lemon/cost_scaling.h
author Antal Nemes <thoneyvazul@gmail.com>
Tue, 30 Nov 2010 20:21:52 +0100
changeset 1056 92a884824429
parent 1049 7bf489cf624e
child 1071 879fcb781086
permissions -rw-r--r--
Port Edmonds-Karp algorithm from svn -r3524 (#177)
     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     /// The \ref CostScalingDefaultTraits "traits class" of the algorithm
   156     typedef TR Traits;
   157 
   158   public:
   159 
   160     /// \brief Problem type constants for the \c run() function.
   161     ///
   162     /// Enum type containing the problem type constants that can be
   163     /// returned by the \ref run() function of the algorithm.
   164     enum ProblemType {
   165       /// The problem has no feasible solution (flow).
   166       INFEASIBLE,
   167       /// The problem has optimal solution (i.e. it is feasible and
   168       /// bounded), and the algorithm has found optimal flow and node
   169       /// potentials (primal and dual solutions).
   170       OPTIMAL,
   171       /// The digraph contains an arc of negative cost and infinite
   172       /// upper bound. It means that the objective function is unbounded
   173       /// on that arc, however, note that it could actually be bounded
   174       /// over the feasible flows, but this algroithm cannot handle
   175       /// these cases.
   176       UNBOUNDED
   177     };
   178 
   179     /// \brief Constants for selecting the internal method.
   180     ///
   181     /// Enum type containing constants for selecting the internal method
   182     /// for the \ref run() function.
   183     ///
   184     /// \ref CostScaling provides three internal methods that differ mainly
   185     /// in their base operations, which are used in conjunction with the
   186     /// relabel operation.
   187     /// By default, the so called \ref PARTIAL_AUGMENT
   188     /// "Partial Augment-Relabel" method is used, which turned out to be
   189     /// the most efficient and the most robust on various test inputs.
   190     /// However, the other methods can be selected using the \ref run()
   191     /// function with the proper parameter.
   192     enum Method {
   193       /// Local push operations are used, i.e. flow is moved only on one
   194       /// admissible arc at once.
   195       PUSH,
   196       /// Augment operations are used, i.e. flow is moved on admissible
   197       /// paths from a node with excess to a node with deficit.
   198       AUGMENT,
   199       /// Partial augment operations are used, i.e. flow is moved on
   200       /// admissible paths started from a node with excess, but the
   201       /// lengths of these paths are limited. This method can be viewed
   202       /// as a combined version of the previous two operations.
   203       PARTIAL_AUGMENT
   204     };
   205 
   206   private:
   207 
   208     TEMPLATE_DIGRAPH_TYPEDEFS(GR);
   209 
   210     typedef std::vector<int> IntVector;
   211     typedef std::vector<Value> ValueVector;
   212     typedef std::vector<Cost> CostVector;
   213     typedef std::vector<LargeCost> LargeCostVector;
   214     typedef std::vector<char> BoolVector;
   215     // Note: vector<char> is used instead of vector<bool> for efficiency reasons
   216 
   217   private:
   218 
   219     template <typename KT, typename VT>
   220     class StaticVectorMap {
   221     public:
   222       typedef KT Key;
   223       typedef VT Value;
   224 
   225       StaticVectorMap(std::vector<Value>& v) : _v(v) {}
   226 
   227       const Value& operator[](const Key& key) const {
   228         return _v[StaticDigraph::id(key)];
   229       }
   230 
   231       Value& operator[](const Key& key) {
   232         return _v[StaticDigraph::id(key)];
   233       }
   234 
   235       void set(const Key& key, const Value& val) {
   236         _v[StaticDigraph::id(key)] = val;
   237       }
   238 
   239     private:
   240       std::vector<Value>& _v;
   241     };
   242 
   243     typedef StaticVectorMap<StaticDigraph::Arc, LargeCost> LargeCostArcMap;
   244 
   245   private:
   246 
   247     // Data related to the underlying digraph
   248     const GR &_graph;
   249     int _node_num;
   250     int _arc_num;
   251     int _res_node_num;
   252     int _res_arc_num;
   253     int _root;
   254 
   255     // Parameters of the problem
   256     bool _have_lower;
   257     Value _sum_supply;
   258     int _sup_node_num;
   259 
   260     // Data structures for storing the digraph
   261     IntNodeMap _node_id;
   262     IntArcMap _arc_idf;
   263     IntArcMap _arc_idb;
   264     IntVector _first_out;
   265     BoolVector _forward;
   266     IntVector _source;
   267     IntVector _target;
   268     IntVector _reverse;
   269 
   270     // Node and arc data
   271     ValueVector _lower;
   272     ValueVector _upper;
   273     CostVector _scost;
   274     ValueVector _supply;
   275 
   276     ValueVector _res_cap;
   277     LargeCostVector _cost;
   278     LargeCostVector _pi;
   279     ValueVector _excess;
   280     IntVector _next_out;
   281     std::deque<int> _active_nodes;
   282 
   283     // Data for scaling
   284     LargeCost _epsilon;
   285     int _alpha;
   286 
   287     IntVector _buckets;
   288     IntVector _bucket_next;
   289     IntVector _bucket_prev;
   290     IntVector _rank;
   291     int _max_rank;
   292 
   293   public:
   294 
   295     /// \brief Constant for infinite upper bounds (capacities).
   296     ///
   297     /// Constant for infinite upper bounds (capacities).
   298     /// It is \c std::numeric_limits<Value>::infinity() if available,
   299     /// \c std::numeric_limits<Value>::max() otherwise.
   300     const Value INF;
   301 
   302   public:
   303 
   304     /// \name Named Template Parameters
   305     /// @{
   306 
   307     template <typename T>
   308     struct SetLargeCostTraits : public Traits {
   309       typedef T LargeCost;
   310     };
   311 
   312     /// \brief \ref named-templ-param "Named parameter" for setting
   313     /// \c LargeCost type.
   314     ///
   315     /// \ref named-templ-param "Named parameter" for setting \c LargeCost
   316     /// type, which is used for internal computations in the algorithm.
   317     /// \c Cost must be convertible to \c LargeCost.
   318     template <typename T>
   319     struct SetLargeCost
   320       : public CostScaling<GR, V, C, SetLargeCostTraits<T> > {
   321       typedef  CostScaling<GR, V, C, SetLargeCostTraits<T> > Create;
   322     };
   323 
   324     /// @}
   325 
   326   protected:
   327 
   328     CostScaling() {}
   329 
   330   public:
   331 
   332     /// \brief Constructor.
   333     ///
   334     /// The constructor of the class.
   335     ///
   336     /// \param graph The digraph the algorithm runs on.
   337     CostScaling(const GR& graph) :
   338       _graph(graph), _node_id(graph), _arc_idf(graph), _arc_idb(graph),
   339       INF(std::numeric_limits<Value>::has_infinity ?
   340           std::numeric_limits<Value>::infinity() :
   341           std::numeric_limits<Value>::max())
   342     {
   343       // Check the number types
   344       LEMON_ASSERT(std::numeric_limits<Value>::is_signed,
   345         "The flow type of CostScaling must be signed");
   346       LEMON_ASSERT(std::numeric_limits<Cost>::is_signed,
   347         "The cost type of CostScaling must be signed");
   348 
   349       // Reset data structures
   350       reset();
   351     }
   352 
   353     /// \name Parameters
   354     /// The parameters of the algorithm can be specified using these
   355     /// functions.
   356 
   357     /// @{
   358 
   359     /// \brief Set the lower bounds on the arcs.
   360     ///
   361     /// This function sets the lower bounds on the arcs.
   362     /// If it is not used before calling \ref run(), the lower bounds
   363     /// will be set to zero on all arcs.
   364     ///
   365     /// \param map An arc map storing the lower bounds.
   366     /// Its \c Value type must be convertible to the \c Value type
   367     /// of the algorithm.
   368     ///
   369     /// \return <tt>(*this)</tt>
   370     template <typename LowerMap>
   371     CostScaling& lowerMap(const LowerMap& map) {
   372       _have_lower = true;
   373       for (ArcIt a(_graph); a != INVALID; ++a) {
   374         _lower[_arc_idf[a]] = map[a];
   375         _lower[_arc_idb[a]] = map[a];
   376       }
   377       return *this;
   378     }
   379 
   380     /// \brief Set the upper bounds (capacities) on the arcs.
   381     ///
   382     /// This function sets the upper bounds (capacities) on the arcs.
   383     /// If it is not used before calling \ref run(), the upper bounds
   384     /// will be set to \ref INF on all arcs (i.e. the flow value will be
   385     /// unbounded from above).
   386     ///
   387     /// \param map An arc map storing the upper bounds.
   388     /// Its \c Value type must be convertible to the \c Value type
   389     /// of the algorithm.
   390     ///
   391     /// \return <tt>(*this)</tt>
   392     template<typename UpperMap>
   393     CostScaling& upperMap(const UpperMap& map) {
   394       for (ArcIt a(_graph); a != INVALID; ++a) {
   395         _upper[_arc_idf[a]] = map[a];
   396       }
   397       return *this;
   398     }
   399 
   400     /// \brief Set the costs of the arcs.
   401     ///
   402     /// This function sets the costs of the arcs.
   403     /// If it is not used before calling \ref run(), the costs
   404     /// will be set to \c 1 on all arcs.
   405     ///
   406     /// \param map An arc map storing the costs.
   407     /// Its \c Value type must be convertible to the \c Cost type
   408     /// of the algorithm.
   409     ///
   410     /// \return <tt>(*this)</tt>
   411     template<typename CostMap>
   412     CostScaling& costMap(const CostMap& map) {
   413       for (ArcIt a(_graph); a != INVALID; ++a) {
   414         _scost[_arc_idf[a]] =  map[a];
   415         _scost[_arc_idb[a]] = -map[a];
   416       }
   417       return *this;
   418     }
   419 
   420     /// \brief Set the supply values of the nodes.
   421     ///
   422     /// This function sets the supply values of the nodes.
   423     /// If neither this function nor \ref stSupply() is used before
   424     /// calling \ref run(), the supply of each node will be set to zero.
   425     ///
   426     /// \param map A node map storing the supply values.
   427     /// Its \c Value type must be convertible to the \c Value type
   428     /// of the algorithm.
   429     ///
   430     /// \return <tt>(*this)</tt>
   431     template<typename SupplyMap>
   432     CostScaling& supplyMap(const SupplyMap& map) {
   433       for (NodeIt n(_graph); n != INVALID; ++n) {
   434         _supply[_node_id[n]] = map[n];
   435       }
   436       return *this;
   437     }
   438 
   439     /// \brief Set single source and target nodes and a supply value.
   440     ///
   441     /// This function sets a single source node and a single target node
   442     /// and the required flow value.
   443     /// If neither this function nor \ref supplyMap() is used before
   444     /// calling \ref run(), the supply of each node will be set to zero.
   445     ///
   446     /// Using this function has the same effect as using \ref supplyMap()
   447     /// with a map in which \c k is assigned to \c s, \c -k is
   448     /// assigned to \c t and all other nodes have zero supply value.
   449     ///
   450     /// \param s The source node.
   451     /// \param t The target node.
   452     /// \param k The required amount of flow from node \c s to node \c t
   453     /// (i.e. the supply of \c s and the demand of \c t).
   454     ///
   455     /// \return <tt>(*this)</tt>
   456     CostScaling& stSupply(const Node& s, const Node& t, Value k) {
   457       for (int i = 0; i != _res_node_num; ++i) {
   458         _supply[i] = 0;
   459       }
   460       _supply[_node_id[s]] =  k;
   461       _supply[_node_id[t]] = -k;
   462       return *this;
   463     }
   464 
   465     /// @}
   466 
   467     /// \name Execution control
   468     /// The algorithm can be executed using \ref run().
   469 
   470     /// @{
   471 
   472     /// \brief Run the algorithm.
   473     ///
   474     /// This function runs the algorithm.
   475     /// The paramters can be specified using functions \ref lowerMap(),
   476     /// \ref upperMap(), \ref costMap(), \ref supplyMap(), \ref stSupply().
   477     /// For example,
   478     /// \code
   479     ///   CostScaling<ListDigraph> cs(graph);
   480     ///   cs.lowerMap(lower).upperMap(upper).costMap(cost)
   481     ///     .supplyMap(sup).run();
   482     /// \endcode
   483     ///
   484     /// This function can be called more than once. All the given parameters
   485     /// are kept for the next call, unless \ref resetParams() or \ref reset()
   486     /// is used, thus only the modified parameters have to be set again.
   487     /// If the underlying digraph was also modified after the construction
   488     /// of the class (or the last \ref reset() call), then the \ref reset()
   489     /// function must be called.
   490     ///
   491     /// \param method The internal method that will be used in the
   492     /// algorithm. For more information, see \ref Method.
   493     /// \param factor The cost scaling factor. It must be at least two.
   494     ///
   495     /// \return \c INFEASIBLE if no feasible flow exists,
   496     /// \n \c OPTIMAL if the problem has optimal solution
   497     /// (i.e. it is feasible and bounded), and the algorithm has found
   498     /// optimal flow and node potentials (primal and dual solutions),
   499     /// \n \c UNBOUNDED if the digraph contains an arc of negative cost
   500     /// and infinite upper bound. It means that the objective function
   501     /// is unbounded on that arc, however, note that it could actually be
   502     /// bounded over the feasible flows, but this algroithm cannot handle
   503     /// these cases.
   504     ///
   505     /// \see ProblemType, Method
   506     /// \see resetParams(), reset()
   507     ProblemType run(Method method = PARTIAL_AUGMENT, int factor = 16) {
   508       LEMON_ASSERT(factor >= 2, "The scaling factor must be at least 2");
   509       _alpha = factor;
   510       ProblemType pt = init();
   511       if (pt != OPTIMAL) return pt;
   512       start(method);
   513       return OPTIMAL;
   514     }
   515 
   516     /// \brief Reset all the parameters that have been given before.
   517     ///
   518     /// This function resets all the paramaters that have been given
   519     /// before using functions \ref lowerMap(), \ref upperMap(),
   520     /// \ref costMap(), \ref supplyMap(), \ref stSupply().
   521     ///
   522     /// It is useful for multiple \ref run() calls. Basically, all the given
   523     /// parameters are kept for the next \ref run() call, unless
   524     /// \ref resetParams() or \ref reset() is used.
   525     /// If the underlying digraph was also modified after the construction
   526     /// of the class or the last \ref reset() call, then the \ref reset()
   527     /// function must be used, otherwise \ref resetParams() is sufficient.
   528     ///
   529     /// For example,
   530     /// \code
   531     ///   CostScaling<ListDigraph> cs(graph);
   532     ///
   533     ///   // First run
   534     ///   cs.lowerMap(lower).upperMap(upper).costMap(cost)
   535     ///     .supplyMap(sup).run();
   536     ///
   537     ///   // Run again with modified cost map (resetParams() is not called,
   538     ///   // so only the cost map have to be set again)
   539     ///   cost[e] += 100;
   540     ///   cs.costMap(cost).run();
   541     ///
   542     ///   // Run again from scratch using resetParams()
   543     ///   // (the lower bounds will be set to zero on all arcs)
   544     ///   cs.resetParams();
   545     ///   cs.upperMap(capacity).costMap(cost)
   546     ///     .supplyMap(sup).run();
   547     /// \endcode
   548     ///
   549     /// \return <tt>(*this)</tt>
   550     ///
   551     /// \see reset(), run()
   552     CostScaling& resetParams() {
   553       for (int i = 0; i != _res_node_num; ++i) {
   554         _supply[i] = 0;
   555       }
   556       int limit = _first_out[_root];
   557       for (int j = 0; j != limit; ++j) {
   558         _lower[j] = 0;
   559         _upper[j] = INF;
   560         _scost[j] = _forward[j] ? 1 : -1;
   561       }
   562       for (int j = limit; j != _res_arc_num; ++j) {
   563         _lower[j] = 0;
   564         _upper[j] = INF;
   565         _scost[j] = 0;
   566         _scost[_reverse[j]] = 0;
   567       }
   568       _have_lower = false;
   569       return *this;
   570     }
   571 
   572     /// \brief Reset the internal data structures and all the parameters
   573     /// that have been given before.
   574     ///
   575     /// This function resets the internal data structures and all the
   576     /// paramaters that have been given before using functions \ref lowerMap(),
   577     /// \ref upperMap(), \ref costMap(), \ref supplyMap(), \ref stSupply().
   578     ///
   579     /// It is useful for multiple \ref run() calls. By default, all the given
   580     /// parameters are kept for the next \ref run() call, unless
   581     /// \ref resetParams() or \ref reset() is used.
   582     /// If the underlying digraph was also modified after the construction
   583     /// of the class or the last \ref reset() call, then the \ref reset()
   584     /// function must be used, otherwise \ref resetParams() is sufficient.
   585     ///
   586     /// See \ref resetParams() for examples.
   587     ///
   588     /// \return <tt>(*this)</tt>
   589     ///
   590     /// \see resetParams(), run()
   591     CostScaling& reset() {
   592       // Resize vectors
   593       _node_num = countNodes(_graph);
   594       _arc_num = countArcs(_graph);
   595       _res_node_num = _node_num + 1;
   596       _res_arc_num = 2 * (_arc_num + _node_num);
   597       _root = _node_num;
   598 
   599       _first_out.resize(_res_node_num + 1);
   600       _forward.resize(_res_arc_num);
   601       _source.resize(_res_arc_num);
   602       _target.resize(_res_arc_num);
   603       _reverse.resize(_res_arc_num);
   604 
   605       _lower.resize(_res_arc_num);
   606       _upper.resize(_res_arc_num);
   607       _scost.resize(_res_arc_num);
   608       _supply.resize(_res_node_num);
   609 
   610       _res_cap.resize(_res_arc_num);
   611       _cost.resize(_res_arc_num);
   612       _pi.resize(_res_node_num);
   613       _excess.resize(_res_node_num);
   614       _next_out.resize(_res_node_num);
   615 
   616       // Copy the graph
   617       int i = 0, j = 0, k = 2 * _arc_num + _node_num;
   618       for (NodeIt n(_graph); n != INVALID; ++n, ++i) {
   619         _node_id[n] = i;
   620       }
   621       i = 0;
   622       for (NodeIt n(_graph); n != INVALID; ++n, ++i) {
   623         _first_out[i] = j;
   624         for (OutArcIt a(_graph, n); a != INVALID; ++a, ++j) {
   625           _arc_idf[a] = j;
   626           _forward[j] = true;
   627           _source[j] = i;
   628           _target[j] = _node_id[_graph.runningNode(a)];
   629         }
   630         for (InArcIt a(_graph, n); a != INVALID; ++a, ++j) {
   631           _arc_idb[a] = j;
   632           _forward[j] = false;
   633           _source[j] = i;
   634           _target[j] = _node_id[_graph.runningNode(a)];
   635         }
   636         _forward[j] = false;
   637         _source[j] = i;
   638         _target[j] = _root;
   639         _reverse[j] = k;
   640         _forward[k] = true;
   641         _source[k] = _root;
   642         _target[k] = i;
   643         _reverse[k] = j;
   644         ++j; ++k;
   645       }
   646       _first_out[i] = j;
   647       _first_out[_res_node_num] = k;
   648       for (ArcIt a(_graph); a != INVALID; ++a) {
   649         int fi = _arc_idf[a];
   650         int bi = _arc_idb[a];
   651         _reverse[fi] = bi;
   652         _reverse[bi] = fi;
   653       }
   654 
   655       // Reset parameters
   656       resetParams();
   657       return *this;
   658     }
   659 
   660     /// @}
   661 
   662     /// \name Query Functions
   663     /// The results of the algorithm can be obtained using these
   664     /// functions.\n
   665     /// The \ref run() function must be called before using them.
   666 
   667     /// @{
   668 
   669     /// \brief Return the total cost of the found flow.
   670     ///
   671     /// This function returns the total cost of the found flow.
   672     /// Its complexity is O(e).
   673     ///
   674     /// \note The return type of the function can be specified as a
   675     /// template parameter. For example,
   676     /// \code
   677     ///   cs.totalCost<double>();
   678     /// \endcode
   679     /// It is useful if the total cost cannot be stored in the \c Cost
   680     /// type of the algorithm, which is the default return type of the
   681     /// function.
   682     ///
   683     /// \pre \ref run() must be called before using this function.
   684     template <typename Number>
   685     Number totalCost() const {
   686       Number c = 0;
   687       for (ArcIt a(_graph); a != INVALID; ++a) {
   688         int i = _arc_idb[a];
   689         c += static_cast<Number>(_res_cap[i]) *
   690              (-static_cast<Number>(_scost[i]));
   691       }
   692       return c;
   693     }
   694 
   695 #ifndef DOXYGEN
   696     Cost totalCost() const {
   697       return totalCost<Cost>();
   698     }
   699 #endif
   700 
   701     /// \brief Return the flow on the given arc.
   702     ///
   703     /// This function returns the flow on the given arc.
   704     ///
   705     /// \pre \ref run() must be called before using this function.
   706     Value flow(const Arc& a) const {
   707       return _res_cap[_arc_idb[a]];
   708     }
   709 
   710     /// \brief Copy the flow values (the primal solution) into the
   711     /// given map.
   712     ///
   713     /// This function copies the flow value on each arc into the given
   714     /// map. The \c Value type of the algorithm must be convertible to
   715     /// the \c Value type of the map.
   716     ///
   717     /// \pre \ref run() must be called before using this function.
   718     template <typename FlowMap>
   719     void flowMap(FlowMap &map) const {
   720       for (ArcIt a(_graph); a != INVALID; ++a) {
   721         map.set(a, _res_cap[_arc_idb[a]]);
   722       }
   723     }
   724 
   725     /// \brief Return the potential (dual value) of the given node.
   726     ///
   727     /// This function returns the potential (dual value) of the
   728     /// given node.
   729     ///
   730     /// \pre \ref run() must be called before using this function.
   731     Cost potential(const Node& n) const {
   732       return static_cast<Cost>(_pi[_node_id[n]]);
   733     }
   734 
   735     /// \brief Copy the potential values (the dual solution) into the
   736     /// given map.
   737     ///
   738     /// This function copies the potential (dual value) of each node
   739     /// into the given map.
   740     /// The \c Cost type of the algorithm must be convertible to the
   741     /// \c Value type of the map.
   742     ///
   743     /// \pre \ref run() must be called before using this function.
   744     template <typename PotentialMap>
   745     void potentialMap(PotentialMap &map) const {
   746       for (NodeIt n(_graph); n != INVALID; ++n) {
   747         map.set(n, static_cast<Cost>(_pi[_node_id[n]]));
   748       }
   749     }
   750 
   751     /// @}
   752 
   753   private:
   754 
   755     // Initialize the algorithm
   756     ProblemType init() {
   757       if (_res_node_num <= 1) return INFEASIBLE;
   758 
   759       // Check the sum of supply values
   760       _sum_supply = 0;
   761       for (int i = 0; i != _root; ++i) {
   762         _sum_supply += _supply[i];
   763       }
   764       if (_sum_supply > 0) return INFEASIBLE;
   765 
   766 
   767       // Initialize vectors
   768       for (int i = 0; i != _res_node_num; ++i) {
   769         _pi[i] = 0;
   770         _excess[i] = _supply[i];
   771       }
   772 
   773       // Remove infinite upper bounds and check negative arcs
   774       const Value MAX = std::numeric_limits<Value>::max();
   775       int last_out;
   776       if (_have_lower) {
   777         for (int i = 0; i != _root; ++i) {
   778           last_out = _first_out[i+1];
   779           for (int j = _first_out[i]; j != last_out; ++j) {
   780             if (_forward[j]) {
   781               Value c = _scost[j] < 0 ? _upper[j] : _lower[j];
   782               if (c >= MAX) return UNBOUNDED;
   783               _excess[i] -= c;
   784               _excess[_target[j]] += c;
   785             }
   786           }
   787         }
   788       } else {
   789         for (int i = 0; i != _root; ++i) {
   790           last_out = _first_out[i+1];
   791           for (int j = _first_out[i]; j != last_out; ++j) {
   792             if (_forward[j] && _scost[j] < 0) {
   793               Value c = _upper[j];
   794               if (c >= MAX) return UNBOUNDED;
   795               _excess[i] -= c;
   796               _excess[_target[j]] += c;
   797             }
   798           }
   799         }
   800       }
   801       Value ex, max_cap = 0;
   802       for (int i = 0; i != _res_node_num; ++i) {
   803         ex = _excess[i];
   804         _excess[i] = 0;
   805         if (ex < 0) max_cap -= ex;
   806       }
   807       for (int j = 0; j != _res_arc_num; ++j) {
   808         if (_upper[j] >= MAX) _upper[j] = max_cap;
   809       }
   810 
   811       // Initialize the large cost vector and the epsilon parameter
   812       _epsilon = 0;
   813       LargeCost lc;
   814       for (int i = 0; i != _root; ++i) {
   815         last_out = _first_out[i+1];
   816         for (int j = _first_out[i]; j != last_out; ++j) {
   817           lc = static_cast<LargeCost>(_scost[j]) * _res_node_num * _alpha;
   818           _cost[j] = lc;
   819           if (lc > _epsilon) _epsilon = lc;
   820         }
   821       }
   822       _epsilon /= _alpha;
   823 
   824       // Initialize maps for Circulation and remove non-zero lower bounds
   825       ConstMap<Arc, Value> low(0);
   826       typedef typename Digraph::template ArcMap<Value> ValueArcMap;
   827       typedef typename Digraph::template NodeMap<Value> ValueNodeMap;
   828       ValueArcMap cap(_graph), flow(_graph);
   829       ValueNodeMap sup(_graph);
   830       for (NodeIt n(_graph); n != INVALID; ++n) {
   831         sup[n] = _supply[_node_id[n]];
   832       }
   833       if (_have_lower) {
   834         for (ArcIt a(_graph); a != INVALID; ++a) {
   835           int j = _arc_idf[a];
   836           Value c = _lower[j];
   837           cap[a] = _upper[j] - c;
   838           sup[_graph.source(a)] -= c;
   839           sup[_graph.target(a)] += c;
   840         }
   841       } else {
   842         for (ArcIt a(_graph); a != INVALID; ++a) {
   843           cap[a] = _upper[_arc_idf[a]];
   844         }
   845       }
   846 
   847       _sup_node_num = 0;
   848       for (NodeIt n(_graph); n != INVALID; ++n) {
   849         if (sup[n] > 0) ++_sup_node_num;
   850       }
   851 
   852       // Find a feasible flow using Circulation
   853       Circulation<Digraph, ConstMap<Arc, Value>, ValueArcMap, ValueNodeMap>
   854         circ(_graph, low, cap, sup);
   855       if (!circ.flowMap(flow).run()) return INFEASIBLE;
   856 
   857       // Set residual capacities and handle GEQ supply type
   858       if (_sum_supply < 0) {
   859         for (ArcIt a(_graph); a != INVALID; ++a) {
   860           Value fa = flow[a];
   861           _res_cap[_arc_idf[a]] = cap[a] - fa;
   862           _res_cap[_arc_idb[a]] = fa;
   863           sup[_graph.source(a)] -= fa;
   864           sup[_graph.target(a)] += fa;
   865         }
   866         for (NodeIt n(_graph); n != INVALID; ++n) {
   867           _excess[_node_id[n]] = sup[n];
   868         }
   869         for (int a = _first_out[_root]; a != _res_arc_num; ++a) {
   870           int u = _target[a];
   871           int ra = _reverse[a];
   872           _res_cap[a] = -_sum_supply + 1;
   873           _res_cap[ra] = -_excess[u];
   874           _cost[a] = 0;
   875           _cost[ra] = 0;
   876           _excess[u] = 0;
   877         }
   878       } else {
   879         for (ArcIt a(_graph); a != INVALID; ++a) {
   880           Value fa = flow[a];
   881           _res_cap[_arc_idf[a]] = cap[a] - fa;
   882           _res_cap[_arc_idb[a]] = fa;
   883         }
   884         for (int a = _first_out[_root]; a != _res_arc_num; ++a) {
   885           int ra = _reverse[a];
   886           _res_cap[a] = 0;
   887           _res_cap[ra] = 0;
   888           _cost[a] = 0;
   889           _cost[ra] = 0;
   890         }
   891       }
   892 
   893       // Initialize data structures for buckets
   894       _max_rank = _alpha * _res_node_num;
   895       _buckets.resize(_max_rank);
   896       _bucket_next.resize(_res_node_num + 1);
   897       _bucket_prev.resize(_res_node_num + 1);
   898       _rank.resize(_res_node_num + 1);
   899 
   900       return OPTIMAL;
   901     }
   902 
   903     // Execute the algorithm and transform the results
   904     void start(Method method) {
   905       const int MAX_PARTIAL_PATH_LENGTH = 4;
   906 
   907       switch (method) {
   908         case PUSH:
   909           startPush();
   910           break;
   911         case AUGMENT:
   912           startAugment(_res_node_num - 1);
   913           break;
   914         case PARTIAL_AUGMENT:
   915           startAugment(MAX_PARTIAL_PATH_LENGTH);
   916           break;
   917       }
   918 
   919       // Compute node potentials (dual solution)
   920       for (int i = 0; i != _res_node_num; ++i) {
   921         _pi[i] = static_cast<Cost>(_pi[i] / (_res_node_num * _alpha));
   922       }
   923       bool optimal = true;
   924       for (int i = 0; optimal && i != _res_node_num; ++i) {
   925         LargeCost pi_i = _pi[i];
   926         int last_out = _first_out[i+1];
   927         for (int j = _first_out[i]; j != last_out; ++j) {
   928           if (_res_cap[j] > 0 && _scost[j] + pi_i - _pi[_target[j]] < 0) {
   929             optimal = false;
   930             break;
   931           }
   932         }
   933       }
   934 
   935       if (!optimal) {
   936         // Compute node potentials for the original costs with BellmanFord
   937         // (if it is necessary)
   938         typedef std::pair<int, int> IntPair;
   939         StaticDigraph sgr;
   940         std::vector<IntPair> arc_vec;
   941         std::vector<LargeCost> cost_vec;
   942         LargeCostArcMap cost_map(cost_vec);
   943 
   944         arc_vec.clear();
   945         cost_vec.clear();
   946         for (int j = 0; j != _res_arc_num; ++j) {
   947           if (_res_cap[j] > 0) {
   948             int u = _source[j], v = _target[j];
   949             arc_vec.push_back(IntPair(u, v));
   950             cost_vec.push_back(_scost[j] + _pi[u] - _pi[v]);
   951           }
   952         }
   953         sgr.build(_res_node_num, arc_vec.begin(), arc_vec.end());
   954 
   955         typename BellmanFord<StaticDigraph, LargeCostArcMap>::Create
   956           bf(sgr, cost_map);
   957         bf.init(0);
   958         bf.start();
   959 
   960         for (int i = 0; i != _res_node_num; ++i) {
   961           _pi[i] += bf.dist(sgr.node(i));
   962         }
   963       }
   964 
   965       // Shift potentials to meet the requirements of the GEQ type
   966       // optimality conditions
   967       LargeCost max_pot = _pi[_root];
   968       for (int i = 0; i != _res_node_num; ++i) {
   969         if (_pi[i] > max_pot) max_pot = _pi[i];
   970       }
   971       if (max_pot != 0) {
   972         for (int i = 0; i != _res_node_num; ++i) {
   973           _pi[i] -= max_pot;
   974         }
   975       }
   976 
   977       // Handle non-zero lower bounds
   978       if (_have_lower) {
   979         int limit = _first_out[_root];
   980         for (int j = 0; j != limit; ++j) {
   981           if (!_forward[j]) _res_cap[j] += _lower[j];
   982         }
   983       }
   984     }
   985 
   986     // Initialize a cost scaling phase
   987     void initPhase() {
   988       // Saturate arcs not satisfying the optimality condition
   989       for (int u = 0; u != _res_node_num; ++u) {
   990         int last_out = _first_out[u+1];
   991         LargeCost pi_u = _pi[u];
   992         for (int a = _first_out[u]; a != last_out; ++a) {
   993           Value delta = _res_cap[a];
   994           if (delta > 0) {
   995             int v = _target[a];
   996             if (_cost[a] + pi_u - _pi[v] < 0) {
   997               _excess[u] -= delta;
   998               _excess[v] += delta;
   999               _res_cap[a] = 0;
  1000               _res_cap[_reverse[a]] += delta;
  1001             }
  1002           }
  1003         }
  1004       }
  1005 
  1006       // Find active nodes (i.e. nodes with positive excess)
  1007       for (int u = 0; u != _res_node_num; ++u) {
  1008         if (_excess[u] > 0) _active_nodes.push_back(u);
  1009       }
  1010 
  1011       // Initialize the next arcs
  1012       for (int u = 0; u != _res_node_num; ++u) {
  1013         _next_out[u] = _first_out[u];
  1014       }
  1015     }
  1016 
  1017     // Price (potential) refinement heuristic
  1018     bool priceRefinement() {
  1019 
  1020       // Stack for stroing the topological order
  1021       IntVector stack(_res_node_num);
  1022       int stack_top;
  1023 
  1024       // Perform phases
  1025       while (topologicalSort(stack, stack_top)) {
  1026 
  1027         // Compute node ranks in the acyclic admissible network and
  1028         // store the nodes in buckets
  1029         for (int i = 0; i != _res_node_num; ++i) {
  1030           _rank[i] = 0;
  1031         }
  1032         const int bucket_end = _root + 1;
  1033         for (int r = 0; r != _max_rank; ++r) {
  1034           _buckets[r] = bucket_end;
  1035         }
  1036         int top_rank = 0;
  1037         for ( ; stack_top >= 0; --stack_top) {
  1038           int u = stack[stack_top], v;
  1039           int rank_u = _rank[u];
  1040 
  1041           LargeCost rc, pi_u = _pi[u];
  1042           int last_out = _first_out[u+1];
  1043           for (int a = _first_out[u]; a != last_out; ++a) {
  1044             if (_res_cap[a] > 0) {
  1045               v = _target[a];
  1046               rc = _cost[a] + pi_u - _pi[v];
  1047               if (rc < 0) {
  1048                 LargeCost nrc = static_cast<LargeCost>((-rc - 0.5) / _epsilon);
  1049                 if (nrc < LargeCost(_max_rank)) {
  1050                   int new_rank_v = rank_u + static_cast<int>(nrc);
  1051                   if (new_rank_v > _rank[v]) {
  1052                     _rank[v] = new_rank_v;
  1053                   }
  1054                 }
  1055               }
  1056             }
  1057           }
  1058 
  1059           if (rank_u > 0) {
  1060             top_rank = std::max(top_rank, rank_u);
  1061             int bfirst = _buckets[rank_u];
  1062             _bucket_next[u] = bfirst;
  1063             _bucket_prev[bfirst] = u;
  1064             _buckets[rank_u] = u;
  1065           }
  1066         }
  1067 
  1068         // Check if the current flow is epsilon-optimal
  1069         if (top_rank == 0) {
  1070           return true;
  1071         }
  1072 
  1073         // Process buckets in top-down order
  1074         for (int rank = top_rank; rank > 0; --rank) {
  1075           while (_buckets[rank] != bucket_end) {
  1076             // Remove the first node from the current bucket
  1077             int u = _buckets[rank];
  1078             _buckets[rank] = _bucket_next[u];
  1079 
  1080             // Search the outgoing arcs of u
  1081             LargeCost rc, pi_u = _pi[u];
  1082             int last_out = _first_out[u+1];
  1083             int v, old_rank_v, new_rank_v;
  1084             for (int a = _first_out[u]; a != last_out; ++a) {
  1085               if (_res_cap[a] > 0) {
  1086                 v = _target[a];
  1087                 old_rank_v = _rank[v];
  1088 
  1089                 if (old_rank_v < rank) {
  1090 
  1091                   // Compute the new rank of node v
  1092                   rc = _cost[a] + pi_u - _pi[v];
  1093                   if (rc < 0) {
  1094                     new_rank_v = rank;
  1095                   } else {
  1096                     LargeCost nrc = rc / _epsilon;
  1097                     new_rank_v = 0;
  1098                     if (nrc < LargeCost(_max_rank)) {
  1099                       new_rank_v = rank - 1 - static_cast<int>(nrc);
  1100                     }
  1101                   }
  1102 
  1103                   // Change the rank of node v
  1104                   if (new_rank_v > old_rank_v) {
  1105                     _rank[v] = new_rank_v;
  1106 
  1107                     // Remove v from its old bucket
  1108                     if (old_rank_v > 0) {
  1109                       if (_buckets[old_rank_v] == v) {
  1110                         _buckets[old_rank_v] = _bucket_next[v];
  1111                       } else {
  1112                         int pv = _bucket_prev[v], nv = _bucket_next[v];
  1113                         _bucket_next[pv] = nv;
  1114                         _bucket_prev[nv] = pv;
  1115                       }
  1116                     }
  1117 
  1118                     // Insert v into its new bucket
  1119                     int nv = _buckets[new_rank_v];
  1120                     _bucket_next[v] = nv;
  1121                     _bucket_prev[nv] = v;
  1122                     _buckets[new_rank_v] = v;
  1123                   }
  1124                 }
  1125               }
  1126             }
  1127 
  1128             // Refine potential of node u
  1129             _pi[u] -= rank * _epsilon;
  1130           }
  1131         }
  1132 
  1133       }
  1134 
  1135       return false;
  1136     }
  1137 
  1138     // Find and cancel cycles in the admissible network and
  1139     // determine topological order using DFS
  1140     bool topologicalSort(IntVector &stack, int &stack_top) {
  1141       const int MAX_CYCLE_CANCEL = 1;
  1142 
  1143       BoolVector reached(_res_node_num, false);
  1144       BoolVector processed(_res_node_num, false);
  1145       IntVector pred(_res_node_num);
  1146       for (int i = 0; i != _res_node_num; ++i) {
  1147         _next_out[i] = _first_out[i];
  1148       }
  1149       stack_top = -1;
  1150 
  1151       int cycle_cnt = 0;
  1152       for (int start = 0; start != _res_node_num; ++start) {
  1153         if (reached[start]) continue;
  1154 
  1155         // Start DFS search from this start node
  1156         pred[start] = -1;
  1157         int tip = start, v;
  1158         while (true) {
  1159           // Check the outgoing arcs of the current tip node
  1160           reached[tip] = true;
  1161           LargeCost pi_tip = _pi[tip];
  1162           int a, last_out = _first_out[tip+1];
  1163           for (a = _next_out[tip]; a != last_out; ++a) {
  1164             if (_res_cap[a] > 0) {
  1165               v = _target[a];
  1166               if (_cost[a] + pi_tip - _pi[v] < 0) {
  1167                 if (!reached[v]) {
  1168                   // A new node is reached
  1169                   reached[v] = true;
  1170                   pred[v] = tip;
  1171                   _next_out[tip] = a;
  1172                   tip = v;
  1173                   a = _next_out[tip];
  1174                   last_out = _first_out[tip+1];
  1175                   break;
  1176                 }
  1177                 else if (!processed[v]) {
  1178                   // A cycle is found
  1179                   ++cycle_cnt;
  1180                   _next_out[tip] = a;
  1181 
  1182                   // Find the minimum residual capacity along the cycle
  1183                   Value d, delta = _res_cap[a];
  1184                   int u, delta_node = tip;
  1185                   for (u = tip; u != v; ) {
  1186                     u = pred[u];
  1187                     d = _res_cap[_next_out[u]];
  1188                     if (d <= delta) {
  1189                       delta = d;
  1190                       delta_node = u;
  1191                     }
  1192                   }
  1193 
  1194                   // Augment along the cycle
  1195                   _res_cap[a] -= delta;
  1196                   _res_cap[_reverse[a]] += delta;
  1197                   for (u = tip; u != v; ) {
  1198                     u = pred[u];
  1199                     int ca = _next_out[u];
  1200                     _res_cap[ca] -= delta;
  1201                     _res_cap[_reverse[ca]] += delta;
  1202                   }
  1203 
  1204                   // Check the maximum number of cycle canceling
  1205                   if (cycle_cnt >= MAX_CYCLE_CANCEL) {
  1206                     return false;
  1207                   }
  1208 
  1209                   // Roll back search to delta_node
  1210                   if (delta_node != tip) {
  1211                     for (u = tip; u != delta_node; u = pred[u]) {
  1212                       reached[u] = false;
  1213                     }
  1214                     tip = delta_node;
  1215                     a = _next_out[tip] + 1;
  1216                     last_out = _first_out[tip+1];
  1217                     break;
  1218                   }
  1219                 }
  1220               }
  1221             }
  1222           }
  1223 
  1224           // Step back to the previous node
  1225           if (a == last_out) {
  1226             processed[tip] = true;
  1227             stack[++stack_top] = tip;
  1228             tip = pred[tip];
  1229             if (tip < 0) {
  1230               // Finish DFS from the current start node
  1231               break;
  1232             }
  1233             ++_next_out[tip];
  1234           }
  1235         }
  1236 
  1237       }
  1238 
  1239       return (cycle_cnt == 0);
  1240     }
  1241 
  1242     // Global potential update heuristic
  1243     void globalUpdate() {
  1244       const int bucket_end = _root + 1;
  1245 
  1246       // Initialize buckets
  1247       for (int r = 0; r != _max_rank; ++r) {
  1248         _buckets[r] = bucket_end;
  1249       }
  1250       Value total_excess = 0;
  1251       int b0 = bucket_end;
  1252       for (int i = 0; i != _res_node_num; ++i) {
  1253         if (_excess[i] < 0) {
  1254           _rank[i] = 0;
  1255           _bucket_next[i] = b0;
  1256           _bucket_prev[b0] = i;
  1257           b0 = i;
  1258         } else {
  1259           total_excess += _excess[i];
  1260           _rank[i] = _max_rank;
  1261         }
  1262       }
  1263       if (total_excess == 0) return;
  1264       _buckets[0] = b0;
  1265 
  1266       // Search the buckets
  1267       int r = 0;
  1268       for ( ; r != _max_rank; ++r) {
  1269         while (_buckets[r] != bucket_end) {
  1270           // Remove the first node from the current bucket
  1271           int u = _buckets[r];
  1272           _buckets[r] = _bucket_next[u];
  1273 
  1274           // Search the incoming arcs of u
  1275           LargeCost pi_u = _pi[u];
  1276           int last_out = _first_out[u+1];
  1277           for (int a = _first_out[u]; a != last_out; ++a) {
  1278             int ra = _reverse[a];
  1279             if (_res_cap[ra] > 0) {
  1280               int v = _source[ra];
  1281               int old_rank_v = _rank[v];
  1282               if (r < old_rank_v) {
  1283                 // Compute the new rank of v
  1284                 LargeCost nrc = (_cost[ra] + _pi[v] - pi_u) / _epsilon;
  1285                 int new_rank_v = old_rank_v;
  1286                 if (nrc < LargeCost(_max_rank)) {
  1287                   new_rank_v = r + 1 + static_cast<int>(nrc);
  1288                 }
  1289 
  1290                 // Change the rank of v
  1291                 if (new_rank_v < old_rank_v) {
  1292                   _rank[v] = new_rank_v;
  1293                   _next_out[v] = _first_out[v];
  1294 
  1295                   // Remove v from its old bucket
  1296                   if (old_rank_v < _max_rank) {
  1297                     if (_buckets[old_rank_v] == v) {
  1298                       _buckets[old_rank_v] = _bucket_next[v];
  1299                     } else {
  1300                       int pv = _bucket_prev[v], nv = _bucket_next[v];
  1301                       _bucket_next[pv] = nv;
  1302                       _bucket_prev[nv] = pv;
  1303                     }
  1304                   }
  1305 
  1306                   // Insert v into its new bucket
  1307                   int nv = _buckets[new_rank_v];
  1308                   _bucket_next[v] = nv;
  1309                   _bucket_prev[nv] = v;
  1310                   _buckets[new_rank_v] = v;
  1311                 }
  1312               }
  1313             }
  1314           }
  1315 
  1316           // Finish search if there are no more active nodes
  1317           if (_excess[u] > 0) {
  1318             total_excess -= _excess[u];
  1319             if (total_excess <= 0) break;
  1320           }
  1321         }
  1322         if (total_excess <= 0) break;
  1323       }
  1324 
  1325       // Relabel nodes
  1326       for (int u = 0; u != _res_node_num; ++u) {
  1327         int k = std::min(_rank[u], r);
  1328         if (k > 0) {
  1329           _pi[u] -= _epsilon * k;
  1330           _next_out[u] = _first_out[u];
  1331         }
  1332       }
  1333     }
  1334 
  1335     /// Execute the algorithm performing augment and relabel operations
  1336     void startAugment(int max_length) {
  1337       // Paramters for heuristics
  1338       const int PRICE_REFINEMENT_LIMIT = 2;
  1339       const double GLOBAL_UPDATE_FACTOR = 1.0;
  1340       const int global_update_skip = static_cast<int>(GLOBAL_UPDATE_FACTOR *
  1341         (_res_node_num + _sup_node_num * _sup_node_num));
  1342       int next_global_update_limit = global_update_skip;
  1343 
  1344       // Perform cost scaling phases
  1345       IntVector path;
  1346       BoolVector path_arc(_res_arc_num, false);
  1347       int relabel_cnt = 0;
  1348       int eps_phase_cnt = 0;
  1349       for ( ; _epsilon >= 1; _epsilon = _epsilon < _alpha && _epsilon > 1 ?
  1350                                         1 : _epsilon / _alpha )
  1351       {
  1352         ++eps_phase_cnt;
  1353 
  1354         // Price refinement heuristic
  1355         if (eps_phase_cnt >= PRICE_REFINEMENT_LIMIT) {
  1356           if (priceRefinement()) continue;
  1357         }
  1358 
  1359         // Initialize current phase
  1360         initPhase();
  1361 
  1362         // Perform partial augment and relabel operations
  1363         while (true) {
  1364           // Select an active node (FIFO selection)
  1365           while (_active_nodes.size() > 0 &&
  1366                  _excess[_active_nodes.front()] <= 0) {
  1367             _active_nodes.pop_front();
  1368           }
  1369           if (_active_nodes.size() == 0) break;
  1370           int start = _active_nodes.front();
  1371 
  1372           // Find an augmenting path from the start node
  1373           int tip = start;
  1374           while (int(path.size()) < max_length && _excess[tip] >= 0) {
  1375             int u;
  1376             LargeCost rc, min_red_cost = std::numeric_limits<LargeCost>::max();
  1377             LargeCost pi_tip = _pi[tip];
  1378             int last_out = _first_out[tip+1];
  1379             for (int a = _next_out[tip]; a != last_out; ++a) {
  1380               if (_res_cap[a] > 0) {
  1381                 u = _target[a];
  1382                 rc = _cost[a] + pi_tip - _pi[u];
  1383                 if (rc < 0) {
  1384                   path.push_back(a);
  1385                   _next_out[tip] = a;
  1386                   if (path_arc[a]) {
  1387                     goto augment;   // a cycle is found, stop path search
  1388                   }
  1389                   tip = u;
  1390                   path_arc[a] = true;
  1391                   goto next_step;
  1392                 }
  1393                 else if (rc < min_red_cost) {
  1394                   min_red_cost = rc;
  1395                 }
  1396               }
  1397             }
  1398 
  1399             // Relabel tip node
  1400             if (tip != start) {
  1401               int ra = _reverse[path.back()];
  1402               min_red_cost =
  1403                 std::min(min_red_cost, _cost[ra] + pi_tip - _pi[_target[ra]]);
  1404             }
  1405             last_out = _next_out[tip];
  1406             for (int a = _first_out[tip]; a != last_out; ++a) {
  1407               if (_res_cap[a] > 0) {
  1408                 rc = _cost[a] + pi_tip - _pi[_target[a]];
  1409                 if (rc < min_red_cost) {
  1410                   min_red_cost = rc;
  1411                 }
  1412               }
  1413             }
  1414             _pi[tip] -= min_red_cost + _epsilon;
  1415             _next_out[tip] = _first_out[tip];
  1416             ++relabel_cnt;
  1417 
  1418             // Step back
  1419             if (tip != start) {
  1420               int pa = path.back();
  1421               path_arc[pa] = false;
  1422               tip = _source[pa];
  1423               path.pop_back();
  1424             }
  1425 
  1426           next_step: ;
  1427           }
  1428 
  1429           // Augment along the found path (as much flow as possible)
  1430         augment:
  1431           Value delta;
  1432           int pa, u, v = start;
  1433           for (int i = 0; i != int(path.size()); ++i) {
  1434             pa = path[i];
  1435             u = v;
  1436             v = _target[pa];
  1437             path_arc[pa] = false;
  1438             delta = std::min(_res_cap[pa], _excess[u]);
  1439             _res_cap[pa] -= delta;
  1440             _res_cap[_reverse[pa]] += delta;
  1441             _excess[u] -= delta;
  1442             _excess[v] += delta;
  1443             if (_excess[v] > 0 && _excess[v] <= delta) {
  1444               _active_nodes.push_back(v);
  1445             }
  1446           }
  1447           path.clear();
  1448 
  1449           // Global update heuristic
  1450           if (relabel_cnt >= next_global_update_limit) {
  1451             globalUpdate();
  1452             next_global_update_limit += global_update_skip;
  1453           }
  1454         }
  1455 
  1456       }
  1457 
  1458     }
  1459 
  1460     /// Execute the algorithm performing push and relabel operations
  1461     void startPush() {
  1462       // Paramters for heuristics
  1463       const int PRICE_REFINEMENT_LIMIT = 2;
  1464       const double GLOBAL_UPDATE_FACTOR = 2.0;
  1465 
  1466       const int global_update_skip = static_cast<int>(GLOBAL_UPDATE_FACTOR *
  1467         (_res_node_num + _sup_node_num * _sup_node_num));
  1468       int next_global_update_limit = global_update_skip;
  1469 
  1470       // Perform cost scaling phases
  1471       BoolVector hyper(_res_node_num, false);
  1472       LargeCostVector hyper_cost(_res_node_num);
  1473       int relabel_cnt = 0;
  1474       int eps_phase_cnt = 0;
  1475       for ( ; _epsilon >= 1; _epsilon = _epsilon < _alpha && _epsilon > 1 ?
  1476                                         1 : _epsilon / _alpha )
  1477       {
  1478         ++eps_phase_cnt;
  1479 
  1480         // Price refinement heuristic
  1481         if (eps_phase_cnt >= PRICE_REFINEMENT_LIMIT) {
  1482           if (priceRefinement()) continue;
  1483         }
  1484 
  1485         // Initialize current phase
  1486         initPhase();
  1487 
  1488         // Perform push and relabel operations
  1489         while (_active_nodes.size() > 0) {
  1490           LargeCost min_red_cost, rc, pi_n;
  1491           Value delta;
  1492           int n, t, a, last_out = _res_arc_num;
  1493 
  1494         next_node:
  1495           // Select an active node (FIFO selection)
  1496           n = _active_nodes.front();
  1497           last_out = _first_out[n+1];
  1498           pi_n = _pi[n];
  1499 
  1500           // Perform push operations if there are admissible arcs
  1501           if (_excess[n] > 0) {
  1502             for (a = _next_out[n]; a != last_out; ++a) {
  1503               if (_res_cap[a] > 0 &&
  1504                   _cost[a] + pi_n - _pi[_target[a]] < 0) {
  1505                 delta = std::min(_res_cap[a], _excess[n]);
  1506                 t = _target[a];
  1507 
  1508                 // Push-look-ahead heuristic
  1509                 Value ahead = -_excess[t];
  1510                 int last_out_t = _first_out[t+1];
  1511                 LargeCost pi_t = _pi[t];
  1512                 for (int ta = _next_out[t]; ta != last_out_t; ++ta) {
  1513                   if (_res_cap[ta] > 0 &&
  1514                       _cost[ta] + pi_t - _pi[_target[ta]] < 0)
  1515                     ahead += _res_cap[ta];
  1516                   if (ahead >= delta) break;
  1517                 }
  1518                 if (ahead < 0) ahead = 0;
  1519 
  1520                 // Push flow along the arc
  1521                 if (ahead < delta && !hyper[t]) {
  1522                   _res_cap[a] -= ahead;
  1523                   _res_cap[_reverse[a]] += ahead;
  1524                   _excess[n] -= ahead;
  1525                   _excess[t] += ahead;
  1526                   _active_nodes.push_front(t);
  1527                   hyper[t] = true;
  1528                   hyper_cost[t] = _cost[a] + pi_n - pi_t;
  1529                   _next_out[n] = a;
  1530                   goto next_node;
  1531                 } else {
  1532                   _res_cap[a] -= delta;
  1533                   _res_cap[_reverse[a]] += delta;
  1534                   _excess[n] -= delta;
  1535                   _excess[t] += delta;
  1536                   if (_excess[t] > 0 && _excess[t] <= delta)
  1537                     _active_nodes.push_back(t);
  1538                 }
  1539 
  1540                 if (_excess[n] == 0) {
  1541                   _next_out[n] = a;
  1542                   goto remove_nodes;
  1543                 }
  1544               }
  1545             }
  1546             _next_out[n] = a;
  1547           }
  1548 
  1549           // Relabel the node if it is still active (or hyper)
  1550           if (_excess[n] > 0 || hyper[n]) {
  1551              min_red_cost = hyper[n] ? -hyper_cost[n] :
  1552                std::numeric_limits<LargeCost>::max();
  1553             for (int a = _first_out[n]; a != last_out; ++a) {
  1554               if (_res_cap[a] > 0) {
  1555                 rc = _cost[a] + pi_n - _pi[_target[a]];
  1556                 if (rc < min_red_cost) {
  1557                   min_red_cost = rc;
  1558                 }
  1559               }
  1560             }
  1561             _pi[n] -= min_red_cost + _epsilon;
  1562             _next_out[n] = _first_out[n];
  1563             hyper[n] = false;
  1564             ++relabel_cnt;
  1565           }
  1566 
  1567           // Remove nodes that are not active nor hyper
  1568         remove_nodes:
  1569           while ( _active_nodes.size() > 0 &&
  1570                   _excess[_active_nodes.front()] <= 0 &&
  1571                   !hyper[_active_nodes.front()] ) {
  1572             _active_nodes.pop_front();
  1573           }
  1574 
  1575           // Global update heuristic
  1576           if (relabel_cnt >= next_global_update_limit) {
  1577             globalUpdate();
  1578             for (int u = 0; u != _res_node_num; ++u)
  1579               hyper[u] = false;
  1580             next_global_update_limit += global_update_skip;
  1581           }
  1582         }
  1583       }
  1584     }
  1585 
  1586   }; //class CostScaling
  1587 
  1588   ///@}
  1589 
  1590 } //namespace lemon
  1591 
  1592 #endif //LEMON_COST_SCALING_H