/* -*- C++ -*-

 *

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

 *

 * Copyright (C) 2003-2007

 * 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

  ///\brief 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 some of init functions.

  ///

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

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

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

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

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

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

  ///calling \c decomposition() 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;

    typedef std::vector<Node> NV;

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

    typedef ExtendFindEnum<EFECrossRef> EFE;

  public:

    ///\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 DecompType \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 DecompType {

      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<DecompType> position;

  public:

    MaxMatching(const Graph& _g) 

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

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

    ///

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

    ///

    void init() {

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

	_mate.set(v,INVALID);      

	position.set(v,C);

      }

    }

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

    ///

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

    void greedyInit() {

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

	_mate.set(v,INVALID);      

	position.set(v,C);

      }

      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;

	    }

	  }

	} 

    }

    ///\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.

    ///

    ///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;

    }

    ///\brief 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(const Node& node) const {

      return _mate[node];

    } 

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

    ///

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

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

    UEdge matchingEdge(const Node& node) const {

      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;

    } 

    /// \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.

    ///

    ///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 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 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 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 UEdge

    ///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 MatchingMap>

    void matching(MatchingMap& 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;

	    }

	  }

	}

      } 

    }

    ///\brief Returns 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 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 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);    

    }

  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);

    }

  };

} //END OF NAMESPACE LEMON

#endif //LEMON_MAX_MATCHING_H