lemon/cycle_canceling.h
author Alpar Juttner <alpar@cs.elte.hu>
Wed, 17 Mar 2010 10:29:57 +0100
changeset 875 07ec2b52e53d
parent 840 2914b6f0fde0
child 877 141f9c0db4a3
permissions -rw-r--r--
Merge #356
     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_CYCLE_CANCELING_H
    20 #define LEMON_CYCLE_CANCELING_H
    21 
    22 /// \ingroup min_cost_flow_algs
    23 /// \file
    24 /// \brief Cycle-canceling algorithms for finding a minimum cost flow.
    25 
    26 #include <vector>
    27 #include <limits>
    28 
    29 #include <lemon/core.h>
    30 #include <lemon/maps.h>
    31 #include <lemon/path.h>
    32 #include <lemon/math.h>
    33 #include <lemon/static_graph.h>
    34 #include <lemon/adaptors.h>
    35 #include <lemon/circulation.h>
    36 #include <lemon/bellman_ford.h>
    37 #include <lemon/howard_mmc.h>
    38 
    39 namespace lemon {
    40 
    41   /// \addtogroup min_cost_flow_algs
    42   /// @{
    43 
    44   /// \brief Implementation of cycle-canceling algorithms for
    45   /// finding a \ref min_cost_flow "minimum cost flow".
    46   ///
    47   /// \ref CycleCanceling implements three different cycle-canceling
    48   /// algorithms for finding a \ref min_cost_flow "minimum cost flow"
    49   /// \ref amo93networkflows, \ref klein67primal,
    50   /// \ref goldberg89cyclecanceling.
    51   /// The most efficent one (both theoretically and practically)
    52   /// is the \ref CANCEL_AND_TIGHTEN "Cancel and Tighten" algorithm,
    53   /// thus it is the default method.
    54   /// It is strongly polynomial, but in practice, it is typically much
    55   /// slower than the scaling algorithms and NetworkSimplex.
    56   ///
    57   /// Most of the parameters of the problem (except for the digraph)
    58   /// can be given using separate functions, and the algorithm can be
    59   /// executed using the \ref run() function. If some parameters are not
    60   /// specified, then default values will be used.
    61   ///
    62   /// \tparam GR The digraph type the algorithm runs on.
    63   /// \tparam V The number type used for flow amounts, capacity bounds
    64   /// and supply values in the algorithm. By default, it is \c int.
    65   /// \tparam C The number type used for costs and potentials in the
    66   /// algorithm. By default, it is the same as \c V.
    67   ///
    68   /// \warning Both number types must be signed and all input data must
    69   /// be integer.
    70   /// \warning This algorithm does not support negative costs for such
    71   /// arcs that have infinite upper bound.
    72   ///
    73   /// \note For more information about the three available methods,
    74   /// see \ref Method.
    75 #ifdef DOXYGEN
    76   template <typename GR, typename V, typename C>
    77 #else
    78   template <typename GR, typename V = int, typename C = V>
    79 #endif
    80   class CycleCanceling
    81   {
    82   public:
    83 
    84     /// The type of the digraph
    85     typedef GR Digraph;
    86     /// The type of the flow amounts, capacity bounds and supply values
    87     typedef V Value;
    88     /// The type of the arc costs
    89     typedef C Cost;
    90 
    91   public:
    92 
    93     /// \brief Problem type constants for the \c run() function.
    94     ///
    95     /// Enum type containing the problem type constants that can be
    96     /// returned by the \ref run() function of the algorithm.
    97     enum ProblemType {
    98       /// The problem has no feasible solution (flow).
    99       INFEASIBLE,
   100       /// The problem has optimal solution (i.e. it is feasible and
   101       /// bounded), and the algorithm has found optimal flow and node
   102       /// potentials (primal and dual solutions).
   103       OPTIMAL,
   104       /// The digraph contains an arc of negative cost and infinite
   105       /// upper bound. It means that the objective function is unbounded
   106       /// on that arc, however, note that it could actually be bounded
   107       /// over the feasible flows, but this algroithm cannot handle
   108       /// these cases.
   109       UNBOUNDED
   110     };
   111 
   112     /// \brief Constants for selecting the used method.
   113     ///
   114     /// Enum type containing constants for selecting the used method
   115     /// for the \ref run() function.
   116     ///
   117     /// \ref CycleCanceling provides three different cycle-canceling
   118     /// methods. By default, \ref CANCEL_AND_TIGHTEN "Cancel and Tighten"
   119     /// is used, which proved to be the most efficient and the most robust
   120     /// on various test inputs.
   121     /// However, the other methods can be selected using the \ref run()
   122     /// function with the proper parameter.
   123     enum Method {
   124       /// A simple cycle-canceling method, which uses the
   125       /// \ref BellmanFord "Bellman-Ford" algorithm with limited iteration
   126       /// number for detecting negative cycles in the residual network.
   127       SIMPLE_CYCLE_CANCELING,
   128       /// The "Minimum Mean Cycle-Canceling" algorithm, which is a
   129       /// well-known strongly polynomial method
   130       /// \ref goldberg89cyclecanceling. It improves along a
   131       /// \ref min_mean_cycle "minimum mean cycle" in each iteration.
   132       /// Its running time complexity is O(n<sup>2</sup>m<sup>3</sup>log(n)).
   133       MINIMUM_MEAN_CYCLE_CANCELING,
   134       /// The "Cancel And Tighten" algorithm, which can be viewed as an
   135       /// improved version of the previous method
   136       /// \ref goldberg89cyclecanceling.
   137       /// It is faster both in theory and in practice, its running time
   138       /// complexity is O(n<sup>2</sup>m<sup>2</sup>log(n)).
   139       CANCEL_AND_TIGHTEN
   140     };
   141 
   142   private:
   143 
   144     TEMPLATE_DIGRAPH_TYPEDEFS(GR);
   145     
   146     typedef std::vector<int> IntVector;
   147     typedef std::vector<double> DoubleVector;
   148     typedef std::vector<Value> ValueVector;
   149     typedef std::vector<Cost> CostVector;
   150     typedef std::vector<char> BoolVector;
   151     // Note: vector<char> is used instead of vector<bool> for efficiency reasons
   152 
   153   private:
   154   
   155     template <typename KT, typename VT>
   156     class StaticVectorMap {
   157     public:
   158       typedef KT Key;
   159       typedef VT Value;
   160       
   161       StaticVectorMap(std::vector<Value>& v) : _v(v) {}
   162       
   163       const Value& operator[](const Key& key) const {
   164         return _v[StaticDigraph::id(key)];
   165       }
   166 
   167       Value& operator[](const Key& key) {
   168         return _v[StaticDigraph::id(key)];
   169       }
   170       
   171       void set(const Key& key, const Value& val) {
   172         _v[StaticDigraph::id(key)] = val;
   173       }
   174 
   175     private:
   176       std::vector<Value>& _v;
   177     };
   178 
   179     typedef StaticVectorMap<StaticDigraph::Node, Cost> CostNodeMap;
   180     typedef StaticVectorMap<StaticDigraph::Arc, Cost> CostArcMap;
   181 
   182   private:
   183 
   184 
   185     // Data related to the underlying digraph
   186     const GR &_graph;
   187     int _node_num;
   188     int _arc_num;
   189     int _res_node_num;
   190     int _res_arc_num;
   191     int _root;
   192 
   193     // Parameters of the problem
   194     bool _have_lower;
   195     Value _sum_supply;
   196 
   197     // Data structures for storing the digraph
   198     IntNodeMap _node_id;
   199     IntArcMap _arc_idf;
   200     IntArcMap _arc_idb;
   201     IntVector _first_out;
   202     BoolVector _forward;
   203     IntVector _source;
   204     IntVector _target;
   205     IntVector _reverse;
   206 
   207     // Node and arc data
   208     ValueVector _lower;
   209     ValueVector _upper;
   210     CostVector _cost;
   211     ValueVector _supply;
   212 
   213     ValueVector _res_cap;
   214     CostVector _pi;
   215 
   216     // Data for a StaticDigraph structure
   217     typedef std::pair<int, int> IntPair;
   218     StaticDigraph _sgr;
   219     std::vector<IntPair> _arc_vec;
   220     std::vector<Cost> _cost_vec;
   221     IntVector _id_vec;
   222     CostArcMap _cost_map;
   223     CostNodeMap _pi_map;
   224   
   225   public:
   226   
   227     /// \brief Constant for infinite upper bounds (capacities).
   228     ///
   229     /// Constant for infinite upper bounds (capacities).
   230     /// It is \c std::numeric_limits<Value>::infinity() if available,
   231     /// \c std::numeric_limits<Value>::max() otherwise.
   232     const Value INF;
   233 
   234   public:
   235 
   236     /// \brief Constructor.
   237     ///
   238     /// The constructor of the class.
   239     ///
   240     /// \param graph The digraph the algorithm runs on.
   241     CycleCanceling(const GR& graph) :
   242       _graph(graph), _node_id(graph), _arc_idf(graph), _arc_idb(graph),
   243       _cost_map(_cost_vec), _pi_map(_pi),
   244       INF(std::numeric_limits<Value>::has_infinity ?
   245           std::numeric_limits<Value>::infinity() :
   246           std::numeric_limits<Value>::max())
   247     {
   248       // Check the number types
   249       LEMON_ASSERT(std::numeric_limits<Value>::is_signed,
   250         "The flow type of CycleCanceling must be signed");
   251       LEMON_ASSERT(std::numeric_limits<Cost>::is_signed,
   252         "The cost type of CycleCanceling must be signed");
   253 
   254       // Reset data structures
   255       reset();
   256     }
   257 
   258     /// \name Parameters
   259     /// The parameters of the algorithm can be specified using these
   260     /// functions.
   261 
   262     /// @{
   263 
   264     /// \brief Set the lower bounds on the arcs.
   265     ///
   266     /// This function sets the lower bounds on the arcs.
   267     /// If it is not used before calling \ref run(), the lower bounds
   268     /// will be set to zero on all arcs.
   269     ///
   270     /// \param map An arc map storing the lower bounds.
   271     /// Its \c Value type must be convertible to the \c Value type
   272     /// of the algorithm.
   273     ///
   274     /// \return <tt>(*this)</tt>
   275     template <typename LowerMap>
   276     CycleCanceling& lowerMap(const LowerMap& map) {
   277       _have_lower = true;
   278       for (ArcIt a(_graph); a != INVALID; ++a) {
   279         _lower[_arc_idf[a]] = map[a];
   280         _lower[_arc_idb[a]] = map[a];
   281       }
   282       return *this;
   283     }
   284 
   285     /// \brief Set the upper bounds (capacities) on the arcs.
   286     ///
   287     /// This function sets the upper bounds (capacities) on the arcs.
   288     /// If it is not used before calling \ref run(), the upper bounds
   289     /// will be set to \ref INF on all arcs (i.e. the flow value will be
   290     /// unbounded from above).
   291     ///
   292     /// \param map An arc map storing the upper bounds.
   293     /// Its \c Value type must be convertible to the \c Value type
   294     /// of the algorithm.
   295     ///
   296     /// \return <tt>(*this)</tt>
   297     template<typename UpperMap>
   298     CycleCanceling& upperMap(const UpperMap& map) {
   299       for (ArcIt a(_graph); a != INVALID; ++a) {
   300         _upper[_arc_idf[a]] = map[a];
   301       }
   302       return *this;
   303     }
   304 
   305     /// \brief Set the costs of the arcs.
   306     ///
   307     /// This function sets the costs of the arcs.
   308     /// If it is not used before calling \ref run(), the costs
   309     /// will be set to \c 1 on all arcs.
   310     ///
   311     /// \param map An arc map storing the costs.
   312     /// Its \c Value type must be convertible to the \c Cost type
   313     /// of the algorithm.
   314     ///
   315     /// \return <tt>(*this)</tt>
   316     template<typename CostMap>
   317     CycleCanceling& costMap(const CostMap& map) {
   318       for (ArcIt a(_graph); a != INVALID; ++a) {
   319         _cost[_arc_idf[a]] =  map[a];
   320         _cost[_arc_idb[a]] = -map[a];
   321       }
   322       return *this;
   323     }
   324 
   325     /// \brief Set the supply values of the nodes.
   326     ///
   327     /// This function sets the supply values of the nodes.
   328     /// If neither this function nor \ref stSupply() is used before
   329     /// calling \ref run(), the supply of each node will be set to zero.
   330     ///
   331     /// \param map A node map storing the supply values.
   332     /// Its \c Value type must be convertible to the \c Value type
   333     /// of the algorithm.
   334     ///
   335     /// \return <tt>(*this)</tt>
   336     template<typename SupplyMap>
   337     CycleCanceling& supplyMap(const SupplyMap& map) {
   338       for (NodeIt n(_graph); n != INVALID; ++n) {
   339         _supply[_node_id[n]] = map[n];
   340       }
   341       return *this;
   342     }
   343 
   344     /// \brief Set single source and target nodes and a supply value.
   345     ///
   346     /// This function sets a single source node and a single target node
   347     /// and the required flow value.
   348     /// If neither this function nor \ref supplyMap() is used before
   349     /// calling \ref run(), the supply of each node will be set to zero.
   350     ///
   351     /// Using this function has the same effect as using \ref supplyMap()
   352     /// with such a map in which \c k is assigned to \c s, \c -k is
   353     /// assigned to \c t and all other nodes have zero supply value.
   354     ///
   355     /// \param s The source node.
   356     /// \param t The target node.
   357     /// \param k The required amount of flow from node \c s to node \c t
   358     /// (i.e. the supply of \c s and the demand of \c t).
   359     ///
   360     /// \return <tt>(*this)</tt>
   361     CycleCanceling& stSupply(const Node& s, const Node& t, Value k) {
   362       for (int i = 0; i != _res_node_num; ++i) {
   363         _supply[i] = 0;
   364       }
   365       _supply[_node_id[s]] =  k;
   366       _supply[_node_id[t]] = -k;
   367       return *this;
   368     }
   369     
   370     /// @}
   371 
   372     /// \name Execution control
   373     /// The algorithm can be executed using \ref run().
   374 
   375     /// @{
   376 
   377     /// \brief Run the algorithm.
   378     ///
   379     /// This function runs the algorithm.
   380     /// The paramters can be specified using functions \ref lowerMap(),
   381     /// \ref upperMap(), \ref costMap(), \ref supplyMap(), \ref stSupply().
   382     /// For example,
   383     /// \code
   384     ///   CycleCanceling<ListDigraph> cc(graph);
   385     ///   cc.lowerMap(lower).upperMap(upper).costMap(cost)
   386     ///     .supplyMap(sup).run();
   387     /// \endcode
   388     ///
   389     /// This function can be called more than once. All the given parameters
   390     /// are kept for the next call, unless \ref resetParams() or \ref reset()
   391     /// is used, thus only the modified parameters have to be set again.
   392     /// If the underlying digraph was also modified after the construction
   393     /// of the class (or the last \ref reset() call), then the \ref reset()
   394     /// function must be called.
   395     ///
   396     /// \param method The cycle-canceling method that will be used.
   397     /// For more information, see \ref Method.
   398     ///
   399     /// \return \c INFEASIBLE if no feasible flow exists,
   400     /// \n \c OPTIMAL if the problem has optimal solution
   401     /// (i.e. it is feasible and bounded), and the algorithm has found
   402     /// optimal flow and node potentials (primal and dual solutions),
   403     /// \n \c UNBOUNDED if the digraph contains an arc of negative cost
   404     /// and infinite upper bound. It means that the objective function
   405     /// is unbounded on that arc, however, note that it could actually be
   406     /// bounded over the feasible flows, but this algroithm cannot handle
   407     /// these cases.
   408     ///
   409     /// \see ProblemType, Method
   410     /// \see resetParams(), reset()
   411     ProblemType run(Method method = CANCEL_AND_TIGHTEN) {
   412       ProblemType pt = init();
   413       if (pt != OPTIMAL) return pt;
   414       start(method);
   415       return OPTIMAL;
   416     }
   417 
   418     /// \brief Reset all the parameters that have been given before.
   419     ///
   420     /// This function resets all the paramaters that have been given
   421     /// before using functions \ref lowerMap(), \ref upperMap(),
   422     /// \ref costMap(), \ref supplyMap(), \ref stSupply().
   423     ///
   424     /// It is useful for multiple \ref run() calls. Basically, all the given
   425     /// parameters are kept for the next \ref run() call, unless
   426     /// \ref resetParams() or \ref reset() is used.
   427     /// If the underlying digraph was also modified after the construction
   428     /// of the class or the last \ref reset() call, then the \ref reset()
   429     /// function must be used, otherwise \ref resetParams() is sufficient.
   430     ///
   431     /// For example,
   432     /// \code
   433     ///   CycleCanceling<ListDigraph> cs(graph);
   434     ///
   435     ///   // First run
   436     ///   cc.lowerMap(lower).upperMap(upper).costMap(cost)
   437     ///     .supplyMap(sup).run();
   438     ///
   439     ///   // Run again with modified cost map (resetParams() is not called,
   440     ///   // so only the cost map have to be set again)
   441     ///   cost[e] += 100;
   442     ///   cc.costMap(cost).run();
   443     ///
   444     ///   // Run again from scratch using resetParams()
   445     ///   // (the lower bounds will be set to zero on all arcs)
   446     ///   cc.resetParams();
   447     ///   cc.upperMap(capacity).costMap(cost)
   448     ///     .supplyMap(sup).run();
   449     /// \endcode
   450     ///
   451     /// \return <tt>(*this)</tt>
   452     ///
   453     /// \see reset(), run()
   454     CycleCanceling& resetParams() {
   455       for (int i = 0; i != _res_node_num; ++i) {
   456         _supply[i] = 0;
   457       }
   458       int limit = _first_out[_root];
   459       for (int j = 0; j != limit; ++j) {
   460         _lower[j] = 0;
   461         _upper[j] = INF;
   462         _cost[j] = _forward[j] ? 1 : -1;
   463       }
   464       for (int j = limit; j != _res_arc_num; ++j) {
   465         _lower[j] = 0;
   466         _upper[j] = INF;
   467         _cost[j] = 0;
   468         _cost[_reverse[j]] = 0;
   469       }      
   470       _have_lower = false;
   471       return *this;
   472     }
   473 
   474     /// \brief Reset the internal data structures and all the parameters
   475     /// that have been given before.
   476     ///
   477     /// This function resets the internal data structures and all the
   478     /// paramaters that have been given before using functions \ref lowerMap(),
   479     /// \ref upperMap(), \ref costMap(), \ref supplyMap(), \ref stSupply().
   480     ///
   481     /// It is useful for multiple \ref run() calls. Basically, all the given
   482     /// parameters are kept for the next \ref run() call, unless
   483     /// \ref resetParams() or \ref reset() is used.
   484     /// If the underlying digraph was also modified after the construction
   485     /// of the class or the last \ref reset() call, then the \ref reset()
   486     /// function must be used, otherwise \ref resetParams() is sufficient.
   487     ///
   488     /// See \ref resetParams() for examples.
   489     ///
   490     /// \return <tt>(*this)</tt>
   491     ///
   492     /// \see resetParams(), run()
   493     CycleCanceling& reset() {
   494       // Resize vectors
   495       _node_num = countNodes(_graph);
   496       _arc_num = countArcs(_graph);
   497       _res_node_num = _node_num + 1;
   498       _res_arc_num = 2 * (_arc_num + _node_num);
   499       _root = _node_num;
   500 
   501       _first_out.resize(_res_node_num + 1);
   502       _forward.resize(_res_arc_num);
   503       _source.resize(_res_arc_num);
   504       _target.resize(_res_arc_num);
   505       _reverse.resize(_res_arc_num);
   506 
   507       _lower.resize(_res_arc_num);
   508       _upper.resize(_res_arc_num);
   509       _cost.resize(_res_arc_num);
   510       _supply.resize(_res_node_num);
   511       
   512       _res_cap.resize(_res_arc_num);
   513       _pi.resize(_res_node_num);
   514 
   515       _arc_vec.reserve(_res_arc_num);
   516       _cost_vec.reserve(_res_arc_num);
   517       _id_vec.reserve(_res_arc_num);
   518 
   519       // Copy the graph
   520       int i = 0, j = 0, k = 2 * _arc_num + _node_num;
   521       for (NodeIt n(_graph); n != INVALID; ++n, ++i) {
   522         _node_id[n] = i;
   523       }
   524       i = 0;
   525       for (NodeIt n(_graph); n != INVALID; ++n, ++i) {
   526         _first_out[i] = j;
   527         for (OutArcIt a(_graph, n); a != INVALID; ++a, ++j) {
   528           _arc_idf[a] = j;
   529           _forward[j] = true;
   530           _source[j] = i;
   531           _target[j] = _node_id[_graph.runningNode(a)];
   532         }
   533         for (InArcIt a(_graph, n); a != INVALID; ++a, ++j) {
   534           _arc_idb[a] = j;
   535           _forward[j] = false;
   536           _source[j] = i;
   537           _target[j] = _node_id[_graph.runningNode(a)];
   538         }
   539         _forward[j] = false;
   540         _source[j] = i;
   541         _target[j] = _root;
   542         _reverse[j] = k;
   543         _forward[k] = true;
   544         _source[k] = _root;
   545         _target[k] = i;
   546         _reverse[k] = j;
   547         ++j; ++k;
   548       }
   549       _first_out[i] = j;
   550       _first_out[_res_node_num] = k;
   551       for (ArcIt a(_graph); a != INVALID; ++a) {
   552         int fi = _arc_idf[a];
   553         int bi = _arc_idb[a];
   554         _reverse[fi] = bi;
   555         _reverse[bi] = fi;
   556       }
   557       
   558       // Reset parameters
   559       resetParams();
   560       return *this;
   561     }
   562 
   563     /// @}
   564 
   565     /// \name Query Functions
   566     /// The results of the algorithm can be obtained using these
   567     /// functions.\n
   568     /// The \ref run() function must be called before using them.
   569 
   570     /// @{
   571 
   572     /// \brief Return the total cost of the found flow.
   573     ///
   574     /// This function returns the total cost of the found flow.
   575     /// Its complexity is O(e).
   576     ///
   577     /// \note The return type of the function can be specified as a
   578     /// template parameter. For example,
   579     /// \code
   580     ///   cc.totalCost<double>();
   581     /// \endcode
   582     /// It is useful if the total cost cannot be stored in the \c Cost
   583     /// type of the algorithm, which is the default return type of the
   584     /// function.
   585     ///
   586     /// \pre \ref run() must be called before using this function.
   587     template <typename Number>
   588     Number totalCost() const {
   589       Number c = 0;
   590       for (ArcIt a(_graph); a != INVALID; ++a) {
   591         int i = _arc_idb[a];
   592         c += static_cast<Number>(_res_cap[i]) *
   593              (-static_cast<Number>(_cost[i]));
   594       }
   595       return c;
   596     }
   597 
   598 #ifndef DOXYGEN
   599     Cost totalCost() const {
   600       return totalCost<Cost>();
   601     }
   602 #endif
   603 
   604     /// \brief Return the flow on the given arc.
   605     ///
   606     /// This function returns the flow on the given arc.
   607     ///
   608     /// \pre \ref run() must be called before using this function.
   609     Value flow(const Arc& a) const {
   610       return _res_cap[_arc_idb[a]];
   611     }
   612 
   613     /// \brief Return the flow map (the primal solution).
   614     ///
   615     /// This function copies the flow value on each arc into the given
   616     /// map. The \c Value type of the algorithm must be convertible to
   617     /// the \c Value type of the map.
   618     ///
   619     /// \pre \ref run() must be called before using this function.
   620     template <typename FlowMap>
   621     void flowMap(FlowMap &map) const {
   622       for (ArcIt a(_graph); a != INVALID; ++a) {
   623         map.set(a, _res_cap[_arc_idb[a]]);
   624       }
   625     }
   626 
   627     /// \brief Return the potential (dual value) of the given node.
   628     ///
   629     /// This function returns the potential (dual value) of the
   630     /// given node.
   631     ///
   632     /// \pre \ref run() must be called before using this function.
   633     Cost potential(const Node& n) const {
   634       return static_cast<Cost>(_pi[_node_id[n]]);
   635     }
   636 
   637     /// \brief Return the potential map (the dual solution).
   638     ///
   639     /// This function copies the potential (dual value) of each node
   640     /// into the given map.
   641     /// The \c Cost type of the algorithm must be convertible to the
   642     /// \c Value type of the map.
   643     ///
   644     /// \pre \ref run() must be called before using this function.
   645     template <typename PotentialMap>
   646     void potentialMap(PotentialMap &map) const {
   647       for (NodeIt n(_graph); n != INVALID; ++n) {
   648         map.set(n, static_cast<Cost>(_pi[_node_id[n]]));
   649       }
   650     }
   651 
   652     /// @}
   653 
   654   private:
   655 
   656     // Initialize the algorithm
   657     ProblemType init() {
   658       if (_res_node_num <= 1) return INFEASIBLE;
   659 
   660       // Check the sum of supply values
   661       _sum_supply = 0;
   662       for (int i = 0; i != _root; ++i) {
   663         _sum_supply += _supply[i];
   664       }
   665       if (_sum_supply > 0) return INFEASIBLE;
   666       
   667 
   668       // Initialize vectors
   669       for (int i = 0; i != _res_node_num; ++i) {
   670         _pi[i] = 0;
   671       }
   672       ValueVector excess(_supply);
   673       
   674       // Remove infinite upper bounds and check negative arcs
   675       const Value MAX = std::numeric_limits<Value>::max();
   676       int last_out;
   677       if (_have_lower) {
   678         for (int i = 0; i != _root; ++i) {
   679           last_out = _first_out[i+1];
   680           for (int j = _first_out[i]; j != last_out; ++j) {
   681             if (_forward[j]) {
   682               Value c = _cost[j] < 0 ? _upper[j] : _lower[j];
   683               if (c >= MAX) return UNBOUNDED;
   684               excess[i] -= c;
   685               excess[_target[j]] += c;
   686             }
   687           }
   688         }
   689       } else {
   690         for (int i = 0; i != _root; ++i) {
   691           last_out = _first_out[i+1];
   692           for (int j = _first_out[i]; j != last_out; ++j) {
   693             if (_forward[j] && _cost[j] < 0) {
   694               Value c = _upper[j];
   695               if (c >= MAX) return UNBOUNDED;
   696               excess[i] -= c;
   697               excess[_target[j]] += c;
   698             }
   699           }
   700         }
   701       }
   702       Value ex, max_cap = 0;
   703       for (int i = 0; i != _res_node_num; ++i) {
   704         ex = excess[i];
   705         if (ex < 0) max_cap -= ex;
   706       }
   707       for (int j = 0; j != _res_arc_num; ++j) {
   708         if (_upper[j] >= MAX) _upper[j] = max_cap;
   709       }
   710 
   711       // Initialize maps for Circulation and remove non-zero lower bounds
   712       ConstMap<Arc, Value> low(0);
   713       typedef typename Digraph::template ArcMap<Value> ValueArcMap;
   714       typedef typename Digraph::template NodeMap<Value> ValueNodeMap;
   715       ValueArcMap cap(_graph), flow(_graph);
   716       ValueNodeMap sup(_graph);
   717       for (NodeIt n(_graph); n != INVALID; ++n) {
   718         sup[n] = _supply[_node_id[n]];
   719       }
   720       if (_have_lower) {
   721         for (ArcIt a(_graph); a != INVALID; ++a) {
   722           int j = _arc_idf[a];
   723           Value c = _lower[j];
   724           cap[a] = _upper[j] - c;
   725           sup[_graph.source(a)] -= c;
   726           sup[_graph.target(a)] += c;
   727         }
   728       } else {
   729         for (ArcIt a(_graph); a != INVALID; ++a) {
   730           cap[a] = _upper[_arc_idf[a]];
   731         }
   732       }
   733 
   734       // Find a feasible flow using Circulation
   735       Circulation<Digraph, ConstMap<Arc, Value>, ValueArcMap, ValueNodeMap>
   736         circ(_graph, low, cap, sup);
   737       if (!circ.flowMap(flow).run()) return INFEASIBLE;
   738 
   739       // Set residual capacities and handle GEQ supply type
   740       if (_sum_supply < 0) {
   741         for (ArcIt a(_graph); a != INVALID; ++a) {
   742           Value fa = flow[a];
   743           _res_cap[_arc_idf[a]] = cap[a] - fa;
   744           _res_cap[_arc_idb[a]] = fa;
   745           sup[_graph.source(a)] -= fa;
   746           sup[_graph.target(a)] += fa;
   747         }
   748         for (NodeIt n(_graph); n != INVALID; ++n) {
   749           excess[_node_id[n]] = sup[n];
   750         }
   751         for (int a = _first_out[_root]; a != _res_arc_num; ++a) {
   752           int u = _target[a];
   753           int ra = _reverse[a];
   754           _res_cap[a] = -_sum_supply + 1;
   755           _res_cap[ra] = -excess[u];
   756           _cost[a] = 0;
   757           _cost[ra] = 0;
   758         }
   759       } else {
   760         for (ArcIt a(_graph); a != INVALID; ++a) {
   761           Value fa = flow[a];
   762           _res_cap[_arc_idf[a]] = cap[a] - fa;
   763           _res_cap[_arc_idb[a]] = fa;
   764         }
   765         for (int a = _first_out[_root]; a != _res_arc_num; ++a) {
   766           int ra = _reverse[a];
   767           _res_cap[a] = 1;
   768           _res_cap[ra] = 0;
   769           _cost[a] = 0;
   770           _cost[ra] = 0;
   771         }
   772       }
   773       
   774       return OPTIMAL;
   775     }
   776     
   777     // Build a StaticDigraph structure containing the current
   778     // residual network
   779     void buildResidualNetwork() {
   780       _arc_vec.clear();
   781       _cost_vec.clear();
   782       _id_vec.clear();
   783       for (int j = 0; j != _res_arc_num; ++j) {
   784         if (_res_cap[j] > 0) {
   785           _arc_vec.push_back(IntPair(_source[j], _target[j]));
   786           _cost_vec.push_back(_cost[j]);
   787           _id_vec.push_back(j);
   788         }
   789       }
   790       _sgr.build(_res_node_num, _arc_vec.begin(), _arc_vec.end());
   791     }
   792 
   793     // Execute the algorithm and transform the results
   794     void start(Method method) {
   795       // Execute the algorithm
   796       switch (method) {
   797         case SIMPLE_CYCLE_CANCELING:
   798           startSimpleCycleCanceling();
   799           break;
   800         case MINIMUM_MEAN_CYCLE_CANCELING:
   801           startMinMeanCycleCanceling();
   802           break;
   803         case CANCEL_AND_TIGHTEN:
   804           startCancelAndTighten();
   805           break;
   806       }
   807 
   808       // Compute node potentials
   809       if (method != SIMPLE_CYCLE_CANCELING) {
   810         buildResidualNetwork();
   811         typename BellmanFord<StaticDigraph, CostArcMap>
   812           ::template SetDistMap<CostNodeMap>::Create bf(_sgr, _cost_map);
   813         bf.distMap(_pi_map);
   814         bf.init(0);
   815         bf.start();
   816       }
   817 
   818       // Handle non-zero lower bounds
   819       if (_have_lower) {
   820         int limit = _first_out[_root];
   821         for (int j = 0; j != limit; ++j) {
   822           if (!_forward[j]) _res_cap[j] += _lower[j];
   823         }
   824       }
   825     }
   826 
   827     // Execute the "Simple Cycle Canceling" method
   828     void startSimpleCycleCanceling() {
   829       // Constants for computing the iteration limits
   830       const int BF_FIRST_LIMIT  = 2;
   831       const double BF_LIMIT_FACTOR = 1.5;
   832       
   833       typedef StaticVectorMap<StaticDigraph::Arc, Value> FilterMap;
   834       typedef FilterArcs<StaticDigraph, FilterMap> ResDigraph;
   835       typedef StaticVectorMap<StaticDigraph::Node, StaticDigraph::Arc> PredMap;
   836       typedef typename BellmanFord<ResDigraph, CostArcMap>
   837         ::template SetDistMap<CostNodeMap>
   838         ::template SetPredMap<PredMap>::Create BF;
   839       
   840       // Build the residual network
   841       _arc_vec.clear();
   842       _cost_vec.clear();
   843       for (int j = 0; j != _res_arc_num; ++j) {
   844         _arc_vec.push_back(IntPair(_source[j], _target[j]));
   845         _cost_vec.push_back(_cost[j]);
   846       }
   847       _sgr.build(_res_node_num, _arc_vec.begin(), _arc_vec.end());
   848 
   849       FilterMap filter_map(_res_cap);
   850       ResDigraph rgr(_sgr, filter_map);
   851       std::vector<int> cycle;
   852       std::vector<StaticDigraph::Arc> pred(_res_arc_num);
   853       PredMap pred_map(pred);
   854       BF bf(rgr, _cost_map);
   855       bf.distMap(_pi_map).predMap(pred_map);
   856 
   857       int length_bound = BF_FIRST_LIMIT;
   858       bool optimal = false;
   859       while (!optimal) {
   860         bf.init(0);
   861         int iter_num = 0;
   862         bool cycle_found = false;
   863         while (!cycle_found) {
   864           // Perform some iterations of the Bellman-Ford algorithm
   865           int curr_iter_num = iter_num + length_bound <= _node_num ?
   866             length_bound : _node_num - iter_num;
   867           iter_num += curr_iter_num;
   868           int real_iter_num = curr_iter_num;
   869           for (int i = 0; i < curr_iter_num; ++i) {
   870             if (bf.processNextWeakRound()) {
   871               real_iter_num = i;
   872               break;
   873             }
   874           }
   875           if (real_iter_num < curr_iter_num) {
   876             // Optimal flow is found
   877             optimal = true;
   878             break;
   879           } else {
   880             // Search for node disjoint negative cycles
   881             std::vector<int> state(_res_node_num, 0);
   882             int id = 0;
   883             for (int u = 0; u != _res_node_num; ++u) {
   884               if (state[u] != 0) continue;
   885               ++id;
   886               int v = u;
   887               for (; v != -1 && state[v] == 0; v = pred[v] == INVALID ?
   888                    -1 : rgr.id(rgr.source(pred[v]))) {
   889                 state[v] = id;
   890               }
   891               if (v != -1 && state[v] == id) {
   892                 // A negative cycle is found
   893                 cycle_found = true;
   894                 cycle.clear();
   895                 StaticDigraph::Arc a = pred[v];
   896                 Value d, delta = _res_cap[rgr.id(a)];
   897                 cycle.push_back(rgr.id(a));
   898                 while (rgr.id(rgr.source(a)) != v) {
   899                   a = pred_map[rgr.source(a)];
   900                   d = _res_cap[rgr.id(a)];
   901                   if (d < delta) delta = d;
   902                   cycle.push_back(rgr.id(a));
   903                 }
   904 
   905                 // Augment along the cycle
   906                 for (int i = 0; i < int(cycle.size()); ++i) {
   907                   int j = cycle[i];
   908                   _res_cap[j] -= delta;
   909                   _res_cap[_reverse[j]] += delta;
   910                 }
   911               }
   912             }
   913           }
   914 
   915           // Increase iteration limit if no cycle is found
   916           if (!cycle_found) {
   917             length_bound = static_cast<int>(length_bound * BF_LIMIT_FACTOR);
   918           }
   919         }
   920       }
   921     }
   922 
   923     // Execute the "Minimum Mean Cycle Canceling" method
   924     void startMinMeanCycleCanceling() {
   925       typedef SimplePath<StaticDigraph> SPath;
   926       typedef typename SPath::ArcIt SPathArcIt;
   927       typedef typename HowardMmc<StaticDigraph, CostArcMap>
   928         ::template SetPath<SPath>::Create MMC;
   929       
   930       SPath cycle;
   931       MMC mmc(_sgr, _cost_map);
   932       mmc.cycle(cycle);
   933       buildResidualNetwork();
   934       while (mmc.findCycleMean() && mmc.cycleCost() < 0) {
   935         // Find the cycle
   936         mmc.findCycle();
   937 
   938         // Compute delta value
   939         Value delta = INF;
   940         for (SPathArcIt a(cycle); a != INVALID; ++a) {
   941           Value d = _res_cap[_id_vec[_sgr.id(a)]];
   942           if (d < delta) delta = d;
   943         }
   944 
   945         // Augment along the cycle
   946         for (SPathArcIt a(cycle); a != INVALID; ++a) {
   947           int j = _id_vec[_sgr.id(a)];
   948           _res_cap[j] -= delta;
   949           _res_cap[_reverse[j]] += delta;
   950         }
   951 
   952         // Rebuild the residual network        
   953         buildResidualNetwork();
   954       }
   955     }
   956 
   957     // Execute the "Cancel And Tighten" method
   958     void startCancelAndTighten() {
   959       // Constants for the min mean cycle computations
   960       const double LIMIT_FACTOR = 1.0;
   961       const int MIN_LIMIT = 5;
   962 
   963       // Contruct auxiliary data vectors
   964       DoubleVector pi(_res_node_num, 0.0);
   965       IntVector level(_res_node_num);
   966       BoolVector reached(_res_node_num);
   967       BoolVector processed(_res_node_num);
   968       IntVector pred_node(_res_node_num);
   969       IntVector pred_arc(_res_node_num);
   970       std::vector<int> stack(_res_node_num);
   971       std::vector<int> proc_vector(_res_node_num);
   972 
   973       // Initialize epsilon
   974       double epsilon = 0;
   975       for (int a = 0; a != _res_arc_num; ++a) {
   976         if (_res_cap[a] > 0 && -_cost[a] > epsilon)
   977           epsilon = -_cost[a];
   978       }
   979 
   980       // Start phases
   981       Tolerance<double> tol;
   982       tol.epsilon(1e-6);
   983       int limit = int(LIMIT_FACTOR * std::sqrt(double(_res_node_num)));
   984       if (limit < MIN_LIMIT) limit = MIN_LIMIT;
   985       int iter = limit;
   986       while (epsilon * _res_node_num >= 1) {
   987         // Find and cancel cycles in the admissible network using DFS
   988         for (int u = 0; u != _res_node_num; ++u) {
   989           reached[u] = false;
   990           processed[u] = false;
   991         }
   992         int stack_head = -1;
   993         int proc_head = -1;
   994         for (int start = 0; start != _res_node_num; ++start) {
   995           if (reached[start]) continue;
   996 
   997           // New start node
   998           reached[start] = true;
   999           pred_arc[start] = -1;
  1000           pred_node[start] = -1;
  1001 
  1002           // Find the first admissible outgoing arc
  1003           double p = pi[start];
  1004           int a = _first_out[start];
  1005           int last_out = _first_out[start+1];
  1006           for (; a != last_out && (_res_cap[a] == 0 ||
  1007                !tol.negative(_cost[a] + p - pi[_target[a]])); ++a) ;
  1008           if (a == last_out) {
  1009             processed[start] = true;
  1010             proc_vector[++proc_head] = start;
  1011             continue;
  1012           }
  1013           stack[++stack_head] = a;
  1014 
  1015           while (stack_head >= 0) {
  1016             int sa = stack[stack_head];
  1017             int u = _source[sa];
  1018             int v = _target[sa];
  1019 
  1020             if (!reached[v]) {
  1021               // A new node is reached
  1022               reached[v] = true;
  1023               pred_node[v] = u;
  1024               pred_arc[v] = sa;
  1025               p = pi[v];
  1026               a = _first_out[v];
  1027               last_out = _first_out[v+1];
  1028               for (; a != last_out && (_res_cap[a] == 0 ||
  1029                    !tol.negative(_cost[a] + p - pi[_target[a]])); ++a) ;
  1030               stack[++stack_head] = a == last_out ? -1 : a;
  1031             } else {
  1032               if (!processed[v]) {
  1033                 // A cycle is found
  1034                 int n, w = u;
  1035                 Value d, delta = _res_cap[sa];
  1036                 for (n = u; n != v; n = pred_node[n]) {
  1037                   d = _res_cap[pred_arc[n]];
  1038                   if (d <= delta) {
  1039                     delta = d;
  1040                     w = pred_node[n];
  1041                   }
  1042                 }
  1043 
  1044                 // Augment along the cycle
  1045                 _res_cap[sa] -= delta;
  1046                 _res_cap[_reverse[sa]] += delta;
  1047                 for (n = u; n != v; n = pred_node[n]) {
  1048                   int pa = pred_arc[n];
  1049                   _res_cap[pa] -= delta;
  1050                   _res_cap[_reverse[pa]] += delta;
  1051                 }
  1052                 for (n = u; stack_head > 0 && n != w; n = pred_node[n]) {
  1053                   --stack_head;
  1054                   reached[n] = false;
  1055                 }
  1056                 u = w;
  1057               }
  1058               v = u;
  1059 
  1060               // Find the next admissible outgoing arc
  1061               p = pi[v];
  1062               a = stack[stack_head] + 1;
  1063               last_out = _first_out[v+1];
  1064               for (; a != last_out && (_res_cap[a] == 0 ||
  1065                    !tol.negative(_cost[a] + p - pi[_target[a]])); ++a) ;
  1066               stack[stack_head] = a == last_out ? -1 : a;
  1067             }
  1068 
  1069             while (stack_head >= 0 && stack[stack_head] == -1) {
  1070               processed[v] = true;
  1071               proc_vector[++proc_head] = v;
  1072               if (--stack_head >= 0) {
  1073                 // Find the next admissible outgoing arc
  1074                 v = _source[stack[stack_head]];
  1075                 p = pi[v];
  1076                 a = stack[stack_head] + 1;
  1077                 last_out = _first_out[v+1];
  1078                 for (; a != last_out && (_res_cap[a] == 0 ||
  1079                      !tol.negative(_cost[a] + p - pi[_target[a]])); ++a) ;
  1080                 stack[stack_head] = a == last_out ? -1 : a;
  1081               }
  1082             }
  1083           }
  1084         }
  1085 
  1086         // Tighten potentials and epsilon
  1087         if (--iter > 0) {
  1088           for (int u = 0; u != _res_node_num; ++u) {
  1089             level[u] = 0;
  1090           }
  1091           for (int i = proc_head; i > 0; --i) {
  1092             int u = proc_vector[i];
  1093             double p = pi[u];
  1094             int l = level[u] + 1;
  1095             int last_out = _first_out[u+1];
  1096             for (int a = _first_out[u]; a != last_out; ++a) {
  1097               int v = _target[a];
  1098               if (_res_cap[a] > 0 && tol.negative(_cost[a] + p - pi[v]) &&
  1099                   l > level[v]) level[v] = l;
  1100             }
  1101           }
  1102 
  1103           // Modify potentials
  1104           double q = std::numeric_limits<double>::max();
  1105           for (int u = 0; u != _res_node_num; ++u) {
  1106             int lu = level[u];
  1107             double p, pu = pi[u];
  1108             int last_out = _first_out[u+1];
  1109             for (int a = _first_out[u]; a != last_out; ++a) {
  1110               if (_res_cap[a] == 0) continue;
  1111               int v = _target[a];
  1112               int ld = lu - level[v];
  1113               if (ld > 0) {
  1114                 p = (_cost[a] + pu - pi[v] + epsilon) / (ld + 1);
  1115                 if (p < q) q = p;
  1116               }
  1117             }
  1118           }
  1119           for (int u = 0; u != _res_node_num; ++u) {
  1120             pi[u] -= q * level[u];
  1121           }
  1122 
  1123           // Modify epsilon
  1124           epsilon = 0;
  1125           for (int u = 0; u != _res_node_num; ++u) {
  1126             double curr, pu = pi[u];
  1127             int last_out = _first_out[u+1];
  1128             for (int a = _first_out[u]; a != last_out; ++a) {
  1129               if (_res_cap[a] == 0) continue;
  1130               curr = _cost[a] + pu - pi[_target[a]];
  1131               if (-curr > epsilon) epsilon = -curr;
  1132             }
  1133           }
  1134         } else {
  1135           typedef HowardMmc<StaticDigraph, CostArcMap> MMC;
  1136           typedef typename BellmanFord<StaticDigraph, CostArcMap>
  1137             ::template SetDistMap<CostNodeMap>::Create BF;
  1138 
  1139           // Set epsilon to the minimum cycle mean
  1140           buildResidualNetwork();
  1141           MMC mmc(_sgr, _cost_map);
  1142           mmc.findCycleMean();
  1143           epsilon = -mmc.cycleMean();
  1144           Cost cycle_cost = mmc.cycleCost();
  1145           int cycle_size = mmc.cycleSize();
  1146           
  1147           // Compute feasible potentials for the current epsilon
  1148           for (int i = 0; i != int(_cost_vec.size()); ++i) {
  1149             _cost_vec[i] = cycle_size * _cost_vec[i] - cycle_cost;
  1150           }
  1151           BF bf(_sgr, _cost_map);
  1152           bf.distMap(_pi_map);
  1153           bf.init(0);
  1154           bf.start();
  1155           for (int u = 0; u != _res_node_num; ++u) {
  1156             pi[u] = static_cast<double>(_pi[u]) / cycle_size;
  1157           }
  1158         
  1159           iter = limit;
  1160         }
  1161       }
  1162     }
  1163 
  1164   }; //class CycleCanceling
  1165 
  1166   ///@}
  1167 
  1168 } //namespace lemon
  1169 
  1170 #endif //LEMON_CYCLE_CANCELING_H