node map. This map

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

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

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

     ///must hold, and \c uv will be an edge of the matching.

     UEdge matchingEdge(const Node& node) const {

     template<typename NMapN>

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

     void readNMapNode(NMapN& map) {

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

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

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

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

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

       } 

 	  return e;

 	}

       }

       return INVALID;

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

     /// \brief Returns the class of the node in the Edmonds-Gallai

     /// decomposition.

     ///

     /// Returns the class of the node in the Edmonds-Gallai

     /// decomposition.

     DecompType decomposition(const Node& n) {

       return position[n] == A;

     }

     /// \brief Returns true when the node is in the barrier.

     ///

     /// Returns true when the node is in the barrier.

     bool barrier(const Node& n) {

       return position[n] == A;

     }

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

     ///

     template<typename NMapN>

     template <typename MateMap>

     void writeNMapNode (NMapN& map) const {

     void mateMap(MateMap& map) const {

     ///Reads a matching from an \c UEdge valued \c Node map.

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

     ///map.

     ///Reads a matching from an \c UEdge valued \c Node map. \c

     ///

     ///map[v] must be an \c UEdge 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 edge

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

     template<typename NMapE>

     void readNMapEdge(NMapE& map) {

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

 	UEdge e=map[v];

 	if ( e!=INVALID )

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

       } 

     } 

     ///Writes the matching stored to an \c UEdge valued \c Node map.

     template<typename NMapE>

     template <typename MatchingMap>

     void writeNMapEdge (NMapE& map)  const {

     void matchingMap(MatchingMap& map)  const {

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

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

     ///Reads a matching from a \c bool valued \c Edge map.

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

     ///map.

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

     ///

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

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

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

     template<typename EMapB>

     void readEMapBool(EMapB& map) {

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

 	if ( map[e] ) {

 	  Node u=g.source(e);	  

 	  Node v=g.target(e);

 	  _mate.set(u,v);

 	  _mate.set(v,u);

 	} 

       } 

     }

     ///Writes the matching stored to a \c bool valued \c Edge map.

     template<typename EMapB>

     template<typename MatchingMap>

     void writeEMapBool (EMapB& map) const {

     void matching(MatchingMap& map) const {

     ///Writes the canonical decomposition of the graph after running

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

     ///

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

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

     template<typename NMapEnum>

     template <typename DecompositionMap>

     void writePos (NMapEnum& map) const {

     void decomposition(DecompositionMap& map) const {

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

       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 graph minus the barrier size is a lower bound for the

     ///uncovered nodes in the graph. 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);    

 		    UFE& blossom, UFE& tree);

 		    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() )

     }

 		    UFE& blossom, UFE& tree);

 		    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;

 	  }

 	}

       }

     }

 		UFE& blossom, UFE& tree,std::queue<Node>& Q);

 		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;

     }

 		    UFE& blossom, UFE& tree, std::queue<Node>& Q);

 		    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);

     }

 		       NodeMap<Node>& ear, UFE& blossom, UFE& tree, 

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

 		       std::queue<Node>& Q);

 		       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;

     }

 		 UFE& blossom, UFE& tree);

 		 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);

     }