/* -*- C++ -*-

  *

  * This file is a part of LEMON, a generic C++ optimization library

  *

  * Copyright (C) 2003-2006

  * Egervary Jeno Kombinatorikus Optimalizalasi Kutatocsoport

  * (Egervary Research Group on Combinatorial Optimization, EGRES).

  *

  * Permission to use, modify and distribute this software is granted

  * provided that this copyright notice appears in all copies. For

  * precise terms see the accompanying LICENSE file.

  *

  * This software is provided "AS IS" with no warranty of any kind,

  * express or implied, and with no claim as to its suitability for any

  * purpose.

  *

  */

 #ifndef LEMON_MAX_MATCHING_H

 #define LEMON_MAX_MATCHING_H

 #include <queue>

 #include <lemon/bits/invalid.h>

 #include <lemon/unionfind.h>

 #include <lemon/graph_utils.h>

 ///\ingroup matching

 ///\file

 ///\brief Maximum matching algorithm in undirected graph.

 namespace lemon {

   /// \ingroup matching

   ///Edmonds' alternating forest maximum matching algorithm.

   ///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 matching 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. 

   ///

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

   ///

   ///\author Jacint Szabo  

   template <typename Graph>

   class MaxMatching {

   protected:

     typedef typename Graph::Node Node;

     typedef typename Graph::Edge Edge;

     typedef typename Graph::UEdge UEdge;

     typedef typename Graph::UEdgeIt UEdgeIt;

     typedef typename Graph::NodeIt NodeIt;

     typedef typename Graph::IncEdgeIt IncEdgeIt;

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

     typedef UnionFindEnum<UFECrossRef> 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 \c D induce a graph with factor-critical

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

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

     enum pos_enum {

       D=0,

       A=1,

       C=2

     }; 

   protected:

     static const int HEUR_density=2;

     const Graph& g;

     typename Graph::template NodeMap<Node> _mate;

     typename Graph::template NodeMap<pos_enum> position;

   public:

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

     ///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) ) {

 	greedyMatching();

 	runEdmonds(0);

       } else runEdmonds(1);

     }

     ///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.  

     void runEdmonds( int heur = 1 ) {

       //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() {

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

 	if ( _mate[v]==INVALID ) {

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

 	    Node y=g.runningNode(e);

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

 	      _mate.set(v,y);

 	      _mate.set(y,v);

 	      break;

 	    }

 	  }

 	} 

     }

     ///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() const {

       int s=0;

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

 	if ( _mate[v]!=INVALID ) {

 	  ++s;

 	}

       }

       return s/2;

     }

     ///Resets the actual matching to the empty 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.

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

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

     Node mate(Node& node) const {

       return _mate[node];

     } 

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

     ///Reads a 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 matching.

     template<typename NMapN>

     void readNMapNode(NMapN& map) {

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

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

       } 

     } 

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

     ///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 edge of the matching.

     template<typename NMapN>

     void writeNMapNode (NMapN& map) const {

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

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

       } 

     } 

     ///Reads a matching from 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.

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

     ///map. \c map[v] will be an \c UEdge 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 edge is an edge

     ///of the matching.

     template<typename NMapE>

     void writeNMapEdge (NMapE& map)  const {

       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(u,e);

 	      map.set(v,e);

 	      todo.set(u,false);

 	      todo.set(v,false);

 	      break;

 	    }

 	  }

 	}

       } 

     }

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

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

     ///map. This map will 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 writeEMapBool (EMapB& map) const {

       for(UEdgeIt 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;

 	    }

 	  }

 	}

       } 

     }

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

     ///the algorithm.

     ///After calling any run methods of the class, 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) const {

       for(NodeIt v(g); v!=INVALID; ++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::template NodeMap<Node>& ear,  

 		    UFE& blossom, UFE& tree);

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

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

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

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

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

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

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

 		       std::queue<Node>& Q);

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

 		 UFE& blossom, UFE& tree);

   };

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

   //  IMPLEMENTATIONS

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

   template <typename Graph>

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

 				      NodeMap<Node>& ear, UFE& blossom, UFE& 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, 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, tree, Q);	

 	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, tree, Q) ) return;

 	  }

 	  R.push(x);

 	}

       } //for e

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

   }

   template <typename Graph>

   void MaxMatching<Graph>::normShrink(Node v,

 				      typename Graph::template

 				      NodeMap<Node>& ear,  

 				      UFE& blossom, UFE& 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, 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, tree, Q) ) return;

 	  break;

 	default: break;

        	}

       }

     }

   }

   template <typename Graph>

     void MaxMatching<Graph>::shrink(Node x,Node y, typename 

 				    Graph::template NodeMap<Node>& ear,  

 				    UFE& blossom, UFE& 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=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];

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

     Node top=y;

     Node middle=blossom.find(top);

     Node bottom=x;

     while ( !path[middle] )

       shrinkStep(top, middle, bottom, ear, blossom, 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=blossom.find(top);

     bottom=y;

     Node blossom_base=blossom.find(base);

     while ( middle!=blossom_base )

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

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

     //and the positions of the odd nodes.

     blossom.makeRep(base);

   }

   template <typename Graph>

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

 				      typename Graph::template

 				      NodeMap<Node>& ear,  

 				      UFE& blossom, UFE& 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=blossom.find(top);

     tree.erase(bottom);

     tree.erase(oldmiddle);

     blossom.insert(bottom);

     blossom.join(bottom, oldmiddle);

     blossom.join(top, oldmiddle);

   }

   template <typename Graph>

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

 					 NodeMap<Node>& ear, UFE& blossom, UFE& 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];

       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>::augment(Node x,

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

 				   UFE& blossom, UFE& 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);

     }

     Node y=blossom.find(x);

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

       if ( position[tit] == D ) {

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

 	  position.set( bit ,C);

 	}

 	blossom.eraseClass(tit);

       } else position.set( tit ,C);

     }

     tree.eraseClass(y);

   }

 } //END OF NAMESPACE LEMON

 #endif //LEMON_MAX_MATCHING_H