/* -*- mode: C++; indent-tabs-mode: nil; -*-

 *

 * This file is a part of LEMON, a generic C++ optimization library.

 *

 * Copyright (C) 2003-2010

 * 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/tolerance.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.

  /// \tparam GR Digraph type.

  /// \tparam CAP Capacity map type.

  template <typename GR, typename CAP>

  struct PreflowDefaultTraits {

    /// \brief The type of the digraph the algorithm runs on.

    typedef GR Digraph;

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

    ///

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

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

    typedef CAP CapacityMap;

    /// \brief The type of the flow values.

    typedef typename CapacityMap::Value Value;

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

    ///

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

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

#ifdef DOXYGEN

    typedef GR::ArcMap<Value> FlowMap;

#else

    typedef typename Digraph::template ArcMap<Value> FlowMap;

#endif

    /// \brief Instantiates a FlowMap.

    ///

    /// This function instantiates a \ref FlowMap.

    /// \param digraph The digraph for which we would like to define

    /// the flow map.

    static FlowMap* createFlowMap(const Digraph& digraph) {

      return new FlowMap(digraph);

    }

    /// \brief The elevator type used by Preflow algorithm.

    ///

    /// The elevator type used by Preflow algorithm.

    ///

    /// \sa Elevator, LinkedElevator

#ifdef DOXYGEN

    typedef lemon::Elevator<GR, GR::Node> Elevator;

#else

    typedef lemon::Elevator<Digraph, typename Digraph::Node> Elevator;

#endif

    /// \brief Instantiates an Elevator.

    ///

    /// This function instantiates an \ref Elevator.

    /// \param digraph The digraph for which we would like to define

    /// the elevator.

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

    static Elevator* createElevator(const Digraph& digraph, int max_level) {

      return new Elevator(digraph, max_level);

    }

    /// \brief The tolerance used by the algorithm

    ///

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

    typedef lemon::Tolerance<Value> Tolerance;

  };

  /// \ingroup max_flow

  ///

  /// \brief %Preflow algorithm class.

  ///

  /// This class provides an implementation of Goldberg-Tarjan's \e preflow

  /// \e push-relabel algorithm producing a \ref max_flow

  /// "flow of maximum value" in a digraph \ref clrs01algorithms,

  /// \ref amo93networkflows, \ref goldberg88newapproach.

  /// The preflow algorithms are the fastest known maximum

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

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

  /// The worst case time complexity of the algorithm is \f$O(n^2\sqrt{e})\f$.

  ///

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

  /// the maximum flow value and the minimum cut is obtained. The

  /// second phase constructs a feasible maximum flow on each arc.

  ///

  /// \warning This implementation cannot handle infinite or very large

  /// capacities (e.g. the maximum value of \c CAP::Value).

  ///

  /// \tparam GR The type of the digraph the algorithm runs on.

  /// \tparam CAP The type of the capacity map. The default map

  /// type is \ref concepts::Digraph::ArcMap "GR::ArcMap<int>".

  /// \tparam TR The traits class that defines various types used by the

  /// algorithm. By default, it is \ref PreflowDefaultTraits

  /// "PreflowDefaultTraits<GR, CAP>".

  /// In most cases, this parameter should not be set directly,

  /// consider to use the named template parameters instead.

#ifdef DOXYGEN

  template <typename GR, typename CAP, typename TR>

#else

  template <typename GR,

            typename CAP = typename GR::template ArcMap<int>,

            typename TR = PreflowDefaultTraits<GR, CAP> >

#endif

  class Preflow {

  public:

    ///The \ref PreflowDefaultTraits "traits class" of the algorithm.

    typedef TR Traits;

    ///The type of the digraph the algorithm runs on.

    typedef typename Traits::Digraph Digraph;

    ///The type of the capacity map.

    typedef typename Traits::CapacityMap CapacityMap;

    ///The type of the flow values.

    typedef typename Traits::Value Value;

    ///The type of the flow map.

    typedef typename Traits::FlowMap FlowMap;

    ///The type of the elevator.

    typedef typename Traits::Elevator Elevator;

    ///The type of the tolerance.

    typedef typename Traits::Tolerance Tolerance;

  private:

    TEMPLATE_DIGRAPH_TYPEDEFS(Digraph);

    const Digraph& _graph;

    const CapacityMap* _capacity;

    int _node_num;

    Node _source, _target;

    FlowMap* _flow;

    bool _local_flow;

    Elevator* _level;

    bool _local_level;

    typedef typename Digraph::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 T>

    struct SetFlowMapTraits : public Traits {

      typedef T FlowMap;

      static FlowMap *createFlowMap(const Digraph&) {

        LEMON_ASSERT(false, "FlowMap is not initialized");

        return 0; // ignore warnings

      }

    };

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

    /// FlowMap type

    ///

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

    /// type.

    template <typename T>

    struct SetFlowMap

      : public Preflow<Digraph, CapacityMap, SetFlowMapTraits<T> > {

      typedef Preflow<Digraph, CapacityMap,

                      SetFlowMapTraits<T> > Create;

    };

    template <typename T>

    struct SetElevatorTraits : public Traits {

      typedef T Elevator;

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

        LEMON_ASSERT(false, "Elevator is not initialized");

        return 0; // ignore warnings

      }

    };

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

    /// Elevator type

    ///

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

    /// type. If this named parameter is used, then an external

    /// elevator object must be passed to the algorithm using the

    /// \ref elevator(Elevator&) "elevator()" function before calling

    /// \ref run() or \ref init().

    /// \sa SetStandardElevator

    template <typename T>

    struct SetElevator

      : public Preflow<Digraph, CapacityMap, SetElevatorTraits<T> > {

      typedef Preflow<Digraph, CapacityMap,

                      SetElevatorTraits<T> > Create;

    };

    template <typename T>

    struct SetStandardElevatorTraits : public Traits {

      typedef T Elevator;

      static Elevator *createElevator(const Digraph& digraph, int max_level) {

        return new Elevator(digraph, max_level);

      }

    };

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

    /// Elevator type with automatic allocation

    ///

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

    /// type with automatic allocation.

    /// The Elevator should have standard constructor interface to be

    /// able to automatically created by the algorithm (i.e. the

    /// digraph and the maximum level should be passed to it).

    /// However, an external elevator object could also be passed to the

    /// algorithm with the \ref elevator(Elevator&) "elevator()" function

    /// before calling \ref run() or \ref init().

    /// \sa SetElevator

    template <typename T>

    struct SetStandardElevator

      : public Preflow<Digraph, CapacityMap,

                       SetStandardElevatorTraits<T> > {

      typedef Preflow<Digraph, CapacityMap,

                      SetStandardElevatorTraits<T> > Create;

    };

    /// @}

  protected:

    Preflow() {}

  public:

    /// \brief The constructor of the class.

    ///

    /// The constructor of the class.

    /// \param digraph The digraph the algorithm runs on.

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

    /// \param source The source node.

    /// \param target The target node.

    Preflow(const Digraph& digraph, const CapacityMap& capacity,

            Node source, Node target)

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

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

        _flow(0), _local_flow(false),

        _level(0), _local_level(false),

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

    /// \brief Destructor.

    ///

    /// Destructor.

    ~Preflow() {

      destroyStructures();

    }

    /// \brief Sets the capacity map.

    ///

    /// Sets the capacity map.

    /// \return <tt>(*this)</tt>

    Preflow& capacityMap(const CapacityMap& map) {

      _capacity = ↦

      return *this;

    }

    /// \brief Sets the flow map.

    ///

    /// Sets the flow map.

    /// If you don't use this function before calling \ref run() or

    /// \ref init(), an instance will be allocated automatically.

    /// The destructor deallocates this automatically allocated map,

    /// of course.

    /// \return <tt>(*this)</tt>

    Preflow& flowMap(FlowMap& map) {

      if (_local_flow) {

        delete _flow;

        _local_flow = false;

      }

      _flow = ↦

      return *this;

    }

    /// \brief Sets the source node.

    ///

    /// Sets the source node.

    /// \return <tt>(*this)</tt>

    Preflow& source(const Node& node) {

      _source = node;

      return *this;

    }

    /// \brief Sets the target node.

    ///

    /// Sets the target node.

    /// \return <tt>(*this)</tt>

    Preflow& target(const Node& node) {

      _target = node;

      return *this;

    }

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

    ///

    /// Sets the elevator used by algorithm.

    /// If you don't use this function before calling \ref run() or

    /// \ref init(), an instance will be allocated automatically.

    /// The destructor deallocates this automatically allocated elevator,

    /// of course.

    /// \return <tt>(*this)</tt>

    Preflow& elevator(Elevator& elevator) {

      if (_local_level) {

        delete _level;

        _local_level = false;

      }

      _level = &elevator;

      return *this;

    }

    /// \brief Returns a const reference to the elevator.

    ///

    /// Returns a const reference to the elevator.

    ///

    /// \pre Either \ref run() or \ref init() must be called before

    /// using this function.

    const Elevator& elevator() const {

      return *_level;

    }

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

    ///

    /// Sets the tolerance object used by the algorithm.

    /// \return <tt>(*this)</tt>

    Preflow& tolerance(const Tolerance& tolerance) {

      _tolerance = tolerance;

      return *this;

    }

    /// \brief Returns a const reference to the tolerance.

    ///

    /// Returns a const reference to the tolerance object used by

    /// the algorithm.

    const Tolerance& tolerance() const {

      return _tolerance;

    }

    /// \name Execution Control

    /// The simplest way to execute the preflow algorithm is to use

    /// \ref run() or \ref runMinCut().\n

    /// If you need better control on the initial solution or the execution,

    /// you have to call one of the \ref init() functions first, then

    /// \ref startFirstPhase() and if you need it \ref startSecondPhase().

    ///@{

    /// \brief Initializes the internal data structures.

    ///

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

    /// flow to zero on each arc.

    void init() {

      createStructures();

      _phase = true;

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

        (*_excess)[n] = 0;

      }

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

        _flow->set(e, 0);

      }

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

      _level->initStart();

      _level->initAddItem(_target);

      std::vector<Node> queue;

      reached[_source] = true;

      queue.push_back(_target);

      reached[_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 (InArcIt e(_graph, n); e != INVALID; ++e) {

            Node u = _graph.source(e);

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

              reached[u] = true;

              _level->initAddItem(u);

              nqueue.push_back(u);

            }

          }

        }

        queue.swap(nqueue);

      }

      _level->initFinish();

      for (OutArcIt 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)[u] += (*_capacity)[e];

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

            _level->activate(u);

          }

        }

      }

    }

    /// \brief Initializes the internal data structures using the

    /// given flow map.

    ///

    /// 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, i.e. at each node excluding the

    /// source node the incoming flow should greater or equal to the

    /// outgoing flow.

    /// \return \c false if the given \c flowMap is not a preflow.

    template <typename FlowMap>

    bool init(const FlowMap& flowMap) {

      createStructures();

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

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

      }

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

        Value excess = 0;

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

          excess += (*_flow)[e];

        }

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

          excess -= (*_flow)[e];

        }

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

        (*_excess)[n] = excess;

      }

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

      _level->initStart();

      _level->initAddItem(_target);

      std::vector<Node> queue;

      reached[_source] = true;

      queue.push_back(_target);

      reached[_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 (InArcIt e(_graph, n); e != INVALID; ++e) {

            Node u = _graph.source(e);

            if (!reached[u] &&

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

              reached[u] = true;

              _level->initAddItem(u);

              nqueue.push_back(u);

            }

          }

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

            Node v = _graph.target(e);

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

              reached[v] = true;

              _level->initAddItem(v);

              nqueue.push_back(v);

            }

          }

        }

        queue.swap(nqueue);

      }

      _level->initFinish();

      for (OutArcIt 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)[u] += rem;

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

            _level->activate(u);

          }

        }

      }

      for (InArcIt 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)[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 must be called before

    /// using this function.

    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 (OutArcIt 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)[v] += excess;

                excess = 0;

                goto no_more_push_1;

              } else {

                excess -= rem;

                (*_excess)[v] += rem;

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

              }

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

              new_level = (*_level)[v];

            }

          }

          for (InArcIt 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)[v] += excess;

                excess = 0;

                goto no_more_push_1;

              } else {

                excess -= rem;

                (*_excess)[v] += rem;

                _flow->set(e, 0);

              }

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

              new_level = (*_level)[v];

            }

          }

        no_more_push_1:

          (*_excess)[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 (OutArcIt 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)[v] += excess;

                excess = 0;

                goto no_more_push_2;

              } else {

                excess -= rem;

                (*_excess)[v] += rem;

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

              }

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

              new_level = (*_level)[v];

            }

          }

          for (InArcIt 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)[v] += excess;

                excess = 0;

                goto no_more_push_2;

              } else {

                excess -= rem;

                (*_excess)[v] += rem;

                _flow->set(e, 0);

              }

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

              new_level = (*_level)[v];

            }

          }

        no_more_push_2:

          (*_excess)[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 one of the \ref init() functions

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

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

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

    /// \pre One of the \ref init() functions and \ref startFirstPhase()

    /// must be called before using this function.

    void startSecondPhase() {

      _phase = false;

      typename Digraph::template NodeMap<bool> reached(_graph);

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

        reached[n] = (*_level)[n] < _level->maxLevel();

      }

      _level->initStart();

      _level->initAddItem(_source);

      std::vector<Node> queue;

      queue.push_back(_source);

      reached[_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 (OutArcIt e(_graph, n); e != INVALID; ++e) {

            Node v = _graph.target(e);

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

              reached[v] = true;

              _level->initAddItem(v);

              nqueue.push_back(v);

            }

          }

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

            Node u = _graph.source(e);

            if (!reached[u] &&

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

              reached[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->dirtyTopButOne(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 (OutArcIt 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)[v] += excess;

              excess = 0;

              goto no_more_push;

            } else {

              excess -= rem;

              (*_excess)[v] += rem;

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

            }

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

            new_level = (*_level)[v];

          }

        }

        for (InArcIt 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)[v] += excess;

              excess = 0;

              goto no_more_push;

            } else {

              excess -= rem;

              (*_excess)[v] += rem;

              _flow->set(e, 0);

            }

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

            new_level = (*_level)[v];

          }

        }

      no_more_push:

        (*_excess)[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 results of the preflow algorithm can be obtained using these

    /// functions.\n

    /// Either one of the \ref run() "run*()" functions or one of the

    /// \ref startFirstPhase() "start*()" functions should be called

    /// before using them.

    ///@{

    /// \brief Returns the value of the maximum flow.

    ///

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

    /// of the target node. This value equals to the value of

    /// the maximum flow already after the first phase of the algorithm.

    ///

    /// \pre Either \ref run() or \ref init() must be called before

    /// using this function.

    Value flowValue() const {

      return (*_excess)[_target];

    }

    /// \brief Returns the flow value on the given arc.

    ///

    /// Returns the flow value on the given arc. This method can

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

    ///

    /// \pre Either \ref run() or \ref init() must be called before

    /// using this function.

    Value flow(const Arc& arc) const {

      return (*_flow)[arc];

    }

    /// \brief Returns a const reference to the flow map.

    ///

    /// Returns a const reference to the arc map storing the found flow.

    /// This method can be called after the second phase of the algorithm.

    ///

    /// \pre Either \ref run() or \ref init() must be called before

    /// using this function.

    const FlowMap& flowMap() const {

      return *_flow;

    }

    /// \brief Returns \c true when the node is on the source side of the

    /// minimum cut.

    ///

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

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

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

    ///

    /// \pre Either \ref run() or \ref init() must be called before

    /// using this function.

    bool minCut(const Node& node) const {

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

    }

    /// \brief Gives back a minimum value cut.

    ///

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

    /// cut. \c cutMap should be a \ref concepts::WriteMap "writable"

    /// node map with \c bool (or convertible) value type.

    ///

    /// This method can be called both after running \ref startFirstPhase()

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

    /// could be slightly different if inexact computation is used.

    ///

    /// \note This function calls \ref minCut() for each node, so it runs in

    /// O(n) time.

    ///

    /// \pre Either \ref run() or \ref init() must be called before

    /// using this function.

    template <typename CutMap>

    void minCutMap(CutMap& cutMap) const {

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

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

      }

    }

    /// @}

  };

}

#endif