/* -*- C++ -*-

  * lemon/preflow.h - Part of LEMON, a generic C++ optimization library

  *

  * Copyright (C) 2005 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_PREFLOW_H

 #define LEMON_PREFLOW_H

 #include <vector>

 #include <queue>

 #include <lemon/error.h>

 #include <lemon/invalid.h>

 #include <lemon/maps.h>

 #include <lemon/graph_utils.h>

 /// \file

 /// \ingroup flowalgs

 /// \brief Implementation of the preflow algorithm.

 namespace lemon {

   /// \addtogroup flowalgs

   /// @{                                                   

   ///%Preflow algorithms class.

   ///This class provides an implementation of the \e preflow \e

   ///algorithm producing a flow of maximum value in a directed

   ///graph. The preflow algorithms are the fastest known max flow algorithms

   ///up to now. The \e source node, the \e target node, the \e

   ///capacity of the edges and the \e starting \e flow value of the

   ///edges should be passed to the algorithm through the

   ///constructor. It is possible to change these quantities using the

   ///functions \ref source, \ref target, \ref capacityMap and \ref

   ///flowMap.

   ///

   ///After running \ref lemon::Preflow::phase1() "phase1()"

   ///or \ref lemon::Preflow::run() "run()", the maximal flow

   ///value can be obtained by calling \ref flowValue(). The minimum

   ///value cut can be written into a <tt>bool</tt> node map by

   ///calling \ref minCut(). (\ref minMinCut() and \ref maxMinCut() writes

   ///the inclusionwise minimum and maximum of the minimum value cuts,

   ///resp.)

   ///

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

   ///\param Num The number type of the capacities and the flow values.

   ///\param CapacityMap The capacity map type.

   ///\param FlowMap The flow map type.

   ///

   ///\author Jacint Szabo 

   ///\todo Second template parameter is superfluous

   template <typename Graph, typename Num,

 	    typename CapacityMap=typename Graph::template EdgeMap<Num>,

             typename FlowMap=typename Graph::template EdgeMap<Num> >

   class Preflow {

   protected:

     typedef typename Graph::Node Node;

     typedef typename Graph::NodeIt NodeIt;

     typedef typename Graph::EdgeIt EdgeIt;

     typedef typename Graph::OutEdgeIt OutEdgeIt;

     typedef typename Graph::InEdgeIt InEdgeIt;

     typedef typename Graph::template NodeMap<Node> NNMap;

     typedef typename std::vector<Node> VecNode;

     const Graph* _g;

     Node _source;

     Node _target;

     const CapacityMap* _capacity;

     FlowMap* _flow;

     int _node_num;      //the number of nodes of G

     typename Graph::template NodeMap<int> level;  

     typename Graph::template NodeMap<Num> excess;

     // constants used for heuristics

     static const int H0=20;

     static const int H1=1;

   public:

     ///\ref Exception for the case when s=t.

     ///\ref Exception for the case when the source equals the target.

     class InvalidArgument : public lemon::LogicError {

     public:

       virtual const char* exceptionName() const {

 	return "lemon::Preflow::InvalidArgument";

       }

     };

     ///Indicates the property of the starting flow map.

     ///Indicates the property of the starting flow map.

     ///The meanings are as follows:

     ///- \c ZERO_FLOW: constant zero flow

     ///- \c GEN_FLOW: any flow, i.e. the sum of the in-flows equals to

     ///the sum of the out-flows in every node except the \e source and

     ///the \e target.

     ///- \c PRE_FLOW: any preflow, i.e. the sum of the in-flows is at 

     ///least the sum of the out-flows in every node except the \e source.

     ///- \c NO_FLOW: indicates an unspecified edge map. \c flow will be 

     ///set to the constant zero flow in the beginning of

     ///the algorithm in this case.

     ///

     enum FlowEnum{

       NO_FLOW,

       ZERO_FLOW,

       GEN_FLOW,

       PRE_FLOW

     };

     ///Indicates the state of the preflow algorithm.

     ///Indicates the state of the preflow algorithm.

     ///The meanings are as follows:

     ///- \c AFTER_NOTHING: before running the algorithm or

     ///  at an unspecified state.

     ///- \c AFTER_PREFLOW_PHASE_1: right after running \c phase1

     ///- \c AFTER_PREFLOW_PHASE_2: after running \ref phase2()

     ///

     enum StatusEnum {

       AFTER_NOTHING,

       AFTER_PREFLOW_PHASE_1,      

       AFTER_PREFLOW_PHASE_2

     };

   protected: 

     FlowEnum flow_prop;

     StatusEnum status; // Do not needle this flag only if necessary.

   public: 

     ///The constructor of the class.

     ///The constructor of the class. 

     ///\param _gr The directed graph the algorithm runs on. 

     ///\param _s The source node.

     ///\param _t The target node.

     ///\param _cap The capacity of the edges. 

     ///\param _f The flow of the edges. 

     ///Except the graph, all of these parameters can be reset by

     ///calling \ref source, \ref target, \ref capacityMap and \ref

     ///flowMap, resp.

       Preflow(const Graph& _gr, Node _s, Node _t, 

 	      const CapacityMap& _cap, FlowMap& _f) :

 	_g(&_gr), _source(_s), _target(_t), _capacity(&_cap),

 	_flow(&_f), _node_num(countNodes(_gr)), level(_gr), excess(_gr,0), 

 	flow_prop(NO_FLOW), status(AFTER_NOTHING) { 

 	if ( _source==_target )

 	  throw InvalidArgument();

       }

     ///Runs the preflow algorithm.  

     ///Runs the preflow algorithm.

     ///

     void run() {

       phase1(flow_prop);

       phase2();

     }

     ///Runs the preflow algorithm.  

     ///Runs the preflow algorithm. 

     ///\pre The starting flow map must be

     /// - a constant zero flow if \c fp is \c ZERO_FLOW,

     /// - an arbitrary flow if \c fp is \c GEN_FLOW,

     /// - an arbitrary preflow if \c fp is \c PRE_FLOW,

     /// - any map if \c fp is NO_FLOW.

     ///If the starting flow map is a flow or a preflow then 

     ///the algorithm terminates faster.

     void run(FlowEnum fp) {

       flow_prop=fp;

       run();

     }

     ///Runs the first phase of the preflow algorithm.

     ///The preflow algorithm consists of two phases, this method runs

     ///the first phase. After the first phase the maximum flow value

     ///and a minimum value cut can already be computed, although a

     ///maximum flow is not yet obtained. So after calling this method

     ///\ref flowValue returns the value of a maximum flow and \ref

     ///minCut returns a minimum cut.     

     ///\warning \ref minMinCut and \ref maxMinCut do not give minimum

     ///value cuts unless calling \ref phase2.  

     ///\pre The starting flow must be 

     ///- a constant zero flow if \c fp is \c ZERO_FLOW, 

     ///- an arbitary flow if \c fp is \c GEN_FLOW, 

     ///- an arbitary preflow if \c fp is \c PRE_FLOW, 

     ///- any map if \c fp is NO_FLOW.

     void phase1(FlowEnum fp)

     {

       flow_prop=fp;

       phase1();

     }

     ///Runs the first phase of the preflow algorithm.

     ///The preflow algorithm consists of two phases, this method runs

     ///the first phase. After the first phase the maximum flow value

     ///and a minimum value cut can already be computed, although a

     ///maximum flow is not yet obtained. So after calling this method

     ///\ref flowValue returns the value of a maximum flow and \ref

     ///minCut returns a minimum cut.

     ///\warning \ref minCut(), \ref minMinCut() and \ref maxMinCut() do not

     ///give minimum value cuts unless calling \ref phase2().

     void phase1()

     {

       int heur0=(int)(H0*_node_num);  //time while running 'bound decrease'

       int heur1=(int)(H1*_node_num);  //time while running 'highest label'

       int heur=heur1;         //starting time interval (#of relabels)

       int numrelabel=0;

       bool what_heur=1;

       //It is 0 in case 'bound decrease' and 1 in case 'highest label'

       bool end=false;

       //Needed for 'bound decrease', true means no active 

       //nodes are above bound b.

       int k=_node_num-2;  //bound on the highest level under n containing a node

       int b=k;    //bound on the highest level under n of an active node

       VecNode first(_node_num, INVALID);

       NNMap next(*_g, INVALID);

       NNMap left(*_g, INVALID);

       NNMap right(*_g, INVALID);

       VecNode level_list(_node_num,INVALID);

       //List of the nodes in level i<n, set to n.

       preflowPreproc(first, next, level_list, left, right);

       //Push/relabel on the highest level active nodes.

       while ( true ) {

 	if ( b == 0 ) {

 	  if ( !what_heur && !end && k > 0 ) {

 	    b=k;

 	    end=true;

 	  } else break;

 	}

 	if ( first[b]==INVALID ) --b;

 	else {

 	  end=false;

 	  Node w=first[b];

 	  first[b]=next[w];

 	  int newlevel=push(w, next, first);

 	  if ( excess[w] > 0 ) relabel(w, newlevel, first, next, level_list, 

 				       left, right, b, k, what_heur);

 	  ++numrelabel;

 	  if ( numrelabel >= heur ) {

 	    numrelabel=0;

 	    if ( what_heur ) {

 	      what_heur=0;

 	      heur=heur0;

 	      end=false;

 	    } else {

 	      what_heur=1;

 	      heur=heur1;

 	      b=k;

 	    }

 	  }

 	}

       }

       flow_prop=PRE_FLOW;

       status=AFTER_PREFLOW_PHASE_1;

     }

     // Heuristics:

     //   2 phase

     //   gap

     //   list 'level_list' on the nodes on level i implemented by hand

     //   stack 'active' on the active nodes on level i      

     //   runs heuristic 'highest label' for H1*n relabels

     //   runs heuristic 'bound decrease' for H0*n relabels,

     //        starts with 'highest label'

     //   Parameters H0 and H1 are initialized to 20 and 1.

     ///Runs the second phase of the preflow algorithm.

     ///The preflow algorithm consists of two phases, this method runs

     ///the second phase. After calling \ref phase1() and then

     ///\ref phase2(),

     /// \ref flowMap() return a maximum flow, \ref flowValue

     ///returns the value of a maximum flow, \ref minCut returns a

     ///minimum cut, while the methods \ref minMinCut and \ref

     ///maxMinCut return the inclusionwise minimum and maximum cuts of

     ///minimum value, resp.  \pre \ref phase1 must be called before.

     void phase2()

     {

       int k=_node_num-2;  //bound on the highest level under n containing a node

       int b=k;    //bound on the highest level under n of an active node

       VecNode first(_node_num, INVALID);

       NNMap next(*_g, INVALID); 

       level.set(_source,0);

       std::queue<Node> bfs_queue;

       bfs_queue.push(_source);

       while ( !bfs_queue.empty() ) {

 	Node v=bfs_queue.front();

 	bfs_queue.pop();

 	int l=level[v]+1;

 	for(InEdgeIt e(*_g,v); e!=INVALID; ++e) {

 	  if ( (*_capacity)[e] <= (*_flow)[e] ) continue;

 	  Node u=_g->source(e);

 	  if ( level[u] >= _node_num ) {

 	    bfs_queue.push(u);

 	    level.set(u, l);

 	    if ( excess[u] > 0 ) {

 	      next.set(u,first[l]);

 	      first[l]=u;

 	    }

 	  }

 	}

 	for(OutEdgeIt e(*_g,v); e!=INVALID; ++e) {

 	  if ( 0 >= (*_flow)[e] ) continue;

 	  Node u=_g->target(e);

 	  if ( level[u] >= _node_num ) {

 	    bfs_queue.push(u);

 	    level.set(u, l);

 	    if ( excess[u] > 0 ) {

 	      next.set(u,first[l]);

 	      first[l]=u;

 	    }

 	  }

 	}

       }

       b=_node_num-2;

       while ( true ) {

 	if ( b == 0 ) break;

 	if ( first[b]==INVALID ) --b;

 	else {

 	  Node w=first[b];

 	  first[b]=next[w];

 	  int newlevel=push(w,next, first);

 	  //relabel

 	  if ( excess[w] > 0 ) {

 	    level.set(w,++newlevel);

 	    next.set(w,first[newlevel]);

 	    first[newlevel]=w;

 	    b=newlevel;

 	  }

 	} 

       } // while(true)

       flow_prop=GEN_FLOW;

       status=AFTER_PREFLOW_PHASE_2;

     }

     /// Returns the value of the maximum flow.

     /// Returns the value of the maximum flow by returning the excess

     /// of the target node \c t. This value equals to the value of

     /// the maximum flow already after running \ref phase1.

     Num flowValue() const {

       return excess[_target];

     }

     ///Returns a minimum value cut.

     ///Sets \c M to the characteristic vector of a minimum value

     ///cut. This method can be called both after running \ref

     ///phase1 and \ref phase2. It is much faster after

     ///\ref phase1.  \pre M should be a bool-valued node-map. \pre

     ///If \ref minCut() is called after \ref phase2() then M should

     ///be initialized to false.

     template<typename _CutMap>

     void minCut(_CutMap& M) const {

       switch ( status ) {

 	case AFTER_PREFLOW_PHASE_1:

 	for(NodeIt v(*_g); v!=INVALID; ++v) {

 	  if (level[v] < _node_num) {

 	    M.set(v, false);

 	  } else {

 	    M.set(v, true);

 	  }

 	}

 	break;

 	case AFTER_PREFLOW_PHASE_2:

 	minMinCut(M);

 	break;

 	case AFTER_NOTHING:

 	break;

       }

     }

     ///Returns the inclusionwise minimum of the minimum value cuts.

     ///Sets \c M to the characteristic vector of the minimum value cut

     ///which is inclusionwise minimum. It is computed by processing a

     ///bfs from the source node \c s in the residual graph.  \pre M

     ///should be a node map of bools initialized to false.  \pre \ref

     ///phase2 should already be run.

     template<typename _CutMap>

     void minMinCut(_CutMap& M) const {

       std::queue<Node> queue;

       M.set(_source,true);

       queue.push(_source);

       while (!queue.empty()) {

 	Node w=queue.front();

 	queue.pop();

 	for(OutEdgeIt e(*_g,w) ; e!=INVALID; ++e) {

 	  Node v=_g->target(e);

 	  if (!M[v] && (*_flow)[e] < (*_capacity)[e] ) {

 	    queue.push(v);

 	    M.set(v, true);

 	  }

 	}

 	for(InEdgeIt e(*_g,w) ; e!=INVALID; ++e) {

 	  Node v=_g->source(e);

 	  if (!M[v] && (*_flow)[e] > 0 ) {

 	    queue.push(v);

 	    M.set(v, true);

 	  }

 	}

       }

     }

     ///Returns the inclusionwise maximum of the minimum value cuts.

     ///Sets \c M to the characteristic vector of the minimum value cut

     ///which is inclusionwise maximum. It is computed by processing a

     ///backward bfs from the target node \c t in the residual graph.

     ///\pre \ref phase2() or run() should already be run.

     template<typename _CutMap>

     void maxMinCut(_CutMap& M) const {

       for(NodeIt v(*_g) ; v!=INVALID; ++v) M.set(v, true);

       std::queue<Node> queue;

       M.set(_target,false);

       queue.push(_target);

       while (!queue.empty()) {

         Node w=queue.front();

 	queue.pop();

 	for(InEdgeIt e(*_g,w) ; e!=INVALID; ++e) {

 	  Node v=_g->source(e);

 	  if (M[v] && (*_flow)[e] < (*_capacity)[e] ) {

 	    queue.push(v);

 	    M.set(v, false);

 	  }

 	}

 	for(OutEdgeIt e(*_g,w) ; e!=INVALID; ++e) {

 	  Node v=_g->target(e);

 	  if (M[v] && (*_flow)[e] > 0 ) {

 	    queue.push(v);

 	    M.set(v, false);

 	  }

 	}

       }

     }

     ///Sets the source node to \c _s.

     ///Sets the source node to \c _s.

     /// 

     void source(Node _s) { 

       _source=_s; 

       if ( flow_prop != ZERO_FLOW ) flow_prop=NO_FLOW;

       status=AFTER_NOTHING; 

     }

     ///Returns the source node.

     ///Returns the source node.

     /// 

     Node source() const { 

       return _source;

     }

     ///Sets the target node to \c _t.

     ///Sets the target node to \c _t.

     ///

     void target(Node _t) { 

       _target=_t; 

       if ( flow_prop == GEN_FLOW ) flow_prop=PRE_FLOW;

       status=AFTER_NOTHING; 

     }

     ///Returns the target node.

     ///Returns the target node.

     /// 

     Node target() const { 

       return _target;

     }

     /// Sets the edge map of the capacities to _cap.

     /// Sets the edge map of the capacities to _cap.

     /// 

     void capacityMap(const CapacityMap& _cap) { 

       _capacity=&_cap; 

       status=AFTER_NOTHING; 

     }

     /// Returns a reference to capacity map.

     /// Returns a reference to capacity map.

     /// 

     const CapacityMap &capacityMap() const { 

       return *_capacity;

     }

     /// Sets the edge map of the flows to _flow.

     /// Sets the edge map of the flows to _flow.

     /// 

     void flowMap(FlowMap& _f) { 

       _flow=&_f; 

       flow_prop=NO_FLOW;

       status=AFTER_NOTHING; 

     }

     /// Returns a reference to flow map.

     /// Returns a reference to flow map.

     /// 

     const FlowMap &flowMap() const { 

       return *_flow;

     }

   private:

     int push(Node w, NNMap& next, VecNode& first) {

       int lev=level[w];

       Num exc=excess[w];

       int newlevel=_node_num;       //bound on the next level of w

       for(OutEdgeIt e(*_g,w) ; e!=INVALID; ++e) {

 	if ( (*_flow)[e] >= (*_capacity)[e] ) continue;

 	Node v=_g->target(e);

 	if( lev > level[v] ) { //Push is allowed now

 	  if ( excess[v]<=0 && v!=_target && v!=_source ) {

 	    next.set(v,first[level[v]]);

 	    first[level[v]]=v;

 	  }

 	  Num cap=(*_capacity)[e];

 	  Num flo=(*_flow)[e];

 	  Num remcap=cap-flo;

 	  if ( remcap >= exc ) { //A nonsaturating push.

 	    _flow->set(e, flo+exc);

 	    excess.set(v, excess[v]+exc);

 	    exc=0;

 	    break;

 	  } else { //A saturating push.

 	    _flow->set(e, cap);

 	    excess.set(v, excess[v]+remcap);

 	    exc-=remcap;

 	  }

 	} else if ( newlevel > level[v] ) newlevel = level[v];

       } //for out edges wv

       if ( exc > 0 ) {

 	for(InEdgeIt e(*_g,w) ; e!=INVALID; ++e) {

 	  if( (*_flow)[e] <= 0 ) continue;

 	  Node v=_g->source(e);

 	  if( lev > level[v] ) { //Push is allowed now

 	    if ( excess[v]<=0 && v!=_target && v!=_source ) {

 	      next.set(v,first[level[v]]);

 	      first[level[v]]=v;

 	    }

 	    Num flo=(*_flow)[e];

 	    if ( flo >= exc ) { //A nonsaturating push.

 	      _flow->set(e, flo-exc);

 	      excess.set(v, excess[v]+exc);

 	      exc=0;

 	      break;

 	    } else {  //A saturating push.

 	      excess.set(v, excess[v]+flo);

 	      exc-=flo;

 	      _flow->set(e,0);

 	    }

 	  } else if ( newlevel > level[v] ) newlevel = level[v];

 	} //for in edges vw

       } // if w still has excess after the out edge for cycle

       excess.set(w, exc);

       return newlevel;

     }

     void preflowPreproc(VecNode& first, NNMap& next, 

 			VecNode& level_list, NNMap& left, NNMap& right)

     {

       for(NodeIt v(*_g); v!=INVALID; ++v) level.set(v,_node_num);

       std::queue<Node> bfs_queue;

       if ( flow_prop == GEN_FLOW || flow_prop == PRE_FLOW ) {

 	//Reverse_bfs from t in the residual graph,

 	//to find the starting level.

 	level.set(_target,0);

 	bfs_queue.push(_target);

 	while ( !bfs_queue.empty() ) {

 	  Node v=bfs_queue.front();

 	  bfs_queue.pop();

 	  int l=level[v]+1;

 	  for(InEdgeIt e(*_g,v) ; e!=INVALID; ++e) {

 	    if ( (*_capacity)[e] <= (*_flow)[e] ) continue;

 	    Node w=_g->source(e);

 	    if ( level[w] == _node_num && w != _source ) {

 	      bfs_queue.push(w);

 	      Node z=level_list[l];

 	      if ( z!=INVALID ) left.set(z,w);

 	      right.set(w,z);

 	      level_list[l]=w;

 	      level.set(w, l);

 	    }

 	  }

 	  for(OutEdgeIt e(*_g,v) ; e!=INVALID; ++e) {

 	    if ( 0 >= (*_flow)[e] ) continue;

 	    Node w=_g->target(e);

 	    if ( level[w] == _node_num && w != _source ) {

 	      bfs_queue.push(w);

 	      Node z=level_list[l];

 	      if ( z!=INVALID ) left.set(z,w);

 	      right.set(w,z);

 	      level_list[l]=w;

 	      level.set(w, l);

 	    }

 	  }

 	} //while

       } //if

       switch (flow_prop) {

 	case NO_FLOW:  

 	for(EdgeIt e(*_g); e!=INVALID; ++e) _flow->set(e,0);

 	case ZERO_FLOW:

 	for(NodeIt v(*_g); v!=INVALID; ++v) excess.set(v,0);

 	//Reverse_bfs from t, to find the starting level.

 	level.set(_target,0);

 	bfs_queue.push(_target);

 	while ( !bfs_queue.empty() ) {

 	  Node v=bfs_queue.front();

 	  bfs_queue.pop();

 	  int l=level[v]+1;

 	  for(InEdgeIt e(*_g,v) ; e!=INVALID; ++e) {

 	    Node w=_g->source(e);

 	    if ( level[w] == _node_num && w != _source ) {

 	      bfs_queue.push(w);

 	      Node z=level_list[l];

 	      if ( z!=INVALID ) left.set(z,w);

 	      right.set(w,z);

 	      level_list[l]=w;

 	      level.set(w, l);

 	    }

 	  }

 	}

 	//the starting flow

 	for(OutEdgeIt e(*_g,_source) ; e!=INVALID; ++e) {

 	  Num c=(*_capacity)[e];

 	  if ( c <= 0 ) continue;

 	  Node w=_g->target(e);

 	  if ( level[w] < _node_num ) {

 	    if ( excess[w] <= 0 && w!=_target ) { //putting into the stack

 	      next.set(w,first[level[w]]);

 	      first[level[w]]=w;

 	    }

 	    _flow->set(e, c);

 	    excess.set(w, excess[w]+c);

 	  }

 	}

 	break;

 	case GEN_FLOW:

 	for(NodeIt v(*_g); v!=INVALID; ++v) excess.set(v,0);

 	{

 	  Num exc=0;

 	  for(InEdgeIt e(*_g,_target) ; e!=INVALID; ++e) exc+=(*_flow)[e];

 	  for(OutEdgeIt e(*_g,_target) ; e!=INVALID; ++e) exc-=(*_flow)[e];

 	  excess.set(_target,exc);

 	}

 	//the starting flow

 	for(OutEdgeIt e(*_g,_source); e!=INVALID; ++e)	{

 	  Num rem=(*_capacity)[e]-(*_flow)[e];

 	  if ( rem <= 0 ) continue;

 	  Node w=_g->target(e);

 	  if ( level[w] < _node_num ) {

 	    if ( excess[w] <= 0 && w!=_target ) { //putting into the stack

 	      next.set(w,first[level[w]]);

 	      first[level[w]]=w;

 	    }   

 	    _flow->set(e, (*_capacity)[e]);

 	    excess.set(w, excess[w]+rem);

 	  }

 	}

 	for(InEdgeIt e(*_g,_source); e!=INVALID; ++e) {

 	  if ( (*_flow)[e] <= 0 ) continue;

 	  Node w=_g->source(e);

 	  if ( level[w] < _node_num ) {

 	    if ( excess[w] <= 0 && w!=_target ) {

 	      next.set(w,first[level[w]]);

 	      first[level[w]]=w;

 	    }  

 	    excess.set(w, excess[w]+(*_flow)[e]);

 	    _flow->set(e, 0);

 	  }

 	}

 	break;

 	case PRE_FLOW:	

 	//the starting flow

 	for(OutEdgeIt e(*_g,_source) ; e!=INVALID; ++e) {

 	  Num rem=(*_capacity)[e]-(*_flow)[e];

 	  if ( rem <= 0 ) continue;

 	  Node w=_g->target(e);

 	  if ( level[w] < _node_num ) _flow->set(e, (*_capacity)[e]);

 	}

 	for(InEdgeIt e(*_g,_source) ; e!=INVALID; ++e) {

 	  if ( (*_flow)[e] <= 0 ) continue;

 	  Node w=_g->source(e);

 	  if ( level[w] < _node_num ) _flow->set(e, 0);

 	}

 	//computing the excess

 	for(NodeIt w(*_g); w!=INVALID; ++w) {

 	  Num exc=0;

 	  for(InEdgeIt e(*_g,w); e!=INVALID; ++e) exc+=(*_flow)[e];

 	  for(OutEdgeIt e(*_g,w); e!=INVALID; ++e) exc-=(*_flow)[e];

 	  excess.set(w,exc);

 	  //putting the active nodes into the stack

 	  int lev=level[w];

 	    if ( exc > 0 && lev < _node_num && Node(w) != _target ) {

 	      next.set(w,first[lev]);

 	      first[lev]=w;

 	    }

 	}

 	break;

       } //switch

     } //preflowPreproc

     void relabel(Node w, int newlevel, VecNode& first, NNMap& next, 

 		 VecNode& level_list, NNMap& left,

 		 NNMap& right, int& b, int& k, bool what_heur )

     {

       int lev=level[w];

       Node right_n=right[w];

       Node left_n=left[w];

       //unlacing starts

       if ( right_n!=INVALID ) {

 	if ( left_n!=INVALID ) {

 	  right.set(left_n, right_n);

 	  left.set(right_n, left_n);

 	} else {

 	  level_list[lev]=right_n;

 	  left.set(right_n, INVALID);

 	}

       } else {

 	if ( left_n!=INVALID ) {

 	  right.set(left_n, INVALID);

 	} else {

 	  level_list[lev]=INVALID;

 	}

       }

       //unlacing ends

       if ( level_list[lev]==INVALID ) {

 	//gapping starts

 	for (int i=lev; i!=k ; ) {

 	  Node v=level_list[++i];

 	  while ( v!=INVALID ) {

 	    level.set(v,_node_num);

 	    v=right[v];

 	  }

 	  level_list[i]=INVALID;

 	  if ( !what_heur ) first[i]=INVALID;

 	}

 	level.set(w,_node_num);

 	b=lev-1;

 	k=b;

 	//gapping ends

       } else {

 	if ( newlevel == _node_num ) level.set(w,_node_num);

 	else {

 	  level.set(w,++newlevel);

 	  next.set(w,first[newlevel]);

 	  first[newlevel]=w;

 	  if ( what_heur ) b=newlevel;

 	  if ( k < newlevel ) ++k;      //now k=newlevel

 	  Node z=level_list[newlevel];

 	  if ( z!=INVALID ) left.set(z,w);

 	  right.set(w,z);

 	  left.set(w,INVALID);

 	  level_list[newlevel]=w;

 	}

       }

     } //relabel

   }; 

   ///Function type interface for Preflow algorithm.

   /// \ingroup flowalgs

   ///Function type interface for Preflow algorithm.

   ///\sa Preflow

   template<class GR, class CM, class FM>

   Preflow<GR,typename CM::Value,CM,FM> preflow(const GR &g,

 			    typename GR::Node source,

 			    typename GR::Node target,

 			    const CM &cap,

 			    FM &flow

 			    )

   {

     return Preflow<GR,typename CM::Value,CM,FM>(g,source,target,cap,flow);

   }

 } //namespace lemon

 #endif //LEMON_PREFLOW_H