COIN-OR::LEMON - Graph Library

source: lemon-main/lemon/howard_mmc.h

Last change on this file was 1093:fb1c7da561ce, checked in by Alpar Juttner <alpar@…>, 6 years ago

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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_HOWARD_MMC_H
20#define LEMON_HOWARD_MMC_H
21
22/// \ingroup min_mean_cycle
23///
24/// \file
25/// \brief Howard's algorithm for finding a minimum mean cycle.
26
27#include <vector>
28#include <limits>
29#include <lemon/core.h>
30#include <lemon/path.h>
31#include <lemon/tolerance.h>
32#include <lemon/connectivity.h>
33
34namespace lemon {
35
36  /// \brief Default traits class of HowardMmc class.
37  ///
38  /// Default traits class of HowardMmc class.
39  /// \tparam GR The type of the digraph.
40  /// \tparam CM The type of the cost map.
41  /// It must conform to the \ref concepts::ReadMap "ReadMap" concept.
42#ifdef DOXYGEN
43  template <typename GR, typename CM>
44#else
45  template <typename GR, typename CM,
46    bool integer = std::numeric_limits<typename CM::Value>::is_integer>
47#endif
48  struct HowardMmcDefaultTraits
49  {
50    /// The type of the digraph
51    typedef GR Digraph;
52    /// The type of the cost map
53    typedef CM CostMap;
54    /// The type of the arc costs
55    typedef typename CostMap::Value Cost;
56
57    /// \brief The large cost type used for internal computations
58    ///
59    /// The large cost type used for internal computations.
60    /// It is \c long \c long if the \c Cost type is integer,
61    /// otherwise it is \c double.
62    /// \c Cost must be convertible to \c LargeCost.
63    typedef double LargeCost;
64
65    /// The tolerance type used for internal computations
66    typedef lemon::Tolerance<LargeCost> Tolerance;
67
68    /// \brief The path type of the found cycles
69    ///
70    /// The path type of the found cycles.
71    /// It must conform to the \ref lemon::concepts::Path "Path" concept
72    /// and it must have an \c addBack() function.
73    typedef lemon::Path<Digraph> Path;
74  };
75
76  // Default traits class for integer cost types
77  template <typename GR, typename CM>
78  struct HowardMmcDefaultTraits<GR, CM, true>
79  {
80    typedef GR Digraph;
81    typedef CM CostMap;
82    typedef typename CostMap::Value Cost;
83#ifdef LEMON_HAVE_LONG_LONG
84    typedef long long LargeCost;
85#else
86    typedef long LargeCost;
87#endif
88    typedef lemon::Tolerance<LargeCost> Tolerance;
89    typedef lemon::Path<Digraph> Path;
90  };
91
92
93  /// \addtogroup min_mean_cycle
94  /// @{
95
96  /// \brief Implementation of Howard's algorithm for finding a minimum
97  /// mean cycle.
98  ///
99  /// This class implements Howard's policy iteration algorithm for finding
100  /// a directed cycle of minimum mean cost in a digraph
101  /// \cite dasdan98minmeancycle, \cite dasdan04experimental.
102  /// This class provides the most efficient algorithm for the
103  /// minimum mean cycle problem, though the best known theoretical
104  /// bound on its running time is exponential.
105  ///
106  /// \tparam GR The type of the digraph the algorithm runs on.
107  /// \tparam CM The type of the cost map. The default
108  /// map type is \ref concepts::Digraph::ArcMap "GR::ArcMap<int>".
109  /// \tparam TR The traits class that defines various types used by the
110  /// algorithm. By default, it is \ref HowardMmcDefaultTraits
111  /// "HowardMmcDefaultTraits<GR, CM>".
112  /// In most cases, this parameter should not be set directly,
113  /// consider to use the named template parameters instead.
114#ifdef DOXYGEN
115  template <typename GR, typename CM, typename TR>
116#else
117  template < typename GR,
118             typename CM = typename GR::template ArcMap<int>,
119             typename TR = HowardMmcDefaultTraits<GR, CM> >
120#endif
121  class HowardMmc
122  {
123  public:
124
125    /// The type of the digraph
126    typedef typename TR::Digraph Digraph;
127    /// The type of the cost map
128    typedef typename TR::CostMap CostMap;
129    /// The type of the arc costs
130    typedef typename TR::Cost Cost;
131
132    /// \brief The large cost type
133    ///
134    /// The large cost type used for internal computations.
135    /// By default, it is \c long \c long if the \c Cost type is integer,
136    /// otherwise it is \c double.
137    typedef typename TR::LargeCost LargeCost;
138
139    /// The tolerance type
140    typedef typename TR::Tolerance Tolerance;
141
142    /// \brief The path type of the found cycles
143    ///
144    /// The path type of the found cycles.
145    /// Using the \ref lemon::HowardMmcDefaultTraits "default traits class",
146    /// it is \ref lemon::Path "Path<Digraph>".
147    typedef typename TR::Path Path;
148
149    /// The \ref lemon::HowardMmcDefaultTraits "traits class" of the algorithm
150    typedef TR Traits;
151
152    /// \brief Constants for the causes of search termination.
153    ///
154    /// Enum type containing constants for the different causes of search
155    /// termination. The \ref findCycleMean() function returns one of
156    /// these values.
157    enum TerminationCause {
158
159      /// No directed cycle can be found in the digraph.
160      NO_CYCLE = 0,
161
162      /// Optimal solution (minimum cycle mean) is found.
163      OPTIMAL = 1,
164
165      /// The iteration count limit is reached.
166      ITERATION_LIMIT
167    };
168
169  private:
170
171    TEMPLATE_DIGRAPH_TYPEDEFS(Digraph);
172
173    // The digraph the algorithm runs on
174    const Digraph &_gr;
175    // The cost of the arcs
176    const CostMap &_cost;
177
178    // Data for the found cycles
179    bool _curr_found, _best_found;
180    LargeCost _curr_cost, _best_cost;
181    int _curr_size, _best_size;
182    Node _curr_node, _best_node;
183
184    Path *_cycle_path;
185    bool _local_path;
186
187    // Internal data used by the algorithm
188    typename Digraph::template NodeMap<Arc> _policy;
189    typename Digraph::template NodeMap<bool> _reached;
190    typename Digraph::template NodeMap<int> _level;
191    typename Digraph::template NodeMap<LargeCost> _dist;
192
193    // Data for storing the strongly connected components
194    int _comp_num;
195    typename Digraph::template NodeMap<int> _comp;
196    std::vector<std::vector<Node> > _comp_nodes;
197    std::vector<Node>* _nodes;
198    typename Digraph::template NodeMap<std::vector<Arc> > _in_arcs;
199
200    // Queue used for BFS search
201    std::vector<Node> _queue;
202    int _qfront, _qback;
203
204    Tolerance _tolerance;
205
206    // Infinite constant
207    const LargeCost INF;
208
209  public:
210
211    /// \name Named Template Parameters
212    /// @{
213
214    template <typename T>
215    struct SetLargeCostTraits : public Traits {
216      typedef T LargeCost;
217      typedef lemon::Tolerance<T> Tolerance;
218    };
219
220    /// \brief \ref named-templ-param "Named parameter" for setting
221    /// \c LargeCost type.
222    ///
223    /// \ref named-templ-param "Named parameter" for setting \c LargeCost
224    /// type. It is used for internal computations in the algorithm.
225    template <typename T>
226    struct SetLargeCost
227      : public HowardMmc<GR, CM, SetLargeCostTraits<T> > {
228      typedef HowardMmc<GR, CM, SetLargeCostTraits<T> > Create;
229    };
230
231    template <typename T>
232    struct SetPathTraits : public Traits {
233      typedef T Path;
234    };
235
236    /// \brief \ref named-templ-param "Named parameter" for setting
237    /// \c %Path type.
238    ///
239    /// \ref named-templ-param "Named parameter" for setting the \c %Path
240    /// type of the found cycles.
241    /// It must conform to the \ref lemon::concepts::Path "Path" concept
242    /// and it must have an \c addBack() function.
243    template <typename T>
244    struct SetPath
245      : public HowardMmc<GR, CM, SetPathTraits<T> > {
246      typedef HowardMmc<GR, CM, SetPathTraits<T> > Create;
247    };
248
249    /// @}
250
251  protected:
252
253    HowardMmc() {}
254
255  public:
256
257    /// \brief Constructor.
258    ///
259    /// The constructor of the class.
260    ///
261    /// \param digraph The digraph the algorithm runs on.
262    /// \param cost The costs of the arcs.
263    HowardMmc( const Digraph &digraph,
264               const CostMap &cost ) :
265      _gr(digraph), _cost(cost), _best_found(false),
266      _best_cost(0), _best_size(1), _cycle_path(NULL), _local_path(false),
267      _policy(digraph), _reached(digraph), _level(digraph), _dist(digraph),
268      _comp(digraph), _in_arcs(digraph),
269      INF(std::numeric_limits<LargeCost>::has_infinity ?
270          std::numeric_limits<LargeCost>::infinity() :
271          std::numeric_limits<LargeCost>::max())
272    {}
273
274    /// Destructor.
275    ~HowardMmc() {
276      if (_local_path) delete _cycle_path;
277    }
278
279    /// \brief Set the path structure for storing the found cycle.
280    ///
281    /// This function sets an external path structure for storing the
282    /// found cycle.
283    ///
284    /// If you don't call this function before calling \ref run() or
285    /// \ref findCycleMean(), a local \ref Path "path" structure
286    /// will be allocated. The destuctor deallocates this automatically
287    /// allocated object, of course.
288    ///
289    /// \note The algorithm calls only the \ref lemon::Path::addBack()
290    /// "addBack()" function of the given path structure.
291    ///
292    /// \return <tt>(*this)</tt>
293    HowardMmc& cycle(Path &path) {
294      if (_local_path) {
295        delete _cycle_path;
296        _local_path = false;
297      }
298      _cycle_path = &path;
299      return *this;
300    }
301
302    /// \brief Set the tolerance used by the algorithm.
303    ///
304    /// This function sets the tolerance object used by the algorithm.
305    ///
306    /// \return <tt>(*this)</tt>
307    HowardMmc& tolerance(const Tolerance& tolerance) {
308      _tolerance = tolerance;
309      return *this;
310    }
311
312    /// \brief Return a const reference to the tolerance.
313    ///
314    /// This function returns a const reference to the tolerance object
315    /// used by the algorithm.
316    const Tolerance& tolerance() const {
317      return _tolerance;
318    }
319
320    /// \name Execution control
321    /// The simplest way to execute the algorithm is to call the \ref run()
322    /// function.\n
323    /// If you only need the minimum mean cost, you may call
324    /// \ref findCycleMean().
325
326    /// @{
327
328    /// \brief Run the algorithm.
329    ///
330    /// This function runs the algorithm.
331    /// It can be called more than once (e.g. if the underlying digraph
332    /// and/or the arc costs have been modified).
333    ///
334    /// \return \c true if a directed cycle exists in the digraph.
335    ///
336    /// \note <tt>mmc.run()</tt> is just a shortcut of the following code.
337    /// \code
338    ///   return mmc.findCycleMean() && mmc.findCycle();
339    /// \endcode
340    bool run() {
341      return findCycleMean() && findCycle();
342    }
343
344    /// \brief Find the minimum cycle mean (or an upper bound).
345    ///
346    /// This function finds the minimum mean cost of the directed
347    /// cycles in the digraph (or an upper bound for it).
348    ///
349    /// By default, the function finds the exact minimum cycle mean,
350    /// but an optional limit can also be specified for the number of
351    /// iterations performed during the search process.
352    /// The return value indicates if the optimal solution is found
353    /// or the iteration limit is reached. In the latter case, an
354    /// approximate solution is provided, which corresponds to a directed
355    /// cycle whose mean cost is relatively small, but not necessarily
356    /// minimal.
357    ///
358    /// \param limit  The maximum allowed number of iterations during
359    /// the search process. Its default value implies that the algorithm
360    /// runs until it finds the exact optimal solution.
361    ///
362    /// \return The termination cause of the search process.
363    /// For more information, see \ref TerminationCause.
364    TerminationCause findCycleMean(int limit =
365                                   std::numeric_limits<int>::max()) {
366      // Initialize and find strongly connected components
367      init();
368      findComponents();
369
370      // Find the minimum cycle mean in the components
371      int iter_count = 0;
372      bool iter_limit_reached = false;
373      for (int comp = 0; comp < _comp_num; ++comp) {
374        // Find the minimum mean cycle in the current component
375        if (!buildPolicyGraph(comp)) continue;
376        while (true) {
377          if (++iter_count > limit) {
378            iter_limit_reached = true;
379            break;
380          }
381          findPolicyCycle();
382          if (!computeNodeDistances()) break;
383        }
384
385        // Update the best cycle (global minimum mean cycle)
386        if ( _curr_found && (!_best_found ||
387             _curr_cost * _best_size < _best_cost * _curr_size) ) {
388          _best_found = true;
389          _best_cost = _curr_cost;
390          _best_size = _curr_size;
391          _best_node = _curr_node;
392        }
393
394        if (iter_limit_reached) break;
395      }
396
397      if (iter_limit_reached) {
398        return ITERATION_LIMIT;
399      } else {
400        return _best_found ? OPTIMAL : NO_CYCLE;
401      }
402    }
403
404    /// \brief Find a minimum mean directed cycle.
405    ///
406    /// This function finds a directed cycle of minimum mean cost
407    /// in the digraph using the data computed by findCycleMean().
408    ///
409    /// \return \c true if a directed cycle exists in the digraph.
410    ///
411    /// \pre \ref findCycleMean() must be called before using this function.
412    bool findCycle() {
413      if (!_best_found) return false;
414      _cycle_path->addBack(_policy[_best_node]);
415      for ( Node v = _best_node;
416            (v = _gr.target(_policy[v])) != _best_node; ) {
417        _cycle_path->addBack(_policy[v]);
418      }
419      return true;
420    }
421
422    /// @}
423
424    /// \name Query Functions
425    /// The results of the algorithm can be obtained using these
426    /// functions.\n
427    /// The algorithm should be executed before using them.
428
429    /// @{
430
431    /// \brief Return the total cost of the found cycle.
432    ///
433    /// This function returns the total cost of the found cycle.
434    ///
435    /// \pre \ref run() or \ref findCycleMean() must be called before
436    /// using this function.
437    Cost cycleCost() const {
438      return static_cast<Cost>(_best_cost);
439    }
440
441    /// \brief Return the number of arcs on the found cycle.
442    ///
443    /// This function returns the number of arcs on the found cycle.
444    ///
445    /// \pre \ref run() or \ref findCycleMean() must be called before
446    /// using this function.
447    int cycleSize() const {
448      return _best_size;
449    }
450
451    /// \brief Return the mean cost of the found cycle.
452    ///
453    /// This function returns the mean cost of the found cycle.
454    ///
455    /// \note <tt>alg.cycleMean()</tt> is just a shortcut of the
456    /// following code.
457    /// \code
458    ///   return static_cast<double>(alg.cycleCost()) / alg.cycleSize();
459    /// \endcode
460    ///
461    /// \pre \ref run() or \ref findCycleMean() must be called before
462    /// using this function.
463    double cycleMean() const {
464      return static_cast<double>(_best_cost) / _best_size;
465    }
466
467    /// \brief Return the found cycle.
468    ///
469    /// This function returns a const reference to the path structure
470    /// storing the found cycle.
471    ///
472    /// \pre \ref run() or \ref findCycle() must be called before using
473    /// this function.
474    const Path& cycle() const {
475      return *_cycle_path;
476    }
477
478    ///@}
479
480  private:
481
482    // Initialize
483    void init() {
484      if (!_cycle_path) {
485        _local_path = true;
486        _cycle_path = new Path;
487      }
488      _queue.resize(countNodes(_gr));
489      _best_found = false;
490      _best_cost = 0;
491      _best_size = 1;
492      _cycle_path->clear();
493    }
494
495    // Find strongly connected components and initialize _comp_nodes
496    // and _in_arcs
497    void findComponents() {
498      _comp_num = stronglyConnectedComponents(_gr, _comp);
499      _comp_nodes.resize(_comp_num);
500      if (_comp_num == 1) {
501        _comp_nodes[0].clear();
502        for (NodeIt n(_gr); n != INVALID; ++n) {
503          _comp_nodes[0].push_back(n);
504          _in_arcs[n].clear();
505          for (InArcIt a(_gr, n); a != INVALID; ++a) {
506            _in_arcs[n].push_back(a);
507          }
508        }
509      } else {
510        for (int i = 0; i < _comp_num; ++i)
511          _comp_nodes[i].clear();
512        for (NodeIt n(_gr); n != INVALID; ++n) {
513          int k = _comp[n];
514          _comp_nodes[k].push_back(n);
515          _in_arcs[n].clear();
516          for (InArcIt a(_gr, n); a != INVALID; ++a) {
517            if (_comp[_gr.source(a)] == k) _in_arcs[n].push_back(a);
518          }
519        }
520      }
521    }
522
523    // Build the policy graph in the given strongly connected component
524    // (the out-degree of every node is 1)
525    bool buildPolicyGraph(int comp) {
526      _nodes = &(_comp_nodes[comp]);
527      if (_nodes->size() < 1 ||
528          (_nodes->size() == 1 && _in_arcs[(*_nodes)[0]].size() == 0)) {
529        return false;
530      }
531      for (int i = 0; i < int(_nodes->size()); ++i) {
532        _dist[(*_nodes)[i]] = INF;
533      }
534      Node u, v;
535      Arc e;
536      for (int i = 0; i < int(_nodes->size()); ++i) {
537        v = (*_nodes)[i];
538        for (int j = 0; j < int(_in_arcs[v].size()); ++j) {
539          e = _in_arcs[v][j];
540          u = _gr.source(e);
541          if (_cost[e] < _dist[u]) {
542            _dist[u] = _cost[e];
543            _policy[u] = e;
544          }
545        }
546      }
547      return true;
548    }
549
550    // Find the minimum mean cycle in the policy graph
551    void findPolicyCycle() {
552      for (int i = 0; i < int(_nodes->size()); ++i) {
553        _level[(*_nodes)[i]] = -1;
554      }
555      LargeCost ccost;
556      int csize;
557      Node u, v;
558      _curr_found = false;
559      for (int i = 0; i < int(_nodes->size()); ++i) {
560        u = (*_nodes)[i];
561        if (_level[u] >= 0) continue;
562        for (; _level[u] < 0; u = _gr.target(_policy[u])) {
563          _level[u] = i;
564        }
565        if (_level[u] == i) {
566          // A cycle is found
567          ccost = _cost[_policy[u]];
568          csize = 1;
569          for (v = u; (v = _gr.target(_policy[v])) != u; ) {
570            ccost += _cost[_policy[v]];
571            ++csize;
572          }
573          if ( !_curr_found ||
574               (ccost * _curr_size < _curr_cost * csize) ) {
575            _curr_found = true;
576            _curr_cost = ccost;
577            _curr_size = csize;
578            _curr_node = u;
579          }
580        }
581      }
582    }
583
584    // Contract the policy graph and compute node distances
585    bool computeNodeDistances() {
586      // Find the component of the main cycle and compute node distances
587      // using reverse BFS
588      for (int i = 0; i < int(_nodes->size()); ++i) {
589        _reached[(*_nodes)[i]] = false;
590      }
591      _qfront = _qback = 0;
592      _queue[0] = _curr_node;
593      _reached[_curr_node] = true;
594      _dist[_curr_node] = 0;
595      Node u, v;
596      Arc e;
597      while (_qfront <= _qback) {
598        v = _queue[_qfront++];
599        for (int j = 0; j < int(_in_arcs[v].size()); ++j) {
600          e = _in_arcs[v][j];
601          u = _gr.source(e);
602          if (_policy[u] == e && !_reached[u]) {
603            _reached[u] = true;
604            _dist[u] = _dist[v] + _cost[e] * _curr_size - _curr_cost;
605            _queue[++_qback] = u;
606          }
607        }
608      }
609
610      // Connect all other nodes to this component and compute node
611      // distances using reverse BFS
612      _qfront = 0;
613      while (_qback < int(_nodes->size())-1) {
614        v = _queue[_qfront++];
615        for (int j = 0; j < int(_in_arcs[v].size()); ++j) {
616          e = _in_arcs[v][j];
617          u = _gr.source(e);
618          if (!_reached[u]) {
619            _reached[u] = true;
620            _policy[u] = e;
621            _dist[u] = _dist[v] + _cost[e] * _curr_size - _curr_cost;
622            _queue[++_qback] = u;
623          }
624        }
625      }
626
627      // Improve node distances
628      bool improved = false;
629      for (int i = 0; i < int(_nodes->size()); ++i) {
630        v = (*_nodes)[i];
631        for (int j = 0; j < int(_in_arcs[v].size()); ++j) {
632          e = _in_arcs[v][j];
633          u = _gr.source(e);
634          LargeCost delta = _dist[v] + _cost[e] * _curr_size - _curr_cost;
635          if (_tolerance.less(delta, _dist[u])) {
636            _dist[u] = delta;
637            _policy[u] = e;
638            improved = true;
639          }
640        }
641      }
642      return improved;
643    }
644
645  }; //class HowardMmc
646
647  ///@}
648
649} //namespace lemon
650
651#endif //LEMON_HOWARD_MMC_H
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