/* -*- 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_PREFLOW_H

 #define LEMON_PREFLOW_H

 #include <lemon/error.h>

 #include <lemon/bits/invalid.h>

 #include <lemon/tolerance.h>

 #include <lemon/maps.h>

 #include <lemon/graph_utils.h>

 #include <lemon/elevator.h>

 /// \file

 /// \ingroup max_flow

 /// \brief Implementation of the preflow algorithm.

 namespace lemon { 

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

   ///

   /// Default traits class of Preflow class.

   /// \param _Graph Graph type.

   /// \param _CapacityMap Type of capacity map.

   template <typename _Graph, typename _CapacityMap>

   struct PreflowDefaultTraits {

     /// \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 Preflow algorithm.

     /// 

     /// The elevator type used by Preflow 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 %Preflow algorithms class.

   ///

   /// This class provides an implementation of the Goldberg's \e

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

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

   /// flow algorithms. The current implementation use a mixture of the

   /// \e "highest label" and the \e "bound decrease" heuristics.

   /// The worst case time complexity of the algorithm is \f$O(n^3)\f$.

   ///

   /// The algorithm consists from two phases. After the first phase

   /// the maximal flow value and the minimum cut can be obtained. The

   /// second phase constructs the feasible maximum flow on each edge.

   ///

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

   /// \param _CapacityMap The flow map type.

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

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

   /// PreflowDefaultTraits.  See \ref PreflowDefaultTraits for the

   /// documentation of a %Preflow traits class. 

   ///

   ///\author Jacint Szabo 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 = PreflowDefaultTraits<_Graph, _CapacityMap> >

 #endif

   class Preflow {

   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;

     /// \brief \ref Exception for uninitialized parameters.

     ///

     /// This error represents problems in the initialization

     /// of the parameters of the algorithms.

     class UninitializedParameter : public lemon::UninitializedParameter {

     public:

       virtual const char* what() const throw() {

 	return "lemon::Preflow::UninitializedParameter";

       }

     };

   private:

     GRAPH_TYPEDEFS(typename Graph);

     const Graph& _graph;

     const CapacityMap* _capacity;

     int _node_num;

     Node _source, _target;

     FlowMap* _flow;

     bool _local_flow;

     Elevator* _level;

     bool _local_level;

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

     ExcessMap* _excess;

     Tolerance _tolerance;

     bool _phase;

     void createStructures() {

       _node_num = countNodes(_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);

       }

     }

     void destroyStructures() {

       if (_local_flow) {

 	delete _flow;

       }

       if (_local_level) {

 	delete _level;

       }

       if (_excess) {

 	delete _excess;

       }

     }

   public:

     typedef Preflow 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 Preflow<Graph, CapacityMap, DefFlowMapTraits<_FlowMap> > {

       typedef Preflow<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 Preflow<Graph, CapacityMap, DefElevatorTraits<_Elevator> > {

       typedef Preflow<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 Preflow<Graph, CapacityMap, 

 		       DefStandardElevatorTraits<_Elevator> > {

       typedef Preflow<Graph, CapacityMap, 

 		      DefStandardElevatorTraits<_Elevator> > Create;

     };    

     /// @}

     /// \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.

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

 	       Node source, Node target) 

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

 	_node_num(0), _source(source), _target(target), 

 	_flow(0), _local_flow(false),

 	_level(0), _local_level(false),

 	_excess(0), _tolerance(), _phase() {}

     /// \brief Destrcutor.

     ///

     /// Destructor.

     ~Preflow() {

       destroyStructures();

     }

     /// \brief Sets the capacity map.

     ///

     /// Sets the capacity map.

     /// \return \c (*this)

     Preflow& capacityMap(const CapacityMap& map) {

       _capacity = ↦

       return *this;

     }

     /// \brief Sets the flow map.

     ///

     /// Sets the flow map.

     /// \return \c (*this)

     Preflow& 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)

     Preflow& 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)

     Preflow& source(const Node& node) {

       _source = node;

       return *this;

     }

     /// \brief Sets the target node.

     ///

     /// Sets the target node.

     /// \return \c (*this)

     Preflow& target(const Node& node) {

       _target = node;

       return *this;

     }

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

     ///

     /// Sets the tolerance used by algorithm.

     Preflow& tolerance(const Tolerance& tolerance) const {

       _tolerance = tolerance;

       return *this;

     } 

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

     ///

     /// Returns the tolerance used by algorithm.

     const Tolerance& tolerance() const {

       return tolerance;

     } 

     /// \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();

       _phase = true;

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

 	_excess->set(n, 0);

       }

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

 	_flow->set(e, 0);

       }

       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()) {

 	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 Initializes the internal data structures.

     ///

     /// Initializes the internal data structures and sets the initial

     /// flow to the given \c flowMap. The \c flowMap should contain a

     /// flow or at least a preflow, ie. in each node excluding the

     /// target the incoming flow should greater or equal to the

     /// outgoing flow.

     /// \return %False when the given \c flowMap is not a preflow. 

     template <typename FlowMap>

     bool flowInit(const FlowMap& flowMap) {

       createStructures();

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

 	_flow->set(e, flowMap[e]);

       }

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

 	Value excess = 0;

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

 	  excess += (*_flow)[e];

 	}

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

 	  excess -= (*_flow)[e];

 	}

 	if (excess < 0 && n != _source) return false;

 	_excess->set(n, excess);

       }

       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] - (*_flow)[e])) {

 	      reached.set(u, true);

 	      _level->initAddItem(u);

 	      nqueue.push_back(u);

 	    }

 	  }

 	  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);

 	    }

 	  }

 	}

 	queue.swap(nqueue);

       }

       _level->initFinish();

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

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

 	if (_tolerance.positive(rem)) {

 	  Node u = _graph.target(e);

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

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

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

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

 	    _level->activate(u);

 	  }

 	}

       }

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

 	Value rem = (*_flow)[e]; 

 	if (_tolerance.positive(rem)) {

 	  Node v = _graph.source(e);

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

 	  _flow->set(e, 0);

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

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

 	    _level->activate(v);

 	  }

 	}

       }

       return true;

     }

     /// \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 = _level->highestActive();

       int level = _level->highestActiveLevel();

       while (n != INVALID) {

 	int num = _node_num;

 	while (num > 0 && n != INVALID) {

 	  Value excess = (*_excess)[n];

 	  int new_level = _level->maxLevel();

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

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

 	    if (!_tolerance.positive(rem)) continue;

 	    Node v = _graph.target(e);

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

 	      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_1;

 	      } 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];

 	    if (!_tolerance.positive(rem)) continue;

 	    Node v = _graph.source(e);

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

 	      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_1;

 	      } 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_1:

 	  _excess->set(n, excess);

 	  if (excess != 0) {

 	    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);

 	  }

 	  n = _level->highestActive();

 	  level = _level->highestActiveLevel();

 	  --num;

 	}

 	num = _node_num * 20;

 	while (num > 0 && n != INVALID) {

 	  Value excess = (*_excess)[n];

 	  int new_level = _level->maxLevel();

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

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

 	    if (!_tolerance.positive(rem)) continue;

 	    Node v = _graph.target(e);

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

 	      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_2;

 	      } 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];

 	    if (!_tolerance.positive(rem)) continue;

 	    Node v = _graph.source(e);

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

 	      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_2;

 	      } 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_2:

 	  _excess->set(n, excess);

 	  if (excess != 0) {

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

 	      _level->liftActiveOn(level, new_level + 1);

 	    } else {

 	      _level->liftActiveToTop(level);

 	    }

 	    if (_level->emptyLevel(level)) {

 	      _level->liftToTop(level);

 	    }

 	  } else {

 	    _level->deactivate(n);

 	  }

 	  while (level >= 0 && _level->activeFree(level)) {

 	    --level;

 	  }

 	  if (level == -1) {

 	    n = _level->highestActive();

 	    level = _level->highestActiveLevel();

 	  } else {

 	    n = _level->activeOn(level);

 	  }

 	  --num;

 	}

       }

     }

     /// \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);

       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();

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

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

 	  if (!_tolerance.positive(rem)) continue;

 	  Node v = _graph.target(e);

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

 	    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];

 	  if (!_tolerance.positive(rem)) continue;

 	  Node v = _graph.source(e);

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

 	    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) {

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

 	    _level->liftHighestActive(new_level + 1);

 	  } else {

 	    // Calculation error 

 	    _level->liftHighestActiveToTop();

 	  }

 	  if (_level->emptyLevel(level)) {

 	    // Calculation error 

 	    _level->liftToTop(level);

 	  }

 	} else {

 	  _level->deactivate(n);

 	}

       }

     }

     /// \brief Runs the preflow algorithm.  

     ///

     /// Runs the preflow 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 preflow algorithm to compute the minimum cut.  

     ///

     /// Runs the preflow 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 %Dijkstra 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];

     }

     /// @}    

   };

 }

 #endif