lemon/suurballe.h
author Peter Kovacs <kpeter@inf.elte.hu>
Mon, 08 Mar 2010 08:33:41 +0100
changeset 941 a93f1a27d831
parent 931 abb95d48e89e
child 956 141f9c0db4a3
permissions -rw-r--r--
Fix gcc 3.3 compilation error (#354)

gcc 3.3 requires that a class has a default constructor if it has
template named parameters. (That constructor can be protected.)
     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   protected:
   407 
   408     Suurballe() {}
   409 
   410   public:
   411 
   412     /// \brief Constructor.
   413     ///
   414     /// Constructor.
   415     ///
   416     /// \param graph The digraph the algorithm runs on.
   417     /// \param length The length (cost) values of the arcs.
   418     Suurballe( const Digraph &graph,
   419                const LengthMap &length ) :
   420       _graph(graph), _length(length), _flow(0), _local_flow(false),
   421       _potential(0), _local_potential(false), _pred(graph),
   422       _init_dist(0), _init_pred(0)
   423     {}
   424 
   425     /// Destructor.
   426     ~Suurballe() {
   427       if (_local_flow) delete _flow;
   428       if (_local_potential) delete _potential;
   429       delete _init_dist;
   430       delete _init_pred;
   431     }
   432 
   433     /// \brief Set the flow map.
   434     ///
   435     /// This function sets the flow map.
   436     /// If it is not used before calling \ref run() or \ref init(),
   437     /// an instance will be allocated automatically. The destructor
   438     /// deallocates this automatically allocated map, of course.
   439     ///
   440     /// The found flow contains only 0 and 1 values, since it is the
   441     /// union of the found arc-disjoint paths.
   442     ///
   443     /// \return <tt>(*this)</tt>
   444     Suurballe& flowMap(FlowMap &map) {
   445       if (_local_flow) {
   446         delete _flow;
   447         _local_flow = false;
   448       }
   449       _flow = &map;
   450       return *this;
   451     }
   452 
   453     /// \brief Set the potential map.
   454     ///
   455     /// This function sets the potential map.
   456     /// If it is not used before calling \ref run() or \ref init(),
   457     /// an instance will be allocated automatically. The destructor
   458     /// deallocates this automatically allocated map, of course.
   459     ///
   460     /// The node potentials provide the dual solution of the underlying
   461     /// \ref min_cost_flow "minimum cost flow problem".
   462     ///
   463     /// \return <tt>(*this)</tt>
   464     Suurballe& potentialMap(PotentialMap &map) {
   465       if (_local_potential) {
   466         delete _potential;
   467         _local_potential = false;
   468       }
   469       _potential = &map;
   470       return *this;
   471     }
   472 
   473     /// \name Execution Control
   474     /// The simplest way to execute the algorithm is to call the run()
   475     /// function.\n
   476     /// If you need to execute the algorithm many times using the same
   477     /// source node, then you may call fullInit() once and start()
   478     /// for each target node.\n
   479     /// If you only need the flow that is the union of the found
   480     /// arc-disjoint paths, then you may call findFlow() instead of
   481     /// start().
   482 
   483     /// @{
   484 
   485     /// \brief Run the algorithm.
   486     ///
   487     /// This function runs the algorithm.
   488     ///
   489     /// \param s The source node.
   490     /// \param t The target node.
   491     /// \param k The number of paths to be found.
   492     ///
   493     /// \return \c k if there are at least \c k arc-disjoint paths from
   494     /// \c s to \c t in the digraph. Otherwise it returns the number of
   495     /// arc-disjoint paths found.
   496     ///
   497     /// \note Apart from the return value, <tt>s.run(s, t, k)</tt> is
   498     /// just a shortcut of the following code.
   499     /// \code
   500     ///   s.init(s);
   501     ///   s.start(t, k);
   502     /// \endcode
   503     int run(const Node& s, const Node& t, int k = 2) {
   504       init(s);
   505       start(t, k);
   506       return _path_num;
   507     }
   508 
   509     /// \brief Initialize the algorithm.
   510     ///
   511     /// This function initializes the algorithm with the given source node.
   512     ///
   513     /// \param s The source node.
   514     void init(const Node& s) {
   515       _s = s;
   516 
   517       // Initialize maps
   518       if (!_flow) {
   519         _flow = new FlowMap(_graph);
   520         _local_flow = true;
   521       }
   522       if (!_potential) {
   523         _potential = new PotentialMap(_graph);
   524         _local_potential = true;
   525       }
   526       _full_init = false;
   527     }
   528 
   529     /// \brief Initialize the algorithm and perform Dijkstra.
   530     ///
   531     /// This function initializes the algorithm and performs a full
   532     /// Dijkstra search from the given source node. It makes consecutive
   533     /// executions of \ref start() "start(t, k)" faster, since they
   534     /// have to perform %Dijkstra only k-1 times.
   535     ///
   536     /// This initialization is usually worth using instead of \ref init()
   537     /// if the algorithm is executed many times using the same source node.
   538     ///
   539     /// \param s The source node.
   540     void fullInit(const Node& s) {
   541       // Initialize maps
   542       init(s);
   543       if (!_init_dist) {
   544         _init_dist = new PotentialMap(_graph);
   545       }
   546       if (!_init_pred) {
   547         _init_pred = new PredMap(_graph);
   548       }
   549 
   550       // Run a full Dijkstra
   551       typename Dijkstra<Digraph, LengthMap>
   552         ::template SetStandardHeap<Heap>
   553         ::template SetDistMap<PotentialMap>
   554         ::template SetPredMap<PredMap>
   555         ::Create dijk(_graph, _length);
   556       dijk.distMap(*_init_dist).predMap(*_init_pred);
   557       dijk.run(s);
   558       
   559       _full_init = true;
   560     }
   561 
   562     /// \brief Execute the algorithm.
   563     ///
   564     /// This function executes the algorithm.
   565     ///
   566     /// \param t The target node.
   567     /// \param k The number of paths to be found.
   568     ///
   569     /// \return \c k if there are at least \c k arc-disjoint paths from
   570     /// \c s to \c t in the digraph. Otherwise it returns the number of
   571     /// arc-disjoint paths found.
   572     ///
   573     /// \note Apart from the return value, <tt>s.start(t, k)</tt> is
   574     /// just a shortcut of the following code.
   575     /// \code
   576     ///   s.findFlow(t, k);
   577     ///   s.findPaths();
   578     /// \endcode
   579     int start(const Node& t, int k = 2) {
   580       findFlow(t, k);
   581       findPaths();
   582       return _path_num;
   583     }
   584 
   585     /// \brief Execute the algorithm to find an optimal flow.
   586     ///
   587     /// This function executes the successive shortest path algorithm to
   588     /// find a minimum cost flow, which is the union of \c k (or less)
   589     /// arc-disjoint paths.
   590     ///
   591     /// \param t The target node.
   592     /// \param k The number of paths to be found.
   593     ///
   594     /// \return \c k if there are at least \c k arc-disjoint paths from
   595     /// the source node to the given node \c t in the digraph.
   596     /// Otherwise it returns the number of arc-disjoint paths found.
   597     ///
   598     /// \pre \ref init() must be called before using this function.
   599     int findFlow(const Node& t, int k = 2) {
   600       _t = t;
   601       ResidualDijkstra dijkstra(*this);
   602       
   603       // Initialization
   604       for (ArcIt e(_graph); e != INVALID; ++e) {
   605         (*_flow)[e] = 0;
   606       }
   607       if (_full_init) {
   608         for (NodeIt n(_graph); n != INVALID; ++n) {
   609           (*_potential)[n] = (*_init_dist)[n];
   610         }
   611         Node u = _t;
   612         Arc e;
   613         while ((e = (*_init_pred)[u]) != INVALID) {
   614           (*_flow)[e] = 1;
   615           u = _graph.source(e);
   616         }        
   617         _path_num = 1;
   618       } else {
   619         for (NodeIt n(_graph); n != INVALID; ++n) {
   620           (*_potential)[n] = 0;
   621         }
   622         _path_num = 0;
   623       }
   624 
   625       // Find shortest paths
   626       while (_path_num < k) {
   627         // Run Dijkstra
   628         if (!dijkstra.run(_path_num)) break;
   629         ++_path_num;
   630 
   631         // Set the flow along the found shortest path
   632         Node u = _t;
   633         Arc e;
   634         while ((e = _pred[u]) != INVALID) {
   635           if (u == _graph.target(e)) {
   636             (*_flow)[e] = 1;
   637             u = _graph.source(e);
   638           } else {
   639             (*_flow)[e] = 0;
   640             u = _graph.target(e);
   641           }
   642         }
   643       }
   644       return _path_num;
   645     }
   646 
   647     /// \brief Compute the paths from the flow.
   648     ///
   649     /// This function computes arc-disjoint paths from the found minimum
   650     /// cost flow, which is the union of them.
   651     ///
   652     /// \pre \ref init() and \ref findFlow() must be called before using
   653     /// this function.
   654     void findPaths() {
   655       FlowMap res_flow(_graph);
   656       for(ArcIt a(_graph); a != INVALID; ++a) res_flow[a] = (*_flow)[a];
   657 
   658       _paths.clear();
   659       _paths.resize(_path_num);
   660       for (int i = 0; i < _path_num; ++i) {
   661         Node n = _s;
   662         while (n != _t) {
   663           OutArcIt e(_graph, n);
   664           for ( ; res_flow[e] == 0; ++e) ;
   665           n = _graph.target(e);
   666           _paths[i].addBack(e);
   667           res_flow[e] = 0;
   668         }
   669       }
   670     }
   671 
   672     /// @}
   673 
   674     /// \name Query Functions
   675     /// The results of the algorithm can be obtained using these
   676     /// functions.
   677     /// \n The algorithm should be executed before using them.
   678 
   679     /// @{
   680 
   681     /// \brief Return the total length of the found paths.
   682     ///
   683     /// This function returns the total length of the found paths, i.e.
   684     /// the total cost of the found flow.
   685     /// The complexity of the function is O(e).
   686     ///
   687     /// \pre \ref run() or \ref findFlow() must be called before using
   688     /// this function.
   689     Length totalLength() const {
   690       Length c = 0;
   691       for (ArcIt e(_graph); e != INVALID; ++e)
   692         c += (*_flow)[e] * _length[e];
   693       return c;
   694     }
   695 
   696     /// \brief Return the flow value on the given arc.
   697     ///
   698     /// This function returns the flow value on the given arc.
   699     /// It is \c 1 if the arc is involved in one of the found arc-disjoint
   700     /// paths, otherwise it is \c 0.
   701     ///
   702     /// \pre \ref run() or \ref findFlow() must be called before using
   703     /// this function.
   704     int flow(const Arc& arc) const {
   705       return (*_flow)[arc];
   706     }
   707 
   708     /// \brief Return a const reference to an arc map storing the
   709     /// found flow.
   710     ///
   711     /// This function returns a const reference to an arc map storing
   712     /// the flow that is the union of the found arc-disjoint paths.
   713     ///
   714     /// \pre \ref run() or \ref findFlow() must be called before using
   715     /// this function.
   716     const FlowMap& flowMap() const {
   717       return *_flow;
   718     }
   719 
   720     /// \brief Return the potential of the given node.
   721     ///
   722     /// This function returns the potential of the given node.
   723     /// The node potentials provide the dual solution of the
   724     /// underlying \ref min_cost_flow "minimum cost flow problem".
   725     ///
   726     /// \pre \ref run() or \ref findFlow() must be called before using
   727     /// this function.
   728     Length potential(const Node& node) const {
   729       return (*_potential)[node];
   730     }
   731 
   732     /// \brief Return a const reference to a node map storing the
   733     /// found potentials (the dual solution).
   734     ///
   735     /// This function returns a const reference to a node map storing
   736     /// the found potentials that provide the dual solution of the
   737     /// underlying \ref min_cost_flow "minimum cost flow problem".
   738     ///
   739     /// \pre \ref run() or \ref findFlow() must be called before using
   740     /// this function.
   741     const PotentialMap& potentialMap() const {
   742       return *_potential;
   743     }
   744 
   745     /// \brief Return the number of the found paths.
   746     ///
   747     /// This function returns the number of the found paths.
   748     ///
   749     /// \pre \ref run() or \ref findFlow() must be called before using
   750     /// this function.
   751     int pathNum() const {
   752       return _path_num;
   753     }
   754 
   755     /// \brief Return a const reference to the specified path.
   756     ///
   757     /// This function returns a const reference to the specified path.
   758     ///
   759     /// \param i The function returns the <tt>i</tt>-th path.
   760     /// \c i must be between \c 0 and <tt>%pathNum()-1</tt>.
   761     ///
   762     /// \pre \ref run() or \ref findPaths() must be called before using
   763     /// this function.
   764     const Path& path(int i) const {
   765       return _paths[i];
   766     }
   767 
   768     /// @}
   769 
   770   }; //class Suurballe
   771 
   772   ///@}
   773 
   774 } //namespace lemon
   775 
   776 #endif //LEMON_SUURBALLE_H