// -*- C++ -*-

 #ifndef HUGO_MAX_MATCHING_H

 #define HUGO_MAX_MATCHING_H

 ///\ingroup galgs

 ///\file

 ///\brief Maximum matching algorithm.

 #include <queue>

 #include <invalid.h>

 #include <unionfind.h>

 namespace hugo {

   /// \addtogroup galgs

   /// @{

   ///Maximum matching algorithms class.

   ///This class provides Edmonds' alternating forest matching

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

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

   ///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. 

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

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

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

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

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

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

   ///writePos after running the algorithm. Before subsequent runs,

   ///the function \ref resetPos() must be called.

   ///\param Graph The undirected graph type the algorithm runs on.

   ///\author Jacint Szabo  

   template <typename Graph>

   class MaxMatching {

     typedef typename Graph::Node Node;

     typedef typename Graph::Edge Edge;

     typedef typename Graph::EdgeIt EdgeIt;

     typedef typename Graph::NodeIt NodeIt;

     typedef typename Graph::OutEdgeIt OutEdgeIt;

     typedef UnionFindEnum<Node, Graph::template NodeMap> UFE;

   public:

     ///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 pos_enum D induce a graph with factor-critical

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

     ///nodes in C induce a graph having a perfect matching. 

     enum pos_enum {

       D=0,

       A=1,

       C=2

     }; 

   private:

     const Graph& G;

     typename Graph::template NodeMap<Node> mate;

     typename Graph::template NodeMap<pos_enum> position;

   public:

     MaxMatching(Graph& _G) : G(_G), mate(_G,INVALID), position(_G,C) {}

     ///Runs Edmonds' algorithm.

     ///Runs Edmonds' algorithm for sparse graphs (edgeNum >=

     ///2*nodeNum), and a heuristical Edmonds' algorithm with a

     ///heuristic of postponing shrinks for dense graphs. \pre Before

     ///the subsequent calls \ref resetPos must be called.

     void run();

     ///Runs Edmonds' algorithm.

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

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

     ///giving a faster algorithm for dense graphs.  \pre Before the

     ///subsequent calls \ref resetPos must be called.

     void runEdmonds( int heur );

     ///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.

     ///Returns the size of the actual matching stored. After \ref

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

     int size();

     ///Resets the map storing the Gallai-Edmonds decomposition.

     ///Resets the map storing the Gallai-Edmonds decomposition of the

     ///graph, making it possible to run the algorithm. Must be called

     ///before all runs of the Edmonds algorithm, except for the first

     ///run.

     void resetPos();

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

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

     ///

     void resetMatching();

     ///Reads a matching from a \c Node map of \c Nodes.

     ///Reads a matching from a \c Node map of \c Nodes. This map must be \e

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

     ///now \c uv is an edge of the matching.

     template<typename NMapN>

     void readNMapNode(NMapN& map) {

       NodeIt v;

       for( G.first(v); G.valid(v); G.next(v)) {

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

       } 

     } 

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

     ///Writes the stored matching to a \c Node map of \c Nodes. 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 edge of the matching.

     template<typename NMapN>

     void writeNMapNode(NMapN& map) {

       NodeIt v;

       for( G.first(v); G.valid(v); G.next(v)) {

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

       } 

     } 

     ///Reads a matching from a \c Node map of \c Edges.

     ///Reads a matching from a \c Node map of incident \c Edges. This

     ///map must have the property that if \c G.bNode(map[u])=v then \c

     ///G.bNode(map[v])=u must hold, and now this edge is an edge of

     ///the matching.

     template<typename NMapE>

     void readNMapEdge(NMapE& map) {

       NodeIt v;

       for( G.first(v); G.valid(v); G.next(v)) {

 	Edge e=map[v];

 	if ( G.valid(e) )

 	  G.tail(e) == v ? mate.set(v,G.head(e)) : mate.set(v,G.tail(e)); 

       } 

     } 

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

     ///Writes the stored matching to a \c Node map of incident \c

     ///Edges. This map will have the property that if \c

     ///G.bNode(map[u])=v then \c G.bNode(map[v])=u holds, and now this

     ///edge is an edge of the matching.

     template<typename NMapE>

     void writeNMapEdge(NMapE& map)  {

       typename Graph::template NodeMap<bool> todo(G,false); 

       NodeIt v;

       for( G.first(v); G.valid(v); G.next(v)) {

 	if ( mate[v]!=INVALID ) todo.set(v,true); 

       }

       NodeIt e;

       for( G.first(e); G.valid(e); G.next(e)) {

 	if ( todo[G.head(e)] && todo[G.tail(e)] ) {

 	  Node u=G.tail(e);

 	  Node v=G.head(e); 

 	  if ( mate[u]=v && mate[v]=u ) {

 	    map.set(u,e);

 	    map.set(v,e);

 	    todo.set(u,false);

 	    todo.set(v,false);

 	  }

 	}

       }

     } 

     ///Reads a matching from an \c Edge map of \c bools.

     ///Reads a matching from an \c Edge map of \c bools. This map must

     ///have the property that there are no two adjacent 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) {

       EdgeIt e;

       for( G.first(e); G.valid(e); G.next(e)) {

 	if ( G.valid(e) ) {

 	  Node u=G.tail(e);	  

 	  Node v=G.head(e);

 	  mate.set(u,v);

 	  mate.set(v,u);

 	} 

       } 

     }

     ///Writes the matching stored to an \c Edge map of \c bools.

     ///Writes the matching stored to an \c Edge map of \c bools. This

     ///map will have the property that there are no two adjacent 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 writeEMapBool(EMapB& map) {

       typename Graph::template NodeMap<bool> todo(G,false); 

       NodeIt v;

       for( G.first(v); G.valid(v); G.next(v)) {

 	if ( mate[v]!=INVALID ) todo.set(v,true); 

       }

       NodeIt e;

       for( G.first(e); G.valid(e); G.next(e)) {

 	map.set(e,false);

 	if ( todo[G.head(e)] && todo[G.tail(e)] ) {

 	  Node u=G.tail(e);

 	  Node v=G.head(e); 

 	  if ( mate[u]=v && mate[v]=u ) {

 	    map.set(e,true);

 	    todo.set(u,false);

 	    todo.set(v,false);

 	  }

 	}

       }

     }

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

     ///the algorithm.

     ///After calling any run methods of the class, and before calling

     ///\ref resetPos(), it writes the Gallai-Edmonds canonical

     ///decomposition of the graph. \c map must be a node map of \ref pos_enum 's.

     template<typename NMapEnum>

     void writePos(NMapEnum& map)  {

       NodeIt v;

       for( G.first(v); G.valid(v); G.next(v)) map.set(v,position[v]);

     }

   private: 

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

 		    UFE& blossom, UFE& tree);

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

 		    UFE& blossom, UFE& tree);

     bool noShrinkStep(Node x, typename Graph::NodeMap<Node>& ear,  

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

     void shrinkStep(Node& top, Node& middle, Node& bottom, typename Graph::NodeMap<Node>& ear,  

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

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

 		 UFE& blossom, UFE& tree);

   };

   // **********************************************************************

   //  IMPLEMENTATIONS

   // **********************************************************************

   template <typename Graph>

   void MaxMatching<Graph>::run() {

     if ( G.edgeNum() > 2*G.nodeNum() ) {

       greedyMatching();

       runEdmonds(1);

     } else runEdmonds(0);

   }

   template <typename Graph>

   void MaxMatching<Graph>::runEdmonds( int heur=1 ) {

     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

     typename UFE::MapType blossom_base(G);

     UFE blossom(blossom_base);

     typename UFE::MapType tree_base(G);

     UFE tree(tree_base);

     NodeIt v;

     for( G.first(v); G.valid(v); G.next(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 );

       }

     }

   }

   template <typename Graph>

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

 				      UFE& blossom, UFE& tree) {

     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();

       if ( noShrinkStep( x, ear, blossom, tree, Q ) ) return;

       else R.push(x);

     }

     while ( !R.empty() ) {

       Node x=R.front();

       R.pop();

       OutEdgeIt e;

       for( G.first(e,x); G.valid(e); G.next(e) ) {

 	Node y=G.bNode(e);

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

 	  //x and y must be in the same tree

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

 	  Node b=blossom.find(x);

 	  path.set(b,true);

 	  b=mate[b];

 	  while ( b!=INVALID ) { 

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

 	    path.set(b,true);

 	    b=mate[b];

 	  } //going till the root

 	  Node top=y;

 	  Node middle=blossom.find(top);

 	  Node bottom=x;

 	  while ( !path[middle] )

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

 	  Node base=middle;

 	  top=x;

 	  middle=blossom.find(top);

 	  bottom=y;

 	  Node blossom_base=blossom.find(base);

 	  while ( middle!=blossom_base )

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

 	  blossom.makeRep(base);

 	} // if shrink is needed

 	while ( !Q.empty() ) {

 	  Node x=Q.front();

 	  Q.pop();

 	  if ( noShrinkStep(x, ear, blossom, tree, Q) ) return;

 	  else R.push(x);

 	}

       } //for e

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

   }

   template <typename Graph>

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

 				      UFE& blossom, UFE& tree) {

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

     Q.push(v);  

     while ( !Q.empty() ) {

       Node x=Q.front();

       Q.pop();

       OutEdgeIt e;

       for( G.first(e,x); G.valid(e); G.next(e) ) {

 	Node y=G.bNode(e);

 	switch ( position[y] ) {

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

 	  if ( blossom.find(x) != blossom.find(y) ) { //shrink

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

 	    Node b=blossom.find(x);

 	    path.set(b,true);

 	    b=mate[b];

 	    while ( b!=INVALID ) { 

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

 	      path.set(b,true);

 	      b=mate[b];

 	    } //going till the root

 	    Node top=y;

 	    Node middle=blossom.find(top);

 	    Node bottom=x;

 	    while ( !path[middle] )

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

 	    Node base=middle;

 	    top=x;

 	    middle=blossom.find(top);

 	    bottom=y;

 	    Node blossom_base=blossom.find(base);

 	    while ( middle!=blossom_base )

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

 	    blossom.makeRep(base);

 	  }

 	  break;

 	case C:

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

 	    ear.set(y,x);

 	    Node w=mate[y];

 	    blossom.insert(w);

 	    position.set(y,A); 

 	    position.set(w,D); 

 	    tree.insert(y);

 	    tree.insert(w);

 	    tree.join(y,blossom.find(x));  

 	    tree.join(w,y);  

 	    Q.push(w);

 	  } else {                 //augment  

 	    augment(x, ear, blossom, tree);

 	    mate.set(x,y);

 	    mate.set(y,x);

 	    return;

 	  } //if 

 	  break;

 	default: break;

 	}

       }

     }

   }

   template <typename Graph>

   void MaxMatching<Graph>::greedyMatching() {

     NodeIt v;

     for( G.first(v); G.valid(v); G.next(v) )

       if ( mate[v]==INVALID ) {

 	OutEdgeIt e;

 	for( G.first(e,v); G.valid(e); G.next(e) ) {

 	  Node y=G.bNode(e);

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

 	    mate.set(v,y);

 	    mate.set(y,v);

 	    break;

 	  }

 	}

       } 

   }

   template <typename Graph>

   int MaxMatching<Graph>::size() {

     int s=0;

     NodeIt v;

     for(G.first(v); G.valid(v); G.next(v) ) {

       if ( G.valid(mate[v]) ) {

 	++s;

       }

     }

     return (int)s/2;

   }

   template <typename Graph>

   void MaxMatching<Graph>::resetPos() {

     NodeIt v;

     for( G.first(v); G.valid(v); G.next(v))

       position.set(v,C);      

   }

   template <typename Graph>

   void MaxMatching<Graph>::resetMatching() {

     NodeIt v;

     for( G.first(v); G.valid(v); G.next(v))

       mate.set(v,INVALID);      

   }

   template <typename Graph>

   bool MaxMatching<Graph>::noShrinkStep(Node x, typename Graph::NodeMap<Node>& ear,  

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

     OutEdgeIt e;

     for( G.first(e,x); G.valid(e); G.next(e) ) {

       Node y=G.bNode(e);

       if ( position[y]==C ) {

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

 	  ear.set(y,x);

 	  Node w=mate[y];

 	  blossom.insert(w);

 	  position.set(y,A);

 	  position.set(w,D);

 	  tree.insert(y);

 	  tree.insert(w);

 	  tree.join(y,blossom.find(x));  

 	  tree.join(w,y);  

 	  Q.push(w);

 	} else {                      //augment 

 	  augment(x, ear, blossom, tree);

 	  mate.set(x,y);

 	  mate.set(y,x);

 	  return true;

 	}

       }

     }

     return false;

   }

   template <typename Graph>

   void MaxMatching<Graph>::shrinkStep(Node& top, Node& middle, Node& bottom, typename Graph::NodeMap<Node>& ear,  

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

     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=blossom.find(top);

     tree.erase(bottom);

     tree.erase(oldmiddle);

     blossom.insert(bottom);

     blossom.join(bottom, oldmiddle);

     blossom.join(top, oldmiddle);

   }

   template <typename Graph>

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

 				   UFE& blossom, UFE& tree) { 

     Node v=mate[x];

     while ( G.valid(v) ) {

       Node u=ear[v];

       mate.set(v,u);

       Node tmp=v;

       v=mate[u];

       mate.set(u,tmp);

     }

     typename UFE::ItemIt it;

     for (tree.first(it,blossom.find(x)); tree.valid(it); tree.next(it)) {   

       if ( position[it] == D ) {

 	typename UFE::ItemIt b_it;

 	for (blossom.first(b_it,it); blossom.valid(b_it); blossom.next(b_it)) {  

 	  position.set( b_it ,C);

 	}

 	blossom.eraseClass(it);

       } else position.set( it ,C);

     }

     tree.eraseClass(x);

   }

   /// @}

 } //END OF NAMESPACE HUGO

 #endif //EDMONDS_H