/* -*- 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_GOLDBERG_TARJAN_H

 #define LEMON_GOLDBERG_TARJAN_H

 #include <vector>

 #include <queue>

 #include <lemon/error.h>

 #include <lemon/bits/invalid.h>

 #include <lemon/tolerance.h>

 #include <lemon/maps.h>

 #include <lemon/graph_utils.h>

 #include <lemon/dynamic_tree.h>

 #include <limits>

 /// \file

 /// \ingroup max_flow

 /// \brief Implementation of the preflow algorithm.

 namespace lemon {

   /// \brief Default traits class of GoldbergTarjan class.

   ///

   /// Default traits class of GoldbergTarjan class.

   /// \param _Graph Graph type.

   /// \param _CapacityMap Type of capacity map.

   template <typename _Graph, typename _CapacityMap>

   struct GoldbergTarjanDefaultTraits {

     /// \brief The graph type the algorithm runs on. 

     typedef _Graph Graph;

     /// \brief The type of the map that stores the edge capacities.

     ///

     /// The type of the map that stores the edge capacities.

     /// It must meet the \ref concepts::ReadMap "ReadMap" concept.

     typedef _CapacityMap CapacityMap;

     /// \brief The type of the length of the edges.

     typedef typename CapacityMap::Value Value;

     /// \brief The map type that stores the flow values.

     ///

     /// The map type that stores the flow values. 

     /// It must meet the \ref concepts::ReadWriteMap "ReadWriteMap" concept.

     typedef typename Graph::template EdgeMap<Value> FlowMap;

     /// \brief Instantiates a FlowMap.

     ///

     /// This function instantiates a \ref FlowMap. 

     /// \param graph The graph, to which we would like to define the flow map.

     static FlowMap* createFlowMap(const Graph& graph) {

       return new FlowMap(graph);

     }

     /// \brief The eleavator type used by GoldbergTarjan algorithm.

     /// 

     /// The elevator type used by GoldbergTarjan algorithm.

     ///

     /// \sa Elevator

     /// \sa LinkedElevator

     typedef LinkedElevator<Graph, typename Graph::Node> Elevator;

     /// \brief Instantiates an Elevator.

     ///

     /// This function instantiates a \ref Elevator. 

     /// \param graph The graph, to which we would like to define the elevator.

     /// \param max_level The maximum level of the elevator.

     static Elevator* createElevator(const Graph& graph, int max_level) {

       return new Elevator(graph, max_level);

     }

     /// \brief The tolerance used by the algorithm

     ///

     /// The tolerance used by the algorithm to handle inexact computation.

     typedef Tolerance<Value> Tolerance;

   };

   /// \ingroup max_flow

   /// \brief Goldberg-Tarjan algorithms class.

   ///

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

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

   /// graph. The GoldbergTarjan algorithm is a theoretical improvement

   /// of the Goldberg's \ref Preflow "preflow" algorithm by using the \ref

   /// DynamicTree "dynamic tree" data structure of Sleator and Tarjan.

   /// 

   /// The original preflow algorithm with \e highest \e label

   /// heuristic has \f$O(n^2\sqrt{e})\f$ or with \e FIFO heuristic has

   /// \f$O(n^3)\f$ time complexity. The current algorithm improved

   /// this complexity to \f$O(nm\log(\frac{n^2}{m}))\f$. The algorithm

   /// builds limited size dynamic trees on the residual graph, which

   /// can be used to make multilevel push operations and gives a

   /// better bound on the number of non-saturating pushes.

   ///

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

   /// \param CapacityMap The capacity map type.

   /// \param _Traits Traits class to set various data types used by

   /// the algorithm.  The default traits class is \ref

   /// GoldbergTarjanDefaultTraits.  See \ref

   /// GoldbergTarjanDefaultTraits for the documentation of a

   /// Goldberg-Tarjan traits class.

   ///

   /// \author Tamas Hamori and Balazs Dezso

 #ifdef DOXYGEN

   template <typename _Graph, typename _CapacityMap, typename _Traits> 

 #else

   template <typename _Graph,

 	    typename _CapacityMap = typename _Graph::template EdgeMap<int>,

             typename _Traits = 

 	    GoldbergTarjanDefaultTraits<_Graph, _CapacityMap> >

 #endif

   class GoldbergTarjan {

   public:

     typedef _Traits Traits;

     typedef typename Traits::Graph Graph;

     typedef typename Traits::CapacityMap CapacityMap;

     typedef typename Traits::Value Value; 

     typedef typename Traits::FlowMap FlowMap;

     typedef typename Traits::Elevator Elevator;

     typedef typename Traits::Tolerance Tolerance;

   protected:

     GRAPH_TYPEDEFS(typename Graph);

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

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

     typedef typename Graph::template NodeMap<Edge> EdgeNodeMap;

     typedef typename Graph::template EdgeMap<Edge> EdgeEdgeMap;

     typedef typename std::vector<Node> VecNode;

     typedef DynamicTree<Value,IntNodeMap,Tolerance> DynTree;

     const Graph& _graph;

     const CapacityMap* _capacity;

     int _node_num; //the number of nodes of G

     Node _source;

     Node _target;

     FlowMap* _flow;

     bool _local_flow;

     Elevator* _level;

     bool _local_level;

     typedef typename Graph::template NodeMap<Value> ExcessMap;

     ExcessMap* _excess;

     Tolerance _tolerance;

     bool _phase;

     // constant for treesize

     static const double _tree_bound = 2;

     int _max_tree_size;

     //tags for the dynamic tree

     DynTree *_dt; 

     //datastructure of dyanamic tree

     IntNodeMap *_dt_index;

     //datastrucure for solution of the communication between the two class

     EdgeNodeMap *_dt_edges; 

     //nodeMap for storing the outgoing edge from the node in the tree  

     //max of the type Value

     const Value _max_value;

   public:

     typedef GoldbergTarjan Create;

     ///\name Named template parameters

     ///@{

     template <typename _FlowMap>

     struct DefFlowMapTraits : public Traits {

       typedef _FlowMap FlowMap;

       static FlowMap *createFlowMap(const Graph&) {

 	throw UninitializedParameter();

       }

     };

     /// \brief \ref named-templ-param "Named parameter" for setting

     /// FlowMap type

     ///

     /// \ref named-templ-param "Named parameter" for setting FlowMap

     /// type

     template <typename _FlowMap>

     struct DefFlowMap 

       : public GoldbergTarjan<Graph, CapacityMap, 

 			      DefFlowMapTraits<_FlowMap> > {

       typedef GoldbergTarjan<Graph, CapacityMap, 

 			     DefFlowMapTraits<_FlowMap> > Create;

     };

     template <typename _Elevator>

     struct DefElevatorTraits : public Traits {

       typedef _Elevator Elevator;

       static Elevator *createElevator(const Graph&, int) {

 	throw UninitializedParameter();

       }

     };

     /// \brief \ref named-templ-param "Named parameter" for setting

     /// Elevator type

     ///

     /// \ref named-templ-param "Named parameter" for setting Elevator

     /// type

     template <typename _Elevator>

     struct DefElevator 

       : public GoldbergTarjan<Graph, CapacityMap, 

 			      DefElevatorTraits<_Elevator> > {

       typedef GoldbergTarjan<Graph, CapacityMap, 

 			     DefElevatorTraits<_Elevator> > Create;

     };

     template <typename _Elevator>

     struct DefStandardElevatorTraits : public Traits {

       typedef _Elevator Elevator;

       static Elevator *createElevator(const Graph& graph, int max_level) {

 	return new Elevator(graph, max_level);

       }

     };

     /// \brief \ref named-templ-param "Named parameter" for setting

     /// Elevator type

     ///

     /// \ref named-templ-param "Named parameter" for setting Elevator

     /// type. The Elevator should be standard constructor interface, ie.

     /// the graph and the maximum level should be passed to it.

     template <typename _Elevator>

     struct DefStandardElevator 

       : public GoldbergTarjan<Graph, CapacityMap, 

 			      DefStandardElevatorTraits<_Elevator> > {

       typedef GoldbergTarjan<Graph, CapacityMap, 

 			     DefStandardElevatorTraits<_Elevator> > Create;

     };    

     ///\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* what() const throw() {

 	return "lemon::GoldbergTarjan::InvalidArgument";

       }

     };

   public: 

     /// \brief The constructor of the class.

     ///

     /// The constructor of the class. 

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

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

     /// \param source The source node.

     /// \param target The target node.

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

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

     /// flowMap, resp.

     GoldbergTarjan(const Graph& graph, const CapacityMap& capacity,

 		   Node source, Node target) 

       : _graph(graph), _capacity(&capacity), 

 	_node_num(), _source(source), _target(target),

 	_flow(0), _local_flow(false),

 	_level(0), _local_level(false),

 	_excess(0), _tolerance(), 

 	_phase(), _max_tree_size(),

 	_dt(0), _dt_index(0), _dt_edges(0), 

 	_max_value(std::numeric_limits<Value>::max()) { 

       if (_source == _target) throw InvalidArgument();

     }

     /// \brief Destrcutor.

     ///

     /// Destructor.

     ~GoldbergTarjan() {

       destroyStructures();

     }

     /// \brief Sets the capacity map.

     ///

     /// Sets the capacity map.

     /// \return \c (*this)

     GoldbergTarjan& capacityMap(const CapacityMap& map) {

       _capacity = ↦

       return *this;

     }

     /// \brief Sets the flow map.

     ///

     /// Sets the flow map.

     /// \return \c (*this)

     GoldbergTarjan& flowMap(FlowMap& map) {

       if (_local_flow) {

 	delete _flow;

 	_local_flow = false;

       }

       _flow = ↦

       return *this;

     }

     /// \brief Returns the flow map.

     ///

     /// \return The flow map.

     const FlowMap& flowMap() {

       return *_flow;

     }

     /// \brief Sets the elevator.

     ///

     /// Sets the elevator.

     /// \return \c (*this)

     GoldbergTarjan& elevator(Elevator& elevator) {

       if (_local_level) {

 	delete _level;

 	_local_level = false;

       }

       _level = &elevator;

       return *this;

     }

     /// \brief Returns the elevator.

     ///

     /// \return The elevator.

     const Elevator& elevator() {

       return *_level;

     }

     /// \brief Sets the source node.

     ///

     /// Sets the source node.

     /// \return \c (*this)

     GoldbergTarjan& source(const Node& node) {

       _source = node;

       return *this;

     }

     /// \brief Sets the target node.

     ///

     /// Sets the target node.

     /// \return \c (*this)

     GoldbergTarjan& target(const Node& node) {

       _target = node;

       return *this;

     }

     /// \brief Sets the tolerance used by algorithm.

     ///

     /// Sets the tolerance used by algorithm.

     GoldbergTarjan& tolerance(const Tolerance& tolerance) const {

       _tolerance = tolerance;

       if (_dt) {

 	_dt->tolerance(_tolerance);

       }

       return *this;

     } 

     /// \brief Returns the tolerance used by algorithm.

     ///

     /// Returns the tolerance used by algorithm.

     const Tolerance& tolerance() const {

       return tolerance;

     } 

   private:

     void createStructures() {

       _node_num = countNodes(_graph);

       _max_tree_size = int((double(_node_num) * double(_node_num)) / 

 			   double(countEdges(_graph)));

       if (!_flow) {

 	_flow = Traits::createFlowMap(_graph);

 	_local_flow = true;

       }

       if (!_level) {

 	_level = Traits::createElevator(_graph, _node_num);

 	_local_level = true;

       }

       if (!_excess) {

 	_excess = new ExcessMap(_graph);

       }

       if (!_dt_index && !_dt) {

 	_dt_index = new IntNodeMap(_graph);

 	_dt = new DynTree(*_dt_index, _tolerance);

       }

       if (!_dt_edges) {

 	_dt_edges = new EdgeNodeMap(_graph);

       }

     }

     void destroyStructures() {

       if (_local_flow) {

 	delete _flow;

       }

       if (_local_level) {

 	delete _level;

       }

       if (_excess) {

 	delete _excess;

       }

       if (_dt) {

 	delete _dt;

       }

       if (_dt_index) {

 	delete _dt_index;

       }

       if (_dt_edges) {

 	delete _dt_edges;

       }

     }

     bool sendOut(Node n, Value& excess) {

       Node w = _dt->findRoot(n);

       while (w != n) {

 	Value rem;      

 	Node u = _dt->findCost(n, rem);

         if (_tolerance.positive(rem) && !_level->active(w) && w != _target) {

 	  _level->activate(w);

         }

         if (!_tolerance.less(rem, excess)) {

 	  _dt->addCost(n, - excess);

 	  _excess->set(w, (*_excess)[w] + excess);

 	  excess = 0;

 	  return true;

 	}

         _dt->addCost(n, - rem);

         excess -= rem;

         _excess->set(w, (*_excess)[w] + rem);

 	_dt->cut(u);

 	_dt->addCost(u, _max_value);

 	Edge e = (*_dt_edges)[u];

 	_dt_edges->set(u, INVALID);

 	if (u == _graph.source(e)) {

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

 	} else {

 	  _flow->set(e, 0);

 	}           

         w = _dt->findRoot(n);

       }      

       return false;

     }

     bool sendIn(Node n, Value& excess) {

       Node w = _dt->findRoot(n);

       while (w != n) {

 	Value rem;      

 	Node u = _dt->findCost(n, rem);

         if (_tolerance.positive(rem) && !_level->active(w) && w != _source) {

 	  _level->activate(w);

         }

         if (!_tolerance.less(rem, excess)) {

 	  _dt->addCost(n, - excess);

 	  _excess->set(w, (*_excess)[w] + excess);

 	  excess = 0;

 	  return true;

 	}

         _dt->addCost(n, - rem);

         excess -= rem;

         _excess->set(w, (*_excess)[w] + rem);

 	_dt->cut(u);

 	_dt->addCost(u, _max_value);

 	Edge e = (*_dt_edges)[u];

 	_dt_edges->set(u, INVALID);

 	if (u == _graph.source(e)) {

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

 	} else {

 	  _flow->set(e, 0);

 	}           

         w = _dt->findRoot(n);

       }      

       return false;

     }

     void cutChildren(Node n) {

       for (OutEdgeIt e(_graph, n); e != INVALID; ++e) {

         Node v = _graph.target(e);

         if ((*_dt_edges)[v] != INVALID && (*_dt_edges)[v] == e) {

           _dt->cut(v);

           _dt_edges->set(v, INVALID);

 	  Value rem;

           _dt->findCost(v, rem);

           _dt->addCost(v, - rem);

           _dt->addCost(v, _max_value);

           _flow->set(e, rem);

         }      

       }

       for (InEdgeIt e(_graph, n); e != INVALID; ++e) {

         Node v = _graph.source(e);

         if ((*_dt_edges)[v] != INVALID && (*_dt_edges)[v] == e) {

           _dt->cut(v);

           _dt_edges->set(v, INVALID);

 	  Value rem;

           _dt->findCost(v, rem);

           _dt->addCost(v, - rem);

           _dt->addCost(v, _max_value);

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

         }      

       }

     }

     void extractTrees() {

       for (NodeIt n(_graph); n != INVALID; ++n) {

 	Node w = _dt->findRoot(n);

 	while (w != n) {

 	  Value rem;      

 	  Node u = _dt->findCost(n, rem);

 	  _dt->cut(u);

 	  _dt->addCost(u, - rem);

 	  _dt->addCost(u, _max_value);

 	  Edge e = (*_dt_edges)[u];

 	  _dt_edges->set(u, INVALID);

 	  if (u == _graph.source(e)) {

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

 	  } else {

 	    _flow->set(e, rem);

 	  }

 	  w = _dt->findRoot(n);

 	}      

       }

     }

   public:    

     /// \name Execution control The simplest way to execute the

     /// algorithm is to use one of the member functions called \c

     /// run().  

     /// \n

     /// If you need more control on initial solution or

     /// execution then you have to call one \ref init() function and then

     /// the startFirstPhase() and if you need the startSecondPhase().  

     ///@{

     /// \brief Initializes the internal data structures.

     ///

     /// Initializes the internal data structures.

     ///

     void init() {

       createStructures();

       for (NodeIt n(_graph); n != INVALID; ++n) {

 	_excess->set(n, 0);

       }

       for (EdgeIt e(_graph); e != INVALID; ++e) {

 	_flow->set(e, 0);

       }

       _dt->clear();

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

         (*_dt_edges)[v] = INVALID;

         _dt->makeTree(v);

         _dt->addCost(v, _max_value);

       }

       typename Graph::template NodeMap<bool> reached(_graph, false);

       _level->initStart();

       _level->initAddItem(_target);

       std::vector<Node> queue;

       reached.set(_source, true);

       queue.push_back(_target);

       reached.set(_target, true);

       while (!queue.empty()) {

 	_level->initNewLevel();

 	std::vector<Node> nqueue;

 	for (int i = 0; i < int(queue.size()); ++i) {

 	  Node n = queue[i];

 	  for (InEdgeIt e(_graph, n); e != INVALID; ++e) {

 	    Node u = _graph.source(e);

 	    if (!reached[u] && _tolerance.positive((*_capacity)[e])) {

 	      reached.set(u, true);

 	      _level->initAddItem(u);

 	      nqueue.push_back(u);

 	    }

 	  }

 	}

 	queue.swap(nqueue);

       }

       _level->initFinish();

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

 	if (_tolerance.positive((*_capacity)[e])) {

 	  Node u = _graph.target(e);

 	  if ((*_level)[u] == _level->maxLevel()) continue;

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

 	  _excess->set(u, (*_excess)[u] + (*_capacity)[e]);

 	  if (u != _target && !_level->active(u)) {

 	    _level->activate(u);

 	  }

 	}

       }

     }

     /// \brief Starts 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.     

     /// \pre One of the \ref init() functions should be called. 

     void startFirstPhase() {

       _phase = true;

       Node n;

       while ((n = _level->highestActive()) != INVALID) {

 	Value excess = (*_excess)[n];

 	int level = _level->highestActiveLevel();

 	int new_level = _level->maxLevel();

 	if (_dt->findRoot(n) != n) {

 	  if (sendOut(n, excess)) goto no_more_push;

 	}

 	for (OutEdgeIt e(_graph, n); e != INVALID; ++e) {

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

 	  Node v = _graph.target(e);

 	  if (!_tolerance.positive(rem) && (*_dt_edges)[v] != e) continue;

 	  if ((*_level)[v] < level) {

 	    if (_dt->findSize(n) + _dt->findSize(v) < 

 		_tree_bound * _max_tree_size) {

 	      _dt->addCost(n, -_max_value);

 	      _dt->addCost(n, rem);

 	      _dt->link(n, v);

 	      _dt_edges->set(n, e);

 	      if (sendOut(n, excess)) goto no_more_push;

 	    } else {

 	      if (!_level->active(v) && v != _target) {

 		_level->activate(v);

 	      }

 	      if (!_tolerance.less(rem, excess)) {

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

 		_excess->set(v, (*_excess)[v] + excess);

 		excess = 0;		  

 		goto no_more_push;

 	      } else {

 		excess -= rem;

 		_excess->set(v, (*_excess)[v] + rem);

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

 	      }		

 	    }

 	  } else if (new_level > (*_level)[v]) {

 	    new_level = (*_level)[v];

 	  }

 	}

 	for (InEdgeIt e(_graph, n); e != INVALID; ++e) {

 	  Value rem = (*_flow)[e];

 	  Node v = _graph.source(e);

 	  if (!_tolerance.positive(rem) && (*_dt_edges)[v] != e) continue;

 	  if ((*_level)[v] < level) {

 	    if (_dt->findSize(n) + _dt->findSize(v) < 

 		_tree_bound * _max_tree_size) {

 	      _dt->addCost(n, - _max_value);

 	      _dt->addCost(n, rem);

 	      _dt->link(n, v);

 	      _dt_edges->set(n, e);

 	      if (sendOut(n, excess)) goto no_more_push;

 	    } else {

 	      if (!_level->active(v) && v != _target) {

 		_level->activate(v);

 	      }

 	      if (!_tolerance.less(rem, excess)) {

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

 		_excess->set(v, (*_excess)[v] + excess);

 		excess = 0;		  

 		goto no_more_push;

 	      } else {

 		excess -= rem;

 		_excess->set(v, (*_excess)[v] + rem);

 		_flow->set(e, 0);

 	      }		

 	    }

 	  } else if (new_level > (*_level)[v]) {

 	    new_level = (*_level)[v];

 	  }

 	}

       no_more_push:

 	_excess->set(n, excess);

 	if (excess != 0) {

 	  cutChildren(n);

 	  if (new_level + 1 < _level->maxLevel()) {

 	    _level->liftHighestActive(new_level + 1);

 	  } else {

 	    _level->liftHighestActiveToTop();

 	  }

 	  if (_level->emptyLevel(level)) {

 	    _level->liftToTop(level);

 	  }

 	} else {

 	  _level->deactivate(n);

 	}	

       }

       extractTrees();

     }

     /// \brief Starts the second phase of the preflow algorithm.

     ///

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

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

     /// startFirstPhase() and then \ref startSecondPhase(), \ref

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

     /// value of a maximum flow, \ref minCut() returns a minimum cut

     /// \pre The \ref init() and startFirstPhase() functions should be

     /// called before.

     void startSecondPhase() {

       _phase = false;

       typename Graph::template NodeMap<bool> reached(_graph, false);

       for (NodeIt n(_graph); n != INVALID; ++n) {

 	reached.set(n, (*_level)[n] < _level->maxLevel());

       }

       _level->initStart();

       _level->initAddItem(_source);

       std::vector<Node> queue;

       queue.push_back(_source);

       reached.set(_source, true);

       while (!queue.empty()) {

 	_level->initNewLevel();

 	std::vector<Node> nqueue;

 	for (int i = 0; i < int(queue.size()); ++i) {

 	  Node n = queue[i];

 	  for (OutEdgeIt e(_graph, n); e != INVALID; ++e) {

 	    Node v = _graph.target(e);

 	    if (!reached[v] && _tolerance.positive((*_flow)[e])) {

 	      reached.set(v, true);

 	      _level->initAddItem(v);

 	      nqueue.push_back(v);

 	    }

 	  }

 	  for (InEdgeIt e(_graph, n); e != INVALID; ++e) {

 	    Node u = _graph.source(e);

 	    if (!reached[u] && 

 		_tolerance.positive((*_capacity)[e] - (*_flow)[e])) {

 	      reached.set(u, true);

 	      _level->initAddItem(u);

 	      nqueue.push_back(u);

 	    }

 	  }

 	}

 	queue.swap(nqueue);

       }

       _level->initFinish();

       for (NodeIt n(_graph); n != INVALID; ++n) {

 	if (!reached[n]) {

 	  _level->markToBottom(n);

 	} else if ((*_excess)[n] > 0 && _target != n) {

 	  _level->activate(n);

 	}

       }

       Node n;

       while ((n = _level->highestActive()) != INVALID) {

 	Value excess = (*_excess)[n];

 	int level = _level->highestActiveLevel();

 	int new_level = _level->maxLevel();

 	if (_dt->findRoot(n) != n) {

 	  if (sendIn(n, excess)) goto no_more_push;

 	}

 	for (OutEdgeIt e(_graph, n); e != INVALID; ++e) {

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

 	  Node v = _graph.target(e);

 	  if (!_tolerance.positive(rem) && (*_dt_edges)[v] != e) continue;

 	  if ((*_level)[v] < level) {

 	    if (_dt->findSize(n) + _dt->findSize(v) < 

 		_tree_bound * _max_tree_size) {

 	      _dt->addCost(n, -_max_value);

 	      _dt->addCost(n, rem);

 	      _dt->link(n, v);

 	      _dt_edges->set(n, e);

 	      if (sendIn(n, excess)) goto no_more_push;

 	    } else {

 	      if (!_level->active(v) && v != _source) {

 		_level->activate(v);

 	      }

 	      if (!_tolerance.less(rem, excess)) {

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

 		_excess->set(v, (*_excess)[v] + excess);

 		excess = 0;		  

 		goto no_more_push;

 	      } else {

 		excess -= rem;

 		_excess->set(v, (*_excess)[v] + rem);

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

 	      }		

 	    }

 	  } else if (new_level > (*_level)[v]) {

 	    new_level = (*_level)[v];

 	  }

 	}

 	for (InEdgeIt e(_graph, n); e != INVALID; ++e) {

 	  Value rem = (*_flow)[e];

 	  Node v = _graph.source(e);

 	  if (!_tolerance.positive(rem) && (*_dt_edges)[v] != e) continue;

 	  if ((*_level)[v] < level) {

 	    if (_dt->findSize(n) + _dt->findSize(v) < 

 		_tree_bound * _max_tree_size) {

 	      _dt->addCost(n, - _max_value);

 	      _dt->addCost(n, rem);

 	      _dt->link(n, v);

 	      _dt_edges->set(n, e);

 	      if (sendIn(n, excess)) goto no_more_push;

 	    } else {

 	      if (!_level->active(v) && v != _source) {

 		_level->activate(v);

 	      }

 	      if (!_tolerance.less(rem, excess)) {

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

 		_excess->set(v, (*_excess)[v] + excess);

 		excess = 0;		  

 		goto no_more_push;

 	      } else {

 		excess -= rem;

 		_excess->set(v, (*_excess)[v] + rem);

 		_flow->set(e, 0);

 	      }		

 	    }

 	  } else if (new_level > (*_level)[v]) {

 	    new_level = (*_level)[v];

 	  }

 	}

       no_more_push:

 	_excess->set(n, excess);

 	if (excess != 0) {

 	  cutChildren(n);

 	  if (new_level + 1 < _level->maxLevel()) {

 	    _level->liftHighestActive(new_level + 1);

 	  } else {

 	    _level->liftHighestActiveToTop();

 	  }

 	  if (_level->emptyLevel(level)) {

 	    _level->liftToTop(level);

 	  }

 	} else {

 	  _level->deactivate(n);

 	}	

       }

       extractTrees();

     }

     /// \brief Runs the Goldberg-Tarjan algorithm.  

     ///

     /// Runs the Goldberg-Tarjan algorithm.

     /// \note pf.run() is just a shortcut of the following code.

     /// \code

     ///   pf.init();

     ///   pf.startFirstPhase();

     ///   pf.startSecondPhase();

     /// \endcode

     void run() {

       init();

       startFirstPhase();

       startSecondPhase();

     }

     /// \brief Runs the Goldberg-Tarjan algorithm to compute the minimum cut.  

     ///

     /// Runs the Goldberg-Tarjan algorithm to compute the minimum cut.

     /// \note pf.runMinCut() is just a shortcut of the following code.

     /// \code

     ///   pf.init();

     ///   pf.startFirstPhase();

     /// \endcode

     void runMinCut() {

       init();

       startFirstPhase();

     }

     /// @}

     /// \name Query Functions 

     /// The result of the Goldberg-Tarjan algorithm can be obtained

     /// using these functions.

     /// \n

     /// Before the use of these functions, either run() or start() must

     /// be called.

     ///@{

     /// \brief 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 the first phase.

     Value flowValue() const {

       return (*_excess)[_target];

     }

     /// \brief Returns true when the node is on the source side of minimum cut.

     ///

     /// Returns true when the node is on the source side of minimum

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

     /// startFirstPhase() and \ref startSecondPhase().

     bool minCut(const Node& node) const {

       return ((*_level)[node] == _level->maxLevel()) == _phase;

     }

     /// \brief Returns a minimum value cut.

     ///

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

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

     /// startFirstPhase() and \ref startSecondPhase(). The result after second

     /// phase could be changed slightly if inexact computation is used.

     /// \pre The \c cutMap should be a bool-valued node-map.

     template <typename CutMap>

     void minCutMap(CutMap& cutMap) const {

       for (NodeIt n(_graph); n != INVALID; ++n) {

 	cutMap.set(n, minCut(n));

       }

     }

     /// \brief Returns the flow on the edge.

     ///

     /// Sets the \c flowMap to the flow on the edges. This method can

     /// be called after the second phase of algorithm.

     Value flow(const Edge& edge) const {

       return (*_flow)[edge];

     }

     /// @}

   }; 

 } //namespace lemon

 #endif