lemon/cost_scaling.h
author Alpar Juttner <alpar@cs.elte.hu>
Mon, 14 Dec 2009 06:07:52 +0100
changeset 822 f903263902f6
parent 820 7ef7a5fbb85d
child 825 75e6020b19b1
child 830 75c97c3786d6
child 839 f3bc4e9b5f3a
permissions -rw-r--r--
Merge #180 and a bugfix in #51
     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_COST_SCALING_H
    20 #define LEMON_COST_SCALING_H
    21 
    22 /// \ingroup min_cost_flow_algs
    23 /// \file
    24 /// \brief Cost scaling algorithm for finding a minimum cost flow.
    25 
    26 #include <vector>
    27 #include <deque>
    28 #include <limits>
    29 
    30 #include <lemon/core.h>
    31 #include <lemon/maps.h>
    32 #include <lemon/math.h>
    33 #include <lemon/static_graph.h>
    34 #include <lemon/circulation.h>
    35 #include <lemon/bellman_ford.h>
    36 
    37 namespace lemon {
    38 
    39   /// \brief Default traits class of CostScaling algorithm.
    40   ///
    41   /// Default traits class of CostScaling algorithm.
    42   /// \tparam GR Digraph type.
    43   /// \tparam V The number type used for flow amounts, capacity bounds
    44   /// and supply values. By default it is \c int.
    45   /// \tparam C The number type used for costs and potentials.
    46   /// By default it is the same as \c V.
    47 #ifdef DOXYGEN
    48   template <typename GR, typename V = int, typename C = V>
    49 #else
    50   template < typename GR, typename V = int, typename C = V,
    51              bool integer = std::numeric_limits<C>::is_integer >
    52 #endif
    53   struct CostScalingDefaultTraits
    54   {
    55     /// The type of the digraph
    56     typedef GR Digraph;
    57     /// The type of the flow amounts, capacity bounds and supply values
    58     typedef V Value;
    59     /// The type of the arc costs
    60     typedef C Cost;
    61 
    62     /// \brief The large cost type used for internal computations
    63     ///
    64     /// The large cost type used for internal computations.
    65     /// It is \c long \c long if the \c Cost type is integer,
    66     /// otherwise it is \c double.
    67     /// \c Cost must be convertible to \c LargeCost.
    68     typedef double LargeCost;
    69   };
    70 
    71   // Default traits class for integer cost types
    72   template <typename GR, typename V, typename C>
    73   struct CostScalingDefaultTraits<GR, V, C, true>
    74   {
    75     typedef GR Digraph;
    76     typedef V Value;
    77     typedef C Cost;
    78 #ifdef LEMON_HAVE_LONG_LONG
    79     typedef long long LargeCost;
    80 #else
    81     typedef long LargeCost;
    82 #endif
    83   };
    84 
    85 
    86   /// \addtogroup min_cost_flow_algs
    87   /// @{
    88 
    89   /// \brief Implementation of the Cost Scaling algorithm for
    90   /// finding a \ref min_cost_flow "minimum cost flow".
    91   ///
    92   /// \ref CostScaling implements a cost scaling algorithm that performs
    93   /// push/augment and relabel operations for finding a \ref min_cost_flow
    94   /// "minimum cost flow" \ref amo93networkflows, \ref goldberg90approximation,
    95   /// \ref goldberg97efficient, \ref bunnagel98efficient. 
    96   /// It is a highly efficient primal-dual solution method, which
    97   /// can be viewed as the generalization of the \ref Preflow
    98   /// "preflow push-relabel" algorithm for the maximum flow problem.
    99   ///
   100   /// Most of the parameters of the problem (except for the digraph)
   101   /// can be given using separate functions, and the algorithm can be
   102   /// executed using the \ref run() function. If some parameters are not
   103   /// specified, then default values will be used.
   104   ///
   105   /// \tparam GR The digraph type the algorithm runs on.
   106   /// \tparam V The number type used for flow amounts, capacity bounds
   107   /// and supply values in the algorithm. By default it is \c int.
   108   /// \tparam C The number type used for costs and potentials in the
   109   /// algorithm. By default it is the same as \c V.
   110   ///
   111   /// \warning Both number types must be signed and all input data must
   112   /// be integer.
   113   /// \warning This algorithm does not support negative costs for such
   114   /// arcs that have infinite upper bound.
   115   ///
   116   /// \note %CostScaling provides three different internal methods,
   117   /// from which the most efficient one is used by default.
   118   /// For more information, see \ref Method.
   119 #ifdef DOXYGEN
   120   template <typename GR, typename V, typename C, typename TR>
   121 #else
   122   template < typename GR, typename V = int, typename C = V,
   123              typename TR = CostScalingDefaultTraits<GR, V, C> >
   124 #endif
   125   class CostScaling
   126   {
   127   public:
   128 
   129     /// The type of the digraph
   130     typedef typename TR::Digraph Digraph;
   131     /// The type of the flow amounts, capacity bounds and supply values
   132     typedef typename TR::Value Value;
   133     /// The type of the arc costs
   134     typedef typename TR::Cost Cost;
   135 
   136     /// \brief The large cost type
   137     ///
   138     /// The large cost type used for internal computations.
   139     /// Using the \ref CostScalingDefaultTraits "default traits class",
   140     /// it is \c long \c long if the \c Cost type is integer,
   141     /// otherwise it is \c double.
   142     typedef typename TR::LargeCost LargeCost;
   143 
   144     /// The \ref CostScalingDefaultTraits "traits class" of the algorithm
   145     typedef TR Traits;
   146 
   147   public:
   148 
   149     /// \brief Problem type constants for the \c run() function.
   150     ///
   151     /// Enum type containing the problem type constants that can be
   152     /// returned by the \ref run() function of the algorithm.
   153     enum ProblemType {
   154       /// The problem has no feasible solution (flow).
   155       INFEASIBLE,
   156       /// The problem has optimal solution (i.e. it is feasible and
   157       /// bounded), and the algorithm has found optimal flow and node
   158       /// potentials (primal and dual solutions).
   159       OPTIMAL,
   160       /// The digraph contains an arc of negative cost and infinite
   161       /// upper bound. It means that the objective function is unbounded
   162       /// on that arc, however, note that it could actually be bounded
   163       /// over the feasible flows, but this algroithm cannot handle
   164       /// these cases.
   165       UNBOUNDED
   166     };
   167 
   168     /// \brief Constants for selecting the internal method.
   169     ///
   170     /// Enum type containing constants for selecting the internal method
   171     /// for the \ref run() function.
   172     ///
   173     /// \ref CostScaling provides three internal methods that differ mainly
   174     /// in their base operations, which are used in conjunction with the
   175     /// relabel operation.
   176     /// By default, the so called \ref PARTIAL_AUGMENT
   177     /// "Partial Augment-Relabel" method is used, which proved to be
   178     /// the most efficient and the most robust on various test inputs.
   179     /// However, the other methods can be selected using the \ref run()
   180     /// function with the proper parameter.
   181     enum Method {
   182       /// Local push operations are used, i.e. flow is moved only on one
   183       /// admissible arc at once.
   184       PUSH,
   185       /// Augment operations are used, i.e. flow is moved on admissible
   186       /// paths from a node with excess to a node with deficit.
   187       AUGMENT,
   188       /// Partial augment operations are used, i.e. flow is moved on 
   189       /// admissible paths started from a node with excess, but the
   190       /// lengths of these paths are limited. This method can be viewed
   191       /// as a combined version of the previous two operations.
   192       PARTIAL_AUGMENT
   193     };
   194 
   195   private:
   196 
   197     TEMPLATE_DIGRAPH_TYPEDEFS(GR);
   198 
   199     typedef std::vector<int> IntVector;
   200     typedef std::vector<char> BoolVector;
   201     typedef std::vector<Value> ValueVector;
   202     typedef std::vector<Cost> CostVector;
   203     typedef std::vector<LargeCost> LargeCostVector;
   204 
   205   private:
   206   
   207     template <typename KT, typename VT>
   208     class StaticVectorMap {
   209     public:
   210       typedef KT Key;
   211       typedef VT Value;
   212       
   213       StaticVectorMap(std::vector<Value>& v) : _v(v) {}
   214       
   215       const Value& operator[](const Key& key) const {
   216         return _v[StaticDigraph::id(key)];
   217       }
   218 
   219       Value& operator[](const Key& key) {
   220         return _v[StaticDigraph::id(key)];
   221       }
   222       
   223       void set(const Key& key, const Value& val) {
   224         _v[StaticDigraph::id(key)] = val;
   225       }
   226 
   227     private:
   228       std::vector<Value>& _v;
   229     };
   230 
   231     typedef StaticVectorMap<StaticDigraph::Node, LargeCost> LargeCostNodeMap;
   232     typedef StaticVectorMap<StaticDigraph::Arc, LargeCost> LargeCostArcMap;
   233 
   234   private:
   235 
   236     // Data related to the underlying digraph
   237     const GR &_graph;
   238     int _node_num;
   239     int _arc_num;
   240     int _res_node_num;
   241     int _res_arc_num;
   242     int _root;
   243 
   244     // Parameters of the problem
   245     bool _have_lower;
   246     Value _sum_supply;
   247 
   248     // Data structures for storing the digraph
   249     IntNodeMap _node_id;
   250     IntArcMap _arc_idf;
   251     IntArcMap _arc_idb;
   252     IntVector _first_out;
   253     BoolVector _forward;
   254     IntVector _source;
   255     IntVector _target;
   256     IntVector _reverse;
   257 
   258     // Node and arc data
   259     ValueVector _lower;
   260     ValueVector _upper;
   261     CostVector _scost;
   262     ValueVector _supply;
   263 
   264     ValueVector _res_cap;
   265     LargeCostVector _cost;
   266     LargeCostVector _pi;
   267     ValueVector _excess;
   268     IntVector _next_out;
   269     std::deque<int> _active_nodes;
   270 
   271     // Data for scaling
   272     LargeCost _epsilon;
   273     int _alpha;
   274 
   275     // Data for a StaticDigraph structure
   276     typedef std::pair<int, int> IntPair;
   277     StaticDigraph _sgr;
   278     std::vector<IntPair> _arc_vec;
   279     std::vector<LargeCost> _cost_vec;
   280     LargeCostArcMap _cost_map;
   281     LargeCostNodeMap _pi_map;
   282   
   283   public:
   284   
   285     /// \brief Constant for infinite upper bounds (capacities).
   286     ///
   287     /// Constant for infinite upper bounds (capacities).
   288     /// It is \c std::numeric_limits<Value>::infinity() if available,
   289     /// \c std::numeric_limits<Value>::max() otherwise.
   290     const Value INF;
   291 
   292   public:
   293 
   294     /// \name Named Template Parameters
   295     /// @{
   296 
   297     template <typename T>
   298     struct SetLargeCostTraits : public Traits {
   299       typedef T LargeCost;
   300     };
   301 
   302     /// \brief \ref named-templ-param "Named parameter" for setting
   303     /// \c LargeCost type.
   304     ///
   305     /// \ref named-templ-param "Named parameter" for setting \c LargeCost
   306     /// type, which is used for internal computations in the algorithm.
   307     /// \c Cost must be convertible to \c LargeCost.
   308     template <typename T>
   309     struct SetLargeCost
   310       : public CostScaling<GR, V, C, SetLargeCostTraits<T> > {
   311       typedef  CostScaling<GR, V, C, SetLargeCostTraits<T> > Create;
   312     };
   313 
   314     /// @}
   315 
   316   public:
   317 
   318     /// \brief Constructor.
   319     ///
   320     /// The constructor of the class.
   321     ///
   322     /// \param graph The digraph the algorithm runs on.
   323     CostScaling(const GR& graph) :
   324       _graph(graph), _node_id(graph), _arc_idf(graph), _arc_idb(graph),
   325       _cost_map(_cost_vec), _pi_map(_pi),
   326       INF(std::numeric_limits<Value>::has_infinity ?
   327           std::numeric_limits<Value>::infinity() :
   328           std::numeric_limits<Value>::max())
   329     {
   330       // Check the number types
   331       LEMON_ASSERT(std::numeric_limits<Value>::is_signed,
   332         "The flow type of CostScaling must be signed");
   333       LEMON_ASSERT(std::numeric_limits<Cost>::is_signed,
   334         "The cost type of CostScaling must be signed");
   335 
   336       // Resize vectors
   337       _node_num = countNodes(_graph);
   338       _arc_num = countArcs(_graph);
   339       _res_node_num = _node_num + 1;
   340       _res_arc_num = 2 * (_arc_num + _node_num);
   341       _root = _node_num;
   342 
   343       _first_out.resize(_res_node_num + 1);
   344       _forward.resize(_res_arc_num);
   345       _source.resize(_res_arc_num);
   346       _target.resize(_res_arc_num);
   347       _reverse.resize(_res_arc_num);
   348 
   349       _lower.resize(_res_arc_num);
   350       _upper.resize(_res_arc_num);
   351       _scost.resize(_res_arc_num);
   352       _supply.resize(_res_node_num);
   353       
   354       _res_cap.resize(_res_arc_num);
   355       _cost.resize(_res_arc_num);
   356       _pi.resize(_res_node_num);
   357       _excess.resize(_res_node_num);
   358       _next_out.resize(_res_node_num);
   359 
   360       _arc_vec.reserve(_res_arc_num);
   361       _cost_vec.reserve(_res_arc_num);
   362 
   363       // Copy the graph
   364       int i = 0, j = 0, k = 2 * _arc_num + _node_num;
   365       for (NodeIt n(_graph); n != INVALID; ++n, ++i) {
   366         _node_id[n] = i;
   367       }
   368       i = 0;
   369       for (NodeIt n(_graph); n != INVALID; ++n, ++i) {
   370         _first_out[i] = j;
   371         for (OutArcIt a(_graph, n); a != INVALID; ++a, ++j) {
   372           _arc_idf[a] = j;
   373           _forward[j] = true;
   374           _source[j] = i;
   375           _target[j] = _node_id[_graph.runningNode(a)];
   376         }
   377         for (InArcIt a(_graph, n); a != INVALID; ++a, ++j) {
   378           _arc_idb[a] = j;
   379           _forward[j] = false;
   380           _source[j] = i;
   381           _target[j] = _node_id[_graph.runningNode(a)];
   382         }
   383         _forward[j] = false;
   384         _source[j] = i;
   385         _target[j] = _root;
   386         _reverse[j] = k;
   387         _forward[k] = true;
   388         _source[k] = _root;
   389         _target[k] = i;
   390         _reverse[k] = j;
   391         ++j; ++k;
   392       }
   393       _first_out[i] = j;
   394       _first_out[_res_node_num] = k;
   395       for (ArcIt a(_graph); a != INVALID; ++a) {
   396         int fi = _arc_idf[a];
   397         int bi = _arc_idb[a];
   398         _reverse[fi] = bi;
   399         _reverse[bi] = fi;
   400       }
   401       
   402       // Reset parameters
   403       reset();
   404     }
   405 
   406     /// \name Parameters
   407     /// The parameters of the algorithm can be specified using these
   408     /// functions.
   409 
   410     /// @{
   411 
   412     /// \brief Set the lower bounds on the arcs.
   413     ///
   414     /// This function sets the lower bounds on the arcs.
   415     /// If it is not used before calling \ref run(), the lower bounds
   416     /// will be set to zero on all arcs.
   417     ///
   418     /// \param map An arc map storing the lower bounds.
   419     /// Its \c Value type must be convertible to the \c Value type
   420     /// of the algorithm.
   421     ///
   422     /// \return <tt>(*this)</tt>
   423     template <typename LowerMap>
   424     CostScaling& lowerMap(const LowerMap& map) {
   425       _have_lower = true;
   426       for (ArcIt a(_graph); a != INVALID; ++a) {
   427         _lower[_arc_idf[a]] = map[a];
   428         _lower[_arc_idb[a]] = map[a];
   429       }
   430       return *this;
   431     }
   432 
   433     /// \brief Set the upper bounds (capacities) on the arcs.
   434     ///
   435     /// This function sets the upper bounds (capacities) on the arcs.
   436     /// If it is not used before calling \ref run(), the upper bounds
   437     /// will be set to \ref INF on all arcs (i.e. the flow value will be
   438     /// unbounded from above).
   439     ///
   440     /// \param map An arc map storing the upper bounds.
   441     /// Its \c Value type must be convertible to the \c Value type
   442     /// of the algorithm.
   443     ///
   444     /// \return <tt>(*this)</tt>
   445     template<typename UpperMap>
   446     CostScaling& upperMap(const UpperMap& map) {
   447       for (ArcIt a(_graph); a != INVALID; ++a) {
   448         _upper[_arc_idf[a]] = map[a];
   449       }
   450       return *this;
   451     }
   452 
   453     /// \brief Set the costs of the arcs.
   454     ///
   455     /// This function sets the costs of the arcs.
   456     /// If it is not used before calling \ref run(), the costs
   457     /// will be set to \c 1 on all arcs.
   458     ///
   459     /// \param map An arc map storing the costs.
   460     /// Its \c Value type must be convertible to the \c Cost type
   461     /// of the algorithm.
   462     ///
   463     /// \return <tt>(*this)</tt>
   464     template<typename CostMap>
   465     CostScaling& costMap(const CostMap& map) {
   466       for (ArcIt a(_graph); a != INVALID; ++a) {
   467         _scost[_arc_idf[a]] =  map[a];
   468         _scost[_arc_idb[a]] = -map[a];
   469       }
   470       return *this;
   471     }
   472 
   473     /// \brief Set the supply values of the nodes.
   474     ///
   475     /// This function sets the supply values of the nodes.
   476     /// If neither this function nor \ref stSupply() is used before
   477     /// calling \ref run(), the supply of each node will be set to zero.
   478     ///
   479     /// \param map A node map storing the supply values.
   480     /// Its \c Value type must be convertible to the \c Value type
   481     /// of the algorithm.
   482     ///
   483     /// \return <tt>(*this)</tt>
   484     template<typename SupplyMap>
   485     CostScaling& supplyMap(const SupplyMap& map) {
   486       for (NodeIt n(_graph); n != INVALID; ++n) {
   487         _supply[_node_id[n]] = map[n];
   488       }
   489       return *this;
   490     }
   491 
   492     /// \brief Set single source and target nodes and a supply value.
   493     ///
   494     /// This function sets a single source node and a single target node
   495     /// and the required flow value.
   496     /// If neither this function nor \ref supplyMap() is used before
   497     /// calling \ref run(), the supply of each node will be set to zero.
   498     ///
   499     /// Using this function has the same effect as using \ref supplyMap()
   500     /// with such a map in which \c k is assigned to \c s, \c -k is
   501     /// assigned to \c t and all other nodes have zero supply value.
   502     ///
   503     /// \param s The source node.
   504     /// \param t The target node.
   505     /// \param k The required amount of flow from node \c s to node \c t
   506     /// (i.e. the supply of \c s and the demand of \c t).
   507     ///
   508     /// \return <tt>(*this)</tt>
   509     CostScaling& stSupply(const Node& s, const Node& t, Value k) {
   510       for (int i = 0; i != _res_node_num; ++i) {
   511         _supply[i] = 0;
   512       }
   513       _supply[_node_id[s]] =  k;
   514       _supply[_node_id[t]] = -k;
   515       return *this;
   516     }
   517     
   518     /// @}
   519 
   520     /// \name Execution control
   521     /// The algorithm can be executed using \ref run().
   522 
   523     /// @{
   524 
   525     /// \brief Run the algorithm.
   526     ///
   527     /// This function runs the algorithm.
   528     /// The paramters can be specified using functions \ref lowerMap(),
   529     /// \ref upperMap(), \ref costMap(), \ref supplyMap(), \ref stSupply().
   530     /// For example,
   531     /// \code
   532     ///   CostScaling<ListDigraph> cs(graph);
   533     ///   cs.lowerMap(lower).upperMap(upper).costMap(cost)
   534     ///     .supplyMap(sup).run();
   535     /// \endcode
   536     ///
   537     /// This function can be called more than once. All the parameters
   538     /// that have been given are kept for the next call, unless
   539     /// \ref reset() is called, thus only the modified parameters
   540     /// have to be set again. See \ref reset() for examples.
   541     /// However, the underlying digraph must not be modified after this
   542     /// class have been constructed, since it copies and extends the graph.
   543     ///
   544     /// \param method The internal method that will be used in the
   545     /// algorithm. For more information, see \ref Method.
   546     /// \param factor The cost scaling factor. It must be larger than one.
   547     ///
   548     /// \return \c INFEASIBLE if no feasible flow exists,
   549     /// \n \c OPTIMAL if the problem has optimal solution
   550     /// (i.e. it is feasible and bounded), and the algorithm has found
   551     /// optimal flow and node potentials (primal and dual solutions),
   552     /// \n \c UNBOUNDED if the digraph contains an arc of negative cost
   553     /// and infinite upper bound. It means that the objective function
   554     /// is unbounded on that arc, however, note that it could actually be
   555     /// bounded over the feasible flows, but this algroithm cannot handle
   556     /// these cases.
   557     ///
   558     /// \see ProblemType, Method
   559     ProblemType run(Method method = PARTIAL_AUGMENT, int factor = 8) {
   560       _alpha = factor;
   561       ProblemType pt = init();
   562       if (pt != OPTIMAL) return pt;
   563       start(method);
   564       return OPTIMAL;
   565     }
   566 
   567     /// \brief Reset all the parameters that have been given before.
   568     ///
   569     /// This function resets all the paramaters that have been given
   570     /// before using functions \ref lowerMap(), \ref upperMap(),
   571     /// \ref costMap(), \ref supplyMap(), \ref stSupply().
   572     ///
   573     /// It is useful for multiple run() calls. If this function is not
   574     /// used, all the parameters given before are kept for the next
   575     /// \ref run() call.
   576     /// However, the underlying digraph must not be modified after this
   577     /// class have been constructed, since it copies and extends the graph.
   578     ///
   579     /// For example,
   580     /// \code
   581     ///   CostScaling<ListDigraph> cs(graph);
   582     ///
   583     ///   // First run
   584     ///   cs.lowerMap(lower).upperMap(upper).costMap(cost)
   585     ///     .supplyMap(sup).run();
   586     ///
   587     ///   // Run again with modified cost map (reset() is not called,
   588     ///   // so only the cost map have to be set again)
   589     ///   cost[e] += 100;
   590     ///   cs.costMap(cost).run();
   591     ///
   592     ///   // Run again from scratch using reset()
   593     ///   // (the lower bounds will be set to zero on all arcs)
   594     ///   cs.reset();
   595     ///   cs.upperMap(capacity).costMap(cost)
   596     ///     .supplyMap(sup).run();
   597     /// \endcode
   598     ///
   599     /// \return <tt>(*this)</tt>
   600     CostScaling& reset() {
   601       for (int i = 0; i != _res_node_num; ++i) {
   602         _supply[i] = 0;
   603       }
   604       int limit = _first_out[_root];
   605       for (int j = 0; j != limit; ++j) {
   606         _lower[j] = 0;
   607         _upper[j] = INF;
   608         _scost[j] = _forward[j] ? 1 : -1;
   609       }
   610       for (int j = limit; j != _res_arc_num; ++j) {
   611         _lower[j] = 0;
   612         _upper[j] = INF;
   613         _scost[j] = 0;
   614         _scost[_reverse[j]] = 0;
   615       }      
   616       _have_lower = false;
   617       return *this;
   618     }
   619 
   620     /// @}
   621 
   622     /// \name Query Functions
   623     /// The results of the algorithm can be obtained using these
   624     /// functions.\n
   625     /// The \ref run() function must be called before using them.
   626 
   627     /// @{
   628 
   629     /// \brief Return the total cost of the found flow.
   630     ///
   631     /// This function returns the total cost of the found flow.
   632     /// Its complexity is O(e).
   633     ///
   634     /// \note The return type of the function can be specified as a
   635     /// template parameter. For example,
   636     /// \code
   637     ///   cs.totalCost<double>();
   638     /// \endcode
   639     /// It is useful if the total cost cannot be stored in the \c Cost
   640     /// type of the algorithm, which is the default return type of the
   641     /// function.
   642     ///
   643     /// \pre \ref run() must be called before using this function.
   644     template <typename Number>
   645     Number totalCost() const {
   646       Number c = 0;
   647       for (ArcIt a(_graph); a != INVALID; ++a) {
   648         int i = _arc_idb[a];
   649         c += static_cast<Number>(_res_cap[i]) *
   650              (-static_cast<Number>(_scost[i]));
   651       }
   652       return c;
   653     }
   654 
   655 #ifndef DOXYGEN
   656     Cost totalCost() const {
   657       return totalCost<Cost>();
   658     }
   659 #endif
   660 
   661     /// \brief Return the flow on the given arc.
   662     ///
   663     /// This function returns the flow on the given arc.
   664     ///
   665     /// \pre \ref run() must be called before using this function.
   666     Value flow(const Arc& a) const {
   667       return _res_cap[_arc_idb[a]];
   668     }
   669 
   670     /// \brief Return the flow map (the primal solution).
   671     ///
   672     /// This function copies the flow value on each arc into the given
   673     /// map. The \c Value type of the algorithm must be convertible to
   674     /// the \c Value type of the map.
   675     ///
   676     /// \pre \ref run() must be called before using this function.
   677     template <typename FlowMap>
   678     void flowMap(FlowMap &map) const {
   679       for (ArcIt a(_graph); a != INVALID; ++a) {
   680         map.set(a, _res_cap[_arc_idb[a]]);
   681       }
   682     }
   683 
   684     /// \brief Return the potential (dual value) of the given node.
   685     ///
   686     /// This function returns the potential (dual value) of the
   687     /// given node.
   688     ///
   689     /// \pre \ref run() must be called before using this function.
   690     Cost potential(const Node& n) const {
   691       return static_cast<Cost>(_pi[_node_id[n]]);
   692     }
   693 
   694     /// \brief Return the potential map (the dual solution).
   695     ///
   696     /// This function copies the potential (dual value) of each node
   697     /// into the given map.
   698     /// The \c Cost type of the algorithm must be convertible to the
   699     /// \c Value type of the map.
   700     ///
   701     /// \pre \ref run() must be called before using this function.
   702     template <typename PotentialMap>
   703     void potentialMap(PotentialMap &map) const {
   704       for (NodeIt n(_graph); n != INVALID; ++n) {
   705         map.set(n, static_cast<Cost>(_pi[_node_id[n]]));
   706       }
   707     }
   708 
   709     /// @}
   710 
   711   private:
   712 
   713     // Initialize the algorithm
   714     ProblemType init() {
   715       if (_res_node_num <= 1) return INFEASIBLE;
   716 
   717       // Check the sum of supply values
   718       _sum_supply = 0;
   719       for (int i = 0; i != _root; ++i) {
   720         _sum_supply += _supply[i];
   721       }
   722       if (_sum_supply > 0) return INFEASIBLE;
   723       
   724 
   725       // Initialize vectors
   726       for (int i = 0; i != _res_node_num; ++i) {
   727         _pi[i] = 0;
   728         _excess[i] = _supply[i];
   729       }
   730       
   731       // Remove infinite upper bounds and check negative arcs
   732       const Value MAX = std::numeric_limits<Value>::max();
   733       int last_out;
   734       if (_have_lower) {
   735         for (int i = 0; i != _root; ++i) {
   736           last_out = _first_out[i+1];
   737           for (int j = _first_out[i]; j != last_out; ++j) {
   738             if (_forward[j]) {
   739               Value c = _scost[j] < 0 ? _upper[j] : _lower[j];
   740               if (c >= MAX) return UNBOUNDED;
   741               _excess[i] -= c;
   742               _excess[_target[j]] += c;
   743             }
   744           }
   745         }
   746       } else {
   747         for (int i = 0; i != _root; ++i) {
   748           last_out = _first_out[i+1];
   749           for (int j = _first_out[i]; j != last_out; ++j) {
   750             if (_forward[j] && _scost[j] < 0) {
   751               Value c = _upper[j];
   752               if (c >= MAX) return UNBOUNDED;
   753               _excess[i] -= c;
   754               _excess[_target[j]] += c;
   755             }
   756           }
   757         }
   758       }
   759       Value ex, max_cap = 0;
   760       for (int i = 0; i != _res_node_num; ++i) {
   761         ex = _excess[i];
   762         _excess[i] = 0;
   763         if (ex < 0) max_cap -= ex;
   764       }
   765       for (int j = 0; j != _res_arc_num; ++j) {
   766         if (_upper[j] >= MAX) _upper[j] = max_cap;
   767       }
   768 
   769       // Initialize the large cost vector and the epsilon parameter
   770       _epsilon = 0;
   771       LargeCost lc;
   772       for (int i = 0; i != _root; ++i) {
   773         last_out = _first_out[i+1];
   774         for (int j = _first_out[i]; j != last_out; ++j) {
   775           lc = static_cast<LargeCost>(_scost[j]) * _res_node_num * _alpha;
   776           _cost[j] = lc;
   777           if (lc > _epsilon) _epsilon = lc;
   778         }
   779       }
   780       _epsilon /= _alpha;
   781 
   782       // Initialize maps for Circulation and remove non-zero lower bounds
   783       ConstMap<Arc, Value> low(0);
   784       typedef typename Digraph::template ArcMap<Value> ValueArcMap;
   785       typedef typename Digraph::template NodeMap<Value> ValueNodeMap;
   786       ValueArcMap cap(_graph), flow(_graph);
   787       ValueNodeMap sup(_graph);
   788       for (NodeIt n(_graph); n != INVALID; ++n) {
   789         sup[n] = _supply[_node_id[n]];
   790       }
   791       if (_have_lower) {
   792         for (ArcIt a(_graph); a != INVALID; ++a) {
   793           int j = _arc_idf[a];
   794           Value c = _lower[j];
   795           cap[a] = _upper[j] - c;
   796           sup[_graph.source(a)] -= c;
   797           sup[_graph.target(a)] += c;
   798         }
   799       } else {
   800         for (ArcIt a(_graph); a != INVALID; ++a) {
   801           cap[a] = _upper[_arc_idf[a]];
   802         }
   803       }
   804 
   805       // Find a feasible flow using Circulation
   806       Circulation<Digraph, ConstMap<Arc, Value>, ValueArcMap, ValueNodeMap>
   807         circ(_graph, low, cap, sup);
   808       if (!circ.flowMap(flow).run()) return INFEASIBLE;
   809 
   810       // Set residual capacities and handle GEQ supply type
   811       if (_sum_supply < 0) {
   812         for (ArcIt a(_graph); a != INVALID; ++a) {
   813           Value fa = flow[a];
   814           _res_cap[_arc_idf[a]] = cap[a] - fa;
   815           _res_cap[_arc_idb[a]] = fa;
   816           sup[_graph.source(a)] -= fa;
   817           sup[_graph.target(a)] += fa;
   818         }
   819         for (NodeIt n(_graph); n != INVALID; ++n) {
   820           _excess[_node_id[n]] = sup[n];
   821         }
   822         for (int a = _first_out[_root]; a != _res_arc_num; ++a) {
   823           int u = _target[a];
   824           int ra = _reverse[a];
   825           _res_cap[a] = -_sum_supply + 1;
   826           _res_cap[ra] = -_excess[u];
   827           _cost[a] = 0;
   828           _cost[ra] = 0;
   829           _excess[u] = 0;
   830         }
   831       } else {
   832         for (ArcIt a(_graph); a != INVALID; ++a) {
   833           Value fa = flow[a];
   834           _res_cap[_arc_idf[a]] = cap[a] - fa;
   835           _res_cap[_arc_idb[a]] = fa;
   836         }
   837         for (int a = _first_out[_root]; a != _res_arc_num; ++a) {
   838           int ra = _reverse[a];
   839           _res_cap[a] = 1;
   840           _res_cap[ra] = 0;
   841           _cost[a] = 0;
   842           _cost[ra] = 0;
   843         }
   844       }
   845       
   846       return OPTIMAL;
   847     }
   848 
   849     // Execute the algorithm and transform the results
   850     void start(Method method) {
   851       // Maximum path length for partial augment
   852       const int MAX_PATH_LENGTH = 4;
   853       
   854       // Execute the algorithm
   855       switch (method) {
   856         case PUSH:
   857           startPush();
   858           break;
   859         case AUGMENT:
   860           startAugment();
   861           break;
   862         case PARTIAL_AUGMENT:
   863           startAugment(MAX_PATH_LENGTH);
   864           break;
   865       }
   866 
   867       // Compute node potentials for the original costs
   868       _arc_vec.clear();
   869       _cost_vec.clear();
   870       for (int j = 0; j != _res_arc_num; ++j) {
   871         if (_res_cap[j] > 0) {
   872           _arc_vec.push_back(IntPair(_source[j], _target[j]));
   873           _cost_vec.push_back(_scost[j]);
   874         }
   875       }
   876       _sgr.build(_res_node_num, _arc_vec.begin(), _arc_vec.end());
   877 
   878       typename BellmanFord<StaticDigraph, LargeCostArcMap>
   879         ::template SetDistMap<LargeCostNodeMap>::Create bf(_sgr, _cost_map);
   880       bf.distMap(_pi_map);
   881       bf.init(0);
   882       bf.start();
   883 
   884       // Handle non-zero lower bounds
   885       if (_have_lower) {
   886         int limit = _first_out[_root];
   887         for (int j = 0; j != limit; ++j) {
   888           if (!_forward[j]) _res_cap[j] += _lower[j];
   889         }
   890       }
   891     }
   892 
   893     /// Execute the algorithm performing augment and relabel operations
   894     void startAugment(int max_length = std::numeric_limits<int>::max()) {
   895       // Paramters for heuristics
   896       const int BF_HEURISTIC_EPSILON_BOUND = 1000;
   897       const int BF_HEURISTIC_BOUND_FACTOR  = 3;
   898 
   899       // Perform cost scaling phases
   900       IntVector pred_arc(_res_node_num);
   901       std::vector<int> path_nodes;
   902       for ( ; _epsilon >= 1; _epsilon = _epsilon < _alpha && _epsilon > 1 ?
   903                                         1 : _epsilon / _alpha )
   904       {
   905         // "Early Termination" heuristic: use Bellman-Ford algorithm
   906         // to check if the current flow is optimal
   907         if (_epsilon <= BF_HEURISTIC_EPSILON_BOUND) {
   908           _arc_vec.clear();
   909           _cost_vec.clear();
   910           for (int j = 0; j != _res_arc_num; ++j) {
   911             if (_res_cap[j] > 0) {
   912               _arc_vec.push_back(IntPair(_source[j], _target[j]));
   913               _cost_vec.push_back(_cost[j] + 1);
   914             }
   915           }
   916           _sgr.build(_res_node_num, _arc_vec.begin(), _arc_vec.end());
   917 
   918           BellmanFord<StaticDigraph, LargeCostArcMap> bf(_sgr, _cost_map);
   919           bf.init(0);
   920           bool done = false;
   921           int K = int(BF_HEURISTIC_BOUND_FACTOR * sqrt(_res_node_num));
   922           for (int i = 0; i < K && !done; ++i)
   923             done = bf.processNextWeakRound();
   924           if (done) break;
   925         }
   926 
   927         // Saturate arcs not satisfying the optimality condition
   928         for (int a = 0; a != _res_arc_num; ++a) {
   929           if (_res_cap[a] > 0 &&
   930               _cost[a] + _pi[_source[a]] - _pi[_target[a]] < 0) {
   931             Value delta = _res_cap[a];
   932             _excess[_source[a]] -= delta;
   933             _excess[_target[a]] += delta;
   934             _res_cap[a] = 0;
   935             _res_cap[_reverse[a]] += delta;
   936           }
   937         }
   938         
   939         // Find active nodes (i.e. nodes with positive excess)
   940         for (int u = 0; u != _res_node_num; ++u) {
   941           if (_excess[u] > 0) _active_nodes.push_back(u);
   942         }
   943 
   944         // Initialize the next arcs
   945         for (int u = 0; u != _res_node_num; ++u) {
   946           _next_out[u] = _first_out[u];
   947         }
   948 
   949         // Perform partial augment and relabel operations
   950         while (true) {
   951           // Select an active node (FIFO selection)
   952           while (_active_nodes.size() > 0 &&
   953                  _excess[_active_nodes.front()] <= 0) {
   954             _active_nodes.pop_front();
   955           }
   956           if (_active_nodes.size() == 0) break;
   957           int start = _active_nodes.front();
   958           path_nodes.clear();
   959           path_nodes.push_back(start);
   960 
   961           // Find an augmenting path from the start node
   962           int tip = start;
   963           while (_excess[tip] >= 0 &&
   964                  int(path_nodes.size()) <= max_length) {
   965             int u;
   966             LargeCost min_red_cost, rc;
   967             int last_out = _sum_supply < 0 ?
   968               _first_out[tip+1] : _first_out[tip+1] - 1;
   969             for (int a = _next_out[tip]; a != last_out; ++a) {
   970               if (_res_cap[a] > 0 &&
   971                   _cost[a] + _pi[_source[a]] - _pi[_target[a]] < 0) {
   972                 u = _target[a];
   973                 pred_arc[u] = a;
   974                 _next_out[tip] = a;
   975                 tip = u;
   976                 path_nodes.push_back(tip);
   977                 goto next_step;
   978               }
   979             }
   980 
   981             // Relabel tip node
   982             min_red_cost = std::numeric_limits<LargeCost>::max() / 2;
   983             for (int a = _first_out[tip]; a != last_out; ++a) {
   984               rc = _cost[a] + _pi[_source[a]] - _pi[_target[a]];
   985               if (_res_cap[a] > 0 && rc < min_red_cost) {
   986                 min_red_cost = rc;
   987               }
   988             }
   989             _pi[tip] -= min_red_cost + _epsilon;
   990 
   991             // Reset the next arc of tip
   992             _next_out[tip] = _first_out[tip];
   993 
   994             // Step back
   995             if (tip != start) {
   996               path_nodes.pop_back();
   997               tip = path_nodes.back();
   998             }
   999 
  1000           next_step: ;
  1001           }
  1002 
  1003           // Augment along the found path (as much flow as possible)
  1004           Value delta;
  1005           int u, v = path_nodes.front(), pa;
  1006           for (int i = 1; i < int(path_nodes.size()); ++i) {
  1007             u = v;
  1008             v = path_nodes[i];
  1009             pa = pred_arc[v];
  1010             delta = std::min(_res_cap[pa], _excess[u]);
  1011             _res_cap[pa] -= delta;
  1012             _res_cap[_reverse[pa]] += delta;
  1013             _excess[u] -= delta;
  1014             _excess[v] += delta;
  1015             if (_excess[v] > 0 && _excess[v] <= delta)
  1016               _active_nodes.push_back(v);
  1017           }
  1018         }
  1019       }
  1020     }
  1021 
  1022     /// Execute the algorithm performing push and relabel operations
  1023     void startPush() {
  1024       // Paramters for heuristics
  1025       const int BF_HEURISTIC_EPSILON_BOUND = 1000;
  1026       const int BF_HEURISTIC_BOUND_FACTOR  = 3;
  1027 
  1028       // Perform cost scaling phases
  1029       BoolVector hyper(_res_node_num, false);
  1030       for ( ; _epsilon >= 1; _epsilon = _epsilon < _alpha && _epsilon > 1 ?
  1031                                         1 : _epsilon / _alpha )
  1032       {
  1033         // "Early Termination" heuristic: use Bellman-Ford algorithm
  1034         // to check if the current flow is optimal
  1035         if (_epsilon <= BF_HEURISTIC_EPSILON_BOUND) {
  1036           _arc_vec.clear();
  1037           _cost_vec.clear();
  1038           for (int j = 0; j != _res_arc_num; ++j) {
  1039             if (_res_cap[j] > 0) {
  1040               _arc_vec.push_back(IntPair(_source[j], _target[j]));
  1041               _cost_vec.push_back(_cost[j] + 1);
  1042             }
  1043           }
  1044           _sgr.build(_res_node_num, _arc_vec.begin(), _arc_vec.end());
  1045 
  1046           BellmanFord<StaticDigraph, LargeCostArcMap> bf(_sgr, _cost_map);
  1047           bf.init(0);
  1048           bool done = false;
  1049           int K = int(BF_HEURISTIC_BOUND_FACTOR * sqrt(_res_node_num));
  1050           for (int i = 0; i < K && !done; ++i)
  1051             done = bf.processNextWeakRound();
  1052           if (done) break;
  1053         }
  1054 
  1055         // Saturate arcs not satisfying the optimality condition
  1056         for (int a = 0; a != _res_arc_num; ++a) {
  1057           if (_res_cap[a] > 0 &&
  1058               _cost[a] + _pi[_source[a]] - _pi[_target[a]] < 0) {
  1059             Value delta = _res_cap[a];
  1060             _excess[_source[a]] -= delta;
  1061             _excess[_target[a]] += delta;
  1062             _res_cap[a] = 0;
  1063             _res_cap[_reverse[a]] += delta;
  1064           }
  1065         }
  1066 
  1067         // Find active nodes (i.e. nodes with positive excess)
  1068         for (int u = 0; u != _res_node_num; ++u) {
  1069           if (_excess[u] > 0) _active_nodes.push_back(u);
  1070         }
  1071 
  1072         // Initialize the next arcs
  1073         for (int u = 0; u != _res_node_num; ++u) {
  1074           _next_out[u] = _first_out[u];
  1075         }
  1076 
  1077         // Perform push and relabel operations
  1078         while (_active_nodes.size() > 0) {
  1079           LargeCost min_red_cost, rc;
  1080           Value delta;
  1081           int n, t, a, last_out = _res_arc_num;
  1082 
  1083           // Select an active node (FIFO selection)
  1084         next_node:
  1085           n = _active_nodes.front();
  1086           last_out = _sum_supply < 0 ?
  1087             _first_out[n+1] : _first_out[n+1] - 1;
  1088 
  1089           // Perform push operations if there are admissible arcs
  1090           if (_excess[n] > 0) {
  1091             for (a = _next_out[n]; a != last_out; ++a) {
  1092               if (_res_cap[a] > 0 &&
  1093                   _cost[a] + _pi[_source[a]] - _pi[_target[a]] < 0) {
  1094                 delta = std::min(_res_cap[a], _excess[n]);
  1095                 t = _target[a];
  1096 
  1097                 // Push-look-ahead heuristic
  1098                 Value ahead = -_excess[t];
  1099                 int last_out_t = _sum_supply < 0 ?
  1100                   _first_out[t+1] : _first_out[t+1] - 1;
  1101                 for (int ta = _next_out[t]; ta != last_out_t; ++ta) {
  1102                   if (_res_cap[ta] > 0 && 
  1103                       _cost[ta] + _pi[_source[ta]] - _pi[_target[ta]] < 0)
  1104                     ahead += _res_cap[ta];
  1105                   if (ahead >= delta) break;
  1106                 }
  1107                 if (ahead < 0) ahead = 0;
  1108 
  1109                 // Push flow along the arc
  1110                 if (ahead < delta) {
  1111                   _res_cap[a] -= ahead;
  1112                   _res_cap[_reverse[a]] += ahead;
  1113                   _excess[n] -= ahead;
  1114                   _excess[t] += ahead;
  1115                   _active_nodes.push_front(t);
  1116                   hyper[t] = true;
  1117                   _next_out[n] = a;
  1118                   goto next_node;
  1119                 } else {
  1120                   _res_cap[a] -= delta;
  1121                   _res_cap[_reverse[a]] += delta;
  1122                   _excess[n] -= delta;
  1123                   _excess[t] += delta;
  1124                   if (_excess[t] > 0 && _excess[t] <= delta)
  1125                     _active_nodes.push_back(t);
  1126                 }
  1127 
  1128                 if (_excess[n] == 0) {
  1129                   _next_out[n] = a;
  1130                   goto remove_nodes;
  1131                 }
  1132               }
  1133             }
  1134             _next_out[n] = a;
  1135           }
  1136 
  1137           // Relabel the node if it is still active (or hyper)
  1138           if (_excess[n] > 0 || hyper[n]) {
  1139             min_red_cost = std::numeric_limits<LargeCost>::max() / 2;
  1140             for (int a = _first_out[n]; a != last_out; ++a) {
  1141               rc = _cost[a] + _pi[_source[a]] - _pi[_target[a]];
  1142               if (_res_cap[a] > 0 && rc < min_red_cost) {
  1143                 min_red_cost = rc;
  1144               }
  1145             }
  1146             _pi[n] -= min_red_cost + _epsilon;
  1147             hyper[n] = false;
  1148 
  1149             // Reset the next arc
  1150             _next_out[n] = _first_out[n];
  1151           }
  1152         
  1153           // Remove nodes that are not active nor hyper
  1154         remove_nodes:
  1155           while ( _active_nodes.size() > 0 &&
  1156                   _excess[_active_nodes.front()] <= 0 &&
  1157                   !hyper[_active_nodes.front()] ) {
  1158             _active_nodes.pop_front();
  1159           }
  1160         }
  1161       }
  1162     }
  1163 
  1164   }; //class CostScaling
  1165 
  1166   ///@}
  1167 
  1168 } //namespace lemon
  1169 
  1170 #endif //LEMON_COST_SCALING_H