lemon/cycle_canceling.h
author Alpar Juttner <alpar@cs.elte.hu>
Fri, 15 Mar 2013 17:19:17 +0100
changeset 1048 dbaf21739390
parent 1003 16f55008c863
child 1049 7bf489cf624e
child 1070 ee9bac10f58e
permissions -rw-r--r--
Merge #411
     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_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 #include <lemon/hartmann_orlin_mmc.h>
    39 
    40 namespace lemon {
    41 
    42   /// \addtogroup min_cost_flow_algs
    43   /// @{
    44 
    45   /// \brief Implementation of cycle-canceling algorithms for
    46   /// finding a \ref min_cost_flow "minimum cost flow".
    47   ///
    48   /// \ref CycleCanceling implements three different cycle-canceling
    49   /// algorithms for finding a \ref min_cost_flow "minimum cost flow"
    50   /// \ref amo93networkflows, \ref klein67primal,
    51   /// \ref goldberg89cyclecanceling.
    52   /// The most efficent one is the \ref CANCEL_AND_TIGHTEN
    53   /// "Cancel-and-Tighten" algorithm, thus it is the default method.
    54   /// It runs in strongly polynomial time, but in practice, it is typically
    55   /// orders of magnitude slower than the scaling algorithms and
    56   /// \ref NetworkSimplex.
    57   /// (For more information, see \ref min_cost_flow_algs "the module page".)
    58   ///
    59   /// Most of the parameters of the problem (except for the digraph)
    60   /// can be given using separate functions, and the algorithm can be
    61   /// executed using the \ref run() function. If some parameters are not
    62   /// specified, then default values will be used.
    63   ///
    64   /// \tparam GR The digraph type the algorithm runs on.
    65   /// \tparam V The number type used for flow amounts, capacity bounds
    66   /// and supply values in the algorithm. By default, it is \c int.
    67   /// \tparam C The number type used for costs and potentials in the
    68   /// algorithm. By default, it is the same as \c V.
    69   ///
    70   /// \warning Both \c V and \c C must be signed number types.
    71   /// \warning All input data (capacities, supply values, and costs) must
    72   /// be integer.
    73   /// \warning This algorithm does not support negative costs for
    74   /// arcs having infinite upper bound.
    75   ///
    76   /// \note For more information about the three available methods,
    77   /// see \ref Method.
    78 #ifdef DOXYGEN
    79   template <typename GR, typename V, typename C>
    80 #else
    81   template <typename GR, typename V = int, typename C = V>
    82 #endif
    83   class CycleCanceling
    84   {
    85   public:
    86 
    87     /// The type of the digraph
    88     typedef GR Digraph;
    89     /// The type of the flow amounts, capacity bounds and supply values
    90     typedef V Value;
    91     /// The type of the arc costs
    92     typedef C Cost;
    93 
    94   public:
    95 
    96     /// \brief Problem type constants for the \c run() function.
    97     ///
    98     /// Enum type containing the problem type constants that can be
    99     /// returned by the \ref run() function of the algorithm.
   100     enum ProblemType {
   101       /// The problem has no feasible solution (flow).
   102       INFEASIBLE,
   103       /// The problem has optimal solution (i.e. it is feasible and
   104       /// bounded), and the algorithm has found optimal flow and node
   105       /// potentials (primal and dual solutions).
   106       OPTIMAL,
   107       /// The digraph contains an arc of negative cost and infinite
   108       /// upper bound. It means that the objective function is unbounded
   109       /// on that arc, however, note that it could actually be bounded
   110       /// over the feasible flows, but this algroithm cannot handle
   111       /// these cases.
   112       UNBOUNDED
   113     };
   114 
   115     /// \brief Constants for selecting the used method.
   116     ///
   117     /// Enum type containing constants for selecting the used method
   118     /// for the \ref run() function.
   119     ///
   120     /// \ref CycleCanceling provides three different cycle-canceling
   121     /// methods. By default, \ref CANCEL_AND_TIGHTEN "Cancel-and-Tighten"
   122     /// is used, which is by far the most efficient and the most robust.
   123     /// However, the other methods can be selected using the \ref run()
   124     /// function with the proper parameter.
   125     enum Method {
   126       /// A simple cycle-canceling method, which uses the
   127       /// \ref BellmanFord "Bellman-Ford" algorithm for detecting negative
   128       /// cycles in the residual network.
   129       /// The number of Bellman-Ford iterations is bounded by a successively
   130       /// increased limit.
   131       SIMPLE_CYCLE_CANCELING,
   132       /// The "Minimum Mean Cycle-Canceling" algorithm, which is a
   133       /// well-known strongly polynomial method
   134       /// \ref goldberg89cyclecanceling. It improves along a
   135       /// \ref min_mean_cycle "minimum mean cycle" in each iteration.
   136       /// Its running time complexity is O(n<sup>2</sup>e<sup>3</sup>log(n)).
   137       MINIMUM_MEAN_CYCLE_CANCELING,
   138       /// The "Cancel-and-Tighten" algorithm, which can be viewed as an
   139       /// improved version of the previous method
   140       /// \ref goldberg89cyclecanceling.
   141       /// It is faster both in theory and in practice, its running time
   142       /// complexity is O(n<sup>2</sup>e<sup>2</sup>log(n)).
   143       CANCEL_AND_TIGHTEN
   144     };
   145 
   146   private:
   147 
   148     TEMPLATE_DIGRAPH_TYPEDEFS(GR);
   149 
   150     typedef std::vector<int> IntVector;
   151     typedef std::vector<double> DoubleVector;
   152     typedef std::vector<Value> ValueVector;
   153     typedef std::vector<Cost> CostVector;
   154     typedef std::vector<char> BoolVector;
   155     // Note: vector<char> is used instead of vector<bool> for efficiency reasons
   156 
   157   private:
   158 
   159     template <typename KT, typename VT>
   160     class StaticVectorMap {
   161     public:
   162       typedef KT Key;
   163       typedef VT Value;
   164 
   165       StaticVectorMap(std::vector<Value>& v) : _v(v) {}
   166 
   167       const Value& operator[](const Key& key) const {
   168         return _v[StaticDigraph::id(key)];
   169       }
   170 
   171       Value& operator[](const Key& key) {
   172         return _v[StaticDigraph::id(key)];
   173       }
   174 
   175       void set(const Key& key, const Value& val) {
   176         _v[StaticDigraph::id(key)] = val;
   177       }
   178 
   179     private:
   180       std::vector<Value>& _v;
   181     };
   182 
   183     typedef StaticVectorMap<StaticDigraph::Node, Cost> CostNodeMap;
   184     typedef StaticVectorMap<StaticDigraph::Arc, Cost> CostArcMap;
   185 
   186   private:
   187 
   188 
   189     // Data related to the underlying digraph
   190     const GR &_graph;
   191     int _node_num;
   192     int _arc_num;
   193     int _res_node_num;
   194     int _res_arc_num;
   195     int _root;
   196 
   197     // Parameters of the problem
   198     bool _have_lower;
   199     Value _sum_supply;
   200 
   201     // Data structures for storing the digraph
   202     IntNodeMap _node_id;
   203     IntArcMap _arc_idf;
   204     IntArcMap _arc_idb;
   205     IntVector _first_out;
   206     BoolVector _forward;
   207     IntVector _source;
   208     IntVector _target;
   209     IntVector _reverse;
   210 
   211     // Node and arc data
   212     ValueVector _lower;
   213     ValueVector _upper;
   214     CostVector _cost;
   215     ValueVector _supply;
   216 
   217     ValueVector _res_cap;
   218     CostVector _pi;
   219 
   220     // Data for a StaticDigraph structure
   221     typedef std::pair<int, int> IntPair;
   222     StaticDigraph _sgr;
   223     std::vector<IntPair> _arc_vec;
   224     std::vector<Cost> _cost_vec;
   225     IntVector _id_vec;
   226     CostArcMap _cost_map;
   227     CostNodeMap _pi_map;
   228 
   229   public:
   230 
   231     /// \brief Constant for infinite upper bounds (capacities).
   232     ///
   233     /// Constant for infinite upper bounds (capacities).
   234     /// It is \c std::numeric_limits<Value>::infinity() if available,
   235     /// \c std::numeric_limits<Value>::max() otherwise.
   236     const Value INF;
   237 
   238   public:
   239 
   240     /// \brief Constructor.
   241     ///
   242     /// The constructor of the class.
   243     ///
   244     /// \param graph The digraph the algorithm runs on.
   245     CycleCanceling(const GR& graph) :
   246       _graph(graph), _node_id(graph), _arc_idf(graph), _arc_idb(graph),
   247       _cost_map(_cost_vec), _pi_map(_pi),
   248       INF(std::numeric_limits<Value>::has_infinity ?
   249           std::numeric_limits<Value>::infinity() :
   250           std::numeric_limits<Value>::max())
   251     {
   252       // Check the number types
   253       LEMON_ASSERT(std::numeric_limits<Value>::is_signed,
   254         "The flow type of CycleCanceling must be signed");
   255       LEMON_ASSERT(std::numeric_limits<Cost>::is_signed,
   256         "The cost type of CycleCanceling must be signed");
   257 
   258       // Reset data structures
   259       reset();
   260     }
   261 
   262     /// \name Parameters
   263     /// The parameters of the algorithm can be specified using these
   264     /// functions.
   265 
   266     /// @{
   267 
   268     /// \brief Set the lower bounds on the arcs.
   269     ///
   270     /// This function sets the lower bounds on the arcs.
   271     /// If it is not used before calling \ref run(), the lower bounds
   272     /// will be set to zero on all arcs.
   273     ///
   274     /// \param map An arc map storing the lower bounds.
   275     /// Its \c Value type must be convertible to the \c Value type
   276     /// of the algorithm.
   277     ///
   278     /// \return <tt>(*this)</tt>
   279     template <typename LowerMap>
   280     CycleCanceling& lowerMap(const LowerMap& map) {
   281       _have_lower = true;
   282       for (ArcIt a(_graph); a != INVALID; ++a) {
   283         _lower[_arc_idf[a]] = map[a];
   284         _lower[_arc_idb[a]] = map[a];
   285       }
   286       return *this;
   287     }
   288 
   289     /// \brief Set the upper bounds (capacities) on the arcs.
   290     ///
   291     /// This function sets the upper bounds (capacities) on the arcs.
   292     /// If it is not used before calling \ref run(), the upper bounds
   293     /// will be set to \ref INF on all arcs (i.e. the flow value will be
   294     /// unbounded from above).
   295     ///
   296     /// \param map An arc map storing the upper bounds.
   297     /// Its \c Value type must be convertible to the \c Value type
   298     /// of the algorithm.
   299     ///
   300     /// \return <tt>(*this)</tt>
   301     template<typename UpperMap>
   302     CycleCanceling& upperMap(const UpperMap& map) {
   303       for (ArcIt a(_graph); a != INVALID; ++a) {
   304         _upper[_arc_idf[a]] = map[a];
   305       }
   306       return *this;
   307     }
   308 
   309     /// \brief Set the costs of the arcs.
   310     ///
   311     /// This function sets the costs of the arcs.
   312     /// If it is not used before calling \ref run(), the costs
   313     /// will be set to \c 1 on all arcs.
   314     ///
   315     /// \param map An arc map storing the costs.
   316     /// Its \c Value type must be convertible to the \c Cost type
   317     /// of the algorithm.
   318     ///
   319     /// \return <tt>(*this)</tt>
   320     template<typename CostMap>
   321     CycleCanceling& costMap(const CostMap& map) {
   322       for (ArcIt a(_graph); a != INVALID; ++a) {
   323         _cost[_arc_idf[a]] =  map[a];
   324         _cost[_arc_idb[a]] = -map[a];
   325       }
   326       return *this;
   327     }
   328 
   329     /// \brief Set the supply values of the nodes.
   330     ///
   331     /// This function sets the supply values of the nodes.
   332     /// If neither this function nor \ref stSupply() is used before
   333     /// calling \ref run(), the supply of each node will be set to zero.
   334     ///
   335     /// \param map A node map storing the supply values.
   336     /// Its \c Value type must be convertible to the \c Value type
   337     /// of the algorithm.
   338     ///
   339     /// \return <tt>(*this)</tt>
   340     template<typename SupplyMap>
   341     CycleCanceling& supplyMap(const SupplyMap& map) {
   342       for (NodeIt n(_graph); n != INVALID; ++n) {
   343         _supply[_node_id[n]] = map[n];
   344       }
   345       return *this;
   346     }
   347 
   348     /// \brief Set single source and target nodes and a supply value.
   349     ///
   350     /// This function sets a single source node and a single target node
   351     /// and the required flow value.
   352     /// If neither this function nor \ref supplyMap() is used before
   353     /// calling \ref run(), the supply of each node will be set to zero.
   354     ///
   355     /// Using this function has the same effect as using \ref supplyMap()
   356     /// with a map in which \c k is assigned to \c s, \c -k is
   357     /// assigned to \c t and all other nodes have zero supply value.
   358     ///
   359     /// \param s The source node.
   360     /// \param t The target node.
   361     /// \param k The required amount of flow from node \c s to node \c t
   362     /// (i.e. the supply of \c s and the demand of \c t).
   363     ///
   364     /// \return <tt>(*this)</tt>
   365     CycleCanceling& stSupply(const Node& s, const Node& t, Value k) {
   366       for (int i = 0; i != _res_node_num; ++i) {
   367         _supply[i] = 0;
   368       }
   369       _supply[_node_id[s]] =  k;
   370       _supply[_node_id[t]] = -k;
   371       return *this;
   372     }
   373 
   374     /// @}
   375 
   376     /// \name Execution control
   377     /// The algorithm can be executed using \ref run().
   378 
   379     /// @{
   380 
   381     /// \brief Run the algorithm.
   382     ///
   383     /// This function runs the algorithm.
   384     /// The paramters can be specified using functions \ref lowerMap(),
   385     /// \ref upperMap(), \ref costMap(), \ref supplyMap(), \ref stSupply().
   386     /// For example,
   387     /// \code
   388     ///   CycleCanceling<ListDigraph> cc(graph);
   389     ///   cc.lowerMap(lower).upperMap(upper).costMap(cost)
   390     ///     .supplyMap(sup).run();
   391     /// \endcode
   392     ///
   393     /// This function can be called more than once. All the given parameters
   394     /// are kept for the next call, unless \ref resetParams() or \ref reset()
   395     /// is used, thus only the modified parameters have to be set again.
   396     /// If the underlying digraph was also modified after the construction
   397     /// of the class (or the last \ref reset() call), then the \ref reset()
   398     /// function must be called.
   399     ///
   400     /// \param method The cycle-canceling method that will be used.
   401     /// For more information, see \ref Method.
   402     ///
   403     /// \return \c INFEASIBLE if no feasible flow exists,
   404     /// \n \c OPTIMAL if the problem has optimal solution
   405     /// (i.e. it is feasible and bounded), and the algorithm has found
   406     /// optimal flow and node potentials (primal and dual solutions),
   407     /// \n \c UNBOUNDED if the digraph contains an arc of negative cost
   408     /// and infinite upper bound. It means that the objective function
   409     /// is unbounded on that arc, however, note that it could actually be
   410     /// bounded over the feasible flows, but this algroithm cannot handle
   411     /// these cases.
   412     ///
   413     /// \see ProblemType, Method
   414     /// \see resetParams(), reset()
   415     ProblemType run(Method method = CANCEL_AND_TIGHTEN) {
   416       ProblemType pt = init();
   417       if (pt != OPTIMAL) return pt;
   418       start(method);
   419       return OPTIMAL;
   420     }
   421 
   422     /// \brief Reset all the parameters that have been given before.
   423     ///
   424     /// This function resets all the paramaters that have been given
   425     /// before using functions \ref lowerMap(), \ref upperMap(),
   426     /// \ref costMap(), \ref supplyMap(), \ref stSupply().
   427     ///
   428     /// It is useful for multiple \ref run() calls. Basically, all the given
   429     /// parameters are kept for the next \ref run() call, unless
   430     /// \ref resetParams() or \ref reset() is used.
   431     /// If the underlying digraph was also modified after the construction
   432     /// of the class or the last \ref reset() call, then the \ref reset()
   433     /// function must be used, otherwise \ref resetParams() is sufficient.
   434     ///
   435     /// For example,
   436     /// \code
   437     ///   CycleCanceling<ListDigraph> cs(graph);
   438     ///
   439     ///   // First run
   440     ///   cc.lowerMap(lower).upperMap(upper).costMap(cost)
   441     ///     .supplyMap(sup).run();
   442     ///
   443     ///   // Run again with modified cost map (resetParams() is not called,
   444     ///   // so only the cost map have to be set again)
   445     ///   cost[e] += 100;
   446     ///   cc.costMap(cost).run();
   447     ///
   448     ///   // Run again from scratch using resetParams()
   449     ///   // (the lower bounds will be set to zero on all arcs)
   450     ///   cc.resetParams();
   451     ///   cc.upperMap(capacity).costMap(cost)
   452     ///     .supplyMap(sup).run();
   453     /// \endcode
   454     ///
   455     /// \return <tt>(*this)</tt>
   456     ///
   457     /// \see reset(), run()
   458     CycleCanceling& resetParams() {
   459       for (int i = 0; i != _res_node_num; ++i) {
   460         _supply[i] = 0;
   461       }
   462       int limit = _first_out[_root];
   463       for (int j = 0; j != limit; ++j) {
   464         _lower[j] = 0;
   465         _upper[j] = INF;
   466         _cost[j] = _forward[j] ? 1 : -1;
   467       }
   468       for (int j = limit; j != _res_arc_num; ++j) {
   469         _lower[j] = 0;
   470         _upper[j] = INF;
   471         _cost[j] = 0;
   472         _cost[_reverse[j]] = 0;
   473       }
   474       _have_lower = false;
   475       return *this;
   476     }
   477 
   478     /// \brief Reset the internal data structures and all the parameters
   479     /// that have been given before.
   480     ///
   481     /// This function resets the internal data structures and all the
   482     /// paramaters that have been given before using functions \ref lowerMap(),
   483     /// \ref upperMap(), \ref costMap(), \ref supplyMap(), \ref stSupply().
   484     ///
   485     /// It is useful for multiple \ref run() calls. Basically, all the given
   486     /// parameters are kept for the next \ref run() call, unless
   487     /// \ref resetParams() or \ref reset() is used.
   488     /// If the underlying digraph was also modified after the construction
   489     /// of the class or the last \ref reset() call, then the \ref reset()
   490     /// function must be used, otherwise \ref resetParams() is sufficient.
   491     ///
   492     /// See \ref resetParams() for examples.
   493     ///
   494     /// \return <tt>(*this)</tt>
   495     ///
   496     /// \see resetParams(), run()
   497     CycleCanceling& reset() {
   498       // Resize vectors
   499       _node_num = countNodes(_graph);
   500       _arc_num = countArcs(_graph);
   501       _res_node_num = _node_num + 1;
   502       _res_arc_num = 2 * (_arc_num + _node_num);
   503       _root = _node_num;
   504 
   505       _first_out.resize(_res_node_num + 1);
   506       _forward.resize(_res_arc_num);
   507       _source.resize(_res_arc_num);
   508       _target.resize(_res_arc_num);
   509       _reverse.resize(_res_arc_num);
   510 
   511       _lower.resize(_res_arc_num);
   512       _upper.resize(_res_arc_num);
   513       _cost.resize(_res_arc_num);
   514       _supply.resize(_res_node_num);
   515 
   516       _res_cap.resize(_res_arc_num);
   517       _pi.resize(_res_node_num);
   518 
   519       _arc_vec.reserve(_res_arc_num);
   520       _cost_vec.reserve(_res_arc_num);
   521       _id_vec.reserve(_res_arc_num);
   522 
   523       // Copy the graph
   524       int i = 0, j = 0, k = 2 * _arc_num + _node_num;
   525       for (NodeIt n(_graph); n != INVALID; ++n, ++i) {
   526         _node_id[n] = i;
   527       }
   528       i = 0;
   529       for (NodeIt n(_graph); n != INVALID; ++n, ++i) {
   530         _first_out[i] = j;
   531         for (OutArcIt a(_graph, n); a != INVALID; ++a, ++j) {
   532           _arc_idf[a] = j;
   533           _forward[j] = true;
   534           _source[j] = i;
   535           _target[j] = _node_id[_graph.runningNode(a)];
   536         }
   537         for (InArcIt a(_graph, n); a != INVALID; ++a, ++j) {
   538           _arc_idb[a] = j;
   539           _forward[j] = false;
   540           _source[j] = i;
   541           _target[j] = _node_id[_graph.runningNode(a)];
   542         }
   543         _forward[j] = false;
   544         _source[j] = i;
   545         _target[j] = _root;
   546         _reverse[j] = k;
   547         _forward[k] = true;
   548         _source[k] = _root;
   549         _target[k] = i;
   550         _reverse[k] = j;
   551         ++j; ++k;
   552       }
   553       _first_out[i] = j;
   554       _first_out[_res_node_num] = k;
   555       for (ArcIt a(_graph); a != INVALID; ++a) {
   556         int fi = _arc_idf[a];
   557         int bi = _arc_idb[a];
   558         _reverse[fi] = bi;
   559         _reverse[bi] = fi;
   560       }
   561 
   562       // Reset parameters
   563       resetParams();
   564       return *this;
   565     }
   566 
   567     /// @}
   568 
   569     /// \name Query Functions
   570     /// The results of the algorithm can be obtained using these
   571     /// functions.\n
   572     /// The \ref run() function must be called before using them.
   573 
   574     /// @{
   575 
   576     /// \brief Return the total cost of the found flow.
   577     ///
   578     /// This function returns the total cost of the found flow.
   579     /// Its complexity is O(e).
   580     ///
   581     /// \note The return type of the function can be specified as a
   582     /// template parameter. For example,
   583     /// \code
   584     ///   cc.totalCost<double>();
   585     /// \endcode
   586     /// It is useful if the total cost cannot be stored in the \c Cost
   587     /// type of the algorithm, which is the default return type of the
   588     /// function.
   589     ///
   590     /// \pre \ref run() must be called before using this function.
   591     template <typename Number>
   592     Number totalCost() const {
   593       Number c = 0;
   594       for (ArcIt a(_graph); a != INVALID; ++a) {
   595         int i = _arc_idb[a];
   596         c += static_cast<Number>(_res_cap[i]) *
   597              (-static_cast<Number>(_cost[i]));
   598       }
   599       return c;
   600     }
   601 
   602 #ifndef DOXYGEN
   603     Cost totalCost() const {
   604       return totalCost<Cost>();
   605     }
   606 #endif
   607 
   608     /// \brief Return the flow on the given arc.
   609     ///
   610     /// This function returns the flow on the given arc.
   611     ///
   612     /// \pre \ref run() must be called before using this function.
   613     Value flow(const Arc& a) const {
   614       return _res_cap[_arc_idb[a]];
   615     }
   616 
   617     /// \brief Copy the flow values (the primal solution) into the
   618     /// given map.
   619     ///
   620     /// This function copies the flow value on each arc into the given
   621     /// map. The \c Value type of the algorithm must be convertible to
   622     /// the \c Value type of the map.
   623     ///
   624     /// \pre \ref run() must be called before using this function.
   625     template <typename FlowMap>
   626     void flowMap(FlowMap &map) const {
   627       for (ArcIt a(_graph); a != INVALID; ++a) {
   628         map.set(a, _res_cap[_arc_idb[a]]);
   629       }
   630     }
   631 
   632     /// \brief Return the potential (dual value) of the given node.
   633     ///
   634     /// This function returns the potential (dual value) of the
   635     /// given node.
   636     ///
   637     /// \pre \ref run() must be called before using this function.
   638     Cost potential(const Node& n) const {
   639       return static_cast<Cost>(_pi[_node_id[n]]);
   640     }
   641 
   642     /// \brief Copy the potential values (the dual solution) into the
   643     /// given map.
   644     ///
   645     /// This function copies the potential (dual value) of each node
   646     /// into the given map.
   647     /// The \c Cost type of the algorithm must be convertible to the
   648     /// \c Value type of the map.
   649     ///
   650     /// \pre \ref run() must be called before using this function.
   651     template <typename PotentialMap>
   652     void potentialMap(PotentialMap &map) const {
   653       for (NodeIt n(_graph); n != INVALID; ++n) {
   654         map.set(n, static_cast<Cost>(_pi[_node_id[n]]));
   655       }
   656     }
   657 
   658     /// @}
   659 
   660   private:
   661 
   662     // Initialize the algorithm
   663     ProblemType init() {
   664       if (_res_node_num <= 1) return INFEASIBLE;
   665 
   666       // Check the sum of supply values
   667       _sum_supply = 0;
   668       for (int i = 0; i != _root; ++i) {
   669         _sum_supply += _supply[i];
   670       }
   671       if (_sum_supply > 0) return INFEASIBLE;
   672 
   673 
   674       // Initialize vectors
   675       for (int i = 0; i != _res_node_num; ++i) {
   676         _pi[i] = 0;
   677       }
   678       ValueVector excess(_supply);
   679 
   680       // Remove infinite upper bounds and check negative arcs
   681       const Value MAX = std::numeric_limits<Value>::max();
   682       int last_out;
   683       if (_have_lower) {
   684         for (int i = 0; i != _root; ++i) {
   685           last_out = _first_out[i+1];
   686           for (int j = _first_out[i]; j != last_out; ++j) {
   687             if (_forward[j]) {
   688               Value c = _cost[j] < 0 ? _upper[j] : _lower[j];
   689               if (c >= MAX) return UNBOUNDED;
   690               excess[i] -= c;
   691               excess[_target[j]] += c;
   692             }
   693           }
   694         }
   695       } else {
   696         for (int i = 0; i != _root; ++i) {
   697           last_out = _first_out[i+1];
   698           for (int j = _first_out[i]; j != last_out; ++j) {
   699             if (_forward[j] && _cost[j] < 0) {
   700               Value c = _upper[j];
   701               if (c >= MAX) return UNBOUNDED;
   702               excess[i] -= c;
   703               excess[_target[j]] += c;
   704             }
   705           }
   706         }
   707       }
   708       Value ex, max_cap = 0;
   709       for (int i = 0; i != _res_node_num; ++i) {
   710         ex = excess[i];
   711         if (ex < 0) max_cap -= ex;
   712       }
   713       for (int j = 0; j != _res_arc_num; ++j) {
   714         if (_upper[j] >= MAX) _upper[j] = max_cap;
   715       }
   716 
   717       // Initialize maps for Circulation and remove non-zero lower bounds
   718       ConstMap<Arc, Value> low(0);
   719       typedef typename Digraph::template ArcMap<Value> ValueArcMap;
   720       typedef typename Digraph::template NodeMap<Value> ValueNodeMap;
   721       ValueArcMap cap(_graph), flow(_graph);
   722       ValueNodeMap sup(_graph);
   723       for (NodeIt n(_graph); n != INVALID; ++n) {
   724         sup[n] = _supply[_node_id[n]];
   725       }
   726       if (_have_lower) {
   727         for (ArcIt a(_graph); a != INVALID; ++a) {
   728           int j = _arc_idf[a];
   729           Value c = _lower[j];
   730           cap[a] = _upper[j] - c;
   731           sup[_graph.source(a)] -= c;
   732           sup[_graph.target(a)] += c;
   733         }
   734       } else {
   735         for (ArcIt a(_graph); a != INVALID; ++a) {
   736           cap[a] = _upper[_arc_idf[a]];
   737         }
   738       }
   739 
   740       // Find a feasible flow using Circulation
   741       Circulation<Digraph, ConstMap<Arc, Value>, ValueArcMap, ValueNodeMap>
   742         circ(_graph, low, cap, sup);
   743       if (!circ.flowMap(flow).run()) return INFEASIBLE;
   744 
   745       // Set residual capacities and handle GEQ supply type
   746       if (_sum_supply < 0) {
   747         for (ArcIt a(_graph); a != INVALID; ++a) {
   748           Value fa = flow[a];
   749           _res_cap[_arc_idf[a]] = cap[a] - fa;
   750           _res_cap[_arc_idb[a]] = fa;
   751           sup[_graph.source(a)] -= fa;
   752           sup[_graph.target(a)] += fa;
   753         }
   754         for (NodeIt n(_graph); n != INVALID; ++n) {
   755           excess[_node_id[n]] = sup[n];
   756         }
   757         for (int a = _first_out[_root]; a != _res_arc_num; ++a) {
   758           int u = _target[a];
   759           int ra = _reverse[a];
   760           _res_cap[a] = -_sum_supply + 1;
   761           _res_cap[ra] = -excess[u];
   762           _cost[a] = 0;
   763           _cost[ra] = 0;
   764         }
   765       } else {
   766         for (ArcIt a(_graph); a != INVALID; ++a) {
   767           Value fa = flow[a];
   768           _res_cap[_arc_idf[a]] = cap[a] - fa;
   769           _res_cap[_arc_idb[a]] = fa;
   770         }
   771         for (int a = _first_out[_root]; a != _res_arc_num; ++a) {
   772           int ra = _reverse[a];
   773           _res_cap[a] = 1;
   774           _res_cap[ra] = 0;
   775           _cost[a] = 0;
   776           _cost[ra] = 0;
   777         }
   778       }
   779 
   780       return OPTIMAL;
   781     }
   782 
   783     // Build a StaticDigraph structure containing the current
   784     // residual network
   785     void buildResidualNetwork() {
   786       _arc_vec.clear();
   787       _cost_vec.clear();
   788       _id_vec.clear();
   789       for (int j = 0; j != _res_arc_num; ++j) {
   790         if (_res_cap[j] > 0) {
   791           _arc_vec.push_back(IntPair(_source[j], _target[j]));
   792           _cost_vec.push_back(_cost[j]);
   793           _id_vec.push_back(j);
   794         }
   795       }
   796       _sgr.build(_res_node_num, _arc_vec.begin(), _arc_vec.end());
   797     }
   798 
   799     // Execute the algorithm and transform the results
   800     void start(Method method) {
   801       // Execute the algorithm
   802       switch (method) {
   803         case SIMPLE_CYCLE_CANCELING:
   804           startSimpleCycleCanceling();
   805           break;
   806         case MINIMUM_MEAN_CYCLE_CANCELING:
   807           startMinMeanCycleCanceling();
   808           break;
   809         case CANCEL_AND_TIGHTEN:
   810           startCancelAndTighten();
   811           break;
   812       }
   813 
   814       // Compute node potentials
   815       if (method != SIMPLE_CYCLE_CANCELING) {
   816         buildResidualNetwork();
   817         typename BellmanFord<StaticDigraph, CostArcMap>
   818           ::template SetDistMap<CostNodeMap>::Create bf(_sgr, _cost_map);
   819         bf.distMap(_pi_map);
   820         bf.init(0);
   821         bf.start();
   822       }
   823 
   824       // Handle non-zero lower bounds
   825       if (_have_lower) {
   826         int limit = _first_out[_root];
   827         for (int j = 0; j != limit; ++j) {
   828           if (!_forward[j]) _res_cap[j] += _lower[j];
   829         }
   830       }
   831     }
   832 
   833     // Execute the "Simple Cycle Canceling" method
   834     void startSimpleCycleCanceling() {
   835       // Constants for computing the iteration limits
   836       const int BF_FIRST_LIMIT  = 2;
   837       const double BF_LIMIT_FACTOR = 1.5;
   838 
   839       typedef StaticVectorMap<StaticDigraph::Arc, Value> FilterMap;
   840       typedef FilterArcs<StaticDigraph, FilterMap> ResDigraph;
   841       typedef StaticVectorMap<StaticDigraph::Node, StaticDigraph::Arc> PredMap;
   842       typedef typename BellmanFord<ResDigraph, CostArcMap>
   843         ::template SetDistMap<CostNodeMap>
   844         ::template SetPredMap<PredMap>::Create BF;
   845 
   846       // Build the residual network
   847       _arc_vec.clear();
   848       _cost_vec.clear();
   849       for (int j = 0; j != _res_arc_num; ++j) {
   850         _arc_vec.push_back(IntPair(_source[j], _target[j]));
   851         _cost_vec.push_back(_cost[j]);
   852       }
   853       _sgr.build(_res_node_num, _arc_vec.begin(), _arc_vec.end());
   854 
   855       FilterMap filter_map(_res_cap);
   856       ResDigraph rgr(_sgr, filter_map);
   857       std::vector<int> cycle;
   858       std::vector<StaticDigraph::Arc> pred(_res_arc_num);
   859       PredMap pred_map(pred);
   860       BF bf(rgr, _cost_map);
   861       bf.distMap(_pi_map).predMap(pred_map);
   862 
   863       int length_bound = BF_FIRST_LIMIT;
   864       bool optimal = false;
   865       while (!optimal) {
   866         bf.init(0);
   867         int iter_num = 0;
   868         bool cycle_found = false;
   869         while (!cycle_found) {
   870           // Perform some iterations of the Bellman-Ford algorithm
   871           int curr_iter_num = iter_num + length_bound <= _node_num ?
   872             length_bound : _node_num - iter_num;
   873           iter_num += curr_iter_num;
   874           int real_iter_num = curr_iter_num;
   875           for (int i = 0; i < curr_iter_num; ++i) {
   876             if (bf.processNextWeakRound()) {
   877               real_iter_num = i;
   878               break;
   879             }
   880           }
   881           if (real_iter_num < curr_iter_num) {
   882             // Optimal flow is found
   883             optimal = true;
   884             break;
   885           } else {
   886             // Search for node disjoint negative cycles
   887             std::vector<int> state(_res_node_num, 0);
   888             int id = 0;
   889             for (int u = 0; u != _res_node_num; ++u) {
   890               if (state[u] != 0) continue;
   891               ++id;
   892               int v = u;
   893               for (; v != -1 && state[v] == 0; v = pred[v] == INVALID ?
   894                    -1 : rgr.id(rgr.source(pred[v]))) {
   895                 state[v] = id;
   896               }
   897               if (v != -1 && state[v] == id) {
   898                 // A negative cycle is found
   899                 cycle_found = true;
   900                 cycle.clear();
   901                 StaticDigraph::Arc a = pred[v];
   902                 Value d, delta = _res_cap[rgr.id(a)];
   903                 cycle.push_back(rgr.id(a));
   904                 while (rgr.id(rgr.source(a)) != v) {
   905                   a = pred_map[rgr.source(a)];
   906                   d = _res_cap[rgr.id(a)];
   907                   if (d < delta) delta = d;
   908                   cycle.push_back(rgr.id(a));
   909                 }
   910 
   911                 // Augment along the cycle
   912                 for (int i = 0; i < int(cycle.size()); ++i) {
   913                   int j = cycle[i];
   914                   _res_cap[j] -= delta;
   915                   _res_cap[_reverse[j]] += delta;
   916                 }
   917               }
   918             }
   919           }
   920 
   921           // Increase iteration limit if no cycle is found
   922           if (!cycle_found) {
   923             length_bound = static_cast<int>(length_bound * BF_LIMIT_FACTOR);
   924           }
   925         }
   926       }
   927     }
   928 
   929     // Execute the "Minimum Mean Cycle Canceling" method
   930     void startMinMeanCycleCanceling() {
   931       typedef Path<StaticDigraph> SPath;
   932       typedef typename SPath::ArcIt SPathArcIt;
   933       typedef typename HowardMmc<StaticDigraph, CostArcMap>
   934         ::template SetPath<SPath>::Create HwMmc;
   935       typedef typename HartmannOrlinMmc<StaticDigraph, CostArcMap>
   936         ::template SetPath<SPath>::Create HoMmc;
   937 
   938       const double HW_ITER_LIMIT_FACTOR = 1.0;
   939       const int HW_ITER_LIMIT_MIN_VALUE = 5;
   940 
   941       const int hw_iter_limit =
   942           std::max(static_cast<int>(HW_ITER_LIMIT_FACTOR * _node_num),
   943                    HW_ITER_LIMIT_MIN_VALUE);
   944 
   945       SPath cycle;
   946       HwMmc hw_mmc(_sgr, _cost_map);
   947       hw_mmc.cycle(cycle);
   948       buildResidualNetwork();
   949       while (true) {
   950         
   951         typename HwMmc::TerminationCause hw_tc =
   952             hw_mmc.findCycleMean(hw_iter_limit);
   953         if (hw_tc == HwMmc::ITERATION_LIMIT) {
   954           // Howard's algorithm reached the iteration limit, start a
   955           // strongly polynomial algorithm instead
   956           HoMmc ho_mmc(_sgr, _cost_map);
   957           ho_mmc.cycle(cycle);
   958           // Find a minimum mean cycle (Hartmann-Orlin algorithm)
   959           if (!(ho_mmc.findCycleMean() && ho_mmc.cycleCost() < 0)) break;
   960           ho_mmc.findCycle();
   961         } else {
   962           // Find a minimum mean cycle (Howard algorithm)
   963           if (!(hw_tc == HwMmc::OPTIMAL && hw_mmc.cycleCost() < 0)) break;
   964           hw_mmc.findCycle();
   965         }
   966         
   967         // Compute delta value
   968         Value delta = INF;
   969         for (SPathArcIt a(cycle); a != INVALID; ++a) {
   970           Value d = _res_cap[_id_vec[_sgr.id(a)]];
   971           if (d < delta) delta = d;
   972         }
   973 
   974         // Augment along the cycle
   975         for (SPathArcIt a(cycle); a != INVALID; ++a) {
   976           int j = _id_vec[_sgr.id(a)];
   977           _res_cap[j] -= delta;
   978           _res_cap[_reverse[j]] += delta;
   979         }
   980 
   981         // Rebuild the residual network
   982         buildResidualNetwork();
   983       }
   984     }
   985 
   986     // Execute the "Cancel-and-Tighten" method
   987     void startCancelAndTighten() {
   988       // Constants for the min mean cycle computations
   989       const double LIMIT_FACTOR = 1.0;
   990       const int MIN_LIMIT = 5;
   991       const double HW_ITER_LIMIT_FACTOR = 1.0;
   992       const int HW_ITER_LIMIT_MIN_VALUE = 5;
   993 
   994       const int hw_iter_limit =
   995           std::max(static_cast<int>(HW_ITER_LIMIT_FACTOR * _node_num),
   996                    HW_ITER_LIMIT_MIN_VALUE);
   997 
   998       // Contruct auxiliary data vectors
   999       DoubleVector pi(_res_node_num, 0.0);
  1000       IntVector level(_res_node_num);
  1001       BoolVector reached(_res_node_num);
  1002       BoolVector processed(_res_node_num);
  1003       IntVector pred_node(_res_node_num);
  1004       IntVector pred_arc(_res_node_num);
  1005       std::vector<int> stack(_res_node_num);
  1006       std::vector<int> proc_vector(_res_node_num);
  1007 
  1008       // Initialize epsilon
  1009       double epsilon = 0;
  1010       for (int a = 0; a != _res_arc_num; ++a) {
  1011         if (_res_cap[a] > 0 && -_cost[a] > epsilon)
  1012           epsilon = -_cost[a];
  1013       }
  1014 
  1015       // Start phases
  1016       Tolerance<double> tol;
  1017       tol.epsilon(1e-6);
  1018       int limit = int(LIMIT_FACTOR * std::sqrt(double(_res_node_num)));
  1019       if (limit < MIN_LIMIT) limit = MIN_LIMIT;
  1020       int iter = limit;
  1021       while (epsilon * _res_node_num >= 1) {
  1022         // Find and cancel cycles in the admissible network using DFS
  1023         for (int u = 0; u != _res_node_num; ++u) {
  1024           reached[u] = false;
  1025           processed[u] = false;
  1026         }
  1027         int stack_head = -1;
  1028         int proc_head = -1;
  1029         for (int start = 0; start != _res_node_num; ++start) {
  1030           if (reached[start]) continue;
  1031 
  1032           // New start node
  1033           reached[start] = true;
  1034           pred_arc[start] = -1;
  1035           pred_node[start] = -1;
  1036 
  1037           // Find the first admissible outgoing arc
  1038           double p = pi[start];
  1039           int a = _first_out[start];
  1040           int last_out = _first_out[start+1];
  1041           for (; a != last_out && (_res_cap[a] == 0 ||
  1042                !tol.negative(_cost[a] + p - pi[_target[a]])); ++a) ;
  1043           if (a == last_out) {
  1044             processed[start] = true;
  1045             proc_vector[++proc_head] = start;
  1046             continue;
  1047           }
  1048           stack[++stack_head] = a;
  1049 
  1050           while (stack_head >= 0) {
  1051             int sa = stack[stack_head];
  1052             int u = _source[sa];
  1053             int v = _target[sa];
  1054 
  1055             if (!reached[v]) {
  1056               // A new node is reached
  1057               reached[v] = true;
  1058               pred_node[v] = u;
  1059               pred_arc[v] = sa;
  1060               p = pi[v];
  1061               a = _first_out[v];
  1062               last_out = _first_out[v+1];
  1063               for (; a != last_out && (_res_cap[a] == 0 ||
  1064                    !tol.negative(_cost[a] + p - pi[_target[a]])); ++a) ;
  1065               stack[++stack_head] = a == last_out ? -1 : a;
  1066             } else {
  1067               if (!processed[v]) {
  1068                 // A cycle is found
  1069                 int n, w = u;
  1070                 Value d, delta = _res_cap[sa];
  1071                 for (n = u; n != v; n = pred_node[n]) {
  1072                   d = _res_cap[pred_arc[n]];
  1073                   if (d <= delta) {
  1074                     delta = d;
  1075                     w = pred_node[n];
  1076                   }
  1077                 }
  1078 
  1079                 // Augment along the cycle
  1080                 _res_cap[sa] -= delta;
  1081                 _res_cap[_reverse[sa]] += delta;
  1082                 for (n = u; n != v; n = pred_node[n]) {
  1083                   int pa = pred_arc[n];
  1084                   _res_cap[pa] -= delta;
  1085                   _res_cap[_reverse[pa]] += delta;
  1086                 }
  1087                 for (n = u; stack_head > 0 && n != w; n = pred_node[n]) {
  1088                   --stack_head;
  1089                   reached[n] = false;
  1090                 }
  1091                 u = w;
  1092               }
  1093               v = u;
  1094 
  1095               // Find the next admissible outgoing arc
  1096               p = pi[v];
  1097               a = stack[stack_head] + 1;
  1098               last_out = _first_out[v+1];
  1099               for (; a != last_out && (_res_cap[a] == 0 ||
  1100                    !tol.negative(_cost[a] + p - pi[_target[a]])); ++a) ;
  1101               stack[stack_head] = a == last_out ? -1 : a;
  1102             }
  1103 
  1104             while (stack_head >= 0 && stack[stack_head] == -1) {
  1105               processed[v] = true;
  1106               proc_vector[++proc_head] = v;
  1107               if (--stack_head >= 0) {
  1108                 // Find the next admissible outgoing arc
  1109                 v = _source[stack[stack_head]];
  1110                 p = pi[v];
  1111                 a = stack[stack_head] + 1;
  1112                 last_out = _first_out[v+1];
  1113                 for (; a != last_out && (_res_cap[a] == 0 ||
  1114                      !tol.negative(_cost[a] + p - pi[_target[a]])); ++a) ;
  1115                 stack[stack_head] = a == last_out ? -1 : a;
  1116               }
  1117             }
  1118           }
  1119         }
  1120 
  1121         // Tighten potentials and epsilon
  1122         if (--iter > 0) {
  1123           for (int u = 0; u != _res_node_num; ++u) {
  1124             level[u] = 0;
  1125           }
  1126           for (int i = proc_head; i > 0; --i) {
  1127             int u = proc_vector[i];
  1128             double p = pi[u];
  1129             int l = level[u] + 1;
  1130             int last_out = _first_out[u+1];
  1131             for (int a = _first_out[u]; a != last_out; ++a) {
  1132               int v = _target[a];
  1133               if (_res_cap[a] > 0 && tol.negative(_cost[a] + p - pi[v]) &&
  1134                   l > level[v]) level[v] = l;
  1135             }
  1136           }
  1137 
  1138           // Modify potentials
  1139           double q = std::numeric_limits<double>::max();
  1140           for (int u = 0; u != _res_node_num; ++u) {
  1141             int lu = level[u];
  1142             double p, pu = pi[u];
  1143             int last_out = _first_out[u+1];
  1144             for (int a = _first_out[u]; a != last_out; ++a) {
  1145               if (_res_cap[a] == 0) continue;
  1146               int v = _target[a];
  1147               int ld = lu - level[v];
  1148               if (ld > 0) {
  1149                 p = (_cost[a] + pu - pi[v] + epsilon) / (ld + 1);
  1150                 if (p < q) q = p;
  1151               }
  1152             }
  1153           }
  1154           for (int u = 0; u != _res_node_num; ++u) {
  1155             pi[u] -= q * level[u];
  1156           }
  1157 
  1158           // Modify epsilon
  1159           epsilon = 0;
  1160           for (int u = 0; u != _res_node_num; ++u) {
  1161             double curr, pu = pi[u];
  1162             int last_out = _first_out[u+1];
  1163             for (int a = _first_out[u]; a != last_out; ++a) {
  1164               if (_res_cap[a] == 0) continue;
  1165               curr = _cost[a] + pu - pi[_target[a]];
  1166               if (-curr > epsilon) epsilon = -curr;
  1167             }
  1168           }
  1169         } else {
  1170           typedef HowardMmc<StaticDigraph, CostArcMap> HwMmc;
  1171           typedef HartmannOrlinMmc<StaticDigraph, CostArcMap> HoMmc;
  1172           typedef typename BellmanFord<StaticDigraph, CostArcMap>
  1173             ::template SetDistMap<CostNodeMap>::Create BF;
  1174 
  1175           // Set epsilon to the minimum cycle mean
  1176           Cost cycle_cost = 0;
  1177           int cycle_size = 1;
  1178           buildResidualNetwork();
  1179           HwMmc hw_mmc(_sgr, _cost_map);
  1180           if (hw_mmc.findCycleMean(hw_iter_limit) == HwMmc::ITERATION_LIMIT) {
  1181             // Howard's algorithm reached the iteration limit, start a
  1182             // strongly polynomial algorithm instead
  1183             HoMmc ho_mmc(_sgr, _cost_map);
  1184             ho_mmc.findCycleMean();
  1185             epsilon = -ho_mmc.cycleMean();
  1186             cycle_cost = ho_mmc.cycleCost();
  1187             cycle_size = ho_mmc.cycleSize();
  1188           } else {
  1189             // Set epsilon
  1190             epsilon = -hw_mmc.cycleMean();
  1191             cycle_cost = hw_mmc.cycleCost();
  1192             cycle_size = hw_mmc.cycleSize();
  1193           }
  1194 
  1195           // Compute feasible potentials for the current epsilon
  1196           for (int i = 0; i != int(_cost_vec.size()); ++i) {
  1197             _cost_vec[i] = cycle_size * _cost_vec[i] - cycle_cost;
  1198           }
  1199           BF bf(_sgr, _cost_map);
  1200           bf.distMap(_pi_map);
  1201           bf.init(0);
  1202           bf.start();
  1203           for (int u = 0; u != _res_node_num; ++u) {
  1204             pi[u] = static_cast<double>(_pi[u]) / cycle_size;
  1205           }
  1206 
  1207           iter = limit;
  1208         }
  1209       }
  1210     }
  1211 
  1212   }; //class CycleCanceling
  1213 
  1214   ///@}
  1215 
  1216 } //namespace lemon
  1217 
  1218 #endif //LEMON_CYCLE_CANCELING_H