COIN-OR::LEMON - Graph Library

source: lemon-0.x/lemon/johnson.h @ 1865:dcefd1d1377f

Last change on this file since 1865:dcefd1d1377f was 1864:1788205e36af, checked in by Balazs Dezso, 14 years ago

Fixing Bellman's name

File size: 23.2 KB
Line 
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/bellman_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 bellman-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    /// \name Named template parameters
311
312    ///@{
313
314    template <class T>
315    struct DefPredMapTraits : public Traits {
316      typedef T PredMap;
317      static PredMap *createPredMap(const Graph& graph) {
318        throw UninitializedParameter();
319      }
320    };
321
322    /// \brief \ref named-templ-param "Named parameter" for setting PredMap
323    /// type
324    /// \ref named-templ-param "Named parameter" for setting PredMap type
325    ///
326    template <class T>
327    struct DefPredMap
328      : public Johnson< Graph, LengthMap, DefPredMapTraits<T> > {
329      typedef Johnson< Graph, LengthMap, DefPredMapTraits<T> > Create;
330    };
331   
332    template <class T>
333    struct DefDistMapTraits : public Traits {
334      typedef T DistMap;
335      static DistMap *createDistMap(const Graph& graph) {
336        throw UninitializedParameter();
337      }
338    };
339    /// \brief \ref named-templ-param "Named parameter" for setting DistMap
340    /// type
341    ///
342    /// \ref named-templ-param "Named parameter" for setting DistMap type
343    ///
344    template <class T>
345    struct DefDistMap
346      : public Johnson< Graph, LengthMap, DefDistMapTraits<T> > {
347      typedef Johnson< Graph, LengthMap, DefDistMapTraits<T> > Create;
348    };
349   
350    template <class T>
351    struct DefOperationTraitsTraits : public Traits {
352      typedef T OperationTraits;
353    };
354   
355    /// \brief \ref named-templ-param "Named parameter" for setting
356    /// OperationTraits type
357    ///
358    /// \ref named-templ-param "Named parameter" for setting
359    /// OperationTraits type
360    template <class T>
361    struct DefOperationTraits
362      : public Johnson< Graph, LengthMap, DefOperationTraitsTraits<T> > {
363      typedef Johnson< Graph, LengthMap, DefOperationTraitsTraits<T> > Create;
364    };
365
366    template <class H, class CR>
367    struct DefHeapTraits : public Traits {
368      typedef CR HeapCrossRef;
369      typedef H Heap;
370      static HeapCrossRef *createHeapCrossRef(const Graph &) {
371        throw UninitializedParameter();
372      }
373      static Heap *createHeap(HeapCrossRef &)
374      {
375        throw UninitializedParameter();
376      }
377    };
378    ///\brief \ref named-templ-param "Named parameter" for setting heap and
379    ///cross reference type
380
381    ///\ref named-templ-param "Named parameter" for setting heap and cross
382    ///reference type
383    ///
384    template <class H, class CR = typename Graph::template NodeMap<int> >
385    struct DefHeap
386      : public Johnson< Graph, LengthMap, DefHeapTraits<H, CR> > {
387      typedef Johnson< Graph, LengthMap, DefHeapTraits<H, CR> > Create;
388    };
389
390    template <class H, class CR>
391    struct DefStandardHeapTraits : public Traits {
392      typedef CR HeapCrossRef;
393      typedef H Heap;
394      static HeapCrossRef *createHeapCrossRef(const Graph &G) {
395        return new HeapCrossRef(G);
396      }
397      static Heap *createHeap(HeapCrossRef &R)
398      {
399        return new Heap(R);
400      }
401    };
402    ///\ref named-templ-param "Named parameter" for setting heap and cross
403    ///reference type with automatic allocation
404
405    ///\ref named-templ-param "Named parameter" for setting heap and cross
406    ///reference type. It can allocate the heap and the cross reference
407    ///object if the cross reference's constructor waits for the graph as
408    ///parameter and the heap's constructor waits for the cross reference.
409    template <class H, class CR = typename Graph::template NodeMap<int> >
410    struct DefStandardHeap
411      : public Johnson< Graph, LengthMap, DefStandardHeapTraits<H, CR> > {
412      typedef Johnson< Graph, LengthMap, DefStandardHeapTraits<H, CR> >
413      Create;
414    };
415   
416    ///@}
417
418  protected:
419
420    Johnson() {}
421
422  public:     
423
424    typedef Johnson Create;
425   
426    /// \brief Constructor.
427    ///
428    /// \param _graph the graph the algorithm will run on.
429    /// \param _length the length map used by the algorithm.
430    Johnson(const Graph& _graph, const LengthMap& _length) :
431      graph(&_graph), length(&_length),
432      _pred(0), local_pred(false),
433      _dist(0), local_dist(false),
434      _heap_cross_ref(0), local_heap_cross_ref(false),
435      _heap(0), local_heap(false) {}
436   
437    ///Destructor.
438    ~Johnson() {
439      if (local_pred) delete _pred;
440      if (local_dist) delete _dist;
441      if (local_heap_cross_ref) delete _heap_cross_ref;
442      if (local_heap) delete _heap;
443    }
444
445    /// \brief Sets the length map.
446    ///
447    /// Sets the length map.
448    /// \return \c (*this)
449    Johnson &lengthMap(const LengthMap &m) {
450      length = &m;
451      return *this;
452    }
453
454    /// \brief Sets the map storing the predecessor edges.
455    ///
456    /// Sets the map storing the predecessor edges.
457    /// If you don't use this function before calling \ref run(),
458    /// it will allocate one. The destuctor deallocates this
459    /// automatically allocated map, of course.
460    /// \return \c (*this)
461    Johnson &predMap(PredMap &m) {
462      if(local_pred) {
463        delete _pred;
464        local_pred=false;
465      }
466      _pred = &m;
467      return *this;
468    }
469
470    /// \brief Sets the map storing the distances calculated by the algorithm.
471    ///
472    /// Sets the map storing the distances calculated by the algorithm.
473    /// If you don't use this function before calling \ref run(),
474    /// it will allocate one. The destuctor deallocates this
475    /// automatically allocated map, of course.
476    /// \return \c (*this)
477    Johnson &distMap(DistMap &m) {
478      if(local_dist) {
479        delete _dist;
480        local_dist=false;
481      }
482      _dist = &m;
483      return *this;
484    }
485
486  protected:
487   
488    template <typename PotentialMap>
489    void shiftedRun(const PotentialMap& potential) {
490     
491      typename Graph::template EdgeMap<Value> shiftlen(*graph);
492      for (EdgeIt it(*graph);  it != INVALID; ++it) {
493        shiftlen[it] = (*length)[it]
494          + potential[graph->source(it)]
495          - potential[graph->target(it)];
496      }
497     
498      typename Dijkstra<Graph, typename Graph::template EdgeMap<Value> >::
499        template DefHeap<Heap, HeapCrossRef>::
500        Create dijkstra(*graph, shiftlen);
501
502      dijkstra.heap(*_heap, *_heap_cross_ref);
503     
504      for (NodeIt it(*graph); it != INVALID; ++it) {
505        dijkstra.run(it);
506        for (NodeIt jt(*graph); jt != INVALID; ++jt) {
507          if (dijkstra.reached(jt)) {
508            _dist->set(it, jt, dijkstra.dist(jt) +
509                       potential[jt] - potential[it]);
510            _pred->set(it, jt, dijkstra.predEdge(jt));
511          } else {
512            _dist->set(it, jt, OperationTraits::infinity());
513            _pred->set(it, jt, INVALID);
514          }
515        }
516      }
517    }
518
519  public:   
520
521    ///\name Execution control
522    /// The simplest way to execute the algorithm is to use
523    /// one of the member functions called \c run(...).
524    /// \n
525    /// If you need more control on the execution,
526    /// Finally \ref start() will perform the actual path
527    /// computation.
528
529    ///@{
530
531    /// \brief Initializes the internal data structures.
532    ///
533    /// Initializes the internal data structures.
534    void init() {
535      create_maps();
536    }
537
538    /// \brief Executes the algorithm.
539    ///
540    /// This method runs the %Johnson algorithm in order to compute
541    /// the shortest path to each node pairs. The algorithm
542    /// computes
543    /// - The shortest path tree for each node.
544    /// - The distance between each node pairs.
545    void start() {
546
547      typedef typename BellmanFord<Graph, LengthMap>::
548      template DefOperationTraits<OperationTraits>::
549      template DefPredMap<NullMap<Node, Edge> >::
550      Create BellmanFordType;
551     
552      BellmanFordType bellmanford(*graph, *length);
553
554      NullMap<Node, Edge> predMap;
555
556      bellmanford.predMap(predMap);
557     
558      bellmanford.init(OperationTraits::zero());
559      bellmanford.start();
560
561      shiftedRun(bellmanford.distMap());
562    }
563
564    /// \brief Executes the algorithm and checks the negatvie cycles.
565    ///
566    /// This method runs the %Johnson algorithm in order to compute
567    /// the shortest path to each node pairs. If the graph contains
568    /// negative cycle it gives back false. The algorithm
569    /// computes
570    /// - The shortest path tree for each node.
571    /// - The distance between each node pairs.
572    bool checkedStart() {
573     
574      typedef typename BellmanFord<Graph, LengthMap>::
575      template DefOperationTraits<OperationTraits>::
576      template DefPredMap<NullMap<Node, Edge> >::
577      Create BellmanFordType;
578
579      BellmanFordType bellmanford(*graph, *length);
580
581      NullMap<Node, Edge> predMap;
582
583      bellmanford.predMap(predMap);
584     
585      bellmanford.init(OperationTraits::zero());
586      if (!bellmanford.checkedStart()) return false;
587
588      shiftedRun(bellmanford.distMap());
589      return true;
590    }
591
592   
593    /// \brief Runs %Johnson algorithm.
594    ///   
595    /// This method runs the %Johnson algorithm from a each node
596    /// in order to compute the shortest path to each node pairs.
597    /// The algorithm computes
598    /// - The shortest path tree for each node.
599    /// - The distance between each node pairs.
600    ///
601    /// \note d.run(s) is just a shortcut of the following code.
602    /// \code
603    ///  d.init();
604    ///  d.start();
605    /// \endcode
606    void run() {
607      init();
608      start();
609    }
610   
611    ///@}
612
613    /// \name Query Functions
614    /// The result of the %Johnson algorithm can be obtained using these
615    /// functions.\n
616    /// Before the use of these functions,
617    /// either run() or start() must be called.
618   
619    ///@{
620
621    /// \brief Copies the shortest path to \c t into \c p
622    ///   
623    /// This function copies the shortest path to \c t into \c p.
624    /// If it \c t is a source itself or unreachable, then it does not
625    /// alter \c p.
626    /// \return Returns \c true if a path to \c t was actually copied to \c p,
627    /// \c false otherwise.
628    /// \sa DirPath
629    template <typename Path>
630    bool getPath(Path &p, Node source, Node target) {
631      if (connected(source, target)) {
632        p.clear();
633        typename Path::Builder b(target);
634        for(b.setStartNode(target); predEdge(source, target) != INVALID;
635            target = predNode(target)) {
636          b.pushFront(predEdge(source, target));
637        }
638        b.commit();
639        return true;
640      }
641      return false;
642    }
643         
644    /// \brief The distance between two nodes.
645    ///
646    /// Returns the distance between two nodes.
647    /// \pre \ref run() must be called before using this function.
648    /// \warning If node \c v in unreachable from the root the return value
649    /// of this funcion is undefined.
650    Value dist(Node source, Node target) const {
651      return (*_dist)(source, target);
652    }
653
654    /// \brief Returns the 'previous edge' of the shortest path tree.
655    ///
656    /// For the node \c node it returns the 'previous edge' of the shortest
657    /// path tree to direction of the node \c root
658    /// i.e. it returns the last edge of a shortest path from the node \c root
659    /// to \c node. It is \ref INVALID if \c node is unreachable from the root
660    /// or if \c node=root. The shortest path tree used here is equal to the
661    /// shortest path tree used in \ref predNode().
662    /// \pre \ref run() must be called before using this function.
663    Edge predEdge(Node root, Node node) const {
664      return (*_pred)(root, node);
665    }
666
667    /// \brief Returns the 'previous node' of the shortest path tree.
668    ///
669    /// For a node \c node it returns the 'previous node' of the shortest path
670    /// tree to direction of the node \c root, i.e. it returns the last but
671    /// one node from a shortest path from the \c root to \c node. It is
672    /// INVALID if \c node is unreachable from the root or if \c node=root.
673    /// The shortest path tree used here is equal to the
674    /// shortest path tree used in \ref predEdge(). 
675    /// \pre \ref run() must be called before using this function.
676    Node predNode(Node root, Node node) const {
677      return (*_pred)(root, node) == INVALID ?
678      INVALID : graph->source((*_pred)(root, node));
679    }
680   
681    /// \brief Returns a reference to the matrix node map of distances.
682    ///
683    /// Returns a reference to the matrix node map of distances.
684    ///
685    /// \pre \ref run() must be called before using this function.
686    const DistMap &distMap() const { return *_dist;}
687 
688    /// \brief Returns a reference to the shortest path tree map.
689    ///
690    /// Returns a reference to the matrix node map of the edges of the
691    /// shortest path tree.
692    /// \pre \ref run() must be called before using this function.
693    const PredMap &predMap() const { return *_pred;}
694 
695    /// \brief Checks if a node is reachable from the root.
696    ///
697    /// Returns \c true if \c v is reachable from the root.
698    /// \pre \ref run() must be called before using this function.
699    ///
700    bool connected(Node source, Node target) {
701      return (*_dist)(source, target) != OperationTraits::infinity();
702    }
703   
704    ///@}
705  };
706 
707} //END OF NAMESPACE LEMON
708
709#endif
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