COIN-OR::LEMON - Graph Library

source: lemon/lemon/suurballe.h @ 1369:9fd86ec2cb81

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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) 2003-2013
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
35namespace 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 lemon::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(m).
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
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