COIN-OR::LEMON - Graph Library

source: lemon/lemon/howard_mmc.h @ 1270:dceba191c00d

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[956]1/* -*- mode: C++; indent-tabs-mode: nil; -*-
[805]2 *
[956]3 * This file is a part of LEMON, a generic C++ optimization library.
[805]4 *
[1270]5 * Copyright (C) 2003-2013
[805]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
[942]19#ifndef LEMON_HOWARD_MMC_H
20#define LEMON_HOWARD_MMC_H
[805]21
[815]22/// \ingroup min_mean_cycle
[805]23///
24/// \file
25/// \brief Howard's algorithm for finding a minimum mean cycle.
26
27#include <vector>
[810]28#include <limits>
[805]29#include <lemon/core.h>
30#include <lemon/path.h>
31#include <lemon/tolerance.h>
32#include <lemon/connectivity.h>
33
34namespace lemon {
35
[942]36  /// \brief Default traits class of HowardMmc class.
[808]37  ///
[942]38  /// Default traits class of HowardMmc class.
[808]39  /// \tparam GR The type of the digraph.
[942]40  /// \tparam CM The type of the cost map.
[808]41  /// It must conform to the \ref concepts::ReadMap "ReadMap" concept.
42#ifdef DOXYGEN
[942]43  template <typename GR, typename CM>
[808]44#else
[942]45  template <typename GR, typename CM,
46    bool integer = std::numeric_limits<typename CM::Value>::is_integer>
[808]47#endif
[942]48  struct HowardMmcDefaultTraits
[808]49  {
50    /// The type of the digraph
51    typedef GR Digraph;
[942]52    /// The type of the cost map
53    typedef CM CostMap;
54    /// The type of the arc costs
55    typedef typename CostMap::Value Cost;
[808]56
[942]57    /// \brief The large cost type used for internal computations
[808]58    ///
[942]59    /// The large cost type used for internal computations.
60    /// It is \c long \c long if the \c Cost type is integer,
[808]61    /// otherwise it is \c double.
[942]62    /// \c Cost must be convertible to \c LargeCost.
63    typedef double LargeCost;
[808]64
65    /// The tolerance type used for internal computations
[942]66    typedef lemon::Tolerance<LargeCost> Tolerance;
[808]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
[942]76  // Default traits class for integer cost types
77  template <typename GR, typename CM>
78  struct HowardMmcDefaultTraits<GR, CM, true>
[808]79  {
80    typedef GR Digraph;
[942]81    typedef CM CostMap;
82    typedef typename CostMap::Value Cost;
[808]83#ifdef LEMON_HAVE_LONG_LONG
[942]84    typedef long long LargeCost;
[808]85#else
[942]86    typedef long LargeCost;
[808]87#endif
[942]88    typedef lemon::Tolerance<LargeCost> Tolerance;
[808]89    typedef lemon::Path<Digraph> Path;
90  };
91
92
[815]93  /// \addtogroup min_mean_cycle
[805]94  /// @{
95
96  /// \brief Implementation of Howard's algorithm for finding a minimum
97  /// mean cycle.
98  ///
[811]99  /// This class implements Howard's policy iteration algorithm for finding
[942]100  /// a directed cycle of minimum mean cost in a digraph
[1221]101  /// \cite dasdan98minmeancycle, \cite dasdan04experimental.
[815]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.
[805]105  ///
106  /// \tparam GR The type of the digraph the algorithm runs on.
[942]107  /// \tparam CM The type of the cost map. The default
[805]108  /// map type is \ref concepts::Digraph::ArcMap "GR::ArcMap<int>".
[891]109  /// \tparam TR The traits class that defines various types used by the
[942]110  /// algorithm. By default, it is \ref HowardMmcDefaultTraits
111  /// "HowardMmcDefaultTraits<GR, CM>".
[891]112  /// In most cases, this parameter should not be set directly,
113  /// consider to use the named template parameters instead.
[805]114#ifdef DOXYGEN
[942]115  template <typename GR, typename CM, typename TR>
[805]116#else
117  template < typename GR,
[942]118             typename CM = typename GR::template ArcMap<int>,
119             typename TR = HowardMmcDefaultTraits<GR, CM> >
[805]120#endif
[942]121  class HowardMmc
[805]122  {
123  public:
[956]124
[808]125    /// The type of the digraph
126    typedef typename TR::Digraph Digraph;
[942]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;
[808]131
[942]132    /// \brief The large cost type
[808]133    ///
[942]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,
[808]136    /// otherwise it is \c double.
[942]137    typedef typename TR::LargeCost LargeCost;
[808]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.
[1250]145    /// Using the \ref lemon::HowardMmcDefaultTraits "default traits class",
[808]146    /// it is \ref lemon::Path "Path<Digraph>".
147    typedef typename TR::Path Path;
148
[1250]149    /// The \ref lemon::HowardMmcDefaultTraits "traits class" of the algorithm
[808]150    typedef TR Traits;
[805]151
[1178]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 {
[1270]158
[1178]159      /// No directed cycle can be found in the digraph.
160      NO_CYCLE = 0,
[1270]161
[1178]162      /// Optimal solution (minimum cycle mean) is found.
163      OPTIMAL = 1,
164
165      /// The iteration count limit is reached.
166      ITERATION_LIMIT
167    };
168
[805]169  private:
170
171    TEMPLATE_DIGRAPH_TYPEDEFS(Digraph);
[956]172
[805]173    // The digraph the algorithm runs on
174    const Digraph &_gr;
[942]175    // The cost of the arcs
176    const CostMap &_cost;
[805]177
[807]178    // Data for the found cycles
179    bool _curr_found, _best_found;
[942]180    LargeCost _curr_cost, _best_cost;
[807]181    int _curr_size, _best_size;
182    Node _curr_node, _best_node;
183
[805]184    Path *_cycle_path;
[807]185    bool _local_path;
[805]186
[807]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;
[942]191    typename Digraph::template NodeMap<LargeCost> _dist;
[805]192
[807]193    // Data for storing the strongly connected components
194    int _comp_num;
[805]195    typename Digraph::template NodeMap<int> _comp;
[807]196    std::vector<std::vector<Node> > _comp_nodes;
197    std::vector<Node>* _nodes;
198    typename Digraph::template NodeMap<std::vector<Arc> > _in_arcs;
[956]199
[807]200    // Queue used for BFS search
201    std::vector<Node> _queue;
202    int _qfront, _qback;
[808]203
204    Tolerance _tolerance;
[956]205
[814]206    // Infinite constant
[942]207    const LargeCost INF;
[814]208
[808]209  public:
[956]210
[808]211    /// \name Named Template Parameters
212    /// @{
213
214    template <typename T>
[942]215    struct SetLargeCostTraits : public Traits {
216      typedef T LargeCost;
[808]217      typedef lemon::Tolerance<T> Tolerance;
218    };
219
220    /// \brief \ref named-templ-param "Named parameter" for setting
[942]221    /// \c LargeCost type.
[808]222    ///
[942]223    /// \ref named-templ-param "Named parameter" for setting \c LargeCost
[808]224    /// type. It is used for internal computations in the algorithm.
225    template <typename T>
[942]226    struct SetLargeCost
227      : public HowardMmc<GR, CM, SetLargeCostTraits<T> > {
228      typedef HowardMmc<GR, CM, SetLargeCostTraits<T> > Create;
[808]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
[942]245      : public HowardMmc<GR, CM, SetPathTraits<T> > {
246      typedef HowardMmc<GR, CM, SetPathTraits<T> > Create;
[808]247    };
[956]248
[808]249    /// @}
[805]250
[941]251  protected:
252
[942]253    HowardMmc() {}
[941]254
[805]255  public:
256
257    /// \brief Constructor.
258    ///
259    /// The constructor of the class.
260    ///
261    /// \param digraph The digraph the algorithm runs on.
[942]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),
[807]267      _policy(digraph), _reached(digraph), _level(digraph), _dist(digraph),
[814]268      _comp(digraph), _in_arcs(digraph),
[942]269      INF(std::numeric_limits<LargeCost>::has_infinity ?
270          std::numeric_limits<LargeCost>::infinity() :
271          std::numeric_limits<LargeCost>::max())
[805]272    {}
273
274    /// Destructor.
[942]275    ~HowardMmc() {
[805]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
[1217]285    /// \ref findCycleMean(), a local \ref Path "path" structure
286    /// will be allocated. The destuctor deallocates this automatically
[805]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>
[942]293    HowardMmc& cycle(Path &path) {
[805]294      if (_local_path) {
295        delete _cycle_path;
296        _local_path = false;
297      }
298      _cycle_path = &path;
299      return *this;
300    }
301
[816]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>
[942]307    HowardMmc& tolerance(const Tolerance& tolerance) {
[816]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
[805]320    /// \name Execution control
321    /// The simplest way to execute the algorithm is to call the \ref run()
322    /// function.\n
[942]323    /// If you only need the minimum mean cost, you may call
324    /// \ref findCycleMean().
[805]325
326    /// @{
327
328    /// \brief Run the algorithm.
329    ///
330    /// This function runs the algorithm.
[806]331    /// It can be called more than once (e.g. if the underlying digraph
[942]332    /// and/or the arc costs have been modified).
[805]333    ///
334    /// \return \c true if a directed cycle exists in the digraph.
335    ///
[806]336    /// \note <tt>mmc.run()</tt> is just a shortcut of the following code.
[805]337    /// \code
[942]338    ///   return mmc.findCycleMean() && mmc.findCycle();
[805]339    /// \endcode
340    bool run() {
[942]341      return findCycleMean() && findCycle();
[805]342    }
343
[1178]344    /// \brief Find the minimum cycle mean (or an upper bound).
[805]345    ///
[942]346    /// This function finds the minimum mean cost of the directed
[1178]347    /// cycles in the digraph (or an upper bound for it).
[805]348    ///
[1178]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
[1270]359    /// the search process. Its default value implies that the algorithm
[1178]360    /// runs until it finds the exact optimal solution.
361    ///
362    /// \return The termination cause of the search process.
[1270]363    /// For more information, see \ref TerminationCause.
[1178]364    TerminationCause findCycleMean(int limit = std::numeric_limits<int>::max()) {
[807]365      // Initialize and find strongly connected components
366      init();
367      findComponents();
[956]368
[806]369      // Find the minimum cycle mean in the components
[1178]370      int iter_count = 0;
371      bool iter_limit_reached = false;
[805]372      for (int comp = 0; comp < _comp_num; ++comp) {
[807]373        // Find the minimum mean cycle in the current component
374        if (!buildPolicyGraph(comp)) continue;
[805]375        while (true) {
[1178]376          if (++iter_count > limit) {
377            iter_limit_reached = true;
378            break;
379          }
[807]380          findPolicyCycle();
[805]381          if (!computeNodeDistances()) break;
382        }
[1178]383
[807]384        // Update the best cycle (global minimum mean cycle)
[814]385        if ( _curr_found && (!_best_found ||
[942]386             _curr_cost * _best_size < _best_cost * _curr_size) ) {
[807]387          _best_found = true;
[942]388          _best_cost = _curr_cost;
[807]389          _best_size = _curr_size;
390          _best_node = _curr_node;
391        }
[1270]392
[1178]393        if (iter_limit_reached) break;
[805]394      }
[1178]395
396      if (iter_limit_reached) {
397        return ITERATION_LIMIT;
398      } else {
399        return _best_found ? OPTIMAL : NO_CYCLE;
400      }
[805]401    }
402
403    /// \brief Find a minimum mean directed cycle.
404    ///
[942]405    /// This function finds a directed cycle of minimum mean cost
406    /// in the digraph using the data computed by findCycleMean().
[805]407    ///
408    /// \return \c true if a directed cycle exists in the digraph.
409    ///
[942]410    /// \pre \ref findCycleMean() must be called before using this function.
[805]411    bool findCycle() {
[807]412      if (!_best_found) return false;
413      _cycle_path->addBack(_policy[_best_node]);
414      for ( Node v = _best_node;
415            (v = _gr.target(_policy[v])) != _best_node; ) {
[805]416        _cycle_path->addBack(_policy[v]);
417      }
418      return true;
419    }
420
421    /// @}
422
423    /// \name Query Functions
[806]424    /// The results of the algorithm can be obtained using these
[805]425    /// functions.\n
426    /// The algorithm should be executed before using them.
427
428    /// @{
429
[942]430    /// \brief Return the total cost of the found cycle.
[805]431    ///
[942]432    /// This function returns the total cost of the found cycle.
[805]433    ///
[942]434    /// \pre \ref run() or \ref findCycleMean() must be called before
[805]435    /// using this function.
[942]436    Cost cycleCost() const {
437      return static_cast<Cost>(_best_cost);
[805]438    }
439
440    /// \brief Return the number of arcs on the found cycle.
441    ///
442    /// This function returns the number of arcs on the found cycle.
443    ///
[942]444    /// \pre \ref run() or \ref findCycleMean() must be called before
[805]445    /// using this function.
[942]446    int cycleSize() const {
[807]447      return _best_size;
[805]448    }
449
[942]450    /// \brief Return the mean cost of the found cycle.
[805]451    ///
[942]452    /// This function returns the mean cost of the found cycle.
[805]453    ///
[807]454    /// \note <tt>alg.cycleMean()</tt> is just a shortcut of the
[805]455    /// following code.
456    /// \code
[942]457    ///   return static_cast<double>(alg.cycleCost()) / alg.cycleSize();
[805]458    /// \endcode
459    ///
[942]460    /// \pre \ref run() or \ref findCycleMean() must be called before
[805]461    /// using this function.
462    double cycleMean() const {
[942]463      return static_cast<double>(_best_cost) / _best_size;
[805]464    }
465
466    /// \brief Return the found cycle.
467    ///
468    /// This function returns a const reference to the path structure
469    /// storing the found cycle.
470    ///
471    /// \pre \ref run() or \ref findCycle() must be called before using
472    /// this function.
473    const Path& cycle() const {
474      return *_cycle_path;
475    }
476
477    ///@}
478
479  private:
480
[807]481    // Initialize
482    void init() {
483      if (!_cycle_path) {
484        _local_path = true;
485        _cycle_path = new Path;
[805]486      }
[807]487      _queue.resize(countNodes(_gr));
488      _best_found = false;
[942]489      _best_cost = 0;
[807]490      _best_size = 1;
491      _cycle_path->clear();
492    }
[956]493
[807]494    // Find strongly connected components and initialize _comp_nodes
495    // and _in_arcs
496    void findComponents() {
497      _comp_num = stronglyConnectedComponents(_gr, _comp);
498      _comp_nodes.resize(_comp_num);
499      if (_comp_num == 1) {
500        _comp_nodes[0].clear();
501        for (NodeIt n(_gr); n != INVALID; ++n) {
502          _comp_nodes[0].push_back(n);
503          _in_arcs[n].clear();
504          for (InArcIt a(_gr, n); a != INVALID; ++a) {
505            _in_arcs[n].push_back(a);
506          }
507        }
508      } else {
509        for (int i = 0; i < _comp_num; ++i)
510          _comp_nodes[i].clear();
511        for (NodeIt n(_gr); n != INVALID; ++n) {
512          int k = _comp[n];
513          _comp_nodes[k].push_back(n);
514          _in_arcs[n].clear();
515          for (InArcIt a(_gr, n); a != INVALID; ++a) {
516            if (_comp[_gr.source(a)] == k) _in_arcs[n].push_back(a);
517          }
518        }
[805]519      }
[807]520    }
521
522    // Build the policy graph in the given strongly connected component
523    // (the out-degree of every node is 1)
524    bool buildPolicyGraph(int comp) {
525      _nodes = &(_comp_nodes[comp]);
526      if (_nodes->size() < 1 ||
527          (_nodes->size() == 1 && _in_arcs[(*_nodes)[0]].size() == 0)) {
528        return false;
[805]529      }
[807]530      for (int i = 0; i < int(_nodes->size()); ++i) {
[814]531        _dist[(*_nodes)[i]] = INF;
[807]532      }
533      Node u, v;
534      Arc e;
535      for (int i = 0; i < int(_nodes->size()); ++i) {
536        v = (*_nodes)[i];
537        for (int j = 0; j < int(_in_arcs[v].size()); ++j) {
538          e = _in_arcs[v][j];
539          u = _gr.source(e);
[942]540          if (_cost[e] < _dist[u]) {
541            _dist[u] = _cost[e];
[807]542            _policy[u] = e;
543          }
[805]544        }
545      }
546      return true;
547    }
548
[807]549    // Find the minimum mean cycle in the policy graph
550    void findPolicyCycle() {
551      for (int i = 0; i < int(_nodes->size()); ++i) {
552        _level[(*_nodes)[i]] = -1;
553      }
[942]554      LargeCost ccost;
[805]555      int csize;
556      Node u, v;
[807]557      _curr_found = false;
558      for (int i = 0; i < int(_nodes->size()); ++i) {
559        u = (*_nodes)[i];
560        if (_level[u] >= 0) continue;
561        for (; _level[u] < 0; u = _gr.target(_policy[u])) {
562          _level[u] = i;
563        }
564        if (_level[u] == i) {
565          // A cycle is found
[942]566          ccost = _cost[_policy[u]];
[807]567          csize = 1;
568          for (v = u; (v = _gr.target(_policy[v])) != u; ) {
[942]569            ccost += _cost[_policy[v]];
[807]570            ++csize;
[805]571          }
[807]572          if ( !_curr_found ||
[942]573               (ccost * _curr_size < _curr_cost * csize) ) {
[807]574            _curr_found = true;
[942]575            _curr_cost = ccost;
[807]576            _curr_size = csize;
577            _curr_node = u;
[805]578          }
579        }
580      }
581    }
582
[807]583    // Contract the policy graph and compute node distances
[805]584    bool computeNodeDistances() {
[807]585      // Find the component of the main cycle and compute node distances
586      // using reverse BFS
587      for (int i = 0; i < int(_nodes->size()); ++i) {
588        _reached[(*_nodes)[i]] = false;
589      }
590      _qfront = _qback = 0;
591      _queue[0] = _curr_node;
592      _reached[_curr_node] = true;
593      _dist[_curr_node] = 0;
[805]594      Node u, v;
[807]595      Arc e;
596      while (_qfront <= _qback) {
597        v = _queue[_qfront++];
598        for (int j = 0; j < int(_in_arcs[v].size()); ++j) {
599          e = _in_arcs[v][j];
[805]600          u = _gr.source(e);
[807]601          if (_policy[u] == e && !_reached[u]) {
602            _reached[u] = true;
[942]603            _dist[u] = _dist[v] + _cost[e] * _curr_size - _curr_cost;
[807]604            _queue[++_qback] = u;
[805]605          }
606        }
607      }
[807]608
609      // Connect all other nodes to this component and compute node
610      // distances using reverse BFS
611      _qfront = 0;
612      while (_qback < int(_nodes->size())-1) {
613        v = _queue[_qfront++];
614        for (int j = 0; j < int(_in_arcs[v].size()); ++j) {
615          e = _in_arcs[v][j];
616          u = _gr.source(e);
617          if (!_reached[u]) {
618            _reached[u] = true;
619            _policy[u] = e;
[942]620            _dist[u] = _dist[v] + _cost[e] * _curr_size - _curr_cost;
[807]621            _queue[++_qback] = u;
622          }
623        }
624      }
625
626      // Improve node distances
[805]627      bool improved = false;
[807]628      for (int i = 0; i < int(_nodes->size()); ++i) {
629        v = (*_nodes)[i];
630        for (int j = 0; j < int(_in_arcs[v].size()); ++j) {
631          e = _in_arcs[v][j];
632          u = _gr.source(e);
[942]633          LargeCost delta = _dist[v] + _cost[e] * _curr_size - _curr_cost;
[808]634          if (_tolerance.less(delta, _dist[u])) {
[807]635            _dist[u] = delta;
636            _policy[u] = e;
637            improved = true;
638          }
[805]639        }
640      }
641      return improved;
642    }
643
[942]644  }; //class HowardMmc
[805]645
646  ///@}
647
648} //namespace lemon
649
[942]650#endif //LEMON_HOWARD_MMC_H
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