lemon/suurballe.h
author Peter Kovacs <kpeter@inf.elte.hu>
Tue, 02 Mar 2010 10:27:47 +0100
changeset 858 9f6ed854d409
parent 854 9a7e4e606f83
child 863 a93f1a27d831
permissions -rw-r--r--
Also test fullInit() in suurballe_test (#181, #323)
     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_SUURBALLE_H
    20 #define LEMON_SUURBALLE_H
    21 
    22 ///\ingroup shortest_path
    23 ///\file
    24 ///\brief An algorithm for finding arc-disjoint paths between two
    25 /// nodes having minimum total length.
    26 
    27 #include <vector>
    28 #include <limits>
    29 #include <lemon/bin_heap.h>
    30 #include <lemon/path.h>
    31 #include <lemon/list_graph.h>
    32 #include <lemon/dijkstra.h>
    33 #include <lemon/maps.h>
    34 
    35 namespace lemon {
    36 
    37   /// \brief Default traits class of Suurballe algorithm.
    38   ///
    39   /// Default traits class of Suurballe algorithm.
    40   /// \tparam GR The digraph type the algorithm runs on.
    41   /// \tparam LEN The type of the length map.
    42   /// The default value is <tt>GR::ArcMap<int></tt>.
    43 #ifdef DOXYGEN
    44   template <typename GR, typename LEN>
    45 #else
    46   template < typename GR,
    47              typename LEN = typename GR::template ArcMap<int> >
    48 #endif
    49   struct SuurballeDefaultTraits
    50   {
    51     /// The type of the digraph.
    52     typedef GR Digraph;
    53     /// The type of the length map.
    54     typedef LEN LengthMap;
    55     /// The type of the lengths.
    56     typedef typename LEN::Value Length;
    57     /// The type of the flow map.
    58     typedef typename GR::template ArcMap<int> FlowMap;
    59     /// The type of the potential map.
    60     typedef typename GR::template NodeMap<Length> PotentialMap;
    61 
    62     /// \brief The path type
    63     ///
    64     /// The type used for storing the found arc-disjoint paths.
    65     /// It must conform to the \ref lemon::concepts::Path "Path" concept
    66     /// and it must have an \c addBack() function.
    67     typedef lemon::Path<Digraph> Path;
    68     
    69     /// The cross reference type used for the heap.
    70     typedef typename GR::template NodeMap<int> HeapCrossRef;
    71 
    72     /// \brief The heap type used for internal Dijkstra computations.
    73     ///
    74     /// The type of the heap used for internal Dijkstra computations.
    75     /// It must conform to the \ref lemon::concepts::Heap "Heap" concept
    76     /// and its priority type must be \c Length.
    77     typedef BinHeap<Length, HeapCrossRef> Heap;
    78   };
    79 
    80   /// \addtogroup shortest_path
    81   /// @{
    82 
    83   /// \brief Algorithm for finding arc-disjoint paths between two nodes
    84   /// having minimum total length.
    85   ///
    86   /// \ref lemon::Suurballe "Suurballe" implements an algorithm for
    87   /// finding arc-disjoint paths having minimum total length (cost)
    88   /// from a given source node to a given target node in a digraph.
    89   ///
    90   /// Note that this problem is a special case of the \ref min_cost_flow
    91   /// "minimum cost flow problem". This implementation is actually an
    92   /// efficient specialized version of the \ref CapacityScaling
    93   /// "successive shortest path" algorithm directly for this problem.
    94   /// Therefore this class provides query functions for flow values and
    95   /// node potentials (the dual solution) just like the minimum cost flow
    96   /// algorithms.
    97   ///
    98   /// \tparam GR The digraph type the algorithm runs on.
    99   /// \tparam LEN The type of the length map.
   100   /// The default value is <tt>GR::ArcMap<int></tt>.
   101   ///
   102   /// \warning Length values should be \e non-negative.
   103   ///
   104   /// \note For finding \e node-disjoint paths, this algorithm can be used
   105   /// along with the \ref SplitNodes adaptor.
   106 #ifdef DOXYGEN
   107   template <typename GR, typename LEN, typename TR>
   108 #else
   109   template < typename GR,
   110              typename LEN = typename GR::template ArcMap<int>,
   111              typename TR = SuurballeDefaultTraits<GR, LEN> >
   112 #endif
   113   class Suurballe
   114   {
   115     TEMPLATE_DIGRAPH_TYPEDEFS(GR);
   116 
   117     typedef ConstMap<Arc, int> ConstArcMap;
   118     typedef typename GR::template NodeMap<Arc> PredMap;
   119 
   120   public:
   121 
   122     /// The type of the digraph.
   123     typedef typename TR::Digraph Digraph;
   124     /// The type of the length map.
   125     typedef typename TR::LengthMap LengthMap;
   126     /// The type of the lengths.
   127     typedef typename TR::Length Length;
   128 
   129     /// The type of the flow map.
   130     typedef typename TR::FlowMap FlowMap;
   131     /// The type of the potential map.
   132     typedef typename TR::PotentialMap PotentialMap;
   133     /// The type of the path structures.
   134     typedef typename TR::Path Path;
   135     /// The cross reference type used for the heap.
   136     typedef typename TR::HeapCrossRef HeapCrossRef;
   137     /// The heap type used for internal Dijkstra computations.
   138     typedef typename TR::Heap Heap;
   139 
   140     /// The \ref SuurballeDefaultTraits "traits class" of the algorithm.
   141     typedef TR Traits;
   142 
   143   private:
   144 
   145     // ResidualDijkstra is a special implementation of the
   146     // Dijkstra algorithm for finding shortest paths in the
   147     // residual network with respect to the reduced arc lengths
   148     // and modifying the node potentials according to the
   149     // distance of the nodes.
   150     class ResidualDijkstra
   151     {
   152     private:
   153 
   154       const Digraph &_graph;
   155       const LengthMap &_length;
   156       const FlowMap &_flow;
   157       PotentialMap &_pi;
   158       PredMap &_pred;
   159       Node _s;
   160       Node _t;
   161       
   162       PotentialMap _dist;
   163       std::vector<Node> _proc_nodes;
   164 
   165     public:
   166 
   167       // Constructor
   168       ResidualDijkstra(Suurballe &srb) :
   169         _graph(srb._graph), _length(srb._length),
   170         _flow(*srb._flow), _pi(*srb._potential), _pred(srb._pred), 
   171         _s(srb._s), _t(srb._t), _dist(_graph) {}
   172         
   173       // Run the algorithm and return true if a path is found
   174       // from the source node to the target node.
   175       bool run(int cnt) {
   176         return cnt == 0 ? startFirst() : start();
   177       }
   178 
   179     private:
   180     
   181       // Execute the algorithm for the first time (the flow and potential
   182       // functions have to be identically zero).
   183       bool startFirst() {
   184         HeapCrossRef heap_cross_ref(_graph, Heap::PRE_HEAP);
   185         Heap heap(heap_cross_ref);
   186         heap.push(_s, 0);
   187         _pred[_s] = INVALID;
   188         _proc_nodes.clear();
   189 
   190         // Process nodes
   191         while (!heap.empty() && heap.top() != _t) {
   192           Node u = heap.top(), v;
   193           Length d = heap.prio(), dn;
   194           _dist[u] = heap.prio();
   195           _proc_nodes.push_back(u);
   196           heap.pop();
   197 
   198           // Traverse outgoing arcs
   199           for (OutArcIt e(_graph, u); e != INVALID; ++e) {
   200             v = _graph.target(e);
   201             switch(heap.state(v)) {
   202               case Heap::PRE_HEAP:
   203                 heap.push(v, d + _length[e]);
   204                 _pred[v] = e;
   205                 break;
   206               case Heap::IN_HEAP:
   207                 dn = d + _length[e];
   208                 if (dn < heap[v]) {
   209                   heap.decrease(v, dn);
   210                   _pred[v] = e;
   211                 }
   212                 break;
   213               case Heap::POST_HEAP:
   214                 break;
   215             }
   216           }
   217         }
   218         if (heap.empty()) return false;
   219 
   220         // Update potentials of processed nodes
   221         Length t_dist = heap.prio();
   222         for (int i = 0; i < int(_proc_nodes.size()); ++i)
   223           _pi[_proc_nodes[i]] = _dist[_proc_nodes[i]] - t_dist;
   224         return true;
   225       }
   226 
   227       // Execute the algorithm.
   228       bool start() {
   229         HeapCrossRef heap_cross_ref(_graph, Heap::PRE_HEAP);
   230         Heap heap(heap_cross_ref);
   231         heap.push(_s, 0);
   232         _pred[_s] = INVALID;
   233         _proc_nodes.clear();
   234 
   235         // Process nodes
   236         while (!heap.empty() && heap.top() != _t) {
   237           Node u = heap.top(), v;
   238           Length d = heap.prio() + _pi[u], dn;
   239           _dist[u] = heap.prio();
   240           _proc_nodes.push_back(u);
   241           heap.pop();
   242 
   243           // Traverse outgoing arcs
   244           for (OutArcIt e(_graph, u); e != INVALID; ++e) {
   245             if (_flow[e] == 0) {
   246               v = _graph.target(e);
   247               switch(heap.state(v)) {
   248                 case Heap::PRE_HEAP:
   249                   heap.push(v, d + _length[e] - _pi[v]);
   250                   _pred[v] = e;
   251                   break;
   252                 case Heap::IN_HEAP:
   253                   dn = d + _length[e] - _pi[v];
   254                   if (dn < heap[v]) {
   255                     heap.decrease(v, dn);
   256                     _pred[v] = e;
   257                   }
   258                   break;
   259                 case Heap::POST_HEAP:
   260                   break;
   261               }
   262             }
   263           }
   264 
   265           // Traverse incoming arcs
   266           for (InArcIt e(_graph, u); e != INVALID; ++e) {
   267             if (_flow[e] == 1) {
   268               v = _graph.source(e);
   269               switch(heap.state(v)) {
   270                 case Heap::PRE_HEAP:
   271                   heap.push(v, d - _length[e] - _pi[v]);
   272                   _pred[v] = e;
   273                   break;
   274                 case Heap::IN_HEAP:
   275                   dn = d - _length[e] - _pi[v];
   276                   if (dn < heap[v]) {
   277                     heap.decrease(v, dn);
   278                     _pred[v] = e;
   279                   }
   280                   break;
   281                 case Heap::POST_HEAP:
   282                   break;
   283               }
   284             }
   285           }
   286         }
   287         if (heap.empty()) return false;
   288 
   289         // Update potentials of processed nodes
   290         Length t_dist = heap.prio();
   291         for (int i = 0; i < int(_proc_nodes.size()); ++i)
   292           _pi[_proc_nodes[i]] += _dist[_proc_nodes[i]] - t_dist;
   293         return true;
   294       }
   295 
   296     }; //class ResidualDijkstra
   297 
   298   public:
   299 
   300     /// \name Named Template Parameters
   301     /// @{
   302 
   303     template <typename T>
   304     struct SetFlowMapTraits : public Traits {
   305       typedef T FlowMap;
   306     };
   307 
   308     /// \brief \ref named-templ-param "Named parameter" for setting
   309     /// \c FlowMap type.
   310     ///
   311     /// \ref named-templ-param "Named parameter" for setting
   312     /// \c FlowMap type.
   313     template <typename T>
   314     struct SetFlowMap
   315       : public Suurballe<GR, LEN, SetFlowMapTraits<T> > {
   316       typedef Suurballe<GR, LEN, SetFlowMapTraits<T> > Create;
   317     };
   318 
   319     template <typename T>
   320     struct SetPotentialMapTraits : public Traits {
   321       typedef T PotentialMap;
   322     };
   323 
   324     /// \brief \ref named-templ-param "Named parameter" for setting
   325     /// \c PotentialMap type.
   326     ///
   327     /// \ref named-templ-param "Named parameter" for setting
   328     /// \c PotentialMap type.
   329     template <typename T>
   330     struct SetPotentialMap
   331       : public Suurballe<GR, LEN, SetPotentialMapTraits<T> > {
   332       typedef Suurballe<GR, LEN, SetPotentialMapTraits<T> > Create;
   333     };
   334 
   335     template <typename T>
   336     struct SetPathTraits : public Traits {
   337       typedef T Path;
   338     };
   339 
   340     /// \brief \ref named-templ-param "Named parameter" for setting
   341     /// \c %Path type.
   342     ///
   343     /// \ref named-templ-param "Named parameter" for setting \c %Path type.
   344     /// It must conform to the \ref lemon::concepts::Path "Path" concept
   345     /// and it must have an \c addBack() function.
   346     template <typename T>
   347     struct SetPath
   348       : public Suurballe<GR, LEN, SetPathTraits<T> > {
   349       typedef Suurballe<GR, LEN, SetPathTraits<T> > Create;
   350     };
   351     
   352     template <typename H, typename CR>
   353     struct SetHeapTraits : public Traits {
   354       typedef H Heap;
   355       typedef CR HeapCrossRef;
   356     };
   357 
   358     /// \brief \ref named-templ-param "Named parameter" for setting
   359     /// \c Heap and \c HeapCrossRef types.
   360     ///
   361     /// \ref named-templ-param "Named parameter" for setting \c Heap
   362     /// and \c HeapCrossRef types with automatic allocation. 
   363     /// They will be used for internal Dijkstra computations.
   364     /// The heap type must conform to the \ref lemon::concepts::Heap "Heap"
   365     /// concept and its priority type must be \c Length.
   366     template <typename H,
   367               typename CR = typename Digraph::template NodeMap<int> >
   368     struct SetHeap
   369       : public Suurballe<GR, LEN, SetHeapTraits<H, CR> > {
   370       typedef Suurballe<GR, LEN, SetHeapTraits<H, CR> > Create;
   371     };
   372 
   373     /// @}
   374 
   375   private:
   376 
   377     // The digraph the algorithm runs on
   378     const Digraph &_graph;
   379     // The length map
   380     const LengthMap &_length;
   381 
   382     // Arc map of the current flow
   383     FlowMap *_flow;
   384     bool _local_flow;
   385     // Node map of the current potentials
   386     PotentialMap *_potential;
   387     bool _local_potential;
   388 
   389     // The source node
   390     Node _s;
   391     // The target node
   392     Node _t;
   393 
   394     // Container to store the found paths
   395     std::vector<Path> _paths;
   396     int _path_num;
   397 
   398     // The pred arc map
   399     PredMap _pred;
   400     
   401     // Data for full init
   402     PotentialMap *_init_dist;
   403     PredMap *_init_pred;
   404     bool _full_init;
   405 
   406   public:
   407 
   408     /// \brief Constructor.
   409     ///
   410     /// Constructor.
   411     ///
   412     /// \param graph The digraph the algorithm runs on.
   413     /// \param length The length (cost) values of the arcs.
   414     Suurballe( const Digraph &graph,
   415                const LengthMap &length ) :
   416       _graph(graph), _length(length), _flow(0), _local_flow(false),
   417       _potential(0), _local_potential(false), _pred(graph),
   418       _init_dist(0), _init_pred(0)
   419     {}
   420 
   421     /// Destructor.
   422     ~Suurballe() {
   423       if (_local_flow) delete _flow;
   424       if (_local_potential) delete _potential;
   425       delete _init_dist;
   426       delete _init_pred;
   427     }
   428 
   429     /// \brief Set the flow map.
   430     ///
   431     /// This function sets the flow map.
   432     /// If it is not used before calling \ref run() or \ref init(),
   433     /// an instance will be allocated automatically. The destructor
   434     /// deallocates this automatically allocated map, of course.
   435     ///
   436     /// The found flow contains only 0 and 1 values, since it is the
   437     /// union of the found arc-disjoint paths.
   438     ///
   439     /// \return <tt>(*this)</tt>
   440     Suurballe& flowMap(FlowMap &map) {
   441       if (_local_flow) {
   442         delete _flow;
   443         _local_flow = false;
   444       }
   445       _flow = &map;
   446       return *this;
   447     }
   448 
   449     /// \brief Set the potential map.
   450     ///
   451     /// This function sets the potential map.
   452     /// If it is not used before calling \ref run() or \ref init(),
   453     /// an instance will be allocated automatically. The destructor
   454     /// deallocates this automatically allocated map, of course.
   455     ///
   456     /// The node potentials provide the dual solution of the underlying
   457     /// \ref min_cost_flow "minimum cost flow problem".
   458     ///
   459     /// \return <tt>(*this)</tt>
   460     Suurballe& potentialMap(PotentialMap &map) {
   461       if (_local_potential) {
   462         delete _potential;
   463         _local_potential = false;
   464       }
   465       _potential = &map;
   466       return *this;
   467     }
   468 
   469     /// \name Execution Control
   470     /// The simplest way to execute the algorithm is to call the run()
   471     /// function.\n
   472     /// If you need to execute the algorithm many times using the same
   473     /// source node, then you may call fullInit() once and start()
   474     /// for each target node.\n
   475     /// If you only need the flow that is the union of the found
   476     /// arc-disjoint paths, then you may call findFlow() instead of
   477     /// start().
   478 
   479     /// @{
   480 
   481     /// \brief Run the algorithm.
   482     ///
   483     /// This function runs the algorithm.
   484     ///
   485     /// \param s The source node.
   486     /// \param t The target node.
   487     /// \param k The number of paths to be found.
   488     ///
   489     /// \return \c k if there are at least \c k arc-disjoint paths from
   490     /// \c s to \c t in the digraph. Otherwise it returns the number of
   491     /// arc-disjoint paths found.
   492     ///
   493     /// \note Apart from the return value, <tt>s.run(s, t, k)</tt> is
   494     /// just a shortcut of the following code.
   495     /// \code
   496     ///   s.init(s);
   497     ///   s.start(t, k);
   498     /// \endcode
   499     int run(const Node& s, const Node& t, int k = 2) {
   500       init(s);
   501       start(t, k);
   502       return _path_num;
   503     }
   504 
   505     /// \brief Initialize the algorithm.
   506     ///
   507     /// This function initializes the algorithm with the given source node.
   508     ///
   509     /// \param s The source node.
   510     void init(const Node& s) {
   511       _s = s;
   512 
   513       // Initialize maps
   514       if (!_flow) {
   515         _flow = new FlowMap(_graph);
   516         _local_flow = true;
   517       }
   518       if (!_potential) {
   519         _potential = new PotentialMap(_graph);
   520         _local_potential = true;
   521       }
   522       _full_init = false;
   523     }
   524 
   525     /// \brief Initialize the algorithm and perform Dijkstra.
   526     ///
   527     /// This function initializes the algorithm and performs a full
   528     /// Dijkstra search from the given source node. It makes consecutive
   529     /// executions of \ref start() "start(t, k)" faster, since they
   530     /// have to perform %Dijkstra only k-1 times.
   531     ///
   532     /// This initialization is usually worth using instead of \ref init()
   533     /// if the algorithm is executed many times using the same source node.
   534     ///
   535     /// \param s The source node.
   536     void fullInit(const Node& s) {
   537       // Initialize maps
   538       init(s);
   539       if (!_init_dist) {
   540         _init_dist = new PotentialMap(_graph);
   541       }
   542       if (!_init_pred) {
   543         _init_pred = new PredMap(_graph);
   544       }
   545 
   546       // Run a full Dijkstra
   547       typename Dijkstra<Digraph, LengthMap>
   548         ::template SetStandardHeap<Heap>
   549         ::template SetDistMap<PotentialMap>
   550         ::template SetPredMap<PredMap>
   551         ::Create dijk(_graph, _length);
   552       dijk.distMap(*_init_dist).predMap(*_init_pred);
   553       dijk.run(s);
   554       
   555       _full_init = true;
   556     }
   557 
   558     /// \brief Execute the algorithm.
   559     ///
   560     /// This function executes the algorithm.
   561     ///
   562     /// \param t The target node.
   563     /// \param k The number of paths to be found.
   564     ///
   565     /// \return \c k if there are at least \c k arc-disjoint paths from
   566     /// \c s to \c t in the digraph. Otherwise it returns the number of
   567     /// arc-disjoint paths found.
   568     ///
   569     /// \note Apart from the return value, <tt>s.start(t, k)</tt> is
   570     /// just a shortcut of the following code.
   571     /// \code
   572     ///   s.findFlow(t, k);
   573     ///   s.findPaths();
   574     /// \endcode
   575     int start(const Node& t, int k = 2) {
   576       findFlow(t, k);
   577       findPaths();
   578       return _path_num;
   579     }
   580 
   581     /// \brief Execute the algorithm to find an optimal flow.
   582     ///
   583     /// This function executes the successive shortest path algorithm to
   584     /// find a minimum cost flow, which is the union of \c k (or less)
   585     /// arc-disjoint paths.
   586     ///
   587     /// \param t The target node.
   588     /// \param k The number of paths to be found.
   589     ///
   590     /// \return \c k if there are at least \c k arc-disjoint paths from
   591     /// the source node to the given node \c t in the digraph.
   592     /// Otherwise it returns the number of arc-disjoint paths found.
   593     ///
   594     /// \pre \ref init() must be called before using this function.
   595     int findFlow(const Node& t, int k = 2) {
   596       _t = t;
   597       ResidualDijkstra dijkstra(*this);
   598       
   599       // Initialization
   600       for (ArcIt e(_graph); e != INVALID; ++e) {
   601         (*_flow)[e] = 0;
   602       }
   603       if (_full_init) {
   604         for (NodeIt n(_graph); n != INVALID; ++n) {
   605           (*_potential)[n] = (*_init_dist)[n];
   606         }
   607         Node u = _t;
   608         Arc e;
   609         while ((e = (*_init_pred)[u]) != INVALID) {
   610           (*_flow)[e] = 1;
   611           u = _graph.source(e);
   612         }        
   613         _path_num = 1;
   614       } else {
   615         for (NodeIt n(_graph); n != INVALID; ++n) {
   616           (*_potential)[n] = 0;
   617         }
   618         _path_num = 0;
   619       }
   620 
   621       // Find shortest paths
   622       while (_path_num < k) {
   623         // Run Dijkstra
   624         if (!dijkstra.run(_path_num)) break;
   625         ++_path_num;
   626 
   627         // Set the flow along the found shortest path
   628         Node u = _t;
   629         Arc e;
   630         while ((e = _pred[u]) != INVALID) {
   631           if (u == _graph.target(e)) {
   632             (*_flow)[e] = 1;
   633             u = _graph.source(e);
   634           } else {
   635             (*_flow)[e] = 0;
   636             u = _graph.target(e);
   637           }
   638         }
   639       }
   640       return _path_num;
   641     }
   642 
   643     /// \brief Compute the paths from the flow.
   644     ///
   645     /// This function computes arc-disjoint paths from the found minimum
   646     /// cost flow, which is the union of them.
   647     ///
   648     /// \pre \ref init() and \ref findFlow() must be called before using
   649     /// this function.
   650     void findPaths() {
   651       FlowMap res_flow(_graph);
   652       for(ArcIt a(_graph); a != INVALID; ++a) res_flow[a] = (*_flow)[a];
   653 
   654       _paths.clear();
   655       _paths.resize(_path_num);
   656       for (int i = 0; i < _path_num; ++i) {
   657         Node n = _s;
   658         while (n != _t) {
   659           OutArcIt e(_graph, n);
   660           for ( ; res_flow[e] == 0; ++e) ;
   661           n = _graph.target(e);
   662           _paths[i].addBack(e);
   663           res_flow[e] = 0;
   664         }
   665       }
   666     }
   667 
   668     /// @}
   669 
   670     /// \name Query Functions
   671     /// The results of the algorithm can be obtained using these
   672     /// functions.
   673     /// \n The algorithm should be executed before using them.
   674 
   675     /// @{
   676 
   677     /// \brief Return the total length of the found paths.
   678     ///
   679     /// This function returns the total length of the found paths, i.e.
   680     /// the total cost of the found flow.
   681     /// The complexity of the function is O(e).
   682     ///
   683     /// \pre \ref run() or \ref findFlow() must be called before using
   684     /// this function.
   685     Length totalLength() const {
   686       Length c = 0;
   687       for (ArcIt e(_graph); e != INVALID; ++e)
   688         c += (*_flow)[e] * _length[e];
   689       return c;
   690     }
   691 
   692     /// \brief Return the flow value on the given arc.
   693     ///
   694     /// This function returns the flow value on the given arc.
   695     /// It is \c 1 if the arc is involved in one of the found arc-disjoint
   696     /// paths, otherwise it is \c 0.
   697     ///
   698     /// \pre \ref run() or \ref findFlow() must be called before using
   699     /// this function.
   700     int flow(const Arc& arc) const {
   701       return (*_flow)[arc];
   702     }
   703 
   704     /// \brief Return a const reference to an arc map storing the
   705     /// found flow.
   706     ///
   707     /// This function returns a const reference to an arc map storing
   708     /// the flow that is the union of the found arc-disjoint paths.
   709     ///
   710     /// \pre \ref run() or \ref findFlow() must be called before using
   711     /// this function.
   712     const FlowMap& flowMap() const {
   713       return *_flow;
   714     }
   715 
   716     /// \brief Return the potential of the given node.
   717     ///
   718     /// This function returns the potential of the given node.
   719     /// The node potentials provide the dual solution of the
   720     /// underlying \ref min_cost_flow "minimum cost flow problem".
   721     ///
   722     /// \pre \ref run() or \ref findFlow() must be called before using
   723     /// this function.
   724     Length potential(const Node& node) const {
   725       return (*_potential)[node];
   726     }
   727 
   728     /// \brief Return a const reference to a node map storing the
   729     /// found potentials (the dual solution).
   730     ///
   731     /// This function returns a const reference to a node map storing
   732     /// the found potentials that provide the dual solution of the
   733     /// underlying \ref min_cost_flow "minimum cost flow problem".
   734     ///
   735     /// \pre \ref run() or \ref findFlow() must be called before using
   736     /// this function.
   737     const PotentialMap& potentialMap() const {
   738       return *_potential;
   739     }
   740 
   741     /// \brief Return the number of the found paths.
   742     ///
   743     /// This function returns the number of the found paths.
   744     ///
   745     /// \pre \ref run() or \ref findFlow() must be called before using
   746     /// this function.
   747     int pathNum() const {
   748       return _path_num;
   749     }
   750 
   751     /// \brief Return a const reference to the specified path.
   752     ///
   753     /// This function returns a const reference to the specified path.
   754     ///
   755     /// \param i The function returns the <tt>i</tt>-th path.
   756     /// \c i must be between \c 0 and <tt>%pathNum()-1</tt>.
   757     ///
   758     /// \pre \ref run() or \ref findPaths() must be called before using
   759     /// this function.
   760     const Path& path(int i) const {
   761       return _paths[i];
   762     }
   763 
   764     /// @}
   765 
   766   }; //class Suurballe
   767 
   768   ///@}
   769 
   770 } //namespace lemon
   771 
   772 #endif //LEMON_SUURBALLE_H