lemon/howard_mmc.h
author Peter Kovacs <kpeter@inf.elte.hu>
Sat, 08 Jan 2011 22:49:09 +0100
changeset 1032 62ba43576f85
parent 864 d3ea191c3412
child 1002 f63ba40a60f4
permissions -rw-r--r--
Doc group for TSP algorithms (#386)
     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) 2003-2010
     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 
    34 namespace 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   /// \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 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 HowardMmcDefaultTraits "default traits class",
   146     /// it is \ref lemon::Path "Path<Digraph>".
   147     typedef typename TR::Path Path;
   148 
   149     /// The \ref HowardMmcDefaultTraits "traits class" of the algorithm
   150     typedef TR Traits;
   151 
   152   private:
   153 
   154     TEMPLATE_DIGRAPH_TYPEDEFS(Digraph);
   155 
   156     // The digraph the algorithm runs on
   157     const Digraph &_gr;
   158     // The cost of the arcs
   159     const CostMap &_cost;
   160 
   161     // Data for the found cycles
   162     bool _curr_found, _best_found;
   163     LargeCost _curr_cost, _best_cost;
   164     int _curr_size, _best_size;
   165     Node _curr_node, _best_node;
   166 
   167     Path *_cycle_path;
   168     bool _local_path;
   169 
   170     // Internal data used by the algorithm
   171     typename Digraph::template NodeMap<Arc> _policy;
   172     typename Digraph::template NodeMap<bool> _reached;
   173     typename Digraph::template NodeMap<int> _level;
   174     typename Digraph::template NodeMap<LargeCost> _dist;
   175 
   176     // Data for storing the strongly connected components
   177     int _comp_num;
   178     typename Digraph::template NodeMap<int> _comp;
   179     std::vector<std::vector<Node> > _comp_nodes;
   180     std::vector<Node>* _nodes;
   181     typename Digraph::template NodeMap<std::vector<Arc> > _in_arcs;
   182 
   183     // Queue used for BFS search
   184     std::vector<Node> _queue;
   185     int _qfront, _qback;
   186 
   187     Tolerance _tolerance;
   188 
   189     // Infinite constant
   190     const LargeCost INF;
   191 
   192   public:
   193 
   194     /// \name Named Template Parameters
   195     /// @{
   196 
   197     template <typename T>
   198     struct SetLargeCostTraits : public Traits {
   199       typedef T LargeCost;
   200       typedef lemon::Tolerance<T> Tolerance;
   201     };
   202 
   203     /// \brief \ref named-templ-param "Named parameter" for setting
   204     /// \c LargeCost type.
   205     ///
   206     /// \ref named-templ-param "Named parameter" for setting \c LargeCost
   207     /// type. It is used for internal computations in the algorithm.
   208     template <typename T>
   209     struct SetLargeCost
   210       : public HowardMmc<GR, CM, SetLargeCostTraits<T> > {
   211       typedef HowardMmc<GR, CM, SetLargeCostTraits<T> > Create;
   212     };
   213 
   214     template <typename T>
   215     struct SetPathTraits : public Traits {
   216       typedef T Path;
   217     };
   218 
   219     /// \brief \ref named-templ-param "Named parameter" for setting
   220     /// \c %Path type.
   221     ///
   222     /// \ref named-templ-param "Named parameter" for setting the \c %Path
   223     /// type of the found cycles.
   224     /// It must conform to the \ref lemon::concepts::Path "Path" concept
   225     /// and it must have an \c addBack() function.
   226     template <typename T>
   227     struct SetPath
   228       : public HowardMmc<GR, CM, SetPathTraits<T> > {
   229       typedef HowardMmc<GR, CM, SetPathTraits<T> > Create;
   230     };
   231 
   232     /// @}
   233 
   234   protected:
   235 
   236     HowardMmc() {}
   237 
   238   public:
   239 
   240     /// \brief Constructor.
   241     ///
   242     /// The constructor of the class.
   243     ///
   244     /// \param digraph The digraph the algorithm runs on.
   245     /// \param cost The costs of the arcs.
   246     HowardMmc( const Digraph &digraph,
   247                const CostMap &cost ) :
   248       _gr(digraph), _cost(cost), _best_found(false),
   249       _best_cost(0), _best_size(1), _cycle_path(NULL), _local_path(false),
   250       _policy(digraph), _reached(digraph), _level(digraph), _dist(digraph),
   251       _comp(digraph), _in_arcs(digraph),
   252       INF(std::numeric_limits<LargeCost>::has_infinity ?
   253           std::numeric_limits<LargeCost>::infinity() :
   254           std::numeric_limits<LargeCost>::max())
   255     {}
   256 
   257     /// Destructor.
   258     ~HowardMmc() {
   259       if (_local_path) delete _cycle_path;
   260     }
   261 
   262     /// \brief Set the path structure for storing the found cycle.
   263     ///
   264     /// This function sets an external path structure for storing the
   265     /// found cycle.
   266     ///
   267     /// If you don't call this function before calling \ref run() or
   268     /// \ref findCycleMean(), it will allocate a local \ref Path "path"
   269     /// structure. The destuctor deallocates this automatically
   270     /// allocated object, of course.
   271     ///
   272     /// \note The algorithm calls only the \ref lemon::Path::addBack()
   273     /// "addBack()" function of the given path structure.
   274     ///
   275     /// \return <tt>(*this)</tt>
   276     HowardMmc& cycle(Path &path) {
   277       if (_local_path) {
   278         delete _cycle_path;
   279         _local_path = false;
   280       }
   281       _cycle_path = &path;
   282       return *this;
   283     }
   284 
   285     /// \brief Set the tolerance used by the algorithm.
   286     ///
   287     /// This function sets the tolerance object used by the algorithm.
   288     ///
   289     /// \return <tt>(*this)</tt>
   290     HowardMmc& tolerance(const Tolerance& tolerance) {
   291       _tolerance = tolerance;
   292       return *this;
   293     }
   294 
   295     /// \brief Return a const reference to the tolerance.
   296     ///
   297     /// This function returns a const reference to the tolerance object
   298     /// used by the algorithm.
   299     const Tolerance& tolerance() const {
   300       return _tolerance;
   301     }
   302 
   303     /// \name Execution control
   304     /// The simplest way to execute the algorithm is to call the \ref run()
   305     /// function.\n
   306     /// If you only need the minimum mean cost, you may call
   307     /// \ref findCycleMean().
   308 
   309     /// @{
   310 
   311     /// \brief Run the algorithm.
   312     ///
   313     /// This function runs the algorithm.
   314     /// It can be called more than once (e.g. if the underlying digraph
   315     /// and/or the arc costs have been modified).
   316     ///
   317     /// \return \c true if a directed cycle exists in the digraph.
   318     ///
   319     /// \note <tt>mmc.run()</tt> is just a shortcut of the following code.
   320     /// \code
   321     ///   return mmc.findCycleMean() && mmc.findCycle();
   322     /// \endcode
   323     bool run() {
   324       return findCycleMean() && findCycle();
   325     }
   326 
   327     /// \brief Find the minimum cycle mean.
   328     ///
   329     /// This function finds the minimum mean cost of the directed
   330     /// cycles in the digraph.
   331     ///
   332     /// \return \c true if a directed cycle exists in the digraph.
   333     bool findCycleMean() {
   334       // Initialize and find strongly connected components
   335       init();
   336       findComponents();
   337 
   338       // Find the minimum cycle mean in the components
   339       for (int comp = 0; comp < _comp_num; ++comp) {
   340         // Find the minimum mean cycle in the current component
   341         if (!buildPolicyGraph(comp)) continue;
   342         while (true) {
   343           findPolicyCycle();
   344           if (!computeNodeDistances()) break;
   345         }
   346         // Update the best cycle (global minimum mean cycle)
   347         if ( _curr_found && (!_best_found ||
   348              _curr_cost * _best_size < _best_cost * _curr_size) ) {
   349           _best_found = true;
   350           _best_cost = _curr_cost;
   351           _best_size = _curr_size;
   352           _best_node = _curr_node;
   353         }
   354       }
   355       return _best_found;
   356     }
   357 
   358     /// \brief Find a minimum mean directed cycle.
   359     ///
   360     /// This function finds a directed cycle of minimum mean cost
   361     /// in the digraph using the data computed by findCycleMean().
   362     ///
   363     /// \return \c true if a directed cycle exists in the digraph.
   364     ///
   365     /// \pre \ref findCycleMean() must be called before using this function.
   366     bool findCycle() {
   367       if (!_best_found) return false;
   368       _cycle_path->addBack(_policy[_best_node]);
   369       for ( Node v = _best_node;
   370             (v = _gr.target(_policy[v])) != _best_node; ) {
   371         _cycle_path->addBack(_policy[v]);
   372       }
   373       return true;
   374     }
   375 
   376     /// @}
   377 
   378     /// \name Query Functions
   379     /// The results of the algorithm can be obtained using these
   380     /// functions.\n
   381     /// The algorithm should be executed before using them.
   382 
   383     /// @{
   384 
   385     /// \brief Return the total cost of the found cycle.
   386     ///
   387     /// This function returns the total cost of the found cycle.
   388     ///
   389     /// \pre \ref run() or \ref findCycleMean() must be called before
   390     /// using this function.
   391     Cost cycleCost() const {
   392       return static_cast<Cost>(_best_cost);
   393     }
   394 
   395     /// \brief Return the number of arcs on the found cycle.
   396     ///
   397     /// This function returns the number of arcs on the found cycle.
   398     ///
   399     /// \pre \ref run() or \ref findCycleMean() must be called before
   400     /// using this function.
   401     int cycleSize() const {
   402       return _best_size;
   403     }
   404 
   405     /// \brief Return the mean cost of the found cycle.
   406     ///
   407     /// This function returns the mean cost of the found cycle.
   408     ///
   409     /// \note <tt>alg.cycleMean()</tt> is just a shortcut of the
   410     /// following code.
   411     /// \code
   412     ///   return static_cast<double>(alg.cycleCost()) / alg.cycleSize();
   413     /// \endcode
   414     ///
   415     /// \pre \ref run() or \ref findCycleMean() must be called before
   416     /// using this function.
   417     double cycleMean() const {
   418       return static_cast<double>(_best_cost) / _best_size;
   419     }
   420 
   421     /// \brief Return the found cycle.
   422     ///
   423     /// This function returns a const reference to the path structure
   424     /// storing the found cycle.
   425     ///
   426     /// \pre \ref run() or \ref findCycle() must be called before using
   427     /// this function.
   428     const Path& cycle() const {
   429       return *_cycle_path;
   430     }
   431 
   432     ///@}
   433 
   434   private:
   435 
   436     // Initialize
   437     void init() {
   438       if (!_cycle_path) {
   439         _local_path = true;
   440         _cycle_path = new Path;
   441       }
   442       _queue.resize(countNodes(_gr));
   443       _best_found = false;
   444       _best_cost = 0;
   445       _best_size = 1;
   446       _cycle_path->clear();
   447     }
   448 
   449     // Find strongly connected components and initialize _comp_nodes
   450     // and _in_arcs
   451     void findComponents() {
   452       _comp_num = stronglyConnectedComponents(_gr, _comp);
   453       _comp_nodes.resize(_comp_num);
   454       if (_comp_num == 1) {
   455         _comp_nodes[0].clear();
   456         for (NodeIt n(_gr); n != INVALID; ++n) {
   457           _comp_nodes[0].push_back(n);
   458           _in_arcs[n].clear();
   459           for (InArcIt a(_gr, n); a != INVALID; ++a) {
   460             _in_arcs[n].push_back(a);
   461           }
   462         }
   463       } else {
   464         for (int i = 0; i < _comp_num; ++i)
   465           _comp_nodes[i].clear();
   466         for (NodeIt n(_gr); n != INVALID; ++n) {
   467           int k = _comp[n];
   468           _comp_nodes[k].push_back(n);
   469           _in_arcs[n].clear();
   470           for (InArcIt a(_gr, n); a != INVALID; ++a) {
   471             if (_comp[_gr.source(a)] == k) _in_arcs[n].push_back(a);
   472           }
   473         }
   474       }
   475     }
   476 
   477     // Build the policy graph in the given strongly connected component
   478     // (the out-degree of every node is 1)
   479     bool buildPolicyGraph(int comp) {
   480       _nodes = &(_comp_nodes[comp]);
   481       if (_nodes->size() < 1 ||
   482           (_nodes->size() == 1 && _in_arcs[(*_nodes)[0]].size() == 0)) {
   483         return false;
   484       }
   485       for (int i = 0; i < int(_nodes->size()); ++i) {
   486         _dist[(*_nodes)[i]] = INF;
   487       }
   488       Node u, v;
   489       Arc e;
   490       for (int i = 0; i < int(_nodes->size()); ++i) {
   491         v = (*_nodes)[i];
   492         for (int j = 0; j < int(_in_arcs[v].size()); ++j) {
   493           e = _in_arcs[v][j];
   494           u = _gr.source(e);
   495           if (_cost[e] < _dist[u]) {
   496             _dist[u] = _cost[e];
   497             _policy[u] = e;
   498           }
   499         }
   500       }
   501       return true;
   502     }
   503 
   504     // Find the minimum mean cycle in the policy graph
   505     void findPolicyCycle() {
   506       for (int i = 0; i < int(_nodes->size()); ++i) {
   507         _level[(*_nodes)[i]] = -1;
   508       }
   509       LargeCost ccost;
   510       int csize;
   511       Node u, v;
   512       _curr_found = false;
   513       for (int i = 0; i < int(_nodes->size()); ++i) {
   514         u = (*_nodes)[i];
   515         if (_level[u] >= 0) continue;
   516         for (; _level[u] < 0; u = _gr.target(_policy[u])) {
   517           _level[u] = i;
   518         }
   519         if (_level[u] == i) {
   520           // A cycle is found
   521           ccost = _cost[_policy[u]];
   522           csize = 1;
   523           for (v = u; (v = _gr.target(_policy[v])) != u; ) {
   524             ccost += _cost[_policy[v]];
   525             ++csize;
   526           }
   527           if ( !_curr_found ||
   528                (ccost * _curr_size < _curr_cost * csize) ) {
   529             _curr_found = true;
   530             _curr_cost = ccost;
   531             _curr_size = csize;
   532             _curr_node = u;
   533           }
   534         }
   535       }
   536     }
   537 
   538     // Contract the policy graph and compute node distances
   539     bool computeNodeDistances() {
   540       // Find the component of the main cycle and compute node distances
   541       // using reverse BFS
   542       for (int i = 0; i < int(_nodes->size()); ++i) {
   543         _reached[(*_nodes)[i]] = false;
   544       }
   545       _qfront = _qback = 0;
   546       _queue[0] = _curr_node;
   547       _reached[_curr_node] = true;
   548       _dist[_curr_node] = 0;
   549       Node u, v;
   550       Arc e;
   551       while (_qfront <= _qback) {
   552         v = _queue[_qfront++];
   553         for (int j = 0; j < int(_in_arcs[v].size()); ++j) {
   554           e = _in_arcs[v][j];
   555           u = _gr.source(e);
   556           if (_policy[u] == e && !_reached[u]) {
   557             _reached[u] = true;
   558             _dist[u] = _dist[v] + _cost[e] * _curr_size - _curr_cost;
   559             _queue[++_qback] = u;
   560           }
   561         }
   562       }
   563 
   564       // Connect all other nodes to this component and compute node
   565       // distances using reverse BFS
   566       _qfront = 0;
   567       while (_qback < int(_nodes->size())-1) {
   568         v = _queue[_qfront++];
   569         for (int j = 0; j < int(_in_arcs[v].size()); ++j) {
   570           e = _in_arcs[v][j];
   571           u = _gr.source(e);
   572           if (!_reached[u]) {
   573             _reached[u] = true;
   574             _policy[u] = e;
   575             _dist[u] = _dist[v] + _cost[e] * _curr_size - _curr_cost;
   576             _queue[++_qback] = u;
   577           }
   578         }
   579       }
   580 
   581       // Improve node distances
   582       bool improved = false;
   583       for (int i = 0; i < int(_nodes->size()); ++i) {
   584         v = (*_nodes)[i];
   585         for (int j = 0; j < int(_in_arcs[v].size()); ++j) {
   586           e = _in_arcs[v][j];
   587           u = _gr.source(e);
   588           LargeCost delta = _dist[v] + _cost[e] * _curr_size - _curr_cost;
   589           if (_tolerance.less(delta, _dist[u])) {
   590             _dist[u] = delta;
   591             _policy[u] = e;
   592             improved = true;
   593           }
   594         }
   595       }
   596       return improved;
   597     }
   598 
   599   }; //class HowardMmc
   600 
   601   ///@}
   602 
   603 } //namespace lemon
   604 
   605 #endif //LEMON_HOWARD_MMC_H