lemon/cost_scaling.h
author Alpar Juttner <alpar@cs.elte.hu>
Tue, 06 Aug 2013 12:28:37 +0200
changeset 1079 5958cc5c0a98
parent 1071 879fcb781086
child 1080 c5cd8960df74
permissions -rw-r--r--
Merge further fixes #470
     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_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" \cite amo93networkflows, \cite goldberg90approximation,
    95   /// \cite goldberg97efficient, \cite 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   /// It is a polynomial algorithm, its running time complexity is
   100   /// \f$O(n^2e\log(nK))\f$, where <i>K</i> denotes the maximum arc cost.
   101   ///
   102   /// In general, \ref NetworkSimplex and \ref CostScaling are the fastest
   103   /// implementations available in LEMON for solving this problem.
   104   /// (For more information, see \ref min_cost_flow_algs "the module page".)
   105   ///
   106   /// Most of the parameters of the problem (except for the digraph)
   107   /// can be given using separate functions, and the algorithm can be
   108   /// executed using the \ref run() function. If some parameters are not
   109   /// specified, then default values will be used.
   110   ///
   111   /// \tparam GR The digraph type the algorithm runs on.
   112   /// \tparam V The number type used for flow amounts, capacity bounds
   113   /// and supply values in the algorithm. By default, it is \c int.
   114   /// \tparam C The number type used for costs and potentials in the
   115   /// algorithm. By default, it is the same as \c V.
   116   /// \tparam TR The traits class that defines various types used by the
   117   /// algorithm. By default, it is \ref CostScalingDefaultTraits
   118   /// "CostScalingDefaultTraits<GR, V, C>".
   119   /// In most cases, this parameter should not be set directly,
   120   /// consider to use the named template parameters instead.
   121   ///
   122   /// \warning Both \c V and \c C must be signed number types.
   123   /// \warning All input data (capacities, supply values, and costs) must
   124   /// be integer.
   125   /// \warning This algorithm does not support negative costs for
   126   /// arcs having infinite upper bound.
   127   ///
   128   /// \note %CostScaling provides three different internal methods,
   129   /// from which the most efficient one is used by default.
   130   /// For more information, see \ref Method.
   131 #ifdef DOXYGEN
   132   template <typename GR, typename V, typename C, typename TR>
   133 #else
   134   template < typename GR, typename V = int, typename C = V,
   135              typename TR = CostScalingDefaultTraits<GR, V, C> >
   136 #endif
   137   class CostScaling
   138   {
   139   public:
   140 
   141     /// The type of the digraph
   142     typedef typename TR::Digraph Digraph;
   143     /// The type of the flow amounts, capacity bounds and supply values
   144     typedef typename TR::Value Value;
   145     /// The type of the arc costs
   146     typedef typename TR::Cost Cost;
   147 
   148     /// \brief The large cost type
   149     ///
   150     /// The large cost type used for internal computations.
   151     /// By default, it is \c long \c long if the \c Cost type is integer,
   152     /// otherwise it is \c double.
   153     typedef typename TR::LargeCost LargeCost;
   154 
   155     /// \brief The \ref lemon::CostScalingDefaultTraits "traits class"
   156     /// of the algorithm
   157     typedef TR Traits;
   158 
   159   public:
   160 
   161     /// \brief Problem type constants for the \c run() function.
   162     ///
   163     /// Enum type containing the problem type constants that can be
   164     /// returned by the \ref run() function of the algorithm.
   165     enum ProblemType {
   166       /// The problem has no feasible solution (flow).
   167       INFEASIBLE,
   168       /// The problem has optimal solution (i.e. it is feasible and
   169       /// bounded), and the algorithm has found optimal flow and node
   170       /// potentials (primal and dual solutions).
   171       OPTIMAL,
   172       /// The digraph contains an arc of negative cost and infinite
   173       /// upper bound. It means that the objective function is unbounded
   174       /// on that arc, however, note that it could actually be bounded
   175       /// over the feasible flows, but this algroithm cannot handle
   176       /// these cases.
   177       UNBOUNDED
   178     };
   179 
   180     /// \brief Constants for selecting the internal method.
   181     ///
   182     /// Enum type containing constants for selecting the internal method
   183     /// for the \ref run() function.
   184     ///
   185     /// \ref CostScaling provides three internal methods that differ mainly
   186     /// in their base operations, which are used in conjunction with the
   187     /// relabel operation.
   188     /// By default, the so called \ref PARTIAL_AUGMENT
   189     /// "Partial Augment-Relabel" method is used, which turned out to be
   190     /// the most efficient and the most robust on various test inputs.
   191     /// However, the other methods can be selected using the \ref run()
   192     /// function with the proper parameter.
   193     enum Method {
   194       /// Local push operations are used, i.e. flow is moved only on one
   195       /// admissible arc at once.
   196       PUSH,
   197       /// Augment operations are used, i.e. flow is moved on admissible
   198       /// paths from a node with excess to a node with deficit.
   199       AUGMENT,
   200       /// Partial augment operations are used, i.e. flow is moved on
   201       /// admissible paths started from a node with excess, but the
   202       /// lengths of these paths are limited. This method can be viewed
   203       /// as a combined version of the previous two operations.
   204       PARTIAL_AUGMENT
   205     };
   206 
   207   private:
   208 
   209     TEMPLATE_DIGRAPH_TYPEDEFS(GR);
   210 
   211     typedef std::vector<int> IntVector;
   212     typedef std::vector<Value> ValueVector;
   213     typedef std::vector<Cost> CostVector;
   214     typedef std::vector<LargeCost> LargeCostVector;
   215     typedef std::vector<char> BoolVector;
   216     // Note: vector<char> is used instead of vector<bool> for efficiency reasons
   217 
   218   private:
   219 
   220     template <typename KT, typename VT>
   221     class StaticVectorMap {
   222     public:
   223       typedef KT Key;
   224       typedef VT Value;
   225 
   226       StaticVectorMap(std::vector<Value>& v) : _v(v) {}
   227 
   228       const Value& operator[](const Key& key) const {
   229         return _v[StaticDigraph::id(key)];
   230       }
   231 
   232       Value& operator[](const Key& key) {
   233         return _v[StaticDigraph::id(key)];
   234       }
   235 
   236       void set(const Key& key, const Value& val) {
   237         _v[StaticDigraph::id(key)] = val;
   238       }
   239 
   240     private:
   241       std::vector<Value>& _v;
   242     };
   243 
   244     typedef StaticVectorMap<StaticDigraph::Arc, LargeCost> LargeCostArcMap;
   245 
   246   private:
   247 
   248     // Data related to the underlying digraph
   249     const GR &_graph;
   250     int _node_num;
   251     int _arc_num;
   252     int _res_node_num;
   253     int _res_arc_num;
   254     int _root;
   255 
   256     // Parameters of the problem
   257     bool _have_lower;
   258     Value _sum_supply;
   259     int _sup_node_num;
   260 
   261     // Data structures for storing the digraph
   262     IntNodeMap _node_id;
   263     IntArcMap _arc_idf;
   264     IntArcMap _arc_idb;
   265     IntVector _first_out;
   266     BoolVector _forward;
   267     IntVector _source;
   268     IntVector _target;
   269     IntVector _reverse;
   270 
   271     // Node and arc data
   272     ValueVector _lower;
   273     ValueVector _upper;
   274     CostVector _scost;
   275     ValueVector _supply;
   276 
   277     ValueVector _res_cap;
   278     LargeCostVector _cost;
   279     LargeCostVector _pi;
   280     ValueVector _excess;
   281     IntVector _next_out;
   282     std::deque<int> _active_nodes;
   283 
   284     // Data for scaling
   285     LargeCost _epsilon;
   286     int _alpha;
   287 
   288     IntVector _buckets;
   289     IntVector _bucket_next;
   290     IntVector _bucket_prev;
   291     IntVector _rank;
   292     int _max_rank;
   293 
   294   public:
   295 
   296     /// \brief Constant for infinite upper bounds (capacities).
   297     ///
   298     /// Constant for infinite upper bounds (capacities).
   299     /// It is \c std::numeric_limits<Value>::infinity() if available,
   300     /// \c std::numeric_limits<Value>::max() otherwise.
   301     const Value INF;
   302 
   303   public:
   304 
   305     /// \name Named Template Parameters
   306     /// @{
   307 
   308     template <typename T>
   309     struct SetLargeCostTraits : public Traits {
   310       typedef T LargeCost;
   311     };
   312 
   313     /// \brief \ref named-templ-param "Named parameter" for setting
   314     /// \c LargeCost type.
   315     ///
   316     /// \ref named-templ-param "Named parameter" for setting \c LargeCost
   317     /// type, which is used for internal computations in the algorithm.
   318     /// \c Cost must be convertible to \c LargeCost.
   319     template <typename T>
   320     struct SetLargeCost
   321       : public CostScaling<GR, V, C, SetLargeCostTraits<T> > {
   322       typedef  CostScaling<GR, V, C, SetLargeCostTraits<T> > Create;
   323     };
   324 
   325     /// @}
   326 
   327   protected:
   328 
   329     CostScaling() {}
   330 
   331   public:
   332 
   333     /// \brief Constructor.
   334     ///
   335     /// The constructor of the class.
   336     ///
   337     /// \param graph The digraph the algorithm runs on.
   338     CostScaling(const GR& graph) :
   339       _graph(graph), _node_id(graph), _arc_idf(graph), _arc_idb(graph),
   340       INF(std::numeric_limits<Value>::has_infinity ?
   341           std::numeric_limits<Value>::infinity() :
   342           std::numeric_limits<Value>::max())
   343     {
   344       // Check the number types
   345       LEMON_ASSERT(std::numeric_limits<Value>::is_signed,
   346         "The flow type of CostScaling must be signed");
   347       LEMON_ASSERT(std::numeric_limits<Cost>::is_signed,
   348         "The cost type of CostScaling must be signed");
   349 
   350       // Reset data structures
   351       reset();
   352     }
   353 
   354     /// \name Parameters
   355     /// The parameters of the algorithm can be specified using these
   356     /// functions.
   357 
   358     /// @{
   359 
   360     /// \brief Set the lower bounds on the arcs.
   361     ///
   362     /// This function sets the lower bounds on the arcs.
   363     /// If it is not used before calling \ref run(), the lower bounds
   364     /// will be set to zero on all arcs.
   365     ///
   366     /// \param map An arc map storing the lower bounds.
   367     /// Its \c Value type must be convertible to the \c Value type
   368     /// of the algorithm.
   369     ///
   370     /// \return <tt>(*this)</tt>
   371     template <typename LowerMap>
   372     CostScaling& lowerMap(const LowerMap& map) {
   373       _have_lower = true;
   374       for (ArcIt a(_graph); a != INVALID; ++a) {
   375         _lower[_arc_idf[a]] = map[a];
   376         _lower[_arc_idb[a]] = map[a];
   377       }
   378       return *this;
   379     }
   380 
   381     /// \brief Set the upper bounds (capacities) on the arcs.
   382     ///
   383     /// This function sets the upper bounds (capacities) on the arcs.
   384     /// If it is not used before calling \ref run(), the upper bounds
   385     /// will be set to \ref INF on all arcs (i.e. the flow value will be
   386     /// unbounded from above).
   387     ///
   388     /// \param map An arc map storing the upper bounds.
   389     /// Its \c Value type must be convertible to the \c Value type
   390     /// of the algorithm.
   391     ///
   392     /// \return <tt>(*this)</tt>
   393     template<typename UpperMap>
   394     CostScaling& upperMap(const UpperMap& map) {
   395       for (ArcIt a(_graph); a != INVALID; ++a) {
   396         _upper[_arc_idf[a]] = map[a];
   397       }
   398       return *this;
   399     }
   400 
   401     /// \brief Set the costs of the arcs.
   402     ///
   403     /// This function sets the costs of the arcs.
   404     /// If it is not used before calling \ref run(), the costs
   405     /// will be set to \c 1 on all arcs.
   406     ///
   407     /// \param map An arc map storing the costs.
   408     /// Its \c Value type must be convertible to the \c Cost type
   409     /// of the algorithm.
   410     ///
   411     /// \return <tt>(*this)</tt>
   412     template<typename CostMap>
   413     CostScaling& costMap(const CostMap& map) {
   414       for (ArcIt a(_graph); a != INVALID; ++a) {
   415         _scost[_arc_idf[a]] =  map[a];
   416         _scost[_arc_idb[a]] = -map[a];
   417       }
   418       return *this;
   419     }
   420 
   421     /// \brief Set the supply values of the nodes.
   422     ///
   423     /// This function sets the supply values of the nodes.
   424     /// If neither this function nor \ref stSupply() is used before
   425     /// calling \ref run(), the supply of each node will be set to zero.
   426     ///
   427     /// \param map A node map storing the supply values.
   428     /// Its \c Value type must be convertible to the \c Value type
   429     /// of the algorithm.
   430     ///
   431     /// \return <tt>(*this)</tt>
   432     template<typename SupplyMap>
   433     CostScaling& supplyMap(const SupplyMap& map) {
   434       for (NodeIt n(_graph); n != INVALID; ++n) {
   435         _supply[_node_id[n]] = map[n];
   436       }
   437       return *this;
   438     }
   439 
   440     /// \brief Set single source and target nodes and a supply value.
   441     ///
   442     /// This function sets a single source node and a single target node
   443     /// and the required flow value.
   444     /// If neither this function nor \ref supplyMap() is used before
   445     /// calling \ref run(), the supply of each node will be set to zero.
   446     ///
   447     /// Using this function has the same effect as using \ref supplyMap()
   448     /// with a map in which \c k is assigned to \c s, \c -k is
   449     /// assigned to \c t and all other nodes have zero supply value.
   450     ///
   451     /// \param s The source node.
   452     /// \param t The target node.
   453     /// \param k The required amount of flow from node \c s to node \c t
   454     /// (i.e. the supply of \c s and the demand of \c t).
   455     ///
   456     /// \return <tt>(*this)</tt>
   457     CostScaling& stSupply(const Node& s, const Node& t, Value k) {
   458       for (int i = 0; i != _res_node_num; ++i) {
   459         _supply[i] = 0;
   460       }
   461       _supply[_node_id[s]] =  k;
   462       _supply[_node_id[t]] = -k;
   463       return *this;
   464     }
   465 
   466     /// @}
   467 
   468     /// \name Execution control
   469     /// The algorithm can be executed using \ref run().
   470 
   471     /// @{
   472 
   473     /// \brief Run the algorithm.
   474     ///
   475     /// This function runs the algorithm.
   476     /// The paramters can be specified using functions \ref lowerMap(),
   477     /// \ref upperMap(), \ref costMap(), \ref supplyMap(), \ref stSupply().
   478     /// For example,
   479     /// \code
   480     ///   CostScaling<ListDigraph> cs(graph);
   481     ///   cs.lowerMap(lower).upperMap(upper).costMap(cost)
   482     ///     .supplyMap(sup).run();
   483     /// \endcode
   484     ///
   485     /// This function can be called more than once. All the given parameters
   486     /// are kept for the next call, unless \ref resetParams() or \ref reset()
   487     /// is used, thus only the modified parameters have to be set again.
   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 called.
   491     ///
   492     /// \param method The internal method that will be used in the
   493     /// algorithm. For more information, see \ref Method.
   494     /// \param factor The cost scaling factor. It must be at least two.
   495     ///
   496     /// \return \c INFEASIBLE if no feasible flow exists,
   497     /// \n \c OPTIMAL if the problem has optimal solution
   498     /// (i.e. it is feasible and bounded), and the algorithm has found
   499     /// optimal flow and node potentials (primal and dual solutions),
   500     /// \n \c UNBOUNDED if the digraph contains an arc of negative cost
   501     /// and infinite upper bound. It means that the objective function
   502     /// is unbounded on that arc, however, note that it could actually be
   503     /// bounded over the feasible flows, but this algroithm cannot handle
   504     /// these cases.
   505     ///
   506     /// \see ProblemType, Method
   507     /// \see resetParams(), reset()
   508     ProblemType run(Method method = PARTIAL_AUGMENT, int factor = 16) {
   509       LEMON_ASSERT(factor >= 2, "The scaling factor must be at least 2");
   510       _alpha = factor;
   511       ProblemType pt = init();
   512       if (pt != OPTIMAL) return pt;
   513       start(method);
   514       return OPTIMAL;
   515     }
   516 
   517     /// \brief Reset all the parameters that have been given before.
   518     ///
   519     /// This function resets all the paramaters that have been given
   520     /// before using functions \ref lowerMap(), \ref upperMap(),
   521     /// \ref costMap(), \ref supplyMap(), \ref stSupply().
   522     ///
   523     /// It is useful for multiple \ref run() calls. Basically, all the given
   524     /// parameters are kept for the next \ref run() call, unless
   525     /// \ref resetParams() or \ref reset() is used.
   526     /// If the underlying digraph was also modified after the construction
   527     /// of the class or the last \ref reset() call, then the \ref reset()
   528     /// function must be used, otherwise \ref resetParams() is sufficient.
   529     ///
   530     /// For example,
   531     /// \code
   532     ///   CostScaling<ListDigraph> cs(graph);
   533     ///
   534     ///   // First run
   535     ///   cs.lowerMap(lower).upperMap(upper).costMap(cost)
   536     ///     .supplyMap(sup).run();
   537     ///
   538     ///   // Run again with modified cost map (resetParams() is not called,
   539     ///   // so only the cost map have to be set again)
   540     ///   cost[e] += 100;
   541     ///   cs.costMap(cost).run();
   542     ///
   543     ///   // Run again from scratch using resetParams()
   544     ///   // (the lower bounds will be set to zero on all arcs)
   545     ///   cs.resetParams();
   546     ///   cs.upperMap(capacity).costMap(cost)
   547     ///     .supplyMap(sup).run();
   548     /// \endcode
   549     ///
   550     /// \return <tt>(*this)</tt>
   551     ///
   552     /// \see reset(), run()
   553     CostScaling& resetParams() {
   554       for (int i = 0; i != _res_node_num; ++i) {
   555         _supply[i] = 0;
   556       }
   557       int limit = _first_out[_root];
   558       for (int j = 0; j != limit; ++j) {
   559         _lower[j] = 0;
   560         _upper[j] = INF;
   561         _scost[j] = _forward[j] ? 1 : -1;
   562       }
   563       for (int j = limit; j != _res_arc_num; ++j) {
   564         _lower[j] = 0;
   565         _upper[j] = INF;
   566         _scost[j] = 0;
   567         _scost[_reverse[j]] = 0;
   568       }
   569       _have_lower = false;
   570       return *this;
   571     }
   572 
   573     /// \brief Reset the internal data structures and all the parameters
   574     /// that have been given before.
   575     ///
   576     /// This function resets the internal data structures and all the
   577     /// paramaters that have been given before using functions \ref lowerMap(),
   578     /// \ref upperMap(), \ref costMap(), \ref supplyMap(), \ref stSupply().
   579     ///
   580     /// It is useful for multiple \ref run() calls. By default, all the given
   581     /// parameters are kept for the next \ref run() call, unless
   582     /// \ref resetParams() or \ref reset() is used.
   583     /// If the underlying digraph was also modified after the construction
   584     /// of the class or the last \ref reset() call, then the \ref reset()
   585     /// function must be used, otherwise \ref resetParams() is sufficient.
   586     ///
   587     /// See \ref resetParams() for examples.
   588     ///
   589     /// \return <tt>(*this)</tt>
   590     ///
   591     /// \see resetParams(), run()
   592     CostScaling& reset() {
   593       // Resize vectors
   594       _node_num = countNodes(_graph);
   595       _arc_num = countArcs(_graph);
   596       _res_node_num = _node_num + 1;
   597       _res_arc_num = 2 * (_arc_num + _node_num);
   598       _root = _node_num;
   599 
   600       _first_out.resize(_res_node_num + 1);
   601       _forward.resize(_res_arc_num);
   602       _source.resize(_res_arc_num);
   603       _target.resize(_res_arc_num);
   604       _reverse.resize(_res_arc_num);
   605 
   606       _lower.resize(_res_arc_num);
   607       _upper.resize(_res_arc_num);
   608       _scost.resize(_res_arc_num);
   609       _supply.resize(_res_node_num);
   610 
   611       _res_cap.resize(_res_arc_num);
   612       _cost.resize(_res_arc_num);
   613       _pi.resize(_res_node_num);
   614       _excess.resize(_res_node_num);
   615       _next_out.resize(_res_node_num);
   616 
   617       // Copy the graph
   618       int i = 0, j = 0, k = 2 * _arc_num + _node_num;
   619       for (NodeIt n(_graph); n != INVALID; ++n, ++i) {
   620         _node_id[n] = i;
   621       }
   622       i = 0;
   623       for (NodeIt n(_graph); n != INVALID; ++n, ++i) {
   624         _first_out[i] = j;
   625         for (OutArcIt a(_graph, n); a != INVALID; ++a, ++j) {
   626           _arc_idf[a] = j;
   627           _forward[j] = true;
   628           _source[j] = i;
   629           _target[j] = _node_id[_graph.runningNode(a)];
   630         }
   631         for (InArcIt a(_graph, n); a != INVALID; ++a, ++j) {
   632           _arc_idb[a] = j;
   633           _forward[j] = false;
   634           _source[j] = i;
   635           _target[j] = _node_id[_graph.runningNode(a)];
   636         }
   637         _forward[j] = false;
   638         _source[j] = i;
   639         _target[j] = _root;
   640         _reverse[j] = k;
   641         _forward[k] = true;
   642         _source[k] = _root;
   643         _target[k] = i;
   644         _reverse[k] = j;
   645         ++j; ++k;
   646       }
   647       _first_out[i] = j;
   648       _first_out[_res_node_num] = k;
   649       for (ArcIt a(_graph); a != INVALID; ++a) {
   650         int fi = _arc_idf[a];
   651         int bi = _arc_idb[a];
   652         _reverse[fi] = bi;
   653         _reverse[bi] = fi;
   654       }
   655 
   656       // Reset parameters
   657       resetParams();
   658       return *this;
   659     }
   660 
   661     /// @}
   662 
   663     /// \name Query Functions
   664     /// The results of the algorithm can be obtained using these
   665     /// functions.\n
   666     /// The \ref run() function must be called before using them.
   667 
   668     /// @{
   669 
   670     /// \brief Return the total cost of the found flow.
   671     ///
   672     /// This function returns the total cost of the found flow.
   673     /// Its complexity is O(e).
   674     ///
   675     /// \note The return type of the function can be specified as a
   676     /// template parameter. For example,
   677     /// \code
   678     ///   cs.totalCost<double>();
   679     /// \endcode
   680     /// It is useful if the total cost cannot be stored in the \c Cost
   681     /// type of the algorithm, which is the default return type of the
   682     /// function.
   683     ///
   684     /// \pre \ref run() must be called before using this function.
   685     template <typename Number>
   686     Number totalCost() const {
   687       Number c = 0;
   688       for (ArcIt a(_graph); a != INVALID; ++a) {
   689         int i = _arc_idb[a];
   690         c += static_cast<Number>(_res_cap[i]) *
   691              (-static_cast<Number>(_scost[i]));
   692       }
   693       return c;
   694     }
   695 
   696 #ifndef DOXYGEN
   697     Cost totalCost() const {
   698       return totalCost<Cost>();
   699     }
   700 #endif
   701 
   702     /// \brief Return the flow on the given arc.
   703     ///
   704     /// This function returns the flow on the given arc.
   705     ///
   706     /// \pre \ref run() must be called before using this function.
   707     Value flow(const Arc& a) const {
   708       return _res_cap[_arc_idb[a]];
   709     }
   710 
   711     /// \brief Copy the flow values (the primal solution) into the
   712     /// given map.
   713     ///
   714     /// This function copies the flow value on each arc into the given
   715     /// map. The \c Value type of the algorithm must be convertible to
   716     /// the \c Value type of the map.
   717     ///
   718     /// \pre \ref run() must be called before using this function.
   719     template <typename FlowMap>
   720     void flowMap(FlowMap &map) const {
   721       for (ArcIt a(_graph); a != INVALID; ++a) {
   722         map.set(a, _res_cap[_arc_idb[a]]);
   723       }
   724     }
   725 
   726     /// \brief Return the potential (dual value) of the given node.
   727     ///
   728     /// This function returns the potential (dual value) of the
   729     /// given node.
   730     ///
   731     /// \pre \ref run() must be called before using this function.
   732     Cost potential(const Node& n) const {
   733       return static_cast<Cost>(_pi[_node_id[n]]);
   734     }
   735 
   736     /// \brief Copy the potential values (the dual solution) into the
   737     /// given map.
   738     ///
   739     /// This function copies the potential (dual value) of each node
   740     /// into the given map.
   741     /// The \c Cost type of the algorithm must be convertible to the
   742     /// \c Value type of the map.
   743     ///
   744     /// \pre \ref run() must be called before using this function.
   745     template <typename PotentialMap>
   746     void potentialMap(PotentialMap &map) const {
   747       for (NodeIt n(_graph); n != INVALID; ++n) {
   748         map.set(n, static_cast<Cost>(_pi[_node_id[n]]));
   749       }
   750     }
   751 
   752     /// @}
   753 
   754   private:
   755 
   756     // Initialize the algorithm
   757     ProblemType init() {
   758       if (_res_node_num <= 1) return INFEASIBLE;
   759 
   760       // Check the sum of supply values
   761       _sum_supply = 0;
   762       for (int i = 0; i != _root; ++i) {
   763         _sum_supply += _supply[i];
   764       }
   765       if (_sum_supply > 0) return INFEASIBLE;
   766 
   767       // Check lower and upper bounds
   768       LEMON_DEBUG(checkBoundMaps(),
   769           "Upper bounds must be greater or equal to the lower bounds");
   770 
   771 
   772       // Initialize vectors
   773       for (int i = 0; i != _res_node_num; ++i) {
   774         _pi[i] = 0;
   775         _excess[i] = _supply[i];
   776       }
   777 
   778       // Remove infinite upper bounds and check negative arcs
   779       const Value MAX = std::numeric_limits<Value>::max();
   780       int last_out;
   781       if (_have_lower) {
   782         for (int i = 0; i != _root; ++i) {
   783           last_out = _first_out[i+1];
   784           for (int j = _first_out[i]; j != last_out; ++j) {
   785             if (_forward[j]) {
   786               Value c = _scost[j] < 0 ? _upper[j] : _lower[j];
   787               if (c >= MAX) return UNBOUNDED;
   788               _excess[i] -= c;
   789               _excess[_target[j]] += c;
   790             }
   791           }
   792         }
   793       } else {
   794         for (int i = 0; i != _root; ++i) {
   795           last_out = _first_out[i+1];
   796           for (int j = _first_out[i]; j != last_out; ++j) {
   797             if (_forward[j] && _scost[j] < 0) {
   798               Value c = _upper[j];
   799               if (c >= MAX) return UNBOUNDED;
   800               _excess[i] -= c;
   801               _excess[_target[j]] += c;
   802             }
   803           }
   804         }
   805       }
   806       Value ex, max_cap = 0;
   807       for (int i = 0; i != _res_node_num; ++i) {
   808         ex = _excess[i];
   809         _excess[i] = 0;
   810         if (ex < 0) max_cap -= ex;
   811       }
   812       for (int j = 0; j != _res_arc_num; ++j) {
   813         if (_upper[j] >= MAX) _upper[j] = max_cap;
   814       }
   815 
   816       // Initialize the large cost vector and the epsilon parameter
   817       _epsilon = 0;
   818       LargeCost lc;
   819       for (int i = 0; i != _root; ++i) {
   820         last_out = _first_out[i+1];
   821         for (int j = _first_out[i]; j != last_out; ++j) {
   822           lc = static_cast<LargeCost>(_scost[j]) * _res_node_num * _alpha;
   823           _cost[j] = lc;
   824           if (lc > _epsilon) _epsilon = lc;
   825         }
   826       }
   827       _epsilon /= _alpha;
   828 
   829       // Initialize maps for Circulation and remove non-zero lower bounds
   830       ConstMap<Arc, Value> low(0);
   831       typedef typename Digraph::template ArcMap<Value> ValueArcMap;
   832       typedef typename Digraph::template NodeMap<Value> ValueNodeMap;
   833       ValueArcMap cap(_graph), flow(_graph);
   834       ValueNodeMap sup(_graph);
   835       for (NodeIt n(_graph); n != INVALID; ++n) {
   836         sup[n] = _supply[_node_id[n]];
   837       }
   838       if (_have_lower) {
   839         for (ArcIt a(_graph); a != INVALID; ++a) {
   840           int j = _arc_idf[a];
   841           Value c = _lower[j];
   842           cap[a] = _upper[j] - c;
   843           sup[_graph.source(a)] -= c;
   844           sup[_graph.target(a)] += c;
   845         }
   846       } else {
   847         for (ArcIt a(_graph); a != INVALID; ++a) {
   848           cap[a] = _upper[_arc_idf[a]];
   849         }
   850       }
   851 
   852       _sup_node_num = 0;
   853       for (NodeIt n(_graph); n != INVALID; ++n) {
   854         if (sup[n] > 0) ++_sup_node_num;
   855       }
   856 
   857       // Find a feasible flow using Circulation
   858       Circulation<Digraph, ConstMap<Arc, Value>, ValueArcMap, ValueNodeMap>
   859         circ(_graph, low, cap, sup);
   860       if (!circ.flowMap(flow).run()) return INFEASIBLE;
   861 
   862       // Set residual capacities and handle GEQ supply type
   863       if (_sum_supply < 0) {
   864         for (ArcIt a(_graph); a != INVALID; ++a) {
   865           Value fa = flow[a];
   866           _res_cap[_arc_idf[a]] = cap[a] - fa;
   867           _res_cap[_arc_idb[a]] = fa;
   868           sup[_graph.source(a)] -= fa;
   869           sup[_graph.target(a)] += fa;
   870         }
   871         for (NodeIt n(_graph); n != INVALID; ++n) {
   872           _excess[_node_id[n]] = sup[n];
   873         }
   874         for (int a = _first_out[_root]; a != _res_arc_num; ++a) {
   875           int u = _target[a];
   876           int ra = _reverse[a];
   877           _res_cap[a] = -_sum_supply + 1;
   878           _res_cap[ra] = -_excess[u];
   879           _cost[a] = 0;
   880           _cost[ra] = 0;
   881           _excess[u] = 0;
   882         }
   883       } else {
   884         for (ArcIt a(_graph); a != INVALID; ++a) {
   885           Value fa = flow[a];
   886           _res_cap[_arc_idf[a]] = cap[a] - fa;
   887           _res_cap[_arc_idb[a]] = fa;
   888         }
   889         for (int a = _first_out[_root]; a != _res_arc_num; ++a) {
   890           int ra = _reverse[a];
   891           _res_cap[a] = 0;
   892           _res_cap[ra] = 0;
   893           _cost[a] = 0;
   894           _cost[ra] = 0;
   895         }
   896       }
   897 
   898       // Initialize data structures for buckets
   899       _max_rank = _alpha * _res_node_num;
   900       _buckets.resize(_max_rank);
   901       _bucket_next.resize(_res_node_num + 1);
   902       _bucket_prev.resize(_res_node_num + 1);
   903       _rank.resize(_res_node_num + 1);
   904 
   905       return OPTIMAL;
   906     }
   907     
   908     // Check if the upper bound is greater or equal to the lower bound
   909     // on each arc.
   910     bool checkBoundMaps() {
   911       for (int j = 0; j != _res_arc_num; ++j) {
   912         if (_upper[j] < _lower[j]) return false;
   913       }
   914       return true;
   915     }
   916 
   917     // Execute the algorithm and transform the results
   918     void start(Method method) {
   919       const int MAX_PARTIAL_PATH_LENGTH = 4;
   920 
   921       switch (method) {
   922         case PUSH:
   923           startPush();
   924           break;
   925         case AUGMENT:
   926           startAugment(_res_node_num - 1);
   927           break;
   928         case PARTIAL_AUGMENT:
   929           startAugment(MAX_PARTIAL_PATH_LENGTH);
   930           break;
   931       }
   932 
   933       // Compute node potentials (dual solution)
   934       for (int i = 0; i != _res_node_num; ++i) {
   935         _pi[i] = static_cast<Cost>(_pi[i] / (_res_node_num * _alpha));
   936       }
   937       bool optimal = true;
   938       for (int i = 0; optimal && i != _res_node_num; ++i) {
   939         LargeCost pi_i = _pi[i];
   940         int last_out = _first_out[i+1];
   941         for (int j = _first_out[i]; j != last_out; ++j) {
   942           if (_res_cap[j] > 0 && _scost[j] + pi_i - _pi[_target[j]] < 0) {
   943             optimal = false;
   944             break;
   945           }
   946         }
   947       }
   948 
   949       if (!optimal) {
   950         // Compute node potentials for the original costs with BellmanFord
   951         // (if it is necessary)
   952         typedef std::pair<int, int> IntPair;
   953         StaticDigraph sgr;
   954         std::vector<IntPair> arc_vec;
   955         std::vector<LargeCost> cost_vec;
   956         LargeCostArcMap cost_map(cost_vec);
   957 
   958         arc_vec.clear();
   959         cost_vec.clear();
   960         for (int j = 0; j != _res_arc_num; ++j) {
   961           if (_res_cap[j] > 0) {
   962             int u = _source[j], v = _target[j];
   963             arc_vec.push_back(IntPair(u, v));
   964             cost_vec.push_back(_scost[j] + _pi[u] - _pi[v]);
   965           }
   966         }
   967         sgr.build(_res_node_num, arc_vec.begin(), arc_vec.end());
   968 
   969         typename BellmanFord<StaticDigraph, LargeCostArcMap>::Create
   970           bf(sgr, cost_map);
   971         bf.init(0);
   972         bf.start();
   973 
   974         for (int i = 0; i != _res_node_num; ++i) {
   975           _pi[i] += bf.dist(sgr.node(i));
   976         }
   977       }
   978 
   979       // Shift potentials to meet the requirements of the GEQ type
   980       // optimality conditions
   981       LargeCost max_pot = _pi[_root];
   982       for (int i = 0; i != _res_node_num; ++i) {
   983         if (_pi[i] > max_pot) max_pot = _pi[i];
   984       }
   985       if (max_pot != 0) {
   986         for (int i = 0; i != _res_node_num; ++i) {
   987           _pi[i] -= max_pot;
   988         }
   989       }
   990 
   991       // Handle non-zero lower bounds
   992       if (_have_lower) {
   993         int limit = _first_out[_root];
   994         for (int j = 0; j != limit; ++j) {
   995           if (!_forward[j]) _res_cap[j] += _lower[j];
   996         }
   997       }
   998     }
   999 
  1000     // Initialize a cost scaling phase
  1001     void initPhase() {
  1002       // Saturate arcs not satisfying the optimality condition
  1003       for (int u = 0; u != _res_node_num; ++u) {
  1004         int last_out = _first_out[u+1];
  1005         LargeCost pi_u = _pi[u];
  1006         for (int a = _first_out[u]; a != last_out; ++a) {
  1007           Value delta = _res_cap[a];
  1008           if (delta > 0) {
  1009             int v = _target[a];
  1010             if (_cost[a] + pi_u - _pi[v] < 0) {
  1011               _excess[u] -= delta;
  1012               _excess[v] += delta;
  1013               _res_cap[a] = 0;
  1014               _res_cap[_reverse[a]] += delta;
  1015             }
  1016           }
  1017         }
  1018       }
  1019 
  1020       // Find active nodes (i.e. nodes with positive excess)
  1021       for (int u = 0; u != _res_node_num; ++u) {
  1022         if (_excess[u] > 0) _active_nodes.push_back(u);
  1023       }
  1024 
  1025       // Initialize the next arcs
  1026       for (int u = 0; u != _res_node_num; ++u) {
  1027         _next_out[u] = _first_out[u];
  1028       }
  1029     }
  1030 
  1031     // Price (potential) refinement heuristic
  1032     bool priceRefinement() {
  1033 
  1034       // Stack for stroing the topological order
  1035       IntVector stack(_res_node_num);
  1036       int stack_top;
  1037 
  1038       // Perform phases
  1039       while (topologicalSort(stack, stack_top)) {
  1040 
  1041         // Compute node ranks in the acyclic admissible network and
  1042         // store the nodes in buckets
  1043         for (int i = 0; i != _res_node_num; ++i) {
  1044           _rank[i] = 0;
  1045         }
  1046         const int bucket_end = _root + 1;
  1047         for (int r = 0; r != _max_rank; ++r) {
  1048           _buckets[r] = bucket_end;
  1049         }
  1050         int top_rank = 0;
  1051         for ( ; stack_top >= 0; --stack_top) {
  1052           int u = stack[stack_top], v;
  1053           int rank_u = _rank[u];
  1054 
  1055           LargeCost rc, pi_u = _pi[u];
  1056           int last_out = _first_out[u+1];
  1057           for (int a = _first_out[u]; a != last_out; ++a) {
  1058             if (_res_cap[a] > 0) {
  1059               v = _target[a];
  1060               rc = _cost[a] + pi_u - _pi[v];
  1061               if (rc < 0) {
  1062                 LargeCost nrc = static_cast<LargeCost>((-rc - 0.5) / _epsilon);
  1063                 if (nrc < LargeCost(_max_rank)) {
  1064                   int new_rank_v = rank_u + static_cast<int>(nrc);
  1065                   if (new_rank_v > _rank[v]) {
  1066                     _rank[v] = new_rank_v;
  1067                   }
  1068                 }
  1069               }
  1070             }
  1071           }
  1072 
  1073           if (rank_u > 0) {
  1074             top_rank = std::max(top_rank, rank_u);
  1075             int bfirst = _buckets[rank_u];
  1076             _bucket_next[u] = bfirst;
  1077             _bucket_prev[bfirst] = u;
  1078             _buckets[rank_u] = u;
  1079           }
  1080         }
  1081 
  1082         // Check if the current flow is epsilon-optimal
  1083         if (top_rank == 0) {
  1084           return true;
  1085         }
  1086 
  1087         // Process buckets in top-down order
  1088         for (int rank = top_rank; rank > 0; --rank) {
  1089           while (_buckets[rank] != bucket_end) {
  1090             // Remove the first node from the current bucket
  1091             int u = _buckets[rank];
  1092             _buckets[rank] = _bucket_next[u];
  1093 
  1094             // Search the outgoing arcs of u
  1095             LargeCost rc, pi_u = _pi[u];
  1096             int last_out = _first_out[u+1];
  1097             int v, old_rank_v, new_rank_v;
  1098             for (int a = _first_out[u]; a != last_out; ++a) {
  1099               if (_res_cap[a] > 0) {
  1100                 v = _target[a];
  1101                 old_rank_v = _rank[v];
  1102 
  1103                 if (old_rank_v < rank) {
  1104 
  1105                   // Compute the new rank of node v
  1106                   rc = _cost[a] + pi_u - _pi[v];
  1107                   if (rc < 0) {
  1108                     new_rank_v = rank;
  1109                   } else {
  1110                     LargeCost nrc = rc / _epsilon;
  1111                     new_rank_v = 0;
  1112                     if (nrc < LargeCost(_max_rank)) {
  1113                       new_rank_v = rank - 1 - static_cast<int>(nrc);
  1114                     }
  1115                   }
  1116 
  1117                   // Change the rank of node v
  1118                   if (new_rank_v > old_rank_v) {
  1119                     _rank[v] = new_rank_v;
  1120 
  1121                     // Remove v from its old bucket
  1122                     if (old_rank_v > 0) {
  1123                       if (_buckets[old_rank_v] == v) {
  1124                         _buckets[old_rank_v] = _bucket_next[v];
  1125                       } else {
  1126                         int pv = _bucket_prev[v], nv = _bucket_next[v];
  1127                         _bucket_next[pv] = nv;
  1128                         _bucket_prev[nv] = pv;
  1129                       }
  1130                     }
  1131 
  1132                     // Insert v into its new bucket
  1133                     int nv = _buckets[new_rank_v];
  1134                     _bucket_next[v] = nv;
  1135                     _bucket_prev[nv] = v;
  1136                     _buckets[new_rank_v] = v;
  1137                   }
  1138                 }
  1139               }
  1140             }
  1141 
  1142             // Refine potential of node u
  1143             _pi[u] -= rank * _epsilon;
  1144           }
  1145         }
  1146 
  1147       }
  1148 
  1149       return false;
  1150     }
  1151 
  1152     // Find and cancel cycles in the admissible network and
  1153     // determine topological order using DFS
  1154     bool topologicalSort(IntVector &stack, int &stack_top) {
  1155       const int MAX_CYCLE_CANCEL = 1;
  1156 
  1157       BoolVector reached(_res_node_num, false);
  1158       BoolVector processed(_res_node_num, false);
  1159       IntVector pred(_res_node_num);
  1160       for (int i = 0; i != _res_node_num; ++i) {
  1161         _next_out[i] = _first_out[i];
  1162       }
  1163       stack_top = -1;
  1164 
  1165       int cycle_cnt = 0;
  1166       for (int start = 0; start != _res_node_num; ++start) {
  1167         if (reached[start]) continue;
  1168 
  1169         // Start DFS search from this start node
  1170         pred[start] = -1;
  1171         int tip = start, v;
  1172         while (true) {
  1173           // Check the outgoing arcs of the current tip node
  1174           reached[tip] = true;
  1175           LargeCost pi_tip = _pi[tip];
  1176           int a, last_out = _first_out[tip+1];
  1177           for (a = _next_out[tip]; a != last_out; ++a) {
  1178             if (_res_cap[a] > 0) {
  1179               v = _target[a];
  1180               if (_cost[a] + pi_tip - _pi[v] < 0) {
  1181                 if (!reached[v]) {
  1182                   // A new node is reached
  1183                   reached[v] = true;
  1184                   pred[v] = tip;
  1185                   _next_out[tip] = a;
  1186                   tip = v;
  1187                   a = _next_out[tip];
  1188                   last_out = _first_out[tip+1];
  1189                   break;
  1190                 }
  1191                 else if (!processed[v]) {
  1192                   // A cycle is found
  1193                   ++cycle_cnt;
  1194                   _next_out[tip] = a;
  1195 
  1196                   // Find the minimum residual capacity along the cycle
  1197                   Value d, delta = _res_cap[a];
  1198                   int u, delta_node = tip;
  1199                   for (u = tip; u != v; ) {
  1200                     u = pred[u];
  1201                     d = _res_cap[_next_out[u]];
  1202                     if (d <= delta) {
  1203                       delta = d;
  1204                       delta_node = u;
  1205                     }
  1206                   }
  1207 
  1208                   // Augment along the cycle
  1209                   _res_cap[a] -= delta;
  1210                   _res_cap[_reverse[a]] += delta;
  1211                   for (u = tip; u != v; ) {
  1212                     u = pred[u];
  1213                     int ca = _next_out[u];
  1214                     _res_cap[ca] -= delta;
  1215                     _res_cap[_reverse[ca]] += delta;
  1216                   }
  1217 
  1218                   // Check the maximum number of cycle canceling
  1219                   if (cycle_cnt >= MAX_CYCLE_CANCEL) {
  1220                     return false;
  1221                   }
  1222 
  1223                   // Roll back search to delta_node
  1224                   if (delta_node != tip) {
  1225                     for (u = tip; u != delta_node; u = pred[u]) {
  1226                       reached[u] = false;
  1227                     }
  1228                     tip = delta_node;
  1229                     a = _next_out[tip] + 1;
  1230                     last_out = _first_out[tip+1];
  1231                     break;
  1232                   }
  1233                 }
  1234               }
  1235             }
  1236           }
  1237 
  1238           // Step back to the previous node
  1239           if (a == last_out) {
  1240             processed[tip] = true;
  1241             stack[++stack_top] = tip;
  1242             tip = pred[tip];
  1243             if (tip < 0) {
  1244               // Finish DFS from the current start node
  1245               break;
  1246             }
  1247             ++_next_out[tip];
  1248           }
  1249         }
  1250 
  1251       }
  1252 
  1253       return (cycle_cnt == 0);
  1254     }
  1255 
  1256     // Global potential update heuristic
  1257     void globalUpdate() {
  1258       const int bucket_end = _root + 1;
  1259 
  1260       // Initialize buckets
  1261       for (int r = 0; r != _max_rank; ++r) {
  1262         _buckets[r] = bucket_end;
  1263       }
  1264       Value total_excess = 0;
  1265       int b0 = bucket_end;
  1266       for (int i = 0; i != _res_node_num; ++i) {
  1267         if (_excess[i] < 0) {
  1268           _rank[i] = 0;
  1269           _bucket_next[i] = b0;
  1270           _bucket_prev[b0] = i;
  1271           b0 = i;
  1272         } else {
  1273           total_excess += _excess[i];
  1274           _rank[i] = _max_rank;
  1275         }
  1276       }
  1277       if (total_excess == 0) return;
  1278       _buckets[0] = b0;
  1279 
  1280       // Search the buckets
  1281       int r = 0;
  1282       for ( ; r != _max_rank; ++r) {
  1283         while (_buckets[r] != bucket_end) {
  1284           // Remove the first node from the current bucket
  1285           int u = _buckets[r];
  1286           _buckets[r] = _bucket_next[u];
  1287 
  1288           // Search the incoming arcs of u
  1289           LargeCost pi_u = _pi[u];
  1290           int last_out = _first_out[u+1];
  1291           for (int a = _first_out[u]; a != last_out; ++a) {
  1292             int ra = _reverse[a];
  1293             if (_res_cap[ra] > 0) {
  1294               int v = _source[ra];
  1295               int old_rank_v = _rank[v];
  1296               if (r < old_rank_v) {
  1297                 // Compute the new rank of v
  1298                 LargeCost nrc = (_cost[ra] + _pi[v] - pi_u) / _epsilon;
  1299                 int new_rank_v = old_rank_v;
  1300                 if (nrc < LargeCost(_max_rank)) {
  1301                   new_rank_v = r + 1 + static_cast<int>(nrc);
  1302                 }
  1303 
  1304                 // Change the rank of v
  1305                 if (new_rank_v < old_rank_v) {
  1306                   _rank[v] = new_rank_v;
  1307                   _next_out[v] = _first_out[v];
  1308 
  1309                   // Remove v from its old bucket
  1310                   if (old_rank_v < _max_rank) {
  1311                     if (_buckets[old_rank_v] == v) {
  1312                       _buckets[old_rank_v] = _bucket_next[v];
  1313                     } else {
  1314                       int pv = _bucket_prev[v], nv = _bucket_next[v];
  1315                       _bucket_next[pv] = nv;
  1316                       _bucket_prev[nv] = pv;
  1317                     }
  1318                   }
  1319 
  1320                   // Insert v into its new bucket
  1321                   int nv = _buckets[new_rank_v];
  1322                   _bucket_next[v] = nv;
  1323                   _bucket_prev[nv] = v;
  1324                   _buckets[new_rank_v] = v;
  1325                 }
  1326               }
  1327             }
  1328           }
  1329 
  1330           // Finish search if there are no more active nodes
  1331           if (_excess[u] > 0) {
  1332             total_excess -= _excess[u];
  1333             if (total_excess <= 0) break;
  1334           }
  1335         }
  1336         if (total_excess <= 0) break;
  1337       }
  1338 
  1339       // Relabel nodes
  1340       for (int u = 0; u != _res_node_num; ++u) {
  1341         int k = std::min(_rank[u], r);
  1342         if (k > 0) {
  1343           _pi[u] -= _epsilon * k;
  1344           _next_out[u] = _first_out[u];
  1345         }
  1346       }
  1347     }
  1348 
  1349     /// Execute the algorithm performing augment and relabel operations
  1350     void startAugment(int max_length) {
  1351       // Paramters for heuristics
  1352       const int PRICE_REFINEMENT_LIMIT = 2;
  1353       const double GLOBAL_UPDATE_FACTOR = 1.0;
  1354       const int global_update_skip = static_cast<int>(GLOBAL_UPDATE_FACTOR *
  1355         (_res_node_num + _sup_node_num * _sup_node_num));
  1356       int next_global_update_limit = global_update_skip;
  1357 
  1358       // Perform cost scaling phases
  1359       IntVector path;
  1360       BoolVector path_arc(_res_arc_num, false);
  1361       int relabel_cnt = 0;
  1362       int eps_phase_cnt = 0;
  1363       for ( ; _epsilon >= 1; _epsilon = _epsilon < _alpha && _epsilon > 1 ?
  1364                                         1 : _epsilon / _alpha )
  1365       {
  1366         ++eps_phase_cnt;
  1367 
  1368         // Price refinement heuristic
  1369         if (eps_phase_cnt >= PRICE_REFINEMENT_LIMIT) {
  1370           if (priceRefinement()) continue;
  1371         }
  1372 
  1373         // Initialize current phase
  1374         initPhase();
  1375 
  1376         // Perform partial augment and relabel operations
  1377         while (true) {
  1378           // Select an active node (FIFO selection)
  1379           while (_active_nodes.size() > 0 &&
  1380                  _excess[_active_nodes.front()] <= 0) {
  1381             _active_nodes.pop_front();
  1382           }
  1383           if (_active_nodes.size() == 0) break;
  1384           int start = _active_nodes.front();
  1385 
  1386           // Find an augmenting path from the start node
  1387           int tip = start;
  1388           while (int(path.size()) < max_length && _excess[tip] >= 0) {
  1389             int u;
  1390             LargeCost rc, min_red_cost = std::numeric_limits<LargeCost>::max();
  1391             LargeCost pi_tip = _pi[tip];
  1392             int last_out = _first_out[tip+1];
  1393             for (int a = _next_out[tip]; a != last_out; ++a) {
  1394               if (_res_cap[a] > 0) {
  1395                 u = _target[a];
  1396                 rc = _cost[a] + pi_tip - _pi[u];
  1397                 if (rc < 0) {
  1398                   path.push_back(a);
  1399                   _next_out[tip] = a;
  1400                   if (path_arc[a]) {
  1401                     goto augment;   // a cycle is found, stop path search
  1402                   }
  1403                   tip = u;
  1404                   path_arc[a] = true;
  1405                   goto next_step;
  1406                 }
  1407                 else if (rc < min_red_cost) {
  1408                   min_red_cost = rc;
  1409                 }
  1410               }
  1411             }
  1412 
  1413             // Relabel tip node
  1414             if (tip != start) {
  1415               int ra = _reverse[path.back()];
  1416               min_red_cost =
  1417                 std::min(min_red_cost, _cost[ra] + pi_tip - _pi[_target[ra]]);
  1418             }
  1419             last_out = _next_out[tip];
  1420             for (int a = _first_out[tip]; a != last_out; ++a) {
  1421               if (_res_cap[a] > 0) {
  1422                 rc = _cost[a] + pi_tip - _pi[_target[a]];
  1423                 if (rc < min_red_cost) {
  1424                   min_red_cost = rc;
  1425                 }
  1426               }
  1427             }
  1428             _pi[tip] -= min_red_cost + _epsilon;
  1429             _next_out[tip] = _first_out[tip];
  1430             ++relabel_cnt;
  1431 
  1432             // Step back
  1433             if (tip != start) {
  1434               int pa = path.back();
  1435               path_arc[pa] = false;
  1436               tip = _source[pa];
  1437               path.pop_back();
  1438             }
  1439 
  1440           next_step: ;
  1441           }
  1442 
  1443           // Augment along the found path (as much flow as possible)
  1444         augment:
  1445           Value delta;
  1446           int pa, u, v = start;
  1447           for (int i = 0; i != int(path.size()); ++i) {
  1448             pa = path[i];
  1449             u = v;
  1450             v = _target[pa];
  1451             path_arc[pa] = false;
  1452             delta = std::min(_res_cap[pa], _excess[u]);
  1453             _res_cap[pa] -= delta;
  1454             _res_cap[_reverse[pa]] += delta;
  1455             _excess[u] -= delta;
  1456             _excess[v] += delta;
  1457             if (_excess[v] > 0 && _excess[v] <= delta) {
  1458               _active_nodes.push_back(v);
  1459             }
  1460           }
  1461           path.clear();
  1462 
  1463           // Global update heuristic
  1464           if (relabel_cnt >= next_global_update_limit) {
  1465             globalUpdate();
  1466             next_global_update_limit += global_update_skip;
  1467           }
  1468         }
  1469 
  1470       }
  1471 
  1472     }
  1473 
  1474     /// Execute the algorithm performing push and relabel operations
  1475     void startPush() {
  1476       // Paramters for heuristics
  1477       const int PRICE_REFINEMENT_LIMIT = 2;
  1478       const double GLOBAL_UPDATE_FACTOR = 2.0;
  1479 
  1480       const int global_update_skip = static_cast<int>(GLOBAL_UPDATE_FACTOR *
  1481         (_res_node_num + _sup_node_num * _sup_node_num));
  1482       int next_global_update_limit = global_update_skip;
  1483 
  1484       // Perform cost scaling phases
  1485       BoolVector hyper(_res_node_num, false);
  1486       LargeCostVector hyper_cost(_res_node_num);
  1487       int relabel_cnt = 0;
  1488       int eps_phase_cnt = 0;
  1489       for ( ; _epsilon >= 1; _epsilon = _epsilon < _alpha && _epsilon > 1 ?
  1490                                         1 : _epsilon / _alpha )
  1491       {
  1492         ++eps_phase_cnt;
  1493 
  1494         // Price refinement heuristic
  1495         if (eps_phase_cnt >= PRICE_REFINEMENT_LIMIT) {
  1496           if (priceRefinement()) continue;
  1497         }
  1498 
  1499         // Initialize current phase
  1500         initPhase();
  1501 
  1502         // Perform push and relabel operations
  1503         while (_active_nodes.size() > 0) {
  1504           LargeCost min_red_cost, rc, pi_n;
  1505           Value delta;
  1506           int n, t, a, last_out = _res_arc_num;
  1507 
  1508         next_node:
  1509           // Select an active node (FIFO selection)
  1510           n = _active_nodes.front();
  1511           last_out = _first_out[n+1];
  1512           pi_n = _pi[n];
  1513 
  1514           // Perform push operations if there are admissible arcs
  1515           if (_excess[n] > 0) {
  1516             for (a = _next_out[n]; a != last_out; ++a) {
  1517               if (_res_cap[a] > 0 &&
  1518                   _cost[a] + pi_n - _pi[_target[a]] < 0) {
  1519                 delta = std::min(_res_cap[a], _excess[n]);
  1520                 t = _target[a];
  1521 
  1522                 // Push-look-ahead heuristic
  1523                 Value ahead = -_excess[t];
  1524                 int last_out_t = _first_out[t+1];
  1525                 LargeCost pi_t = _pi[t];
  1526                 for (int ta = _next_out[t]; ta != last_out_t; ++ta) {
  1527                   if (_res_cap[ta] > 0 &&
  1528                       _cost[ta] + pi_t - _pi[_target[ta]] < 0)
  1529                     ahead += _res_cap[ta];
  1530                   if (ahead >= delta) break;
  1531                 }
  1532                 if (ahead < 0) ahead = 0;
  1533 
  1534                 // Push flow along the arc
  1535                 if (ahead < delta && !hyper[t]) {
  1536                   _res_cap[a] -= ahead;
  1537                   _res_cap[_reverse[a]] += ahead;
  1538                   _excess[n] -= ahead;
  1539                   _excess[t] += ahead;
  1540                   _active_nodes.push_front(t);
  1541                   hyper[t] = true;
  1542                   hyper_cost[t] = _cost[a] + pi_n - pi_t;
  1543                   _next_out[n] = a;
  1544                   goto next_node;
  1545                 } else {
  1546                   _res_cap[a] -= delta;
  1547                   _res_cap[_reverse[a]] += delta;
  1548                   _excess[n] -= delta;
  1549                   _excess[t] += delta;
  1550                   if (_excess[t] > 0 && _excess[t] <= delta)
  1551                     _active_nodes.push_back(t);
  1552                 }
  1553 
  1554                 if (_excess[n] == 0) {
  1555                   _next_out[n] = a;
  1556                   goto remove_nodes;
  1557                 }
  1558               }
  1559             }
  1560             _next_out[n] = a;
  1561           }
  1562 
  1563           // Relabel the node if it is still active (or hyper)
  1564           if (_excess[n] > 0 || hyper[n]) {
  1565              min_red_cost = hyper[n] ? -hyper_cost[n] :
  1566                std::numeric_limits<LargeCost>::max();
  1567             for (int a = _first_out[n]; a != last_out; ++a) {
  1568               if (_res_cap[a] > 0) {
  1569                 rc = _cost[a] + pi_n - _pi[_target[a]];
  1570                 if (rc < min_red_cost) {
  1571                   min_red_cost = rc;
  1572                 }
  1573               }
  1574             }
  1575             _pi[n] -= min_red_cost + _epsilon;
  1576             _next_out[n] = _first_out[n];
  1577             hyper[n] = false;
  1578             ++relabel_cnt;
  1579           }
  1580 
  1581           // Remove nodes that are not active nor hyper
  1582         remove_nodes:
  1583           while ( _active_nodes.size() > 0 &&
  1584                   _excess[_active_nodes.front()] <= 0 &&
  1585                   !hyper[_active_nodes.front()] ) {
  1586             _active_nodes.pop_front();
  1587           }
  1588 
  1589           // Global update heuristic
  1590           if (relabel_cnt >= next_global_update_limit) {
  1591             globalUpdate();
  1592             for (int u = 0; u != _res_node_num; ++u)
  1593               hyper[u] = false;
  1594             next_global_update_limit += global_update_skip;
  1595           }
  1596         }
  1597       }
  1598     }
  1599 
  1600   }; //class CostScaling
  1601 
  1602   ///@}
  1603 
  1604 } //namespace lemon
  1605 
  1606 #endif //LEMON_COST_SCALING_H