COIN-OR::LEMON - Graph Library

source: lemon-0.x/lemon/johnson.h @ 1757:bd4199049036

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1/* -*- C++ -*-
2 * lemon/johnson.h - Part of LEMON, a generic C++ optimization library
3 *
4 * Copyright (C) 2005 Egervary Jeno Kombinatorikus Optimalizalasi Kutatocsoport
5 * (Egervary Research Group on Combinatorial Optimization, EGRES).
6 *
7 * Permission to use, modify and distribute this software is granted
8 * provided that this copyright notice appears in all copies. For
9 * precise terms see the accompanying LICENSE file.
10 *
11 * This software is provided "AS IS" with no warranty of any kind,
12 * express or implied, and with no claim as to its suitability for any
13 * purpose.
14 *
15 */
16
17#ifndef LEMON_JOHNSON_H
18#define LEMON_JOHNSON_H
19
20///\ingroup flowalgs
21/// \file
22/// \brief Johnson algorithm.
23///
24
25#include <lemon/list_graph.h>
26#include <lemon/graph_utils.h>
27#include <lemon/dijkstra.h>
28#include <lemon/belmann_ford.h>
29#include <lemon/invalid.h>
30#include <lemon/error.h>
31#include <lemon/maps.h>
32#include <lemon/matrix_maps.h>
33
34#include <limits>
35
36namespace lemon {
37
38  /// \brief Default OperationTraits for the Johnson algorithm class.
39  /// 
40  /// It defines all computational operations and constants which are
41  /// used in the Floyd-Warshall algorithm. The default implementation
42  /// is based on the numeric_limits class. If the numeric type does not
43  /// have infinity value then the maximum value is used as extremal
44  /// infinity value.
45  template <
46    typename Value,
47    bool has_infinity = std::numeric_limits<Value>::has_infinity>
48  struct JohnsonDefaultOperationTraits {
49    /// \brief Gives back the zero value of the type.
50    static Value zero() {
51      return static_cast<Value>(0);
52    }
53    /// \brief Gives back the positive infinity value of the type.
54    static Value infinity() {
55      return std::numeric_limits<Value>::infinity();
56    }
57    /// \brief Gives back the sum of the given two elements.
58    static Value plus(const Value& left, const Value& right) {
59      return left + right;
60    }
61    /// \brief Gives back true only if the first value less than the second.
62    static bool less(const Value& left, const Value& right) {
63      return left < right;
64    }
65  };
66
67  template <typename Value>
68  struct JohnsonDefaultOperationTraits<Value, false> {
69    static Value zero() {
70      return static_cast<Value>(0);
71    }
72    static Value infinity() {
73      return std::numeric_limits<Value>::max();
74    }
75    static Value plus(const Value& left, const Value& right) {
76      if (left == infinity() || right == infinity()) return infinity();
77      return left + right;
78    }
79    static bool less(const Value& left, const Value& right) {
80      return left < right;
81    }
82  };
83 
84  /// \brief Default traits class of Johnson class.
85  ///
86  /// Default traits class of Johnson class.
87  /// \param _Graph Graph type.
88  /// \param _LegthMap Type of length map.
89  template<class _Graph, class _LengthMap>
90  struct JohnsonDefaultTraits {
91    /// The graph type the algorithm runs on.
92    typedef _Graph Graph;
93
94    /// \brief The type of the map that stores the edge lengths.
95    ///
96    /// The type of the map that stores the edge lengths.
97    /// It must meet the \ref concept::ReadMap "ReadMap" concept.
98    typedef _LengthMap LengthMap;
99
100    // The type of the length of the edges.
101    typedef typename _LengthMap::Value Value;
102
103    /// \brief Operation traits for belmann-ford algorithm.
104    ///
105    /// It defines the infinity type on the given Value type
106    /// and the used operation.
107    /// \see JohnsonDefaultOperationTraits
108    typedef JohnsonDefaultOperationTraits<Value> OperationTraits;
109
110    /// The cross reference type used by heap.
111
112    /// The cross reference type used by heap.
113    /// Usually it is \c Graph::NodeMap<int>.
114    typedef typename Graph::template NodeMap<int> HeapCrossRef;
115
116    ///Instantiates a HeapCrossRef.
117
118    ///This function instantiates a \ref HeapCrossRef.
119    /// \param graph is the graph, to which we would like to define the
120    /// HeapCrossRef.
121    static HeapCrossRef *createHeapCrossRef(const Graph& graph) {
122      return new HeapCrossRef(graph);
123    }
124   
125    ///The heap type used by Dijkstra algorithm.
126
127    ///The heap type used by Dijkstra algorithm.
128    ///
129    ///\sa BinHeap
130    ///\sa Dijkstra
131    typedef BinHeap<typename Graph::Node, typename LengthMap::Value,
132                    HeapCrossRef, std::less<Value> > Heap;
133
134    ///Instantiates a Heap.
135
136    ///This function instantiates a \ref Heap.
137    /// \param crossRef The cross reference for the heap.
138    static Heap *createHeap(HeapCrossRef& crossRef) {
139      return new Heap(crossRef);
140    }
141 
142    /// \brief The type of the matrix map that stores the last edges of the
143    /// shortest paths.
144    ///
145    /// The type of the map that stores the last edges of the shortest paths.
146    /// It must be a matrix map with \c Graph::Edge value type.
147    ///
148    typedef DynamicMatrixMap<Graph, typename Graph::Node,
149                             typename Graph::Edge> PredMap;
150
151    /// \brief Instantiates a PredMap.
152    ///
153    /// This function instantiates a \ref PredMap.
154    /// \param G is the graph, to which we would like to define the PredMap.
155    /// \todo The graph alone may be insufficient for the initialization
156    static PredMap *createPredMap(const Graph& graph) {
157      return new PredMap(graph);
158    }
159
160    /// \brief The type of the matrix map that stores the dists of the nodes.
161    ///
162    /// The type of the matrix map that stores the dists of the nodes.
163    /// It must meet the \ref concept::WriteMatrixMap "WriteMatrixMap" concept.
164    ///
165    typedef DynamicMatrixMap<Graph, typename Graph::Node, Value> DistMap;
166   
167    /// \brief Instantiates a DistMap.
168    ///
169    /// This function instantiates a \ref DistMap.
170    /// \param G is the graph, to which we would like to define the
171    /// \ref DistMap
172    static DistMap *createDistMap(const _Graph& graph) {
173      return new DistMap(graph);
174    }
175
176  };
177
178  /// \brief %Johnson algorithm class.
179  ///
180  /// \ingroup flowalgs
181  /// This class provides an efficient implementation of \c %Johnson
182  /// algorithm. The edge lengths are passed to the algorithm using a
183  /// \ref concept::ReadMap "ReadMap", so it is easy to change it to any
184  /// kind of length.
185  ///
186  /// The algorithm solves the shortest path problem for each pair
187  /// of node when the edges can have negative length but the graph should
188  /// not contain cycles with negative sum of length. If we can assume
189  /// that all edge is non-negative in the graph then the dijkstra algorithm
190  /// should be used from each node.
191  ///
192  /// The complexity of this algorithm is $O(n^2 * log(n) + n * log(n) * e)$ or
193  /// with fibonacci heap O(n^2 * log(n) + n * e). Usually the fibonacci heap
194  /// implementation is slower than either binary heap implementation or the
195  /// Floyd-Warshall algorithm.
196  ///
197  /// The type of the length is determined by the
198  /// \ref concept::ReadMap::Value "Value" of the length map.
199  ///
200  /// \param _Graph The graph type the algorithm runs on. The default value
201  /// is \ref ListGraph. The value of _Graph is not used directly by
202  /// Johnson, it is only passed to \ref JohnsonDefaultTraits.
203  /// \param _LengthMap This read-only EdgeMap determines the lengths of the
204  /// edges. It is read once for each edge, so the map may involve in
205  /// relatively time consuming process to compute the edge length if
206  /// it is necessary. The default map type is \ref
207  /// concept::StaticGraph::EdgeMap "Graph::EdgeMap<int>".  The value
208  /// of _LengthMap is not used directly by Johnson, it is only passed
209  /// to \ref JohnsonDefaultTraits.  \param _Traits Traits class to set
210  /// various data types used by the algorithm.  The default traits
211  /// class is \ref JohnsonDefaultTraits
212  /// "JohnsonDefaultTraits<_Graph,_LengthMap>".  See \ref
213  /// JohnsonDefaultTraits for the documentation of a Johnson traits
214  /// class.
215  ///
216  /// \author Balazs Dezso
217
218#ifdef DOXYGEN
219  template <typename _Graph, typename _LengthMap, typename _Traits>
220#else
221  template <typename _Graph=ListGraph,
222            typename _LengthMap=typename _Graph::template EdgeMap<int>,
223            typename _Traits=JohnsonDefaultTraits<_Graph,_LengthMap> >
224#endif
225  class Johnson {
226  public:
227   
228    /// \brief \ref Exception for uninitialized parameters.
229    ///
230    /// This error represents problems in the initialization
231    /// of the parameters of the algorithms.
232
233    class UninitializedParameter : public lemon::UninitializedParameter {
234    public:
235      virtual const char* exceptionName() const {
236        return "lemon::Johnson::UninitializedParameter";
237      }
238    };
239
240    typedef _Traits Traits;
241    ///The type of the underlying graph.
242    typedef typename _Traits::Graph Graph;
243
244    typedef typename Graph::Node Node;
245    typedef typename Graph::NodeIt NodeIt;
246    typedef typename Graph::Edge Edge;
247    typedef typename Graph::EdgeIt EdgeIt;
248   
249    /// \brief The type of the length of the edges.
250    typedef typename _Traits::LengthMap::Value Value;
251    /// \brief The type of the map that stores the edge lengths.
252    typedef typename _Traits::LengthMap LengthMap;
253    /// \brief The type of the map that stores the last
254    /// edges of the shortest paths. The type of the PredMap
255    /// is a matrix map for Edges
256    typedef typename _Traits::PredMap PredMap;
257    /// \brief The type of the map that stores the dists of the nodes.
258    /// The type of the DistMap is a matrix map for Values
259    typedef typename _Traits::DistMap DistMap;
260    /// \brief The operation traits.
261    typedef typename _Traits::OperationTraits OperationTraits;
262    ///The cross reference type used for the current heap.
263    typedef typename _Traits::HeapCrossRef HeapCrossRef;
264    ///The heap type used by the dijkstra algorithm.
265    typedef typename _Traits::Heap Heap;
266  private:
267    /// Pointer to the underlying graph.
268    const Graph *graph;
269    /// Pointer to the length map
270    const LengthMap *length;
271    ///Pointer to the map of predecessors edges.
272    PredMap *_pred;
273    ///Indicates if \ref _pred is locally allocated (\c true) or not.
274    bool local_pred;
275    ///Pointer to the map of distances.
276    DistMap *_dist;
277    ///Indicates if \ref _dist is locally allocated (\c true) or not.
278    bool local_dist;
279    ///Pointer to the heap cross references.
280    HeapCrossRef *_heap_cross_ref;
281    ///Indicates if \ref _heap_cross_ref is locally allocated (\c true) or not.
282    bool local_heap_cross_ref;
283    ///Pointer to the heap.
284    Heap *_heap;
285    ///Indicates if \ref _heap is locally allocated (\c true) or not.
286    bool local_heap;
287
288    /// Creates the maps if necessary.
289    void create_maps() {
290      if(!_pred) {
291        local_pred = true;
292        _pred = Traits::createPredMap(*graph);
293      }
294      if(!_dist) {
295        local_dist = true;
296        _dist = Traits::createDistMap(*graph);
297      }
298      if (!_heap_cross_ref) {
299        local_heap_cross_ref = true;
300        _heap_cross_ref = Traits::createHeapCrossRef(*graph);
301      }
302      if (!_heap) {
303        local_heap = true;
304        _heap = Traits::createHeap(*_heap_cross_ref);
305      }
306    }
307
308  public :
309
310    typedef Johnson Create;
311 
312    /// \name Named template parameters
313
314    ///@{
315
316    template <class T>
317    struct DefPredMapTraits : public Traits {
318      typedef T PredMap;
319      static PredMap *createPredMap(const Graph& graph) {
320        throw UninitializedParameter();
321      }
322    };
323
324    /// \brief \ref named-templ-param "Named parameter" for setting PredMap
325    /// type
326    /// \ref named-templ-param "Named parameter" for setting PredMap type
327    ///
328    template <class T>
329    struct DefPredMap
330      : public Johnson< Graph, LengthMap, DefPredMapTraits<T> > {
331      typedef Johnson< Graph, LengthMap, DefPredMapTraits<T> > Create;
332    };
333   
334    template <class T>
335    struct DefDistMapTraits : public Traits {
336      typedef T DistMap;
337      static DistMap *createDistMap(const Graph& graph) {
338        throw UninitializedParameter();
339      }
340    };
341    /// \brief \ref named-templ-param "Named parameter" for setting DistMap
342    /// type
343    ///
344    /// \ref named-templ-param "Named parameter" for setting DistMap type
345    ///
346    template <class T>
347    struct DefDistMap
348      : public Johnson< Graph, LengthMap, DefDistMapTraits<T> > {
349      typedef Johnson< Graph, LengthMap, DefDistMapTraits<T> > Create;
350    };
351   
352    template <class T>
353    struct DefOperationTraitsTraits : public Traits {
354      typedef T OperationTraits;
355    };
356   
357    /// \brief \ref named-templ-param "Named parameter" for setting
358    /// OperationTraits type
359    ///
360    /// \ref named-templ-param "Named parameter" for setting
361    /// OperationTraits type
362    template <class T>
363    struct DefOperationTraits
364      : public Johnson< Graph, LengthMap, DefOperationTraitsTraits<T> > {
365      typedef Johnson< Graph, LengthMap, DefOperationTraitsTraits<T> > Create;
366    };
367
368    template <class H, class CR>
369    struct DefHeapTraits : public Traits {
370      typedef CR HeapCrossRef;
371      typedef H Heap;
372      static HeapCrossRef *createHeapCrossRef(const Graph &) {
373        throw UninitializedParameter();
374      }
375      static Heap *createHeap(HeapCrossRef &)
376      {
377        throw UninitializedParameter();
378      }
379    };
380    ///\brief \ref named-templ-param "Named parameter" for setting heap and
381    ///cross reference type
382
383    ///\ref named-templ-param "Named parameter" for setting heap and cross
384    ///reference type
385    ///
386    template <class H, class CR = typename Graph::template NodeMap<int> >
387    struct DefHeap
388      : public Johnson< Graph, LengthMap, DefHeapTraits<H, CR> > {
389      typedef Johnson< Graph, LengthMap, DefHeapTraits<H, CR> > Create;
390    };
391
392    template <class H, class CR>
393    struct DefStandardHeapTraits : public Traits {
394      typedef CR HeapCrossRef;
395      typedef H Heap;
396      static HeapCrossRef *createHeapCrossRef(const Graph &G) {
397        return new HeapCrossRef(G);
398      }
399      static Heap *createHeap(HeapCrossRef &R)
400      {
401        return new Heap(R);
402      }
403    };
404    ///\ref named-templ-param "Named parameter" for setting heap and cross
405    ///reference type with automatic allocation
406
407    ///\ref named-templ-param "Named parameter" for setting heap and cross
408    ///reference type. It can allocate the heap and the cross reference
409    ///object if the cross reference's constructor waits for the graph as
410    ///parameter and the heap's constructor waits for the cross reference.
411    template <class H, class CR = typename Graph::template NodeMap<int> >
412    struct DefStandardHeap
413      : public Johnson< Graph, LengthMap, DefStandardHeapTraits<H, CR> > {
414      typedef Johnson< Graph, LengthMap, DefStandardHeapTraits<H, CR> >
415      Create;
416    };
417   
418    ///@}
419
420  protected:
421
422    Johnson() {}
423
424  public:     
425
426    typedef Johnson Create;
427   
428    /// \brief Constructor.
429    ///
430    /// \param _graph the graph the algorithm will run on.
431    /// \param _length the length map used by the algorithm.
432    Johnson(const Graph& _graph, const LengthMap& _length) :
433      graph(&_graph), length(&_length),
434      _pred(0), local_pred(false),
435      _dist(0), local_dist(false),
436      _heap_cross_ref(0), local_heap_cross_ref(false),
437      _heap(0), local_heap(false) {}
438   
439    ///Destructor.
440    ~Johnson() {
441      if (local_pred) delete _pred;
442      if (local_dist) delete _dist;
443      if (local_heap_cross_ref) delete _heap_cross_ref;
444      if (local_heap) delete _heap;
445    }
446
447    /// \brief Sets the length map.
448    ///
449    /// Sets the length map.
450    /// \return \c (*this)
451    Johnson &lengthMap(const LengthMap &m) {
452      length = &m;
453      return *this;
454    }
455
456    /// \brief Sets the map storing the predecessor edges.
457    ///
458    /// Sets the map storing the predecessor edges.
459    /// If you don't use this function before calling \ref run(),
460    /// it will allocate one. The destuctor deallocates this
461    /// automatically allocated map, of course.
462    /// \return \c (*this)
463    Johnson &predMap(PredMap &m) {
464      if(local_pred) {
465        delete _pred;
466        local_pred=false;
467      }
468      _pred = &m;
469      return *this;
470    }
471
472    /// \brief Sets the map storing the distances calculated by the algorithm.
473    ///
474    /// Sets the map storing the distances calculated by the algorithm.
475    /// If you don't use this function before calling \ref run(),
476    /// it will allocate one. The destuctor deallocates this
477    /// automatically allocated map, of course.
478    /// \return \c (*this)
479    Johnson &distMap(DistMap &m) {
480      if(local_dist) {
481        delete _dist;
482        local_dist=false;
483      }
484      _dist = &m;
485      return *this;
486    }
487
488  protected:
489   
490    template <typename PotentialMap>
491    void shiftedRun(const PotentialMap& potential) {
492     
493      typename Graph::template EdgeMap<Value> shiftlen(*graph);
494      for (EdgeIt it(*graph);  it != INVALID; ++it) {
495        shiftlen[it] = (*length)[it]
496          + potential[graph->source(it)]
497          - potential[graph->target(it)];
498      }
499     
500      typename Dijkstra<Graph, typename Graph::template EdgeMap<Value> >::
501        template DefHeap<Heap, HeapCrossRef>::
502        Create dijkstra(*graph, shiftlen);
503
504      dijkstra.heap(*_heap, *_heap_cross_ref);
505     
506      for (NodeIt it(*graph); it != INVALID; ++it) {
507        dijkstra.run(it);
508        for (NodeIt jt(*graph); jt != INVALID; ++jt) {
509          if (dijkstra.reached(jt)) {
510            _dist->set(it, jt, dijkstra.dist(jt) +
511                       potential[jt] - potential[it]);
512            _pred->set(it, jt, dijkstra.pred(jt));
513          } else {
514            _dist->set(it, jt, OperationTraits::infinity());
515            _pred->set(it, jt, INVALID);
516          }
517        }
518      }
519    }
520
521  public:   
522
523    ///\name Execution control
524    /// The simplest way to execute the algorithm is to use
525    /// one of the member functions called \c run(...).
526    /// \n
527    /// If you need more control on the execution,
528    /// Finally \ref start() will perform the actual path
529    /// computation.
530
531    ///@{
532
533    /// \brief Initializes the internal data structures.
534    ///
535    /// Initializes the internal data structures.
536    void init() {
537      create_maps();
538    }
539
540    /// \brief Executes the algorithm.
541    ///
542    /// This method runs the %Johnson algorithm in order to compute
543    /// the shortest path to each node pairs. The algorithm
544    /// computes
545    /// - The shortest path tree for each node.
546    /// - The distance between each node pairs.
547    void start() {
548
549      typedef typename BelmannFord<Graph, LengthMap>::
550      template DefOperationTraits<OperationTraits>::
551      template DefPredMap<NullMap<Node, Edge> >::
552      Create BelmannFordType;
553     
554      BelmannFordType belmannford(*graph, *length);
555
556      NullMap<Node, Edge> predMap;
557
558      belmannford.predMap(predMap);
559     
560      belmannford.init(OperationTraits::zero());
561      belmannford.start();
562
563      shiftedRun(belmannford.distMap());
564    }
565
566    /// \brief Executes the algorithm and checks the negatvie cycles.
567    ///
568    /// This method runs the %Johnson algorithm in order to compute
569    /// the shortest path to each node pairs. If the graph contains
570    /// negative cycle it gives back false. The algorithm
571    /// computes
572    /// - The shortest path tree for each node.
573    /// - The distance between each node pairs.
574    bool checkedStart() {
575     
576      typedef typename BelmannFord<Graph, LengthMap>::
577      template DefOperationTraits<OperationTraits>::
578      template DefPredMap<NullMap<Node, Edge> >::
579      Create BelmannFordType;
580
581      BelmannFordType belmannford(*graph, *length);
582
583      NullMap<Node, Edge> predMap;
584
585      belmannford.predMap(predMap);
586     
587      belmannford.init(OperationTraits::zero());
588      if (!belmannford.checkedStart()) return false;
589
590      shiftedRun(belmannford.distMap());
591      return true;
592    }
593
594   
595    /// \brief Runs %Johnson algorithm.
596    ///   
597    /// This method runs the %Johnson algorithm from a each node
598    /// in order to compute the shortest path to each node pairs.
599    /// The algorithm computes
600    /// - The shortest path tree for each node.
601    /// - The distance between each node pairs.
602    ///
603    /// \note d.run(s) is just a shortcut of the following code.
604    /// \code
605    ///  d.init();
606    ///  d.start();
607    /// \endcode
608    void run() {
609      init();
610      start();
611    }
612   
613    ///@}
614
615    /// \name Query Functions
616    /// The result of the %Johnson algorithm can be obtained using these
617    /// functions.\n
618    /// Before the use of these functions,
619    /// either run() or start() must be called.
620   
621    ///@{
622
623    /// \brief Copies the shortest path to \c t into \c p
624    ///   
625    /// This function copies the shortest path to \c t into \c p.
626    /// If it \c t is a source itself or unreachable, then it does not
627    /// alter \c p.
628    /// \todo Is it the right way to handle unreachable nodes?
629    /// \return Returns \c true if a path to \c t was actually copied to \c p,
630    /// \c false otherwise.
631    /// \sa DirPath
632    template <typename Path>
633    bool getPath(Path &p, Node source, Node target) {
634      if (connected(source, target)) {
635        p.clear();
636        typename Path::Builder b(target);
637        for(b.setStartNode(target); pred(source, target) != INVALID;
638            target = predNode(target)) {
639          b.pushFront(pred(source, target));
640        }
641        b.commit();
642        return true;
643      }
644      return false;
645    }
646         
647    /// \brief The distance between two nodes.
648    ///
649    /// Returns the distance between two nodes.
650    /// \pre \ref run() must be called before using this function.
651    /// \warning If node \c v in unreachable from the root the return value
652    /// of this funcion is undefined.
653    Value dist(Node source, Node target) const {
654      return (*_dist)(source, target);
655    }
656
657    /// \brief Returns the 'previous edge' of the shortest path tree.
658    ///
659    /// For the node \c node it returns the 'previous edge' of the shortest
660    /// path tree to direction of the node \c root
661    /// i.e. it returns the last edge of a shortest path from the node \c root
662    /// to \c node. It is \ref INVALID if \c node is unreachable from the root
663    /// or if \c node=root. The shortest path tree used here is equal to the
664    /// shortest path tree used in \ref predNode().
665    /// \pre \ref run() must be called before using this function.
666    /// \todo predEdge could be a better name.
667    Edge pred(Node root, Node node) const {
668      return (*_pred)(root, node);
669    }
670
671    /// \brief Returns the 'previous node' of the shortest path tree.
672    ///
673    /// For a node \c node it returns the 'previous node' of the shortest path
674    /// tree to direction of the node \c root, i.e. it returns the last but
675    /// one node from a shortest path from the \c root to \c node. It is
676    /// INVALID if \c node is unreachable from the root or if \c node=root.
677    /// The shortest path tree used here is equal to the
678    /// shortest path tree used in \ref pred(). 
679    /// \pre \ref run() must be called before using this function.
680    Node predNode(Node root, Node node) const {
681      return (*_pred)(root, node) == INVALID ?
682      INVALID : graph->source((*_pred)(root, node));
683    }
684   
685    /// \brief Returns a reference to the matrix node map of distances.
686    ///
687    /// Returns a reference to the matrix node map of distances.
688    ///
689    /// \pre \ref run() must be called before using this function.
690    const DistMap &distMap() const { return *_dist;}
691 
692    /// \brief Returns a reference to the shortest path tree map.
693    ///
694    /// Returns a reference to the matrix node map of the edges of the
695    /// shortest path tree.
696    /// \pre \ref run() must be called before using this function.
697    const PredMap &predMap() const { return *_pred;}
698 
699    /// \brief Checks if a node is reachable from the root.
700    ///
701    /// Returns \c true if \c v is reachable from the root.
702    /// \pre \ref run() must be called before using this function.
703    ///
704    bool connected(Node source, Node target) {
705      return (*_dist)(source, target) != OperationTraits::infinity();
706    }
707   
708    ///@}
709  };
710 
711} //END OF NAMESPACE LEMON
712
713#endif
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