COIN-OR::LEMON - Graph Library

source: lemon-1.2/lemon/howard.h @ 814:0643a9c2c3ae

Last change on this file since 814:0643a9c2c3ae was 771:8452ca46e29a, checked in by Peter Kovacs <kpeter@…>, 10 years ago

Add citations to the min mean cycle classes (#179, #184)

File size: 17.8 KB
Line 
1/* -*- C++ -*-
2 *
3 * This file is a part of LEMON, a generic C++ optimization library
4 *
5 * Copyright (C) 2003-2008
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_H
20#define LEMON_HOWARD_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 Howard class.
37  ///
38  /// Default traits class of Howard class.
39  /// \tparam GR The type of the digraph.
40  /// \tparam LEN The type of the length map.
41  /// It must conform to the \ref concepts::ReadMap "ReadMap" concept.
42#ifdef DOXYGEN
43  template <typename GR, typename LEN>
44#else
45  template <typename GR, typename LEN,
46    bool integer = std::numeric_limits<typename LEN::Value>::is_integer>
47#endif
48  struct HowardDefaultTraits
49  {
50    /// The type of the digraph
51    typedef GR Digraph;
52    /// The type of the length map
53    typedef LEN LengthMap;
54    /// The type of the arc lengths
55    typedef typename LengthMap::Value Value;
56
57    /// \brief The large value type used for internal computations
58    ///
59    /// The large value type used for internal computations.
60    /// It is \c long \c long if the \c Value type is integer,
61    /// otherwise it is \c double.
62    /// \c Value must be convertible to \c LargeValue.
63    typedef double LargeValue;
64
65    /// The tolerance type used for internal computations
66    typedef lemon::Tolerance<LargeValue> 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 value types
77  template <typename GR, typename LEN>
78  struct HowardDefaultTraits<GR, LEN, true>
79  {
80    typedef GR Digraph;
81    typedef LEN LengthMap;
82    typedef typename LengthMap::Value Value;
83#ifdef LEMON_HAVE_LONG_LONG
84    typedef long long LargeValue;
85#else
86    typedef long LargeValue;
87#endif
88    typedef lemon::Tolerance<LargeValue> 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 length (cost) in a digraph
101  /// \ref amo93networkflows, \ref dasdan98minmeancycle.
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 LEN The type of the length map. The default
108  /// map type is \ref concepts::Digraph::ArcMap "GR::ArcMap<int>".
109#ifdef DOXYGEN
110  template <typename GR, typename LEN, typename TR>
111#else
112  template < typename GR,
113             typename LEN = typename GR::template ArcMap<int>,
114             typename TR = HowardDefaultTraits<GR, LEN> >
115#endif
116  class Howard
117  {
118  public:
119 
120    /// The type of the digraph
121    typedef typename TR::Digraph Digraph;
122    /// The type of the length map
123    typedef typename TR::LengthMap LengthMap;
124    /// The type of the arc lengths
125    typedef typename TR::Value Value;
126
127    /// \brief The large value type
128    ///
129    /// The large value type used for internal computations.
130    /// Using the \ref HowardDefaultTraits "default traits class",
131    /// it is \c long \c long if the \c Value type is integer,
132    /// otherwise it is \c double.
133    typedef typename TR::LargeValue LargeValue;
134
135    /// The tolerance type
136    typedef typename TR::Tolerance Tolerance;
137
138    /// \brief The path type of the found cycles
139    ///
140    /// The path type of the found cycles.
141    /// Using the \ref HowardDefaultTraits "default traits class",
142    /// it is \ref lemon::Path "Path<Digraph>".
143    typedef typename TR::Path Path;
144
145    /// The \ref HowardDefaultTraits "traits class" of the algorithm
146    typedef TR Traits;
147
148  private:
149
150    TEMPLATE_DIGRAPH_TYPEDEFS(Digraph);
151 
152    // The digraph the algorithm runs on
153    const Digraph &_gr;
154    // The length of the arcs
155    const LengthMap &_length;
156
157    // Data for the found cycles
158    bool _curr_found, _best_found;
159    LargeValue _curr_length, _best_length;
160    int _curr_size, _best_size;
161    Node _curr_node, _best_node;
162
163    Path *_cycle_path;
164    bool _local_path;
165
166    // Internal data used by the algorithm
167    typename Digraph::template NodeMap<Arc> _policy;
168    typename Digraph::template NodeMap<bool> _reached;
169    typename Digraph::template NodeMap<int> _level;
170    typename Digraph::template NodeMap<LargeValue> _dist;
171
172    // Data for storing the strongly connected components
173    int _comp_num;
174    typename Digraph::template NodeMap<int> _comp;
175    std::vector<std::vector<Node> > _comp_nodes;
176    std::vector<Node>* _nodes;
177    typename Digraph::template NodeMap<std::vector<Arc> > _in_arcs;
178   
179    // Queue used for BFS search
180    std::vector<Node> _queue;
181    int _qfront, _qback;
182
183    Tolerance _tolerance;
184 
185    // Infinite constant
186    const LargeValue INF;
187
188  public:
189 
190    /// \name Named Template Parameters
191    /// @{
192
193    template <typename T>
194    struct SetLargeValueTraits : public Traits {
195      typedef T LargeValue;
196      typedef lemon::Tolerance<T> Tolerance;
197    };
198
199    /// \brief \ref named-templ-param "Named parameter" for setting
200    /// \c LargeValue type.
201    ///
202    /// \ref named-templ-param "Named parameter" for setting \c LargeValue
203    /// type. It is used for internal computations in the algorithm.
204    template <typename T>
205    struct SetLargeValue
206      : public Howard<GR, LEN, SetLargeValueTraits<T> > {
207      typedef Howard<GR, LEN, SetLargeValueTraits<T> > Create;
208    };
209
210    template <typename T>
211    struct SetPathTraits : public Traits {
212      typedef T Path;
213    };
214
215    /// \brief \ref named-templ-param "Named parameter" for setting
216    /// \c %Path type.
217    ///
218    /// \ref named-templ-param "Named parameter" for setting the \c %Path
219    /// type of the found cycles.
220    /// It must conform to the \ref lemon::concepts::Path "Path" concept
221    /// and it must have an \c addBack() function.
222    template <typename T>
223    struct SetPath
224      : public Howard<GR, LEN, SetPathTraits<T> > {
225      typedef Howard<GR, LEN, SetPathTraits<T> > Create;
226    };
227   
228    /// @}
229
230  public:
231
232    /// \brief Constructor.
233    ///
234    /// The constructor of the class.
235    ///
236    /// \param digraph The digraph the algorithm runs on.
237    /// \param length The lengths (costs) of the arcs.
238    Howard( const Digraph &digraph,
239            const LengthMap &length ) :
240      _gr(digraph), _length(length), _best_found(false),
241      _best_length(0), _best_size(1), _cycle_path(NULL), _local_path(false),
242      _policy(digraph), _reached(digraph), _level(digraph), _dist(digraph),
243      _comp(digraph), _in_arcs(digraph),
244      INF(std::numeric_limits<LargeValue>::has_infinity ?
245          std::numeric_limits<LargeValue>::infinity() :
246          std::numeric_limits<LargeValue>::max())
247    {}
248
249    /// Destructor.
250    ~Howard() {
251      if (_local_path) delete _cycle_path;
252    }
253
254    /// \brief Set the path structure for storing the found cycle.
255    ///
256    /// This function sets an external path structure for storing the
257    /// found cycle.
258    ///
259    /// If you don't call this function before calling \ref run() or
260    /// \ref findMinMean(), it will allocate a local \ref Path "path"
261    /// structure. The destuctor deallocates this automatically
262    /// allocated object, of course.
263    ///
264    /// \note The algorithm calls only the \ref lemon::Path::addBack()
265    /// "addBack()" function of the given path structure.
266    ///
267    /// \return <tt>(*this)</tt>
268    Howard& cycle(Path &path) {
269      if (_local_path) {
270        delete _cycle_path;
271        _local_path = false;
272      }
273      _cycle_path = &path;
274      return *this;
275    }
276
277    /// \brief Set the tolerance used by the algorithm.
278    ///
279    /// This function sets the tolerance object used by the algorithm.
280    ///
281    /// \return <tt>(*this)</tt>
282    Howard& tolerance(const Tolerance& tolerance) {
283      _tolerance = tolerance;
284      return *this;
285    }
286
287    /// \brief Return a const reference to the tolerance.
288    ///
289    /// This function returns a const reference to the tolerance object
290    /// used by the algorithm.
291    const Tolerance& tolerance() const {
292      return _tolerance;
293    }
294
295    /// \name Execution control
296    /// The simplest way to execute the algorithm is to call the \ref run()
297    /// function.\n
298    /// If you only need the minimum mean length, you may call
299    /// \ref findMinMean().
300
301    /// @{
302
303    /// \brief Run the algorithm.
304    ///
305    /// This function runs the algorithm.
306    /// It can be called more than once (e.g. if the underlying digraph
307    /// and/or the arc lengths have been modified).
308    ///
309    /// \return \c true if a directed cycle exists in the digraph.
310    ///
311    /// \note <tt>mmc.run()</tt> is just a shortcut of the following code.
312    /// \code
313    ///   return mmc.findMinMean() && mmc.findCycle();
314    /// \endcode
315    bool run() {
316      return findMinMean() && findCycle();
317    }
318
319    /// \brief Find the minimum cycle mean.
320    ///
321    /// This function finds the minimum mean length of the directed
322    /// cycles in the digraph.
323    ///
324    /// \return \c true if a directed cycle exists in the digraph.
325    bool findMinMean() {
326      // Initialize and find strongly connected components
327      init();
328      findComponents();
329     
330      // Find the minimum cycle mean in the components
331      for (int comp = 0; comp < _comp_num; ++comp) {
332        // Find the minimum mean cycle in the current component
333        if (!buildPolicyGraph(comp)) continue;
334        while (true) {
335          findPolicyCycle();
336          if (!computeNodeDistances()) break;
337        }
338        // Update the best cycle (global minimum mean cycle)
339        if ( _curr_found && (!_best_found ||
340             _curr_length * _best_size < _best_length * _curr_size) ) {
341          _best_found = true;
342          _best_length = _curr_length;
343          _best_size = _curr_size;
344          _best_node = _curr_node;
345        }
346      }
347      return _best_found;
348    }
349
350    /// \brief Find a minimum mean directed cycle.
351    ///
352    /// This function finds a directed cycle of minimum mean length
353    /// in the digraph using the data computed by findMinMean().
354    ///
355    /// \return \c true if a directed cycle exists in the digraph.
356    ///
357    /// \pre \ref findMinMean() must be called before using this function.
358    bool findCycle() {
359      if (!_best_found) return false;
360      _cycle_path->addBack(_policy[_best_node]);
361      for ( Node v = _best_node;
362            (v = _gr.target(_policy[v])) != _best_node; ) {
363        _cycle_path->addBack(_policy[v]);
364      }
365      return true;
366    }
367
368    /// @}
369
370    /// \name Query Functions
371    /// The results of the algorithm can be obtained using these
372    /// functions.\n
373    /// The algorithm should be executed before using them.
374
375    /// @{
376
377    /// \brief Return the total length of the found cycle.
378    ///
379    /// This function returns the total length of the found cycle.
380    ///
381    /// \pre \ref run() or \ref findMinMean() must be called before
382    /// using this function.
383    LargeValue cycleLength() const {
384      return _best_length;
385    }
386
387    /// \brief Return the number of arcs on the found cycle.
388    ///
389    /// This function returns the number of arcs on the found cycle.
390    ///
391    /// \pre \ref run() or \ref findMinMean() must be called before
392    /// using this function.
393    int cycleArcNum() const {
394      return _best_size;
395    }
396
397    /// \brief Return the mean length of the found cycle.
398    ///
399    /// This function returns the mean length of the found cycle.
400    ///
401    /// \note <tt>alg.cycleMean()</tt> is just a shortcut of the
402    /// following code.
403    /// \code
404    ///   return static_cast<double>(alg.cycleLength()) / alg.cycleArcNum();
405    /// \endcode
406    ///
407    /// \pre \ref run() or \ref findMinMean() must be called before
408    /// using this function.
409    double cycleMean() const {
410      return static_cast<double>(_best_length) / _best_size;
411    }
412
413    /// \brief Return the found cycle.
414    ///
415    /// This function returns a const reference to the path structure
416    /// storing the found cycle.
417    ///
418    /// \pre \ref run() or \ref findCycle() must be called before using
419    /// this function.
420    const Path& cycle() const {
421      return *_cycle_path;
422    }
423
424    ///@}
425
426  private:
427
428    // Initialize
429    void init() {
430      if (!_cycle_path) {
431        _local_path = true;
432        _cycle_path = new Path;
433      }
434      _queue.resize(countNodes(_gr));
435      _best_found = false;
436      _best_length = 0;
437      _best_size = 1;
438      _cycle_path->clear();
439    }
440   
441    // Find strongly connected components and initialize _comp_nodes
442    // and _in_arcs
443    void findComponents() {
444      _comp_num = stronglyConnectedComponents(_gr, _comp);
445      _comp_nodes.resize(_comp_num);
446      if (_comp_num == 1) {
447        _comp_nodes[0].clear();
448        for (NodeIt n(_gr); n != INVALID; ++n) {
449          _comp_nodes[0].push_back(n);
450          _in_arcs[n].clear();
451          for (InArcIt a(_gr, n); a != INVALID; ++a) {
452            _in_arcs[n].push_back(a);
453          }
454        }
455      } else {
456        for (int i = 0; i < _comp_num; ++i)
457          _comp_nodes[i].clear();
458        for (NodeIt n(_gr); n != INVALID; ++n) {
459          int k = _comp[n];
460          _comp_nodes[k].push_back(n);
461          _in_arcs[n].clear();
462          for (InArcIt a(_gr, n); a != INVALID; ++a) {
463            if (_comp[_gr.source(a)] == k) _in_arcs[n].push_back(a);
464          }
465        }
466      }
467    }
468
469    // Build the policy graph in the given strongly connected component
470    // (the out-degree of every node is 1)
471    bool buildPolicyGraph(int comp) {
472      _nodes = &(_comp_nodes[comp]);
473      if (_nodes->size() < 1 ||
474          (_nodes->size() == 1 && _in_arcs[(*_nodes)[0]].size() == 0)) {
475        return false;
476      }
477      for (int i = 0; i < int(_nodes->size()); ++i) {
478        _dist[(*_nodes)[i]] = INF;
479      }
480      Node u, v;
481      Arc e;
482      for (int i = 0; i < int(_nodes->size()); ++i) {
483        v = (*_nodes)[i];
484        for (int j = 0; j < int(_in_arcs[v].size()); ++j) {
485          e = _in_arcs[v][j];
486          u = _gr.source(e);
487          if (_length[e] < _dist[u]) {
488            _dist[u] = _length[e];
489            _policy[u] = e;
490          }
491        }
492      }
493      return true;
494    }
495
496    // Find the minimum mean cycle in the policy graph
497    void findPolicyCycle() {
498      for (int i = 0; i < int(_nodes->size()); ++i) {
499        _level[(*_nodes)[i]] = -1;
500      }
501      LargeValue clength;
502      int csize;
503      Node u, v;
504      _curr_found = false;
505      for (int i = 0; i < int(_nodes->size()); ++i) {
506        u = (*_nodes)[i];
507        if (_level[u] >= 0) continue;
508        for (; _level[u] < 0; u = _gr.target(_policy[u])) {
509          _level[u] = i;
510        }
511        if (_level[u] == i) {
512          // A cycle is found
513          clength = _length[_policy[u]];
514          csize = 1;
515          for (v = u; (v = _gr.target(_policy[v])) != u; ) {
516            clength += _length[_policy[v]];
517            ++csize;
518          }
519          if ( !_curr_found ||
520               (clength * _curr_size < _curr_length * csize) ) {
521            _curr_found = true;
522            _curr_length = clength;
523            _curr_size = csize;
524            _curr_node = u;
525          }
526        }
527      }
528    }
529
530    // Contract the policy graph and compute node distances
531    bool computeNodeDistances() {
532      // Find the component of the main cycle and compute node distances
533      // using reverse BFS
534      for (int i = 0; i < int(_nodes->size()); ++i) {
535        _reached[(*_nodes)[i]] = false;
536      }
537      _qfront = _qback = 0;
538      _queue[0] = _curr_node;
539      _reached[_curr_node] = true;
540      _dist[_curr_node] = 0;
541      Node u, v;
542      Arc e;
543      while (_qfront <= _qback) {
544        v = _queue[_qfront++];
545        for (int j = 0; j < int(_in_arcs[v].size()); ++j) {
546          e = _in_arcs[v][j];
547          u = _gr.source(e);
548          if (_policy[u] == e && !_reached[u]) {
549            _reached[u] = true;
550            _dist[u] = _dist[v] + _length[e] * _curr_size - _curr_length;
551            _queue[++_qback] = u;
552          }
553        }
554      }
555
556      // Connect all other nodes to this component and compute node
557      // distances using reverse BFS
558      _qfront = 0;
559      while (_qback < int(_nodes->size())-1) {
560        v = _queue[_qfront++];
561        for (int j = 0; j < int(_in_arcs[v].size()); ++j) {
562          e = _in_arcs[v][j];
563          u = _gr.source(e);
564          if (!_reached[u]) {
565            _reached[u] = true;
566            _policy[u] = e;
567            _dist[u] = _dist[v] + _length[e] * _curr_size - _curr_length;
568            _queue[++_qback] = u;
569          }
570        }
571      }
572
573      // Improve node distances
574      bool improved = false;
575      for (int i = 0; i < int(_nodes->size()); ++i) {
576        v = (*_nodes)[i];
577        for (int j = 0; j < int(_in_arcs[v].size()); ++j) {
578          e = _in_arcs[v][j];
579          u = _gr.source(e);
580          LargeValue delta = _dist[v] + _length[e] * _curr_size - _curr_length;
581          if (_tolerance.less(delta, _dist[u])) {
582            _dist[u] = delta;
583            _policy[u] = e;
584            improved = true;
585          }
586        }
587      }
588      return improved;
589    }
590
591  }; //class Howard
592
593  ///@}
594
595} //namespace lemon
596
597#endif //LEMON_HOWARD_H
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