/* -*- 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;

    }