lemon/network_simplex.h
author Peter Kovacs <kpeter@inf.elte.hu>
Sun, 09 Jan 2011 23:20:56 +0100
changeset 920 745312f9b441
parent 896 fb932bcfd803
child 922 9312d6c89d02
permissions -rw-r--r--
Improve the doc of path structures (#406)
     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_NETWORK_SIMPLEX_H
    20 #define LEMON_NETWORK_SIMPLEX_H
    21 
    22 /// \ingroup min_cost_flow_algs
    23 ///
    24 /// \file
    25 /// \brief Network Simplex algorithm for finding a minimum cost flow.
    26 
    27 #include <vector>
    28 #include <limits>
    29 #include <algorithm>
    30 
    31 #include <lemon/core.h>
    32 #include <lemon/math.h>
    33 
    34 namespace lemon {
    35 
    36   /// \addtogroup min_cost_flow_algs
    37   /// @{
    38 
    39   /// \brief Implementation of the primal Network Simplex algorithm
    40   /// for finding a \ref min_cost_flow "minimum cost flow".
    41   ///
    42   /// \ref NetworkSimplex implements the primal Network Simplex algorithm
    43   /// for finding a \ref min_cost_flow "minimum cost flow"
    44   /// \ref amo93networkflows, \ref dantzig63linearprog,
    45   /// \ref kellyoneill91netsimplex.
    46   /// This algorithm is a highly efficient specialized version of the
    47   /// linear programming simplex method directly for the minimum cost
    48   /// flow problem.
    49   ///
    50   /// In general, \ref NetworkSimplex and \ref CostScaling are the fastest
    51   /// implementations available in LEMON for this problem.
    52   /// Furthermore, this class supports both directions of the supply/demand
    53   /// inequality constraints. For more information, see \ref SupplyType.
    54   ///
    55   /// Most of the parameters of the problem (except for the digraph)
    56   /// can be given using separate functions, and the algorithm can be
    57   /// executed using the \ref run() function. If some parameters are not
    58   /// specified, then default values will be used.
    59   ///
    60   /// \tparam GR The digraph type the algorithm runs on.
    61   /// \tparam V The number type used for flow amounts, capacity bounds
    62   /// and supply values in the algorithm. By default, it is \c int.
    63   /// \tparam C The number type used for costs and potentials in the
    64   /// algorithm. By default, it is the same as \c V.
    65   ///
    66   /// \warning Both number types must be signed and all input data must
    67   /// be integer.
    68   ///
    69   /// \note %NetworkSimplex provides five different pivot rule
    70   /// implementations, from which the most efficient one is used
    71   /// by default. For more information, see \ref PivotRule.
    72   template <typename GR, typename V = int, typename C = V>
    73   class NetworkSimplex
    74   {
    75   public:
    76 
    77     /// The type of the flow amounts, capacity bounds and supply values
    78     typedef V Value;
    79     /// The type of the arc costs
    80     typedef C Cost;
    81 
    82   public:
    83 
    84     /// \brief Problem type constants for the \c run() function.
    85     ///
    86     /// Enum type containing the problem type constants that can be
    87     /// returned by the \ref run() function of the algorithm.
    88     enum ProblemType {
    89       /// The problem has no feasible solution (flow).
    90       INFEASIBLE,
    91       /// The problem has optimal solution (i.e. it is feasible and
    92       /// bounded), and the algorithm has found optimal flow and node
    93       /// potentials (primal and dual solutions).
    94       OPTIMAL,
    95       /// The objective function of the problem is unbounded, i.e.
    96       /// there is a directed cycle having negative total cost and
    97       /// infinite upper bound.
    98       UNBOUNDED
    99     };
   100 
   101     /// \brief Constants for selecting the type of the supply constraints.
   102     ///
   103     /// Enum type containing constants for selecting the supply type,
   104     /// i.e. the direction of the inequalities in the supply/demand
   105     /// constraints of the \ref min_cost_flow "minimum cost flow problem".
   106     ///
   107     /// The default supply type is \c GEQ, the \c LEQ type can be
   108     /// selected using \ref supplyType().
   109     /// The equality form is a special case of both supply types.
   110     enum SupplyType {
   111       /// This option means that there are <em>"greater or equal"</em>
   112       /// supply/demand constraints in the definition of the problem.
   113       GEQ,
   114       /// This option means that there are <em>"less or equal"</em>
   115       /// supply/demand constraints in the definition of the problem.
   116       LEQ
   117     };
   118 
   119     /// \brief Constants for selecting the pivot rule.
   120     ///
   121     /// Enum type containing constants for selecting the pivot rule for
   122     /// the \ref run() function.
   123     ///
   124     /// \ref NetworkSimplex provides five different pivot rule
   125     /// implementations that significantly affect the running time
   126     /// of the algorithm.
   127     /// By default, \ref BLOCK_SEARCH "Block Search" is used, which
   128     /// turend out to be the most efficient and the most robust on various
   129     /// test inputs.
   130     /// However, another pivot rule can be selected using the \ref run()
   131     /// function with the proper parameter.
   132     enum PivotRule {
   133 
   134       /// The \e First \e Eligible pivot rule.
   135       /// The next eligible arc is selected in a wraparound fashion
   136       /// in every iteration.
   137       FIRST_ELIGIBLE,
   138 
   139       /// The \e Best \e Eligible pivot rule.
   140       /// The best eligible arc is selected in every iteration.
   141       BEST_ELIGIBLE,
   142 
   143       /// The \e Block \e Search pivot rule.
   144       /// A specified number of arcs are examined in every iteration
   145       /// in a wraparound fashion and the best eligible arc is selected
   146       /// from this block.
   147       BLOCK_SEARCH,
   148 
   149       /// The \e Candidate \e List pivot rule.
   150       /// In a major iteration a candidate list is built from eligible arcs
   151       /// in a wraparound fashion and in the following minor iterations
   152       /// the best eligible arc is selected from this list.
   153       CANDIDATE_LIST,
   154 
   155       /// The \e Altering \e Candidate \e List pivot rule.
   156       /// It is a modified version of the Candidate List method.
   157       /// It keeps only the several best eligible arcs from the former
   158       /// candidate list and extends this list in every iteration.
   159       ALTERING_LIST
   160     };
   161 
   162   private:
   163 
   164     TEMPLATE_DIGRAPH_TYPEDEFS(GR);
   165 
   166     typedef std::vector<int> IntVector;
   167     typedef std::vector<Value> ValueVector;
   168     typedef std::vector<Cost> CostVector;
   169     typedef std::vector<signed char> CharVector;
   170     // Note: vector<signed char> is used instead of vector<ArcState> and
   171     // vector<ArcDirection> for efficiency reasons
   172 
   173     // State constants for arcs
   174     enum ArcState {
   175       STATE_UPPER = -1,
   176       STATE_TREE  =  0,
   177       STATE_LOWER =  1
   178     };
   179 
   180     // Direction constants for tree arcs
   181     enum ArcDirection {
   182       DIR_DOWN = -1,
   183       DIR_UP   =  1
   184     };
   185 
   186   private:
   187 
   188     // Data related to the underlying digraph
   189     const GR &_graph;
   190     int _node_num;
   191     int _arc_num;
   192     int _all_arc_num;
   193     int _search_arc_num;
   194 
   195     // Parameters of the problem
   196     bool _have_lower;
   197     SupplyType _stype;
   198     Value _sum_supply;
   199 
   200     // Data structures for storing the digraph
   201     IntNodeMap _node_id;
   202     IntArcMap _arc_id;
   203     IntVector _source;
   204     IntVector _target;
   205     bool _arc_mixing;
   206 
   207     // Node and arc data
   208     ValueVector _lower;
   209     ValueVector _upper;
   210     ValueVector _cap;
   211     CostVector _cost;
   212     ValueVector _supply;
   213     ValueVector _flow;
   214     CostVector _pi;
   215 
   216     // Data for storing the spanning tree structure
   217     IntVector _parent;
   218     IntVector _pred;
   219     IntVector _thread;
   220     IntVector _rev_thread;
   221     IntVector _succ_num;
   222     IntVector _last_succ;
   223     CharVector _pred_dir;
   224     CharVector _state;
   225     IntVector _dirty_revs;
   226     int _root;
   227 
   228     // Temporary data used in the current pivot iteration
   229     int in_arc, join, u_in, v_in, u_out, v_out;
   230     Value delta;
   231 
   232     const Value MAX;
   233 
   234   public:
   235 
   236     /// \brief Constant for infinite upper bounds (capacities).
   237     ///
   238     /// Constant for infinite upper bounds (capacities).
   239     /// It is \c std::numeric_limits<Value>::infinity() if available,
   240     /// \c std::numeric_limits<Value>::max() otherwise.
   241     const Value INF;
   242 
   243   private:
   244 
   245     // Implementation of the First Eligible pivot rule
   246     class FirstEligiblePivotRule
   247     {
   248     private:
   249 
   250       // References to the NetworkSimplex class
   251       const IntVector  &_source;
   252       const IntVector  &_target;
   253       const CostVector &_cost;
   254       const CharVector &_state;
   255       const CostVector &_pi;
   256       int &_in_arc;
   257       int _search_arc_num;
   258 
   259       // Pivot rule data
   260       int _next_arc;
   261 
   262     public:
   263 
   264       // Constructor
   265       FirstEligiblePivotRule(NetworkSimplex &ns) :
   266         _source(ns._source), _target(ns._target),
   267         _cost(ns._cost), _state(ns._state), _pi(ns._pi),
   268         _in_arc(ns.in_arc), _search_arc_num(ns._search_arc_num),
   269         _next_arc(0)
   270       {}
   271 
   272       // Find next entering arc
   273       bool findEnteringArc() {
   274         Cost c;
   275         for (int e = _next_arc; e != _search_arc_num; ++e) {
   276           c = _state[e] * (_cost[e] + _pi[_source[e]] - _pi[_target[e]]);
   277           if (c < 0) {
   278             _in_arc = e;
   279             _next_arc = e + 1;
   280             return true;
   281           }
   282         }
   283         for (int e = 0; e != _next_arc; ++e) {
   284           c = _state[e] * (_cost[e] + _pi[_source[e]] - _pi[_target[e]]);
   285           if (c < 0) {
   286             _in_arc = e;
   287             _next_arc = e + 1;
   288             return true;
   289           }
   290         }
   291         return false;
   292       }
   293 
   294     }; //class FirstEligiblePivotRule
   295 
   296 
   297     // Implementation of the Best Eligible pivot rule
   298     class BestEligiblePivotRule
   299     {
   300     private:
   301 
   302       // References to the NetworkSimplex class
   303       const IntVector  &_source;
   304       const IntVector  &_target;
   305       const CostVector &_cost;
   306       const CharVector &_state;
   307       const CostVector &_pi;
   308       int &_in_arc;
   309       int _search_arc_num;
   310 
   311     public:
   312 
   313       // Constructor
   314       BestEligiblePivotRule(NetworkSimplex &ns) :
   315         _source(ns._source), _target(ns._target),
   316         _cost(ns._cost), _state(ns._state), _pi(ns._pi),
   317         _in_arc(ns.in_arc), _search_arc_num(ns._search_arc_num)
   318       {}
   319 
   320       // Find next entering arc
   321       bool findEnteringArc() {
   322         Cost c, min = 0;
   323         for (int e = 0; e != _search_arc_num; ++e) {
   324           c = _state[e] * (_cost[e] + _pi[_source[e]] - _pi[_target[e]]);
   325           if (c < min) {
   326             min = c;
   327             _in_arc = e;
   328           }
   329         }
   330         return min < 0;
   331       }
   332 
   333     }; //class BestEligiblePivotRule
   334 
   335 
   336     // Implementation of the Block Search pivot rule
   337     class BlockSearchPivotRule
   338     {
   339     private:
   340 
   341       // References to the NetworkSimplex class
   342       const IntVector  &_source;
   343       const IntVector  &_target;
   344       const CostVector &_cost;
   345       const CharVector &_state;
   346       const CostVector &_pi;
   347       int &_in_arc;
   348       int _search_arc_num;
   349 
   350       // Pivot rule data
   351       int _block_size;
   352       int _next_arc;
   353 
   354     public:
   355 
   356       // Constructor
   357       BlockSearchPivotRule(NetworkSimplex &ns) :
   358         _source(ns._source), _target(ns._target),
   359         _cost(ns._cost), _state(ns._state), _pi(ns._pi),
   360         _in_arc(ns.in_arc), _search_arc_num(ns._search_arc_num),
   361         _next_arc(0)
   362       {
   363         // The main parameters of the pivot rule
   364         const double BLOCK_SIZE_FACTOR = 1.0;
   365         const int MIN_BLOCK_SIZE = 10;
   366 
   367         _block_size = std::max( int(BLOCK_SIZE_FACTOR *
   368                                     std::sqrt(double(_search_arc_num))),
   369                                 MIN_BLOCK_SIZE );
   370       }
   371 
   372       // Find next entering arc
   373       bool findEnteringArc() {
   374         Cost c, min = 0;
   375         int cnt = _block_size;
   376         int e;
   377         for (e = _next_arc; e != _search_arc_num; ++e) {
   378           c = _state[e] * (_cost[e] + _pi[_source[e]] - _pi[_target[e]]);
   379           if (c < min) {
   380             min = c;
   381             _in_arc = e;
   382           }
   383           if (--cnt == 0) {
   384             if (min < 0) goto search_end;
   385             cnt = _block_size;
   386           }
   387         }
   388         for (e = 0; e != _next_arc; ++e) {
   389           c = _state[e] * (_cost[e] + _pi[_source[e]] - _pi[_target[e]]);
   390           if (c < min) {
   391             min = c;
   392             _in_arc = e;
   393           }
   394           if (--cnt == 0) {
   395             if (min < 0) goto search_end;
   396             cnt = _block_size;
   397           }
   398         }
   399         if (min >= 0) return false;
   400 
   401       search_end:
   402         _next_arc = e;
   403         return true;
   404       }
   405 
   406     }; //class BlockSearchPivotRule
   407 
   408 
   409     // Implementation of the Candidate List pivot rule
   410     class CandidateListPivotRule
   411     {
   412     private:
   413 
   414       // References to the NetworkSimplex class
   415       const IntVector  &_source;
   416       const IntVector  &_target;
   417       const CostVector &_cost;
   418       const CharVector &_state;
   419       const CostVector &_pi;
   420       int &_in_arc;
   421       int _search_arc_num;
   422 
   423       // Pivot rule data
   424       IntVector _candidates;
   425       int _list_length, _minor_limit;
   426       int _curr_length, _minor_count;
   427       int _next_arc;
   428 
   429     public:
   430 
   431       /// Constructor
   432       CandidateListPivotRule(NetworkSimplex &ns) :
   433         _source(ns._source), _target(ns._target),
   434         _cost(ns._cost), _state(ns._state), _pi(ns._pi),
   435         _in_arc(ns.in_arc), _search_arc_num(ns._search_arc_num),
   436         _next_arc(0)
   437       {
   438         // The main parameters of the pivot rule
   439         const double LIST_LENGTH_FACTOR = 0.25;
   440         const int MIN_LIST_LENGTH = 10;
   441         const double MINOR_LIMIT_FACTOR = 0.1;
   442         const int MIN_MINOR_LIMIT = 3;
   443 
   444         _list_length = std::max( int(LIST_LENGTH_FACTOR *
   445                                      std::sqrt(double(_search_arc_num))),
   446                                  MIN_LIST_LENGTH );
   447         _minor_limit = std::max( int(MINOR_LIMIT_FACTOR * _list_length),
   448                                  MIN_MINOR_LIMIT );
   449         _curr_length = _minor_count = 0;
   450         _candidates.resize(_list_length);
   451       }
   452 
   453       /// Find next entering arc
   454       bool findEnteringArc() {
   455         Cost min, c;
   456         int e;
   457         if (_curr_length > 0 && _minor_count < _minor_limit) {
   458           // Minor iteration: select the best eligible arc from the
   459           // current candidate list
   460           ++_minor_count;
   461           min = 0;
   462           for (int i = 0; i < _curr_length; ++i) {
   463             e = _candidates[i];
   464             c = _state[e] * (_cost[e] + _pi[_source[e]] - _pi[_target[e]]);
   465             if (c < min) {
   466               min = c;
   467               _in_arc = e;
   468             }
   469             else if (c >= 0) {
   470               _candidates[i--] = _candidates[--_curr_length];
   471             }
   472           }
   473           if (min < 0) return true;
   474         }
   475 
   476         // Major iteration: build a new candidate list
   477         min = 0;
   478         _curr_length = 0;
   479         for (e = _next_arc; e != _search_arc_num; ++e) {
   480           c = _state[e] * (_cost[e] + _pi[_source[e]] - _pi[_target[e]]);
   481           if (c < 0) {
   482             _candidates[_curr_length++] = e;
   483             if (c < min) {
   484               min = c;
   485               _in_arc = e;
   486             }
   487             if (_curr_length == _list_length) goto search_end;
   488           }
   489         }
   490         for (e = 0; e != _next_arc; ++e) {
   491           c = _state[e] * (_cost[e] + _pi[_source[e]] - _pi[_target[e]]);
   492           if (c < 0) {
   493             _candidates[_curr_length++] = e;
   494             if (c < min) {
   495               min = c;
   496               _in_arc = e;
   497             }
   498             if (_curr_length == _list_length) goto search_end;
   499           }
   500         }
   501         if (_curr_length == 0) return false;
   502 
   503       search_end:
   504         _minor_count = 1;
   505         _next_arc = e;
   506         return true;
   507       }
   508 
   509     }; //class CandidateListPivotRule
   510 
   511 
   512     // Implementation of the Altering Candidate List pivot rule
   513     class AlteringListPivotRule
   514     {
   515     private:
   516 
   517       // References to the NetworkSimplex class
   518       const IntVector  &_source;
   519       const IntVector  &_target;
   520       const CostVector &_cost;
   521       const CharVector &_state;
   522       const CostVector &_pi;
   523       int &_in_arc;
   524       int _search_arc_num;
   525 
   526       // Pivot rule data
   527       int _block_size, _head_length, _curr_length;
   528       int _next_arc;
   529       IntVector _candidates;
   530       CostVector _cand_cost;
   531 
   532       // Functor class to compare arcs during sort of the candidate list
   533       class SortFunc
   534       {
   535       private:
   536         const CostVector &_map;
   537       public:
   538         SortFunc(const CostVector &map) : _map(map) {}
   539         bool operator()(int left, int right) {
   540           return _map[left] > _map[right];
   541         }
   542       };
   543 
   544       SortFunc _sort_func;
   545 
   546     public:
   547 
   548       // Constructor
   549       AlteringListPivotRule(NetworkSimplex &ns) :
   550         _source(ns._source), _target(ns._target),
   551         _cost(ns._cost), _state(ns._state), _pi(ns._pi),
   552         _in_arc(ns.in_arc), _search_arc_num(ns._search_arc_num),
   553         _next_arc(0), _cand_cost(ns._search_arc_num), _sort_func(_cand_cost)
   554       {
   555         // The main parameters of the pivot rule
   556         const double BLOCK_SIZE_FACTOR = 1.0;
   557         const int MIN_BLOCK_SIZE = 10;
   558         const double HEAD_LENGTH_FACTOR = 0.1;
   559         const int MIN_HEAD_LENGTH = 3;
   560 
   561         _block_size = std::max( int(BLOCK_SIZE_FACTOR *
   562                                     std::sqrt(double(_search_arc_num))),
   563                                 MIN_BLOCK_SIZE );
   564         _head_length = std::max( int(HEAD_LENGTH_FACTOR * _block_size),
   565                                  MIN_HEAD_LENGTH );
   566         _candidates.resize(_head_length + _block_size);
   567         _curr_length = 0;
   568       }
   569 
   570       // Find next entering arc
   571       bool findEnteringArc() {
   572         // Check the current candidate list
   573         int e;
   574         Cost c;
   575         for (int i = 0; i != _curr_length; ++i) {
   576           e = _candidates[i];
   577           c = _state[e] * (_cost[e] + _pi[_source[e]] - _pi[_target[e]]);
   578           if (c < 0) {
   579             _cand_cost[e] = c;
   580           } else {
   581             _candidates[i--] = _candidates[--_curr_length];
   582           }
   583         }
   584 
   585         // Extend the list
   586         int cnt = _block_size;
   587         int limit = _head_length;
   588 
   589         for (e = _next_arc; e != _search_arc_num; ++e) {
   590           c = _state[e] * (_cost[e] + _pi[_source[e]] - _pi[_target[e]]);
   591           if (c < 0) {
   592             _cand_cost[e] = c;
   593             _candidates[_curr_length++] = e;
   594           }
   595           if (--cnt == 0) {
   596             if (_curr_length > limit) goto search_end;
   597             limit = 0;
   598             cnt = _block_size;
   599           }
   600         }
   601         for (e = 0; e != _next_arc; ++e) {
   602           _cand_cost[e] = _state[e] *
   603             (_cost[e] + _pi[_source[e]] - _pi[_target[e]]);
   604           if (_cand_cost[e] < 0) {
   605             _candidates[_curr_length++] = e;
   606           }
   607           if (--cnt == 0) {
   608             if (_curr_length > limit) goto search_end;
   609             limit = 0;
   610             cnt = _block_size;
   611           }
   612         }
   613         if (_curr_length == 0) return false;
   614 
   615       search_end:
   616 
   617         // Make heap of the candidate list (approximating a partial sort)
   618         make_heap( _candidates.begin(), _candidates.begin() + _curr_length,
   619                    _sort_func );
   620 
   621         // Pop the first element of the heap
   622         _in_arc = _candidates[0];
   623         _next_arc = e;
   624         pop_heap( _candidates.begin(), _candidates.begin() + _curr_length,
   625                   _sort_func );
   626         _curr_length = std::min(_head_length, _curr_length - 1);
   627         return true;
   628       }
   629 
   630     }; //class AlteringListPivotRule
   631 
   632   public:
   633 
   634     /// \brief Constructor.
   635     ///
   636     /// The constructor of the class.
   637     ///
   638     /// \param graph The digraph the algorithm runs on.
   639     /// \param arc_mixing Indicate if the arcs will be stored in a
   640     /// mixed order in the internal data structure.
   641     /// In general, it leads to similar performance as using the original
   642     /// arc order, but it makes the algorithm more robust and in special
   643     /// cases, even significantly faster. Therefore, it is enabled by default.
   644     NetworkSimplex(const GR& graph, bool arc_mixing = true) :
   645       _graph(graph), _node_id(graph), _arc_id(graph),
   646       _arc_mixing(arc_mixing),
   647       MAX(std::numeric_limits<Value>::max()),
   648       INF(std::numeric_limits<Value>::has_infinity ?
   649           std::numeric_limits<Value>::infinity() : MAX)
   650     {
   651       // Check the number types
   652       LEMON_ASSERT(std::numeric_limits<Value>::is_signed,
   653         "The flow type of NetworkSimplex must be signed");
   654       LEMON_ASSERT(std::numeric_limits<Cost>::is_signed,
   655         "The cost type of NetworkSimplex must be signed");
   656 
   657       // Reset data structures
   658       reset();
   659     }
   660 
   661     /// \name Parameters
   662     /// The parameters of the algorithm can be specified using these
   663     /// functions.
   664 
   665     /// @{
   666 
   667     /// \brief Set the lower bounds on the arcs.
   668     ///
   669     /// This function sets the lower bounds on the arcs.
   670     /// If it is not used before calling \ref run(), the lower bounds
   671     /// will be set to zero on all arcs.
   672     ///
   673     /// \param map An arc map storing the lower bounds.
   674     /// Its \c Value type must be convertible to the \c Value type
   675     /// of the algorithm.
   676     ///
   677     /// \return <tt>(*this)</tt>
   678     template <typename LowerMap>
   679     NetworkSimplex& lowerMap(const LowerMap& map) {
   680       _have_lower = true;
   681       for (ArcIt a(_graph); a != INVALID; ++a) {
   682         _lower[_arc_id[a]] = map[a];
   683       }
   684       return *this;
   685     }
   686 
   687     /// \brief Set the upper bounds (capacities) on the arcs.
   688     ///
   689     /// This function sets the upper bounds (capacities) on the arcs.
   690     /// If it is not used before calling \ref run(), the upper bounds
   691     /// will be set to \ref INF on all arcs (i.e. the flow value will be
   692     /// unbounded from above).
   693     ///
   694     /// \param map An arc map storing the upper bounds.
   695     /// Its \c Value type must be convertible to the \c Value type
   696     /// of the algorithm.
   697     ///
   698     /// \return <tt>(*this)</tt>
   699     template<typename UpperMap>
   700     NetworkSimplex& upperMap(const UpperMap& map) {
   701       for (ArcIt a(_graph); a != INVALID; ++a) {
   702         _upper[_arc_id[a]] = map[a];
   703       }
   704       return *this;
   705     }
   706 
   707     /// \brief Set the costs of the arcs.
   708     ///
   709     /// This function sets the costs of the arcs.
   710     /// If it is not used before calling \ref run(), the costs
   711     /// will be set to \c 1 on all arcs.
   712     ///
   713     /// \param map An arc map storing the costs.
   714     /// Its \c Value type must be convertible to the \c Cost type
   715     /// of the algorithm.
   716     ///
   717     /// \return <tt>(*this)</tt>
   718     template<typename CostMap>
   719     NetworkSimplex& costMap(const CostMap& map) {
   720       for (ArcIt a(_graph); a != INVALID; ++a) {
   721         _cost[_arc_id[a]] = map[a];
   722       }
   723       return *this;
   724     }
   725 
   726     /// \brief Set the supply values of the nodes.
   727     ///
   728     /// This function sets the supply values of the nodes.
   729     /// If neither this function nor \ref stSupply() is used before
   730     /// calling \ref run(), the supply of each node will be set to zero.
   731     ///
   732     /// \param map A node map storing the supply values.
   733     /// Its \c Value type must be convertible to the \c Value type
   734     /// of the algorithm.
   735     ///
   736     /// \return <tt>(*this)</tt>
   737     ///
   738     /// \sa supplyType()
   739     template<typename SupplyMap>
   740     NetworkSimplex& supplyMap(const SupplyMap& map) {
   741       for (NodeIt n(_graph); n != INVALID; ++n) {
   742         _supply[_node_id[n]] = map[n];
   743       }
   744       return *this;
   745     }
   746 
   747     /// \brief Set single source and target nodes and a supply value.
   748     ///
   749     /// This function sets a single source node and a single target node
   750     /// and the required flow value.
   751     /// If neither this function nor \ref supplyMap() is used before
   752     /// calling \ref run(), the supply of each node will be set to zero.
   753     ///
   754     /// Using this function has the same effect as using \ref supplyMap()
   755     /// with a map in which \c k is assigned to \c s, \c -k is
   756     /// assigned to \c t and all other nodes have zero supply value.
   757     ///
   758     /// \param s The source node.
   759     /// \param t The target node.
   760     /// \param k The required amount of flow from node \c s to node \c t
   761     /// (i.e. the supply of \c s and the demand of \c t).
   762     ///
   763     /// \return <tt>(*this)</tt>
   764     NetworkSimplex& stSupply(const Node& s, const Node& t, Value k) {
   765       for (int i = 0; i != _node_num; ++i) {
   766         _supply[i] = 0;
   767       }
   768       _supply[_node_id[s]] =  k;
   769       _supply[_node_id[t]] = -k;
   770       return *this;
   771     }
   772 
   773     /// \brief Set the type of the supply constraints.
   774     ///
   775     /// This function sets the type of the supply/demand constraints.
   776     /// If it is not used before calling \ref run(), the \ref GEQ supply
   777     /// type will be used.
   778     ///
   779     /// For more information, see \ref SupplyType.
   780     ///
   781     /// \return <tt>(*this)</tt>
   782     NetworkSimplex& supplyType(SupplyType supply_type) {
   783       _stype = supply_type;
   784       return *this;
   785     }
   786 
   787     /// @}
   788 
   789     /// \name Execution Control
   790     /// The algorithm can be executed using \ref run().
   791 
   792     /// @{
   793 
   794     /// \brief Run the algorithm.
   795     ///
   796     /// This function runs the algorithm.
   797     /// The paramters can be specified using functions \ref lowerMap(),
   798     /// \ref upperMap(), \ref costMap(), \ref supplyMap(), \ref stSupply(),
   799     /// \ref supplyType().
   800     /// For example,
   801     /// \code
   802     ///   NetworkSimplex<ListDigraph> ns(graph);
   803     ///   ns.lowerMap(lower).upperMap(upper).costMap(cost)
   804     ///     .supplyMap(sup).run();
   805     /// \endcode
   806     ///
   807     /// This function can be called more than once. All the given parameters
   808     /// are kept for the next call, unless \ref resetParams() or \ref reset()
   809     /// is used, thus only the modified parameters have to be set again.
   810     /// If the underlying digraph was also modified after the construction
   811     /// of the class (or the last \ref reset() call), then the \ref reset()
   812     /// function must be called.
   813     ///
   814     /// \param pivot_rule The pivot rule that will be used during the
   815     /// algorithm. For more information, see \ref PivotRule.
   816     ///
   817     /// \return \c INFEASIBLE if no feasible flow exists,
   818     /// \n \c OPTIMAL if the problem has optimal solution
   819     /// (i.e. it is feasible and bounded), and the algorithm has found
   820     /// optimal flow and node potentials (primal and dual solutions),
   821     /// \n \c UNBOUNDED if the objective function of the problem is
   822     /// unbounded, i.e. there is a directed cycle having negative total
   823     /// cost and infinite upper bound.
   824     ///
   825     /// \see ProblemType, PivotRule
   826     /// \see resetParams(), reset()
   827     ProblemType run(PivotRule pivot_rule = BLOCK_SEARCH) {
   828       if (!init()) return INFEASIBLE;
   829       return start(pivot_rule);
   830     }
   831 
   832     /// \brief Reset all the parameters that have been given before.
   833     ///
   834     /// This function resets all the paramaters that have been given
   835     /// before using functions \ref lowerMap(), \ref upperMap(),
   836     /// \ref costMap(), \ref supplyMap(), \ref stSupply(), \ref supplyType().
   837     ///
   838     /// It is useful for multiple \ref run() calls. Basically, all the given
   839     /// parameters are kept for the next \ref run() call, unless
   840     /// \ref resetParams() or \ref reset() is used.
   841     /// If the underlying digraph was also modified after the construction
   842     /// of the class or the last \ref reset() call, then the \ref reset()
   843     /// function must be used, otherwise \ref resetParams() is sufficient.
   844     ///
   845     /// For example,
   846     /// \code
   847     ///   NetworkSimplex<ListDigraph> ns(graph);
   848     ///
   849     ///   // First run
   850     ///   ns.lowerMap(lower).upperMap(upper).costMap(cost)
   851     ///     .supplyMap(sup).run();
   852     ///
   853     ///   // Run again with modified cost map (resetParams() is not called,
   854     ///   // so only the cost map have to be set again)
   855     ///   cost[e] += 100;
   856     ///   ns.costMap(cost).run();
   857     ///
   858     ///   // Run again from scratch using resetParams()
   859     ///   // (the lower bounds will be set to zero on all arcs)
   860     ///   ns.resetParams();
   861     ///   ns.upperMap(capacity).costMap(cost)
   862     ///     .supplyMap(sup).run();
   863     /// \endcode
   864     ///
   865     /// \return <tt>(*this)</tt>
   866     ///
   867     /// \see reset(), run()
   868     NetworkSimplex& resetParams() {
   869       for (int i = 0; i != _node_num; ++i) {
   870         _supply[i] = 0;
   871       }
   872       for (int i = 0; i != _arc_num; ++i) {
   873         _lower[i] = 0;
   874         _upper[i] = INF;
   875         _cost[i] = 1;
   876       }
   877       _have_lower = false;
   878       _stype = GEQ;
   879       return *this;
   880     }
   881 
   882     /// \brief Reset the internal data structures and all the parameters
   883     /// that have been given before.
   884     ///
   885     /// This function resets the internal data structures and all the
   886     /// paramaters that have been given before using functions \ref lowerMap(),
   887     /// \ref upperMap(), \ref costMap(), \ref supplyMap(), \ref stSupply(),
   888     /// \ref supplyType().
   889     ///
   890     /// It is useful for multiple \ref run() calls. Basically, all the given
   891     /// parameters are kept for the next \ref run() call, unless
   892     /// \ref resetParams() or \ref reset() is used.
   893     /// If the underlying digraph was also modified after the construction
   894     /// of the class or the last \ref reset() call, then the \ref reset()
   895     /// function must be used, otherwise \ref resetParams() is sufficient.
   896     ///
   897     /// See \ref resetParams() for examples.
   898     ///
   899     /// \return <tt>(*this)</tt>
   900     ///
   901     /// \see resetParams(), run()
   902     NetworkSimplex& reset() {
   903       // Resize vectors
   904       _node_num = countNodes(_graph);
   905       _arc_num = countArcs(_graph);
   906       int all_node_num = _node_num + 1;
   907       int max_arc_num = _arc_num + 2 * _node_num;
   908 
   909       _source.resize(max_arc_num);
   910       _target.resize(max_arc_num);
   911 
   912       _lower.resize(_arc_num);
   913       _upper.resize(_arc_num);
   914       _cap.resize(max_arc_num);
   915       _cost.resize(max_arc_num);
   916       _supply.resize(all_node_num);
   917       _flow.resize(max_arc_num);
   918       _pi.resize(all_node_num);
   919 
   920       _parent.resize(all_node_num);
   921       _pred.resize(all_node_num);
   922       _pred_dir.resize(all_node_num);
   923       _thread.resize(all_node_num);
   924       _rev_thread.resize(all_node_num);
   925       _succ_num.resize(all_node_num);
   926       _last_succ.resize(all_node_num);
   927       _state.resize(max_arc_num);
   928 
   929       // Copy the graph
   930       int i = 0;
   931       for (NodeIt n(_graph); n != INVALID; ++n, ++i) {
   932         _node_id[n] = i;
   933       }
   934       if (_arc_mixing) {
   935         // Store the arcs in a mixed order
   936         const int skip = std::max(_arc_num / _node_num, 3);
   937         int i = 0, j = 0;
   938         for (ArcIt a(_graph); a != INVALID; ++a) {
   939           _arc_id[a] = i;
   940           _source[i] = _node_id[_graph.source(a)];
   941           _target[i] = _node_id[_graph.target(a)];
   942           if ((i += skip) >= _arc_num) i = ++j;
   943         }
   944       } else {
   945         // Store the arcs in the original order
   946         int i = 0;
   947         for (ArcIt a(_graph); a != INVALID; ++a, ++i) {
   948           _arc_id[a] = i;
   949           _source[i] = _node_id[_graph.source(a)];
   950           _target[i] = _node_id[_graph.target(a)];
   951         }
   952       }
   953 
   954       // Reset parameters
   955       resetParams();
   956       return *this;
   957     }
   958 
   959     /// @}
   960 
   961     /// \name Query Functions
   962     /// The results of the algorithm can be obtained using these
   963     /// functions.\n
   964     /// The \ref run() function must be called before using them.
   965 
   966     /// @{
   967 
   968     /// \brief Return the total cost of the found flow.
   969     ///
   970     /// This function returns the total cost of the found flow.
   971     /// Its complexity is O(e).
   972     ///
   973     /// \note The return type of the function can be specified as a
   974     /// template parameter. For example,
   975     /// \code
   976     ///   ns.totalCost<double>();
   977     /// \endcode
   978     /// It is useful if the total cost cannot be stored in the \c Cost
   979     /// type of the algorithm, which is the default return type of the
   980     /// function.
   981     ///
   982     /// \pre \ref run() must be called before using this function.
   983     template <typename Number>
   984     Number totalCost() const {
   985       Number c = 0;
   986       for (ArcIt a(_graph); a != INVALID; ++a) {
   987         int i = _arc_id[a];
   988         c += Number(_flow[i]) * Number(_cost[i]);
   989       }
   990       return c;
   991     }
   992 
   993 #ifndef DOXYGEN
   994     Cost totalCost() const {
   995       return totalCost<Cost>();
   996     }
   997 #endif
   998 
   999     /// \brief Return the flow on the given arc.
  1000     ///
  1001     /// This function returns the flow on the given arc.
  1002     ///
  1003     /// \pre \ref run() must be called before using this function.
  1004     Value flow(const Arc& a) const {
  1005       return _flow[_arc_id[a]];
  1006     }
  1007 
  1008     /// \brief Return the flow map (the primal solution).
  1009     ///
  1010     /// This function copies the flow value on each arc into the given
  1011     /// map. The \c Value type of the algorithm must be convertible to
  1012     /// the \c Value type of the map.
  1013     ///
  1014     /// \pre \ref run() must be called before using this function.
  1015     template <typename FlowMap>
  1016     void flowMap(FlowMap &map) const {
  1017       for (ArcIt a(_graph); a != INVALID; ++a) {
  1018         map.set(a, _flow[_arc_id[a]]);
  1019       }
  1020     }
  1021 
  1022     /// \brief Return the potential (dual value) of the given node.
  1023     ///
  1024     /// This function returns the potential (dual value) of the
  1025     /// given node.
  1026     ///
  1027     /// \pre \ref run() must be called before using this function.
  1028     Cost potential(const Node& n) const {
  1029       return _pi[_node_id[n]];
  1030     }
  1031 
  1032     /// \brief Return the potential map (the dual solution).
  1033     ///
  1034     /// This function copies the potential (dual value) of each node
  1035     /// into the given map.
  1036     /// The \c Cost type of the algorithm must be convertible to the
  1037     /// \c Value type of the map.
  1038     ///
  1039     /// \pre \ref run() must be called before using this function.
  1040     template <typename PotentialMap>
  1041     void potentialMap(PotentialMap &map) const {
  1042       for (NodeIt n(_graph); n != INVALID; ++n) {
  1043         map.set(n, _pi[_node_id[n]]);
  1044       }
  1045     }
  1046 
  1047     /// @}
  1048 
  1049   private:
  1050 
  1051     // Initialize internal data structures
  1052     bool init() {
  1053       if (_node_num == 0) return false;
  1054 
  1055       // Check the sum of supply values
  1056       _sum_supply = 0;
  1057       for (int i = 0; i != _node_num; ++i) {
  1058         _sum_supply += _supply[i];
  1059       }
  1060       if ( !((_stype == GEQ && _sum_supply <= 0) ||
  1061              (_stype == LEQ && _sum_supply >= 0)) ) return false;
  1062 
  1063       // Remove non-zero lower bounds
  1064       if (_have_lower) {
  1065         for (int i = 0; i != _arc_num; ++i) {
  1066           Value c = _lower[i];
  1067           if (c >= 0) {
  1068             _cap[i] = _upper[i] < MAX ? _upper[i] - c : INF;
  1069           } else {
  1070             _cap[i] = _upper[i] < MAX + c ? _upper[i] - c : INF;
  1071           }
  1072           _supply[_source[i]] -= c;
  1073           _supply[_target[i]] += c;
  1074         }
  1075       } else {
  1076         for (int i = 0; i != _arc_num; ++i) {
  1077           _cap[i] = _upper[i];
  1078         }
  1079       }
  1080 
  1081       // Initialize artifical cost
  1082       Cost ART_COST;
  1083       if (std::numeric_limits<Cost>::is_exact) {
  1084         ART_COST = std::numeric_limits<Cost>::max() / 2 + 1;
  1085       } else {
  1086         ART_COST = 0;
  1087         for (int i = 0; i != _arc_num; ++i) {
  1088           if (_cost[i] > ART_COST) ART_COST = _cost[i];
  1089         }
  1090         ART_COST = (ART_COST + 1) * _node_num;
  1091       }
  1092 
  1093       // Initialize arc maps
  1094       for (int i = 0; i != _arc_num; ++i) {
  1095         _flow[i] = 0;
  1096         _state[i] = STATE_LOWER;
  1097       }
  1098 
  1099       // Set data for the artificial root node
  1100       _root = _node_num;
  1101       _parent[_root] = -1;
  1102       _pred[_root] = -1;
  1103       _thread[_root] = 0;
  1104       _rev_thread[0] = _root;
  1105       _succ_num[_root] = _node_num + 1;
  1106       _last_succ[_root] = _root - 1;
  1107       _supply[_root] = -_sum_supply;
  1108       _pi[_root] = 0;
  1109 
  1110       // Add artificial arcs and initialize the spanning tree data structure
  1111       if (_sum_supply == 0) {
  1112         // EQ supply constraints
  1113         _search_arc_num = _arc_num;
  1114         _all_arc_num = _arc_num + _node_num;
  1115         for (int u = 0, e = _arc_num; u != _node_num; ++u, ++e) {
  1116           _parent[u] = _root;
  1117           _pred[u] = e;
  1118           _thread[u] = u + 1;
  1119           _rev_thread[u + 1] = u;
  1120           _succ_num[u] = 1;
  1121           _last_succ[u] = u;
  1122           _cap[e] = INF;
  1123           _state[e] = STATE_TREE;
  1124           if (_supply[u] >= 0) {
  1125             _pred_dir[u] = DIR_UP;
  1126             _pi[u] = 0;
  1127             _source[e] = u;
  1128             _target[e] = _root;
  1129             _flow[e] = _supply[u];
  1130             _cost[e] = 0;
  1131           } else {
  1132             _pred_dir[u] = DIR_DOWN;
  1133             _pi[u] = ART_COST;
  1134             _source[e] = _root;
  1135             _target[e] = u;
  1136             _flow[e] = -_supply[u];
  1137             _cost[e] = ART_COST;
  1138           }
  1139         }
  1140       }
  1141       else if (_sum_supply > 0) {
  1142         // LEQ supply constraints
  1143         _search_arc_num = _arc_num + _node_num;
  1144         int f = _arc_num + _node_num;
  1145         for (int u = 0, e = _arc_num; u != _node_num; ++u, ++e) {
  1146           _parent[u] = _root;
  1147           _thread[u] = u + 1;
  1148           _rev_thread[u + 1] = u;
  1149           _succ_num[u] = 1;
  1150           _last_succ[u] = u;
  1151           if (_supply[u] >= 0) {
  1152             _pred_dir[u] = DIR_UP;
  1153             _pi[u] = 0;
  1154             _pred[u] = e;
  1155             _source[e] = u;
  1156             _target[e] = _root;
  1157             _cap[e] = INF;
  1158             _flow[e] = _supply[u];
  1159             _cost[e] = 0;
  1160             _state[e] = STATE_TREE;
  1161           } else {
  1162             _pred_dir[u] = DIR_DOWN;
  1163             _pi[u] = ART_COST;
  1164             _pred[u] = f;
  1165             _source[f] = _root;
  1166             _target[f] = u;
  1167             _cap[f] = INF;
  1168             _flow[f] = -_supply[u];
  1169             _cost[f] = ART_COST;
  1170             _state[f] = STATE_TREE;
  1171             _source[e] = u;
  1172             _target[e] = _root;
  1173             _cap[e] = INF;
  1174             _flow[e] = 0;
  1175             _cost[e] = 0;
  1176             _state[e] = STATE_LOWER;
  1177             ++f;
  1178           }
  1179         }
  1180         _all_arc_num = f;
  1181       }
  1182       else {
  1183         // GEQ supply constraints
  1184         _search_arc_num = _arc_num + _node_num;
  1185         int f = _arc_num + _node_num;
  1186         for (int u = 0, e = _arc_num; u != _node_num; ++u, ++e) {
  1187           _parent[u] = _root;
  1188           _thread[u] = u + 1;
  1189           _rev_thread[u + 1] = u;
  1190           _succ_num[u] = 1;
  1191           _last_succ[u] = u;
  1192           if (_supply[u] <= 0) {
  1193             _pred_dir[u] = DIR_DOWN;
  1194             _pi[u] = 0;
  1195             _pred[u] = e;
  1196             _source[e] = _root;
  1197             _target[e] = u;
  1198             _cap[e] = INF;
  1199             _flow[e] = -_supply[u];
  1200             _cost[e] = 0;
  1201             _state[e] = STATE_TREE;
  1202           } else {
  1203             _pred_dir[u] = DIR_UP;
  1204             _pi[u] = -ART_COST;
  1205             _pred[u] = f;
  1206             _source[f] = u;
  1207             _target[f] = _root;
  1208             _cap[f] = INF;
  1209             _flow[f] = _supply[u];
  1210             _state[f] = STATE_TREE;
  1211             _cost[f] = ART_COST;
  1212             _source[e] = _root;
  1213             _target[e] = u;
  1214             _cap[e] = INF;
  1215             _flow[e] = 0;
  1216             _cost[e] = 0;
  1217             _state[e] = STATE_LOWER;
  1218             ++f;
  1219           }
  1220         }
  1221         _all_arc_num = f;
  1222       }
  1223 
  1224       return true;
  1225     }
  1226 
  1227     // Find the join node
  1228     void findJoinNode() {
  1229       int u = _source[in_arc];
  1230       int v = _target[in_arc];
  1231       while (u != v) {
  1232         if (_succ_num[u] < _succ_num[v]) {
  1233           u = _parent[u];
  1234         } else {
  1235           v = _parent[v];
  1236         }
  1237       }
  1238       join = u;
  1239     }
  1240 
  1241     // Find the leaving arc of the cycle and returns true if the
  1242     // leaving arc is not the same as the entering arc
  1243     bool findLeavingArc() {
  1244       // Initialize first and second nodes according to the direction
  1245       // of the cycle
  1246       int first, second;
  1247       if (_state[in_arc] == STATE_LOWER) {
  1248         first  = _source[in_arc];
  1249         second = _target[in_arc];
  1250       } else {
  1251         first  = _target[in_arc];
  1252         second = _source[in_arc];
  1253       }
  1254       delta = _cap[in_arc];
  1255       int result = 0;
  1256       Value c, d;
  1257       int e;
  1258 
  1259       // Search the cycle form the first node to the join node
  1260       for (int u = first; u != join; u = _parent[u]) {
  1261         e = _pred[u];
  1262         d = _flow[e];
  1263         if (_pred_dir[u] == DIR_DOWN) {
  1264           c = _cap[e];
  1265           d = c >= MAX ? INF : c - d;
  1266         }
  1267         if (d < delta) {
  1268           delta = d;
  1269           u_out = u;
  1270           result = 1;
  1271         }
  1272       }
  1273 
  1274       // Search the cycle form the second node to the join node
  1275       for (int u = second; u != join; u = _parent[u]) {
  1276         e = _pred[u];
  1277         d = _flow[e];
  1278         if (_pred_dir[u] == DIR_UP) {
  1279           c = _cap[e];
  1280           d = c >= MAX ? INF : c - d;
  1281         }
  1282         if (d <= delta) {
  1283           delta = d;
  1284           u_out = u;
  1285           result = 2;
  1286         }
  1287       }
  1288 
  1289       if (result == 1) {
  1290         u_in = first;
  1291         v_in = second;
  1292       } else {
  1293         u_in = second;
  1294         v_in = first;
  1295       }
  1296       return result != 0;
  1297     }
  1298 
  1299     // Change _flow and _state vectors
  1300     void changeFlow(bool change) {
  1301       // Augment along the cycle
  1302       if (delta > 0) {
  1303         Value val = _state[in_arc] * delta;
  1304         _flow[in_arc] += val;
  1305         for (int u = _source[in_arc]; u != join; u = _parent[u]) {
  1306           _flow[_pred[u]] -= _pred_dir[u] * val;
  1307         }
  1308         for (int u = _target[in_arc]; u != join; u = _parent[u]) {
  1309           _flow[_pred[u]] += _pred_dir[u] * val;
  1310         }
  1311       }
  1312       // Update the state of the entering and leaving arcs
  1313       if (change) {
  1314         _state[in_arc] = STATE_TREE;
  1315         _state[_pred[u_out]] =
  1316           (_flow[_pred[u_out]] == 0) ? STATE_LOWER : STATE_UPPER;
  1317       } else {
  1318         _state[in_arc] = -_state[in_arc];
  1319       }
  1320     }
  1321 
  1322     // Update the tree structure
  1323     void updateTreeStructure() {
  1324       int old_rev_thread = _rev_thread[u_out];
  1325       int old_succ_num = _succ_num[u_out];
  1326       int old_last_succ = _last_succ[u_out];
  1327       v_out = _parent[u_out];
  1328 
  1329       // Check if u_in and u_out coincide
  1330       if (u_in == u_out) {
  1331         // Update _parent, _pred, _pred_dir
  1332         _parent[u_in] = v_in;
  1333         _pred[u_in] = in_arc;
  1334         _pred_dir[u_in] = u_in == _source[in_arc] ? DIR_UP : DIR_DOWN;
  1335 
  1336         // Update _thread and _rev_thread
  1337         if (_thread[v_in] != u_out) {
  1338           int after = _thread[old_last_succ];
  1339           _thread[old_rev_thread] = after;
  1340           _rev_thread[after] = old_rev_thread;
  1341           after = _thread[v_in];
  1342           _thread[v_in] = u_out;
  1343           _rev_thread[u_out] = v_in;
  1344           _thread[old_last_succ] = after;
  1345           _rev_thread[after] = old_last_succ;
  1346         }
  1347       } else {
  1348         // Handle the case when old_rev_thread equals to v_in
  1349         // (it also means that join and v_out coincide)
  1350         int thread_continue = old_rev_thread == v_in ?
  1351           _thread[old_last_succ] : _thread[v_in];
  1352 
  1353         // Update _thread and _parent along the stem nodes (i.e. the nodes
  1354         // between u_in and u_out, whose parent have to be changed)
  1355         int stem = u_in;              // the current stem node
  1356         int par_stem = v_in;          // the new parent of stem
  1357         int next_stem;                // the next stem node
  1358         int last = _last_succ[u_in];  // the last successor of stem
  1359         int before, after = _thread[last];
  1360         _thread[v_in] = u_in;
  1361         _dirty_revs.clear();
  1362         _dirty_revs.push_back(v_in);
  1363         while (stem != u_out) {
  1364           // Insert the next stem node into the thread list
  1365           next_stem = _parent[stem];
  1366           _thread[last] = next_stem;
  1367           _dirty_revs.push_back(last);
  1368 
  1369           // Remove the subtree of stem from the thread list
  1370           before = _rev_thread[stem];
  1371           _thread[before] = after;
  1372           _rev_thread[after] = before;
  1373 
  1374           // Change the parent node and shift stem nodes
  1375           _parent[stem] = par_stem;
  1376           par_stem = stem;
  1377           stem = next_stem;
  1378 
  1379           // Update last and after
  1380           last = _last_succ[stem] == _last_succ[par_stem] ?
  1381             _rev_thread[par_stem] : _last_succ[stem];
  1382           after = _thread[last];
  1383         }
  1384         _parent[u_out] = par_stem;
  1385         _thread[last] = thread_continue;
  1386         _rev_thread[thread_continue] = last;
  1387         _last_succ[u_out] = last;
  1388 
  1389         // Remove the subtree of u_out from the thread list except for
  1390         // the case when old_rev_thread equals to v_in
  1391         if (old_rev_thread != v_in) {
  1392           _thread[old_rev_thread] = after;
  1393           _rev_thread[after] = old_rev_thread;
  1394         }
  1395 
  1396         // Update _rev_thread using the new _thread values
  1397         for (int i = 0; i != int(_dirty_revs.size()); ++i) {
  1398           int u = _dirty_revs[i];
  1399           _rev_thread[_thread[u]] = u;
  1400         }
  1401 
  1402         // Update _pred, _pred_dir, _last_succ and _succ_num for the
  1403         // stem nodes from u_out to u_in
  1404         int tmp_sc = 0, tmp_ls = _last_succ[u_out];
  1405         for (int u = u_out, p = _parent[u]; u != u_in; u = p, p = _parent[u]) {
  1406           _pred[u] = _pred[p];
  1407           _pred_dir[u] = -_pred_dir[p];
  1408           tmp_sc += _succ_num[u] - _succ_num[p];
  1409           _succ_num[u] = tmp_sc;
  1410           _last_succ[p] = tmp_ls;
  1411         }
  1412         _pred[u_in] = in_arc;
  1413         _pred_dir[u_in] = u_in == _source[in_arc] ? DIR_UP : DIR_DOWN;
  1414         _succ_num[u_in] = old_succ_num;
  1415       }
  1416 
  1417       // Update _last_succ from v_in towards the root
  1418       int up_limit_out = _last_succ[join] == v_in ? join : -1;
  1419       int last_succ_out = _last_succ[u_out];
  1420       for (int u = v_in; u != -1 && _last_succ[u] == v_in; u = _parent[u]) {
  1421         _last_succ[u] = last_succ_out;
  1422       }
  1423 
  1424       // Update _last_succ from v_out towards the root
  1425       if (join != old_rev_thread && v_in != old_rev_thread) {
  1426         for (int u = v_out; u != up_limit_out && _last_succ[u] == old_last_succ;
  1427              u = _parent[u]) {
  1428           _last_succ[u] = old_rev_thread;
  1429         }
  1430       }
  1431       else if (last_succ_out != old_last_succ) {
  1432         for (int u = v_out; u != up_limit_out && _last_succ[u] == old_last_succ;
  1433              u = _parent[u]) {
  1434           _last_succ[u] = last_succ_out;
  1435         }
  1436       }
  1437 
  1438       // Update _succ_num from v_in to join
  1439       for (int u = v_in; u != join; u = _parent[u]) {
  1440         _succ_num[u] += old_succ_num;
  1441       }
  1442       // Update _succ_num from v_out to join
  1443       for (int u = v_out; u != join; u = _parent[u]) {
  1444         _succ_num[u] -= old_succ_num;
  1445       }
  1446     }
  1447 
  1448     // Update potentials in the subtree that has been moved
  1449     void updatePotential() {
  1450       Cost sigma = _pi[v_in] - _pi[u_in] -
  1451                    _pred_dir[u_in] * _cost[in_arc];
  1452       int end = _thread[_last_succ[u_in]];
  1453       for (int u = u_in; u != end; u = _thread[u]) {
  1454         _pi[u] += sigma;
  1455       }
  1456     }
  1457 
  1458     // Heuristic initial pivots
  1459     bool initialPivots() {
  1460       Value curr, total = 0;
  1461       std::vector<Node> supply_nodes, demand_nodes;
  1462       for (NodeIt u(_graph); u != INVALID; ++u) {
  1463         curr = _supply[_node_id[u]];
  1464         if (curr > 0) {
  1465           total += curr;
  1466           supply_nodes.push_back(u);
  1467         }
  1468         else if (curr < 0) {
  1469           demand_nodes.push_back(u);
  1470         }
  1471       }
  1472       if (_sum_supply > 0) total -= _sum_supply;
  1473       if (total <= 0) return true;
  1474 
  1475       IntVector arc_vector;
  1476       if (_sum_supply >= 0) {
  1477         if (supply_nodes.size() == 1 && demand_nodes.size() == 1) {
  1478           // Perform a reverse graph search from the sink to the source
  1479           typename GR::template NodeMap<bool> reached(_graph, false);
  1480           Node s = supply_nodes[0], t = demand_nodes[0];
  1481           std::vector<Node> stack;
  1482           reached[t] = true;
  1483           stack.push_back(t);
  1484           while (!stack.empty()) {
  1485             Node u, v = stack.back();
  1486             stack.pop_back();
  1487             if (v == s) break;
  1488             for (InArcIt a(_graph, v); a != INVALID; ++a) {
  1489               if (reached[u = _graph.source(a)]) continue;
  1490               int j = _arc_id[a];
  1491               if (_cap[j] >= total) {
  1492                 arc_vector.push_back(j);
  1493                 reached[u] = true;
  1494                 stack.push_back(u);
  1495               }
  1496             }
  1497           }
  1498         } else {
  1499           // Find the min. cost incomming arc for each demand node
  1500           for (int i = 0; i != int(demand_nodes.size()); ++i) {
  1501             Node v = demand_nodes[i];
  1502             Cost c, min_cost = std::numeric_limits<Cost>::max();
  1503             Arc min_arc = INVALID;
  1504             for (InArcIt a(_graph, v); a != INVALID; ++a) {
  1505               c = _cost[_arc_id[a]];
  1506               if (c < min_cost) {
  1507                 min_cost = c;
  1508                 min_arc = a;
  1509               }
  1510             }
  1511             if (min_arc != INVALID) {
  1512               arc_vector.push_back(_arc_id[min_arc]);
  1513             }
  1514           }
  1515         }
  1516       } else {
  1517         // Find the min. cost outgoing arc for each supply node
  1518         for (int i = 0; i != int(supply_nodes.size()); ++i) {
  1519           Node u = supply_nodes[i];
  1520           Cost c, min_cost = std::numeric_limits<Cost>::max();
  1521           Arc min_arc = INVALID;
  1522           for (OutArcIt a(_graph, u); a != INVALID; ++a) {
  1523             c = _cost[_arc_id[a]];
  1524             if (c < min_cost) {
  1525               min_cost = c;
  1526               min_arc = a;
  1527             }
  1528           }
  1529           if (min_arc != INVALID) {
  1530             arc_vector.push_back(_arc_id[min_arc]);
  1531           }
  1532         }
  1533       }
  1534 
  1535       // Perform heuristic initial pivots
  1536       for (int i = 0; i != int(arc_vector.size()); ++i) {
  1537         in_arc = arc_vector[i];
  1538         if (_state[in_arc] * (_cost[in_arc] + _pi[_source[in_arc]] -
  1539             _pi[_target[in_arc]]) >= 0) continue;
  1540         findJoinNode();
  1541         bool change = findLeavingArc();
  1542         if (delta >= MAX) return false;
  1543         changeFlow(change);
  1544         if (change) {
  1545           updateTreeStructure();
  1546           updatePotential();
  1547         }
  1548       }
  1549       return true;
  1550     }
  1551 
  1552     // Execute the algorithm
  1553     ProblemType start(PivotRule pivot_rule) {
  1554       // Select the pivot rule implementation
  1555       switch (pivot_rule) {
  1556         case FIRST_ELIGIBLE:
  1557           return start<FirstEligiblePivotRule>();
  1558         case BEST_ELIGIBLE:
  1559           return start<BestEligiblePivotRule>();
  1560         case BLOCK_SEARCH:
  1561           return start<BlockSearchPivotRule>();
  1562         case CANDIDATE_LIST:
  1563           return start<CandidateListPivotRule>();
  1564         case ALTERING_LIST:
  1565           return start<AlteringListPivotRule>();
  1566       }
  1567       return INFEASIBLE; // avoid warning
  1568     }
  1569 
  1570     template <typename PivotRuleImpl>
  1571     ProblemType start() {
  1572       PivotRuleImpl pivot(*this);
  1573 
  1574       // Perform heuristic initial pivots
  1575       if (!initialPivots()) return UNBOUNDED;
  1576 
  1577       // Execute the Network Simplex algorithm
  1578       while (pivot.findEnteringArc()) {
  1579         findJoinNode();
  1580         bool change = findLeavingArc();
  1581         if (delta >= MAX) return UNBOUNDED;
  1582         changeFlow(change);
  1583         if (change) {
  1584           updateTreeStructure();
  1585           updatePotential();
  1586         }
  1587       }
  1588 
  1589       // Check feasibility
  1590       for (int e = _search_arc_num; e != _all_arc_num; ++e) {
  1591         if (_flow[e] != 0) return INFEASIBLE;
  1592       }
  1593 
  1594       // Transform the solution and the supply map to the original form
  1595       if (_have_lower) {
  1596         for (int i = 0; i != _arc_num; ++i) {
  1597           Value c = _lower[i];
  1598           if (c != 0) {
  1599             _flow[i] += c;
  1600             _supply[_source[i]] += c;
  1601             _supply[_target[i]] -= c;
  1602           }
  1603         }
  1604       }
  1605 
  1606       // Shift potentials to meet the requirements of the GEQ/LEQ type
  1607       // optimality conditions
  1608       if (_sum_supply == 0) {
  1609         if (_stype == GEQ) {
  1610           Cost max_pot = -std::numeric_limits<Cost>::max();
  1611           for (int i = 0; i != _node_num; ++i) {
  1612             if (_pi[i] > max_pot) max_pot = _pi[i];
  1613           }
  1614           if (max_pot > 0) {
  1615             for (int i = 0; i != _node_num; ++i)
  1616               _pi[i] -= max_pot;
  1617           }
  1618         } else {
  1619           Cost min_pot = std::numeric_limits<Cost>::max();
  1620           for (int i = 0; i != _node_num; ++i) {
  1621             if (_pi[i] < min_pot) min_pot = _pi[i];
  1622           }
  1623           if (min_pot < 0) {
  1624             for (int i = 0; i != _node_num; ++i)
  1625               _pi[i] -= min_pot;
  1626           }
  1627         }
  1628       }
  1629 
  1630       return OPTIMAL;
  1631     }
  1632 
  1633   }; //class NetworkSimplex
  1634 
  1635   ///@}
  1636 
  1637 } //namespace lemon
  1638 
  1639 #endif //LEMON_NETWORK_SIMPLEX_H