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

  *

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

  *

  * Copyright (C) 2003-2013

  * 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 \cite clrs01algorithms,

   /// \cite amo93networkflows, \cite 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$O(n^2\sqrt{m})\f$.

   ///

   /// 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 lemon::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;

         }

       }

       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;

         }

       }

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

         if(n!=_source && n!=_target && _tolerance.positive((*_excess)[n]))

           _level->activate(n);

       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;

       while (true) {

         int num = _node_num;

         Node n = INVALID;

         int level = -1;

         while (num > 0) {

           n = _level->highestActive();

           if (n == INVALID) goto first_phase_done;

           level = _level->highestActiveLevel();

           --num;

           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);

           }

         }

         num = _node_num * 20;

         while (num > 0) {

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

             --level;

           }

           if (level == -1) {

             n = _level->highestActive();

             level = _level->highestActiveLevel();

             if (n == INVALID) goto first_phase_done;

           } else {

             n = _level->activeOn(level);

           }

           --num;

           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);

           }

         }

       }

     first_phase_done:;

     }

     /// \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