#ifndef LEMON_BP_MATCHING

 #ifndef LEMON_PR_BIPARTITE_MATCHING

 #define LEMON_BP_MATCHING

 #define LEMON_PR_BIPARTITE_MATCHING

 #define BIPARTITE_TYPEDEFS(Graph)		\

   ///Max cardinality matching algorithm based on push-relabel principle

   GRAPH_TYPEDEFS(Graph)				\

     typedef Graph::ANodeIt ANodeIt;	\

   ///\ingroup matching

     typedef Graph::BNodeIt BNodeIt;

   ///Bipartite Max Cardinality Matching algorithm. This class uses the

   ///push-relabel principle which in several cases has better runtime

 #define UNDIRBIPARTITE_TYPEDEFS(Graph)		\

   ///performance than the augmenting path solutions.

   UNDIRGRAPH_TYPEDEFS(Graph)			\

   ///

     typedef Graph::ANodeIt ANodeIt;	\

   ///\author Alpar Juttner

     typedef Graph::BNodeIt BNodeIt;

   template<class Graph>

   class PrBipartiteMatching {

   template<class Graph,

 	   class MT=typename Graph::template ANodeMap<typename Graph::UEdge> >

   class BpMatching {

     MT &_matching;

     int _matching_size;

     int _empty_level;

     typename Graph::template ANodeMap<typename Graph::UEdge> _matching;

     BpMatching(const Graph &g, MT &matching) :

     PrBipartiteMatching(const Graph &g) :

       _matching(matching),

       _matching(g),

   private:

     /// \name Execution control 

     void init() 

     /// The simplest way to execute the algorithm is to use one of the

     {

     /// member functions called \c run(). \n

 //     for(BNodeIt n(g);n!=INVALID;++n) cov[n]=0;

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

     /// you must call \ref init() and then one variant of the start()

     /// member. 

     /// @{

     ///Initialize the data structures

     ///This function constructs a prematching first, which is a

     ///regular matching on the A-side of the graph, but on the B-side

     ///each node could cover more matching edges. After that, the

     ///B-nodes which multiple matched, will be pushed into the lowest

     ///level of the Elevator. The remaning B-nodes will be pushed to

     ///the consequent levels respect to a Bfs on following graph: the

     ///nodes are the B-nodes of the original bipartite graph and two

     ///nodes are adjacent if a node can pass over a matching edge to

     ///an other node. The source of the Bfs are the lowest level

     ///nodes. Last, the reached B-nodes without covered matching edge

     ///becomes active.

     void init() {

       _matching_size=0;

       _empty_level=_node_num;

 	  ++_cov[_g.oppositeNode(n,_matching[n])];

 	  ++_cov[_g.bNode(_matching[n])];

   public:

     int run() 

     ///Start the main phase of the algorithm

     {

       init();

     ///This algorithm calculates the maximum matching with the

     ///push-relabel principle. This function should be called just

       Node act;

     ///after the init() function which already set the initial

       Node bact=INVALID;

     ///prematching, the level function on the B-nodes and the active,

       Node last_activated=INVALID;

     ///ie. unmatched B-nodes.

 //       while((act=last_activated!=INVALID?

     ///

 // 	     last_activated:_levels.highestActive())

     ///The algorithm always takes highest active B-node, and it try to

 // 	    !=INVALID)

     ///find a B-node which is eligible to pass over one of it's

       while((act=_levels.highestActive())!=INVALID) {

     ///matching edge. This condition holds when the B-node is one

 	last_activated=INVALID;

     ///level lower, and the opposite node of it's matching edge is

 	int actlevel=_levels[act];

     ///adjacent to the highest active node. In this case the current

     ///node steals the matching edge and becomes inactive. If there is

 	UEdge bedge=INVALID;

     ///not eligible node then the highest active node should be lift

 	int nlevel=_node_num;

     ///to the next proper level.

 	{

     ///

 	  int nnlevel;

     ///The nodes should not lift higher than the number of the

 	  for(IncEdgeIt tbedge(_g,act);

     ///B-nodes, if a node reach this level it remains unmatched. If

 	      tbedge!=INVALID && nlevel>=actlevel;

     ///during the execution one level becomes empty the nodes above it

 	      ++tbedge)

     ///can be deactivated and lift to the highest level.

 	    if((nnlevel=_levels[_g.bNode(_matching[_g.runningNode(tbedge)])])<

     void start() {

 	       nlevel)

 	      {

 		nlevel=nnlevel;

 		bedge=tbedge;

 	      }

 	}

 	if(nlevel<_node_num) {

 	  if(nlevel>=actlevel)

 	    _levels.liftHighestActiveTo(nlevel+1);

 // 	    _levels.liftTo(act,nlevel+1);

 	  bact=_g.bNode(_matching[_g.aNode(bedge)]);

 	  if(--_cov[bact]<1) {

 	    _levels.activate(bact);

 	    last_activated=bact;

 	  }

 	  _matching[_g.aNode(bedge)]=bedge;

 	  _cov[act]=1;

 	  _levels.deactivate(act);

 	}

 	else {

 	  if(_node_num>actlevel) 

 	    _levels.liftHighestActiveTo(_node_num);

 //  	    _levels.liftTo(act,_node_num);

 	  _levels.deactivate(act); 

 	}

 	if(_levels.onLevel(actlevel)==0)

 	  _levels.liftToTop(actlevel);

       }

       int ret=_node_num;

       for(ANodeIt n(_g);n!=INVALID;++n)

 	if(_matching[n]==INVALID) ret--;

 	else if (_cov[_g.bNode(_matching[n])]>1) {

 	  _cov[_g.bNode(_matching[n])]--;

 	  ret--;

 	  _matching[n]=INVALID;

 	}

       return ret;

     }

     ///\returns -1 if there is a perfect matching, or an empty level

     ///if it doesn't exists

     int runPerfect() 

     {

       init();

 	  return actlevel;

 	  _levels.liftToTop(actlevel);

       }

       }

       return -1;

     }

       _matching_size = _node_num;

       for(ANodeIt n(_g);n!=INVALID;++n)

     template<class GT>

 	if(_matching[n]==INVALID) _matching_size--;

     void aBarrier(GT &bar,int empty_level=-1) 

 	else if (_cov[_g.bNode(_matching[n])]>1) {

 	  _cov[_g.bNode(_matching[n])]--;

 	  _matching_size--;

 	  _matching[n]=INVALID;

 	}

     }

     ///Start the algorithm to find a perfect matching

     ///This function is close to identical to the simple start()

     ///member function but it calculates just perfect matching.

     ///However, the perfect property is only checked on the B-side of

     ///the graph

     ///

     ///The main difference between the two function is the handling of

     ///the empty levels. The simple start() function let the nodes

     ///above the empty levels unmatched while this variant if it find

     ///an empty level immediately terminates and gives back false

     ///return value.

     bool startPerfect() {

       Node act;

       Node bact=INVALID;

       Node last_activated=INVALID;

       while((act=_levels.highestActive())!=INVALID) {

 	last_activated=INVALID;

 	int actlevel=_levels[act];

 	UEdge bedge=INVALID;

 	int nlevel=_node_num;

 	{

 	  int nnlevel;

 	  for(IncEdgeIt tbedge(_g,act);

 	      tbedge!=INVALID && nlevel>=actlevel;

 	      ++tbedge)

 	    if((nnlevel=_levels[_g.bNode(_matching[_g.runningNode(tbedge)])])<

 	       nlevel)

 	      {

 		nlevel=nnlevel;

 		bedge=tbedge;

 	      }

 	}

 	if(nlevel<_node_num) {

 	  if(nlevel>=actlevel)

 	    _levels.liftHighestActiveTo(nlevel+1);

 	  bact=_g.bNode(_matching[_g.aNode(bedge)]);

 	  if(--_cov[bact]<1) {

 	    _levels.activate(bact);

 	    last_activated=bact;

 	  }

 	  _matching[_g.aNode(bedge)]=bedge;

 	  _cov[act]=1;

 	  _levels.deactivate(act);

 	}

 	else {

 	  if(_node_num>actlevel) 

 	    _levels.liftHighestActiveTo(_node_num);

 	  _levels.deactivate(act); 

 	}

 	if(_levels.onLevel(actlevel)==0)

 	  _empty_level=actlevel;

 	  return false;

       }

       return true;

     }

     ///Runs the algorithm

     ///Just a shortcut for the next code:

     ///\code

     /// init();

     /// start();

     ///\endcode

     void run() {

       init();

       start();

     }

     ///Runs the algorithm to find a perfect matching

     ///Just a shortcut for the next code:

     ///\code

     /// init();

     /// startPerfect();

     ///\endcode

     ///

     ///\note If the two nodesets of the graph have different size then

     ///this algorithm checks the perfect property on the B-side.

     bool runPerfect() {

       init();

       return startPerfect();

     }

     ///Runs the algorithm to find a perfect matching

     ///Just a shortcut for the next code:

     ///\code

     /// init();

     /// startPerfect();

     ///\endcode

     ///

     ///\note It checks that the size of the two nodesets are equal.

     bool checkedRunPerfect() {

       if (countANodes(_g) != _node_num) return false;

       init();

       return startPerfect();

     }

     ///@}

     /// \name Query Functions

     /// The result of the %Matching algorithm can be obtained using these

     /// functions.\n

     /// Before the use of these functions,

     /// either run() or start() must be called.

     ///@{

     /// \brief Set true all matching uedge in the map.

     /// 

     /// Set true all matching uedge in the map. It does not change the

     /// value mapped to the other uedges.

     /// \return The number of the matching edges.

     template <typename MatchingMap>

     int quickMatching(MatchingMap& mm) const {

       for (ANodeIt n(_g);n!=INVALID;++n) {

         if (_matching[n]!=INVALID) mm.set(_matching[n],true);

       }

       return _matching_size;

     }

     ///\brief Set true all matching uedge in the map and the others to false.

     ///Set true all matching uedge in the map and the others to false.

     ///\return The number of the matching edges.

     template<class MatchingMap>

     int matching(MatchingMap& mm) const {

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

         mm.set(e,e==_matching[_g.aNode(e)]);

       }

       return _matching_size;

     }

     ///Returns true if the given uedge is in the matching.

     ///It returns true if the given uedge is in the matching.

     ///

     bool matchingEdge(const UEdge& e) const {

       return _matching[_g.aNode(e)]==e;

     }

     ///Returns the matching edge from the node.

     ///Returns the matching edge from the node. If there is not such

     ///edge it gives back \c INVALID.  

     ///\note If the parameter node is a B-node then the running time is

     ///propotional to the degree of the node.

     UEdge matchingEdge(const Node& n) const {

       if (_g.aNode(n)) {

         return _matching[n];

       } else {

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

 	  if (e==_matching[_g.aNode(e)]) return e;

 	return INVALID;

       }

     }

     ///Gives back the number of the matching edges.

     ///Gives back the number of the matching edges.

     int matchingSize() const {

       return _matching_size;

     }

     ///Gives back a barrier on the A-nodes

     ///The barrier is s subset of the nodes on the same side of the

     ///graph. If we tried to find a perfect matching and it failed

     ///then the barrier size will be greater than its neighbours. If

     ///the maximum matching searched then the barrier size minus its

     ///neighbours will be exactly the unmatched nodes on the A-side.

     ///\retval bar A WriteMap on the ANodes with bool value.

     template<class BarrierMap>

     void aBarrier(BarrierMap &bar) const 

 	bar[n] = _matching[n]==INVALID ||

 	bar.set(n,_matching[n]==INVALID ||

 	  _levels[_g.bNode(_matching[n])]<empty_level;  

 	  _levels[_g.bNode(_matching[n])]<_empty_level);  

     template<class GT>

     void bBarrier(GT &bar, int empty_level=-1) 

     ///Gives back a barrier on the B-nodes

     ///The barrier is s subset of the nodes on the same side of the

     ///graph. If we tried to find a perfect matching and it failed

     ///then the barrier size will be greater than its neighbours. If

     ///the maximum matching searched then the barrier size minus its

     ///neighbours will be exactly the unmatched nodes on the B-side.

     ///\retval bar A WriteMap on the BNodes with bool value.

     template<class BarrierMap>

     void bBarrier(BarrierMap &bar) const

       if(empty_level==-1)

       for(BNodeIt n(_g);n!=INVALID;++n) bar.set(n,_levels[n]>=_empty_level);  

 	for(empty_level=0;_levels.onLevel(empty_level);empty_level++) ;

     }

       for(BNodeIt n(_g);n!=INVALID;++n) bar[n]=(_levels[n]>empty_level);  

     }  

     ///Returns a minimum covering of the nodes.

     ///The minimum covering set problem is the dual solution of the

     ///maximum bipartite matching. It provides a solution for this

     ///problem what is proof of the optimality of the matching.

     ///\param covering NodeMap of bool values, the nodes of the cover

     ///set will set true while the others false.  

     ///\return The size of the cover set.

     ///\note This function can be called just after the algorithm have

     ///already found a matching. 

     template<class CoverMap>

     int coverSet(CoverMap& covering) const {

       int ret=0;

       for(BNodeIt n(_g);n!=INVALID;++n) {

 	if (_levels[n]<_empty_level) { covering.set(n,true); ++ret; }

 	else covering.set(n,false);

       }

       for(ANodeIt n(_g);n!=INVALID;++n) {

 	if (_matching[n]!=INVALID &&

 	    _levels[_g.bNode(_matching[n])]>=_empty_level) 

 	  { covering.set(n,true); ++ret; }

 	else covering.set(n,false);

       }

       return ret;

     }

     /// @}

   int maxBpMatching(const Graph &g)

   int prBipartiteMatching(const Graph &g)

     typename Graph::template ANodeMap<typename Graph::UEdge> matching(g);

     PrBipartiteMatching<Graph> bpm(g);

     return maxBpMatching(g,matching);

     return bpm.matchingSize();

   ///\retval matching A readwrite ANodeMap of value type \c Edge.

   ///\retval matching A write UEdgeMap of value type \c bool.

   /// The found edges will be returned in this map,

   /// The found edges will be returned in this map.

   /// i.e. for an \c ANode \c n,

   /// the edge <tt>matching[n]</tt> is the one that covers the node \c n, or

   /// \ref INVALID if it is uncovered.

   int maxBpMatching(const Graph &g,MT &matching) 

   int prBipartiteMatching(const Graph &g,MT &matching) 

     return BpMatching<Graph,MT>(g,matching).run();

     PrBipartiteMatching<Graph> bpm(g);

     bpm.run();

     bpm.matching(matching);

     return bpm.matchingSize();

   ///\retval matching A readwrite ANodeMap of value type \c Edge.

   ///\retval matching A write UEdgeMap of value type \c bool.

   /// The found edges will be returned in this map,

   /// The found edges will be returned in this map.

   /// i.e. for an \c ANode \c n,

   /// the edge <tt>matching[n]</tt> is the one that covers the node \c n, or

   /// \ref INVALID if it is uncovered.

   int maxBpMatching(const Graph &g,MT &matching,GT &barrier) 

   int prBipartiteMatching(const Graph &g,MT &matching,GT &barrier) 

     BpMatching<Graph,MT> bpm(g,matching);

     PrBipartiteMatching<Graph> bpm(g);

     int ret=bpm.run();

     bpm.run();

     bpm.barrier(barrier);

     bpm.matching(matching);

     return ret;

     bpm.bBarrier(barrier);

     return bpm.matchingSize();

   bool perfectBpMatching(const Graph &g)

   bool prPerfectBipartiteMatching(const Graph &g)

     typename Graph::template ANodeMap<typename Graph::UEdge> matching(g);

     PrBipartiteMatching<Graph> bpm(g);

     return perfectBpMatching(g,matching);

     return bpm.runPerfect();

   ///\retval matching A readwrite ANodeMap of value type \c Edge.

   ///\retval matching A write UEdgeMap of value type \c bool.

   /// The found edges will be returned in this map,

   /// The found edges will be returned in this map.

   /// i.e. for an \c ANode \c n,

   /// The values are unchanged if the graph

   /// the edge <tt>matching[n]</tt> is the one that covers the node \c n.

   /// The values are unspecified if the graph

   bool perfectBpMatching(const Graph &g,MT &matching) 

   bool prPerfectBipartiteMatching(const Graph &g,MT &matching) 

     return BpMatching<Graph,MT>(g,matching).runPerfect()<0;

     PrBipartiteMatching<Graph> bpm(g);

     bool ret = bpm.runPerfect();

     if (ret) bpm.matching(matching);

     return ret;

   ///\retval matching A readwrite ANodeMap of value type \c Edge.

   ///\retval matching A readwrite UEdgeMap of value type \c bool.

   /// The found edges will be returned in this map,

   /// The found edges will be returned in this map.

   /// i.e. for an \c ANode \c n,

   /// The values are unchanged if the graph

   /// the edge <tt>matching[n]</tt> is the one that covers the node \c n.

   /// The values are unspecified if the graph

   /// the cardinality of \e B is greater than the numner of its neighbors.

   /// the cardinality of \e B is greater than the number of its neighbors.

   int perfectBpMatching(const Graph &g,MT &matching,GT &barrier) 

   int prPerfectBipartiteMatching(const Graph &g,MT &matching,GT &barrier) 

     BpMatching<Graph,MT> bpm(g,matching);

     PrBipartiteMatching<Graph> bpm(g);

     int ret=bpm.run();

     bool ret=bpm.runPerfect();

     if(ret>=0)

     if(ret)

       bpm.barrier(barrier,ret);

       bpm.matching(matching);

     return ret<0;

     else

       bpm.bBarrier(barrier);

     return ret;