lemon/preflow.h
author Peter Kovacs <kpeter@inf.elte.hu>
Tue, 14 Apr 2009 10:33:17 +0200
changeset 579 d11bf7998905
parent 492 9605e051942f
child 581 aa1804409f29
permissions -rw-r--r--
Various improvements and fixes (mainly in the doc) (#190)
     1 /* -*- mode: C++; indent-tabs-mode: nil; -*-
     2  *
     3  * This file is a part of LEMON, a generic C++ optimization library.
     4  *
     5  * Copyright (C) 2003-2009
     6  * Egervary Jeno Kombinatorikus Optimalizalasi Kutatocsoport
     7  * (Egervary Research Group on Combinatorial Optimization, EGRES).
     8  *
     9  * Permission to use, modify and distribute this software is granted
    10  * provided that this copyright notice appears in all copies. For
    11  * precise terms see the accompanying LICENSE file.
    12  *
    13  * This software is provided "AS IS" with no warranty of any kind,
    14  * express or implied, and with no claim as to its suitability for any
    15  * purpose.
    16  *
    17  */
    18 
    19 #ifndef LEMON_PREFLOW_H
    20 #define LEMON_PREFLOW_H
    21 
    22 #include <lemon/tolerance.h>
    23 #include <lemon/elevator.h>
    24 
    25 /// \file
    26 /// \ingroup max_flow
    27 /// \brief Implementation of the preflow algorithm.
    28 
    29 namespace lemon {
    30 
    31   /// \brief Default traits class of Preflow class.
    32   ///
    33   /// Default traits class of Preflow class.
    34   /// \tparam GR Digraph type.
    35   /// \tparam CAP Capacity map type.
    36   template <typename GR, typename CAP>
    37   struct PreflowDefaultTraits {
    38 
    39     /// \brief The type of the digraph the algorithm runs on.
    40     typedef GR Digraph;
    41 
    42     /// \brief The type of the map that stores the arc capacities.
    43     ///
    44     /// The type of the map that stores the arc capacities.
    45     /// It must meet the \ref concepts::ReadMap "ReadMap" concept.
    46     typedef CAP CapacityMap;
    47 
    48     /// \brief The type of the flow values.
    49     typedef typename CapacityMap::Value Value;
    50 
    51     /// \brief The type of the map that stores the flow values.
    52     ///
    53     /// The type of the map that stores the flow values.
    54     /// It must meet the \ref concepts::ReadWriteMap "ReadWriteMap" concept.
    55     typedef typename Digraph::template ArcMap<Value> FlowMap;
    56 
    57     /// \brief Instantiates a FlowMap.
    58     ///
    59     /// This function instantiates a \ref FlowMap.
    60     /// \param digraph The digraph, to which we would like to define
    61     /// the flow map.
    62     static FlowMap* createFlowMap(const Digraph& digraph) {
    63       return new FlowMap(digraph);
    64     }
    65 
    66     /// \brief The elevator type used by Preflow algorithm.
    67     ///
    68     /// The elevator type used by Preflow algorithm.
    69     ///
    70     /// \sa Elevator
    71     /// \sa LinkedElevator
    72     typedef LinkedElevator<Digraph, typename Digraph::Node> Elevator;
    73 
    74     /// \brief Instantiates an Elevator.
    75     ///
    76     /// This function instantiates an \ref Elevator.
    77     /// \param digraph The digraph, to which we would like to define
    78     /// the elevator.
    79     /// \param max_level The maximum level of the elevator.
    80     static Elevator* createElevator(const Digraph& digraph, int max_level) {
    81       return new Elevator(digraph, max_level);
    82     }
    83 
    84     /// \brief The tolerance used by the algorithm
    85     ///
    86     /// The tolerance used by the algorithm to handle inexact computation.
    87     typedef lemon::Tolerance<Value> Tolerance;
    88 
    89   };
    90 
    91 
    92   /// \ingroup max_flow
    93   ///
    94   /// \brief %Preflow algorithm class.
    95   ///
    96   /// This class provides an implementation of Goldberg-Tarjan's \e preflow
    97   /// \e push-relabel algorithm producing a \ref max_flow
    98   /// "flow of maximum value" in a digraph.
    99   /// The preflow algorithms are the fastest known maximum
   100   /// flow algorithms. The current implementation use a mixture of the
   101   /// \e "highest label" and the \e "bound decrease" heuristics.
   102   /// The worst case time complexity of the algorithm is \f$O(n^2\sqrt{e})\f$.
   103   ///
   104   /// The algorithm consists of two phases. After the first phase
   105   /// the maximum flow value and the minimum cut is obtained. The
   106   /// second phase constructs a feasible maximum flow on each arc.
   107   ///
   108   /// \tparam GR The type of the digraph the algorithm runs on.
   109   /// \tparam CAP The type of the capacity map. The default map
   110   /// type is \ref concepts::Digraph::ArcMap "GR::ArcMap<int>".
   111 #ifdef DOXYGEN
   112   template <typename GR, typename CAP, typename TR>
   113 #else
   114   template <typename GR,
   115             typename CAP = typename GR::template ArcMap<int>,
   116             typename TR = PreflowDefaultTraits<GR, CAP> >
   117 #endif
   118   class Preflow {
   119   public:
   120 
   121     ///The \ref PreflowDefaultTraits "traits class" of the algorithm.
   122     typedef TR Traits;
   123     ///The type of the digraph the algorithm runs on.
   124     typedef typename Traits::Digraph Digraph;
   125     ///The type of the capacity map.
   126     typedef typename Traits::CapacityMap CapacityMap;
   127     ///The type of the flow values.
   128     typedef typename Traits::Value Value;
   129 
   130     ///The type of the flow map.
   131     typedef typename Traits::FlowMap FlowMap;
   132     ///The type of the elevator.
   133     typedef typename Traits::Elevator Elevator;
   134     ///The type of the tolerance.
   135     typedef typename Traits::Tolerance Tolerance;
   136 
   137   private:
   138 
   139     TEMPLATE_DIGRAPH_TYPEDEFS(Digraph);
   140 
   141     const Digraph& _graph;
   142     const CapacityMap* _capacity;
   143 
   144     int _node_num;
   145 
   146     Node _source, _target;
   147 
   148     FlowMap* _flow;
   149     bool _local_flow;
   150 
   151     Elevator* _level;
   152     bool _local_level;
   153 
   154     typedef typename Digraph::template NodeMap<Value> ExcessMap;
   155     ExcessMap* _excess;
   156 
   157     Tolerance _tolerance;
   158 
   159     bool _phase;
   160 
   161 
   162     void createStructures() {
   163       _node_num = countNodes(_graph);
   164 
   165       if (!_flow) {
   166         _flow = Traits::createFlowMap(_graph);
   167         _local_flow = true;
   168       }
   169       if (!_level) {
   170         _level = Traits::createElevator(_graph, _node_num);
   171         _local_level = true;
   172       }
   173       if (!_excess) {
   174         _excess = new ExcessMap(_graph);
   175       }
   176     }
   177 
   178     void destroyStructures() {
   179       if (_local_flow) {
   180         delete _flow;
   181       }
   182       if (_local_level) {
   183         delete _level;
   184       }
   185       if (_excess) {
   186         delete _excess;
   187       }
   188     }
   189 
   190   public:
   191 
   192     typedef Preflow Create;
   193 
   194     ///\name Named Template Parameters
   195 
   196     ///@{
   197 
   198     template <typename T>
   199     struct SetFlowMapTraits : public Traits {
   200       typedef T FlowMap;
   201       static FlowMap *createFlowMap(const Digraph&) {
   202         LEMON_ASSERT(false, "FlowMap is not initialized");
   203         return 0; // ignore warnings
   204       }
   205     };
   206 
   207     /// \brief \ref named-templ-param "Named parameter" for setting
   208     /// FlowMap type
   209     ///
   210     /// \ref named-templ-param "Named parameter" for setting FlowMap
   211     /// type.
   212     template <typename T>
   213     struct SetFlowMap
   214       : public Preflow<Digraph, CapacityMap, SetFlowMapTraits<T> > {
   215       typedef Preflow<Digraph, CapacityMap,
   216                       SetFlowMapTraits<T> > Create;
   217     };
   218 
   219     template <typename T>
   220     struct SetElevatorTraits : public Traits {
   221       typedef T Elevator;
   222       static Elevator *createElevator(const Digraph&, int) {
   223         LEMON_ASSERT(false, "Elevator is not initialized");
   224         return 0; // ignore warnings
   225       }
   226     };
   227 
   228     /// \brief \ref named-templ-param "Named parameter" for setting
   229     /// Elevator type
   230     ///
   231     /// \ref named-templ-param "Named parameter" for setting Elevator
   232     /// type. If this named parameter is used, then an external
   233     /// elevator object must be passed to the algorithm using the
   234     /// \ref elevator(Elevator&) "elevator()" function before calling
   235     /// \ref run() or \ref init().
   236     /// \sa SetStandardElevator
   237     template <typename T>
   238     struct SetElevator
   239       : public Preflow<Digraph, CapacityMap, SetElevatorTraits<T> > {
   240       typedef Preflow<Digraph, CapacityMap,
   241                       SetElevatorTraits<T> > Create;
   242     };
   243 
   244     template <typename T>
   245     struct SetStandardElevatorTraits : public Traits {
   246       typedef T Elevator;
   247       static Elevator *createElevator(const Digraph& digraph, int max_level) {
   248         return new Elevator(digraph, max_level);
   249       }
   250     };
   251 
   252     /// \brief \ref named-templ-param "Named parameter" for setting
   253     /// Elevator type with automatic allocation
   254     ///
   255     /// \ref named-templ-param "Named parameter" for setting Elevator
   256     /// type with automatic allocation.
   257     /// The Elevator should have standard constructor interface to be
   258     /// able to automatically created by the algorithm (i.e. the
   259     /// digraph and the maximum level should be passed to it).
   260     /// However an external elevator object could also be passed to the
   261     /// algorithm with the \ref elevator(Elevator&) "elevator()" function
   262     /// before calling \ref run() or \ref init().
   263     /// \sa SetElevator
   264     template <typename T>
   265     struct SetStandardElevator
   266       : public Preflow<Digraph, CapacityMap,
   267                        SetStandardElevatorTraits<T> > {
   268       typedef Preflow<Digraph, CapacityMap,
   269                       SetStandardElevatorTraits<T> > Create;
   270     };
   271 
   272     /// @}
   273 
   274   protected:
   275 
   276     Preflow() {}
   277 
   278   public:
   279 
   280 
   281     /// \brief The constructor of the class.
   282     ///
   283     /// The constructor of the class.
   284     /// \param digraph The digraph the algorithm runs on.
   285     /// \param capacity The capacity of the arcs.
   286     /// \param source The source node.
   287     /// \param target The target node.
   288     Preflow(const Digraph& digraph, const CapacityMap& capacity,
   289             Node source, Node target)
   290       : _graph(digraph), _capacity(&capacity),
   291         _node_num(0), _source(source), _target(target),
   292         _flow(0), _local_flow(false),
   293         _level(0), _local_level(false),
   294         _excess(0), _tolerance(), _phase() {}
   295 
   296     /// \brief Destructor.
   297     ///
   298     /// Destructor.
   299     ~Preflow() {
   300       destroyStructures();
   301     }
   302 
   303     /// \brief Sets the capacity map.
   304     ///
   305     /// Sets the capacity map.
   306     /// \return <tt>(*this)</tt>
   307     Preflow& capacityMap(const CapacityMap& map) {
   308       _capacity = &map;
   309       return *this;
   310     }
   311 
   312     /// \brief Sets the flow map.
   313     ///
   314     /// Sets the flow map.
   315     /// If you don't use this function before calling \ref run() or
   316     /// \ref init(), an instance will be allocated automatically.
   317     /// The destructor deallocates this automatically allocated map,
   318     /// of course.
   319     /// \return <tt>(*this)</tt>
   320     Preflow& flowMap(FlowMap& map) {
   321       if (_local_flow) {
   322         delete _flow;
   323         _local_flow = false;
   324       }
   325       _flow = &map;
   326       return *this;
   327     }
   328 
   329     /// \brief Sets the source node.
   330     ///
   331     /// Sets the source node.
   332     /// \return <tt>(*this)</tt>
   333     Preflow& source(const Node& node) {
   334       _source = node;
   335       return *this;
   336     }
   337 
   338     /// \brief Sets the target node.
   339     ///
   340     /// Sets the target node.
   341     /// \return <tt>(*this)</tt>
   342     Preflow& target(const Node& node) {
   343       _target = node;
   344       return *this;
   345     }
   346 
   347     /// \brief Sets the elevator used by algorithm.
   348     ///
   349     /// Sets the elevator used by algorithm.
   350     /// If you don't use this function before calling \ref run() or
   351     /// \ref init(), an instance will be allocated automatically.
   352     /// The destructor deallocates this automatically allocated elevator,
   353     /// of course.
   354     /// \return <tt>(*this)</tt>
   355     Preflow& elevator(Elevator& elevator) {
   356       if (_local_level) {
   357         delete _level;
   358         _local_level = false;
   359       }
   360       _level = &elevator;
   361       return *this;
   362     }
   363 
   364     /// \brief Returns a const reference to the elevator.
   365     ///
   366     /// Returns a const reference to the elevator.
   367     ///
   368     /// \pre Either \ref run() or \ref init() must be called before
   369     /// using this function.
   370     const Elevator& elevator() const {
   371       return *_level;
   372     }
   373 
   374     /// \brief Sets the tolerance used by algorithm.
   375     ///
   376     /// Sets the tolerance used by algorithm.
   377     Preflow& tolerance(const Tolerance& tolerance) const {
   378       _tolerance = tolerance;
   379       return *this;
   380     }
   381 
   382     /// \brief Returns a const reference to the tolerance.
   383     ///
   384     /// Returns a const reference to the tolerance.
   385     const Tolerance& tolerance() const {
   386       return tolerance;
   387     }
   388 
   389     /// \name Execution Control
   390     /// The simplest way to execute the preflow algorithm is to use
   391     /// \ref run() or \ref runMinCut().\n
   392     /// If you need more control on the initial solution or the execution,
   393     /// first you have to call one of the \ref init() functions, then
   394     /// \ref startFirstPhase() and if you need it \ref startSecondPhase().
   395 
   396     ///@{
   397 
   398     /// \brief Initializes the internal data structures.
   399     ///
   400     /// Initializes the internal data structures and sets the initial
   401     /// flow to zero on each arc.
   402     void init() {
   403       createStructures();
   404 
   405       _phase = true;
   406       for (NodeIt n(_graph); n != INVALID; ++n) {
   407         _excess->set(n, 0);
   408       }
   409 
   410       for (ArcIt e(_graph); e != INVALID; ++e) {
   411         _flow->set(e, 0);
   412       }
   413 
   414       typename Digraph::template NodeMap<bool> reached(_graph, false);
   415 
   416       _level->initStart();
   417       _level->initAddItem(_target);
   418 
   419       std::vector<Node> queue;
   420       reached.set(_source, true);
   421 
   422       queue.push_back(_target);
   423       reached.set(_target, true);
   424       while (!queue.empty()) {
   425         _level->initNewLevel();
   426         std::vector<Node> nqueue;
   427         for (int i = 0; i < int(queue.size()); ++i) {
   428           Node n = queue[i];
   429           for (InArcIt e(_graph, n); e != INVALID; ++e) {
   430             Node u = _graph.source(e);
   431             if (!reached[u] && _tolerance.positive((*_capacity)[e])) {
   432               reached.set(u, true);
   433               _level->initAddItem(u);
   434               nqueue.push_back(u);
   435             }
   436           }
   437         }
   438         queue.swap(nqueue);
   439       }
   440       _level->initFinish();
   441 
   442       for (OutArcIt e(_graph, _source); e != INVALID; ++e) {
   443         if (_tolerance.positive((*_capacity)[e])) {
   444           Node u = _graph.target(e);
   445           if ((*_level)[u] == _level->maxLevel()) continue;
   446           _flow->set(e, (*_capacity)[e]);
   447           _excess->set(u, (*_excess)[u] + (*_capacity)[e]);
   448           if (u != _target && !_level->active(u)) {
   449             _level->activate(u);
   450           }
   451         }
   452       }
   453     }
   454 
   455     /// \brief Initializes the internal data structures using the
   456     /// given flow map.
   457     ///
   458     /// Initializes the internal data structures and sets the initial
   459     /// flow to the given \c flowMap. The \c flowMap should contain a
   460     /// flow or at least a preflow, i.e. at each node excluding the
   461     /// source node the incoming flow should greater or equal to the
   462     /// outgoing flow.
   463     /// \return \c false if the given \c flowMap is not a preflow.
   464     template <typename FlowMap>
   465     bool init(const FlowMap& flowMap) {
   466       createStructures();
   467 
   468       for (ArcIt e(_graph); e != INVALID; ++e) {
   469         _flow->set(e, flowMap[e]);
   470       }
   471 
   472       for (NodeIt n(_graph); n != INVALID; ++n) {
   473         Value excess = 0;
   474         for (InArcIt e(_graph, n); e != INVALID; ++e) {
   475           excess += (*_flow)[e];
   476         }
   477         for (OutArcIt e(_graph, n); e != INVALID; ++e) {
   478           excess -= (*_flow)[e];
   479         }
   480         if (excess < 0 && n != _source) return false;
   481         _excess->set(n, excess);
   482       }
   483 
   484       typename Digraph::template NodeMap<bool> reached(_graph, false);
   485 
   486       _level->initStart();
   487       _level->initAddItem(_target);
   488 
   489       std::vector<Node> queue;
   490       reached.set(_source, true);
   491 
   492       queue.push_back(_target);
   493       reached.set(_target, true);
   494       while (!queue.empty()) {
   495         _level->initNewLevel();
   496         std::vector<Node> nqueue;
   497         for (int i = 0; i < int(queue.size()); ++i) {
   498           Node n = queue[i];
   499           for (InArcIt e(_graph, n); e != INVALID; ++e) {
   500             Node u = _graph.source(e);
   501             if (!reached[u] &&
   502                 _tolerance.positive((*_capacity)[e] - (*_flow)[e])) {
   503               reached.set(u, true);
   504               _level->initAddItem(u);
   505               nqueue.push_back(u);
   506             }
   507           }
   508           for (OutArcIt e(_graph, n); e != INVALID; ++e) {
   509             Node v = _graph.target(e);
   510             if (!reached[v] && _tolerance.positive((*_flow)[e])) {
   511               reached.set(v, true);
   512               _level->initAddItem(v);
   513               nqueue.push_back(v);
   514             }
   515           }
   516         }
   517         queue.swap(nqueue);
   518       }
   519       _level->initFinish();
   520 
   521       for (OutArcIt e(_graph, _source); e != INVALID; ++e) {
   522         Value rem = (*_capacity)[e] - (*_flow)[e];
   523         if (_tolerance.positive(rem)) {
   524           Node u = _graph.target(e);
   525           if ((*_level)[u] == _level->maxLevel()) continue;
   526           _flow->set(e, (*_capacity)[e]);
   527           _excess->set(u, (*_excess)[u] + rem);
   528           if (u != _target && !_level->active(u)) {
   529             _level->activate(u);
   530           }
   531         }
   532       }
   533       for (InArcIt e(_graph, _source); e != INVALID; ++e) {
   534         Value rem = (*_flow)[e];
   535         if (_tolerance.positive(rem)) {
   536           Node v = _graph.source(e);
   537           if ((*_level)[v] == _level->maxLevel()) continue;
   538           _flow->set(e, 0);
   539           _excess->set(v, (*_excess)[v] + rem);
   540           if (v != _target && !_level->active(v)) {
   541             _level->activate(v);
   542           }
   543         }
   544       }
   545       return true;
   546     }
   547 
   548     /// \brief Starts the first phase of the preflow algorithm.
   549     ///
   550     /// The preflow algorithm consists of two phases, this method runs
   551     /// the first phase. After the first phase the maximum flow value
   552     /// and a minimum value cut can already be computed, although a
   553     /// maximum flow is not yet obtained. So after calling this method
   554     /// \ref flowValue() returns the value of a maximum flow and \ref
   555     /// minCut() returns a minimum cut.
   556     /// \pre One of the \ref init() functions must be called before
   557     /// using this function.
   558     void startFirstPhase() {
   559       _phase = true;
   560 
   561       Node n = _level->highestActive();
   562       int level = _level->highestActiveLevel();
   563       while (n != INVALID) {
   564         int num = _node_num;
   565 
   566         while (num > 0 && n != INVALID) {
   567           Value excess = (*_excess)[n];
   568           int new_level = _level->maxLevel();
   569 
   570           for (OutArcIt e(_graph, n); e != INVALID; ++e) {
   571             Value rem = (*_capacity)[e] - (*_flow)[e];
   572             if (!_tolerance.positive(rem)) continue;
   573             Node v = _graph.target(e);
   574             if ((*_level)[v] < level) {
   575               if (!_level->active(v) && v != _target) {
   576                 _level->activate(v);
   577               }
   578               if (!_tolerance.less(rem, excess)) {
   579                 _flow->set(e, (*_flow)[e] + excess);
   580                 _excess->set(v, (*_excess)[v] + excess);
   581                 excess = 0;
   582                 goto no_more_push_1;
   583               } else {
   584                 excess -= rem;
   585                 _excess->set(v, (*_excess)[v] + rem);
   586                 _flow->set(e, (*_capacity)[e]);
   587               }
   588             } else if (new_level > (*_level)[v]) {
   589               new_level = (*_level)[v];
   590             }
   591           }
   592 
   593           for (InArcIt e(_graph, n); e != INVALID; ++e) {
   594             Value rem = (*_flow)[e];
   595             if (!_tolerance.positive(rem)) continue;
   596             Node v = _graph.source(e);
   597             if ((*_level)[v] < level) {
   598               if (!_level->active(v) && v != _target) {
   599                 _level->activate(v);
   600               }
   601               if (!_tolerance.less(rem, excess)) {
   602                 _flow->set(e, (*_flow)[e] - excess);
   603                 _excess->set(v, (*_excess)[v] + excess);
   604                 excess = 0;
   605                 goto no_more_push_1;
   606               } else {
   607                 excess -= rem;
   608                 _excess->set(v, (*_excess)[v] + rem);
   609                 _flow->set(e, 0);
   610               }
   611             } else if (new_level > (*_level)[v]) {
   612               new_level = (*_level)[v];
   613             }
   614           }
   615 
   616         no_more_push_1:
   617 
   618           _excess->set(n, excess);
   619 
   620           if (excess != 0) {
   621             if (new_level + 1 < _level->maxLevel()) {
   622               _level->liftHighestActive(new_level + 1);
   623             } else {
   624               _level->liftHighestActiveToTop();
   625             }
   626             if (_level->emptyLevel(level)) {
   627               _level->liftToTop(level);
   628             }
   629           } else {
   630             _level->deactivate(n);
   631           }
   632 
   633           n = _level->highestActive();
   634           level = _level->highestActiveLevel();
   635           --num;
   636         }
   637 
   638         num = _node_num * 20;
   639         while (num > 0 && n != INVALID) {
   640           Value excess = (*_excess)[n];
   641           int new_level = _level->maxLevel();
   642 
   643           for (OutArcIt e(_graph, n); e != INVALID; ++e) {
   644             Value rem = (*_capacity)[e] - (*_flow)[e];
   645             if (!_tolerance.positive(rem)) continue;
   646             Node v = _graph.target(e);
   647             if ((*_level)[v] < level) {
   648               if (!_level->active(v) && v != _target) {
   649                 _level->activate(v);
   650               }
   651               if (!_tolerance.less(rem, excess)) {
   652                 _flow->set(e, (*_flow)[e] + excess);
   653                 _excess->set(v, (*_excess)[v] + excess);
   654                 excess = 0;
   655                 goto no_more_push_2;
   656               } else {
   657                 excess -= rem;
   658                 _excess->set(v, (*_excess)[v] + rem);
   659                 _flow->set(e, (*_capacity)[e]);
   660               }
   661             } else if (new_level > (*_level)[v]) {
   662               new_level = (*_level)[v];
   663             }
   664           }
   665 
   666           for (InArcIt e(_graph, n); e != INVALID; ++e) {
   667             Value rem = (*_flow)[e];
   668             if (!_tolerance.positive(rem)) continue;
   669             Node v = _graph.source(e);
   670             if ((*_level)[v] < level) {
   671               if (!_level->active(v) && v != _target) {
   672                 _level->activate(v);
   673               }
   674               if (!_tolerance.less(rem, excess)) {
   675                 _flow->set(e, (*_flow)[e] - excess);
   676                 _excess->set(v, (*_excess)[v] + excess);
   677                 excess = 0;
   678                 goto no_more_push_2;
   679               } else {
   680                 excess -= rem;
   681                 _excess->set(v, (*_excess)[v] + rem);
   682                 _flow->set(e, 0);
   683               }
   684             } else if (new_level > (*_level)[v]) {
   685               new_level = (*_level)[v];
   686             }
   687           }
   688 
   689         no_more_push_2:
   690 
   691           _excess->set(n, excess);
   692 
   693           if (excess != 0) {
   694             if (new_level + 1 < _level->maxLevel()) {
   695               _level->liftActiveOn(level, new_level + 1);
   696             } else {
   697               _level->liftActiveToTop(level);
   698             }
   699             if (_level->emptyLevel(level)) {
   700               _level->liftToTop(level);
   701             }
   702           } else {
   703             _level->deactivate(n);
   704           }
   705 
   706           while (level >= 0 && _level->activeFree(level)) {
   707             --level;
   708           }
   709           if (level == -1) {
   710             n = _level->highestActive();
   711             level = _level->highestActiveLevel();
   712           } else {
   713             n = _level->activeOn(level);
   714           }
   715           --num;
   716         }
   717       }
   718     }
   719 
   720     /// \brief Starts the second phase of the preflow algorithm.
   721     ///
   722     /// The preflow algorithm consists of two phases, this method runs
   723     /// the second phase. After calling one of the \ref init() functions
   724     /// and \ref startFirstPhase() and then \ref startSecondPhase(),
   725     /// \ref flowMap() returns a maximum flow, \ref flowValue() returns the
   726     /// value of a maximum flow, \ref minCut() returns a minimum cut
   727     /// \pre One of the \ref init() functions and \ref startFirstPhase()
   728     /// must be called before using this function.
   729     void startSecondPhase() {
   730       _phase = false;
   731 
   732       typename Digraph::template NodeMap<bool> reached(_graph);
   733       for (NodeIt n(_graph); n != INVALID; ++n) {
   734         reached.set(n, (*_level)[n] < _level->maxLevel());
   735       }
   736 
   737       _level->initStart();
   738       _level->initAddItem(_source);
   739 
   740       std::vector<Node> queue;
   741       queue.push_back(_source);
   742       reached.set(_source, true);
   743 
   744       while (!queue.empty()) {
   745         _level->initNewLevel();
   746         std::vector<Node> nqueue;
   747         for (int i = 0; i < int(queue.size()); ++i) {
   748           Node n = queue[i];
   749           for (OutArcIt e(_graph, n); e != INVALID; ++e) {
   750             Node v = _graph.target(e);
   751             if (!reached[v] && _tolerance.positive((*_flow)[e])) {
   752               reached.set(v, true);
   753               _level->initAddItem(v);
   754               nqueue.push_back(v);
   755             }
   756           }
   757           for (InArcIt e(_graph, n); e != INVALID; ++e) {
   758             Node u = _graph.source(e);
   759             if (!reached[u] &&
   760                 _tolerance.positive((*_capacity)[e] - (*_flow)[e])) {
   761               reached.set(u, true);
   762               _level->initAddItem(u);
   763               nqueue.push_back(u);
   764             }
   765           }
   766         }
   767         queue.swap(nqueue);
   768       }
   769       _level->initFinish();
   770 
   771       for (NodeIt n(_graph); n != INVALID; ++n) {
   772         if (!reached[n]) {
   773           _level->dirtyTopButOne(n);
   774         } else if ((*_excess)[n] > 0 && _target != n) {
   775           _level->activate(n);
   776         }
   777       }
   778 
   779       Node n;
   780       while ((n = _level->highestActive()) != INVALID) {
   781         Value excess = (*_excess)[n];
   782         int level = _level->highestActiveLevel();
   783         int new_level = _level->maxLevel();
   784 
   785         for (OutArcIt e(_graph, n); e != INVALID; ++e) {
   786           Value rem = (*_capacity)[e] - (*_flow)[e];
   787           if (!_tolerance.positive(rem)) continue;
   788           Node v = _graph.target(e);
   789           if ((*_level)[v] < level) {
   790             if (!_level->active(v) && v != _source) {
   791               _level->activate(v);
   792             }
   793             if (!_tolerance.less(rem, excess)) {
   794               _flow->set(e, (*_flow)[e] + excess);
   795               _excess->set(v, (*_excess)[v] + excess);
   796               excess = 0;
   797               goto no_more_push;
   798             } else {
   799               excess -= rem;
   800               _excess->set(v, (*_excess)[v] + rem);
   801               _flow->set(e, (*_capacity)[e]);
   802             }
   803           } else if (new_level > (*_level)[v]) {
   804             new_level = (*_level)[v];
   805           }
   806         }
   807 
   808         for (InArcIt e(_graph, n); e != INVALID; ++e) {
   809           Value rem = (*_flow)[e];
   810           if (!_tolerance.positive(rem)) continue;
   811           Node v = _graph.source(e);
   812           if ((*_level)[v] < level) {
   813             if (!_level->active(v) && v != _source) {
   814               _level->activate(v);
   815             }
   816             if (!_tolerance.less(rem, excess)) {
   817               _flow->set(e, (*_flow)[e] - excess);
   818               _excess->set(v, (*_excess)[v] + excess);
   819               excess = 0;
   820               goto no_more_push;
   821             } else {
   822               excess -= rem;
   823               _excess->set(v, (*_excess)[v] + rem);
   824               _flow->set(e, 0);
   825             }
   826           } else if (new_level > (*_level)[v]) {
   827             new_level = (*_level)[v];
   828           }
   829         }
   830 
   831       no_more_push:
   832 
   833         _excess->set(n, excess);
   834 
   835         if (excess != 0) {
   836           if (new_level + 1 < _level->maxLevel()) {
   837             _level->liftHighestActive(new_level + 1);
   838           } else {
   839             // Calculation error
   840             _level->liftHighestActiveToTop();
   841           }
   842           if (_level->emptyLevel(level)) {
   843             // Calculation error
   844             _level->liftToTop(level);
   845           }
   846         } else {
   847           _level->deactivate(n);
   848         }
   849 
   850       }
   851     }
   852 
   853     /// \brief Runs the preflow algorithm.
   854     ///
   855     /// Runs the preflow algorithm.
   856     /// \note pf.run() is just a shortcut of the following code.
   857     /// \code
   858     ///   pf.init();
   859     ///   pf.startFirstPhase();
   860     ///   pf.startSecondPhase();
   861     /// \endcode
   862     void run() {
   863       init();
   864       startFirstPhase();
   865       startSecondPhase();
   866     }
   867 
   868     /// \brief Runs the preflow algorithm to compute the minimum cut.
   869     ///
   870     /// Runs the preflow algorithm to compute the minimum cut.
   871     /// \note pf.runMinCut() is just a shortcut of the following code.
   872     /// \code
   873     ///   pf.init();
   874     ///   pf.startFirstPhase();
   875     /// \endcode
   876     void runMinCut() {
   877       init();
   878       startFirstPhase();
   879     }
   880 
   881     /// @}
   882 
   883     /// \name Query Functions
   884     /// The results of the preflow algorithm can be obtained using these
   885     /// functions.\n
   886     /// Either one of the \ref run() "run*()" functions or one of the
   887     /// \ref startFirstPhase() "start*()" functions should be called
   888     /// before using them.
   889 
   890     ///@{
   891 
   892     /// \brief Returns the value of the maximum flow.
   893     ///
   894     /// Returns the value of the maximum flow by returning the excess
   895     /// of the target node. This value equals to the value of
   896     /// the maximum flow already after the first phase of the algorithm.
   897     ///
   898     /// \pre Either \ref run() or \ref init() must be called before
   899     /// using this function.
   900     Value flowValue() const {
   901       return (*_excess)[_target];
   902     }
   903 
   904     /// \brief Returns the flow on the given arc.
   905     ///
   906     /// Returns the flow on the given arc. This method can
   907     /// be called after the second phase of the algorithm.
   908     ///
   909     /// \pre Either \ref run() or \ref init() must be called before
   910     /// using this function.
   911     Value flow(const Arc& arc) const {
   912       return (*_flow)[arc];
   913     }
   914 
   915     /// \brief Returns a const reference to the flow map.
   916     ///
   917     /// Returns a const reference to the arc map storing the found flow.
   918     /// This method can be called after the second phase of the algorithm.
   919     ///
   920     /// \pre Either \ref run() or \ref init() must be called before
   921     /// using this function.
   922     const FlowMap& flowMap() const {
   923       return *_flow;
   924     }
   925 
   926     /// \brief Returns \c true when the node is on the source side of the
   927     /// minimum cut.
   928     ///
   929     /// Returns true when the node is on the source side of the found
   930     /// minimum cut. This method can be called both after running \ref
   931     /// startFirstPhase() and \ref startSecondPhase().
   932     ///
   933     /// \pre Either \ref run() or \ref init() must be called before
   934     /// using this function.
   935     bool minCut(const Node& node) const {
   936       return ((*_level)[node] == _level->maxLevel()) == _phase;
   937     }
   938 
   939     /// \brief Gives back a minimum value cut.
   940     ///
   941     /// Sets \c cutMap to the characteristic vector of a minimum value
   942     /// cut. \c cutMap should be a \ref concepts::WriteMap "writable"
   943     /// node map with \c bool (or convertible) value type.
   944     ///
   945     /// This method can be called both after running \ref startFirstPhase()
   946     /// and \ref startSecondPhase(). The result after the second phase
   947     /// could be slightly different if inexact computation is used.
   948     ///
   949     /// \note This function calls \ref minCut() for each node, so it runs in
   950     /// O(n) time.
   951     ///
   952     /// \pre Either \ref run() or \ref init() must be called before
   953     /// using this function.
   954     template <typename CutMap>
   955     void minCutMap(CutMap& cutMap) const {
   956       for (NodeIt n(_graph); n != INVALID; ++n) {
   957         cutMap.set(n, minCut(n));
   958       }
   959     }
   960 
   961     /// @}
   962   };
   963 }
   964 
   965 #endif