lemon/network_simplex.h
author Alpar Juttner <alpar@cs.elte.hu>
Fri, 12 Feb 2010 22:24:26 +0100
changeset 831 cc9e0c15d747
parent 812 4b1b378823dc
child 840 2914b6f0fde0
permissions -rw-r--r--
Merge
     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) 2003-2009
     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, %NetworkSimplex is the fastest implementation available
    51   /// in LEMON for this problem.
    52   /// Moreover, it supports both directions of the supply/demand inequality
    53   /// 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     /// proved 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<char> CharVector;
   168     typedef std::vector<Value> ValueVector;
   169     typedef std::vector<Cost> CostVector;
   170 
   171     // State constants for arcs
   172     enum ArcStateEnum {
   173       STATE_UPPER = -1,
   174       STATE_TREE  =  0,
   175       STATE_LOWER =  1
   176     };
   177 
   178   private:
   179 
   180     // Data related to the underlying digraph
   181     const GR &_graph;
   182     int _node_num;
   183     int _arc_num;
   184     int _all_arc_num;
   185     int _search_arc_num;
   186 
   187     // Parameters of the problem
   188     bool _have_lower;
   189     SupplyType _stype;
   190     Value _sum_supply;
   191 
   192     // Data structures for storing the digraph
   193     IntNodeMap _node_id;
   194     IntArcMap _arc_id;
   195     IntVector _source;
   196     IntVector _target;
   197     bool _arc_mixing;
   198 
   199     // Node and arc data
   200     ValueVector _lower;
   201     ValueVector _upper;
   202     ValueVector _cap;
   203     CostVector _cost;
   204     ValueVector _supply;
   205     ValueVector _flow;
   206     CostVector _pi;
   207 
   208     // Data for storing the spanning tree structure
   209     IntVector _parent;
   210     IntVector _pred;
   211     IntVector _thread;
   212     IntVector _rev_thread;
   213     IntVector _succ_num;
   214     IntVector _last_succ;
   215     IntVector _dirty_revs;
   216     CharVector _forward;
   217     CharVector _state;
   218     int _root;
   219 
   220     // Temporary data used in the current pivot iteration
   221     int in_arc, join, u_in, v_in, u_out, v_out;
   222     int first, second, right, last;
   223     int stem, par_stem, new_stem;
   224     Value delta;
   225     
   226     const Value MAX;
   227 
   228   public:
   229   
   230     /// \brief Constant for infinite upper bounds (capacities).
   231     ///
   232     /// Constant for infinite upper bounds (capacities).
   233     /// It is \c std::numeric_limits<Value>::infinity() if available,
   234     /// \c std::numeric_limits<Value>::max() otherwise.
   235     const Value INF;
   236 
   237   private:
   238 
   239     // Implementation of the First Eligible pivot rule
   240     class FirstEligiblePivotRule
   241     {
   242     private:
   243 
   244       // References to the NetworkSimplex class
   245       const IntVector  &_source;
   246       const IntVector  &_target;
   247       const CostVector &_cost;
   248       const CharVector &_state;
   249       const CostVector &_pi;
   250       int &_in_arc;
   251       int _search_arc_num;
   252 
   253       // Pivot rule data
   254       int _next_arc;
   255 
   256     public:
   257 
   258       // Constructor
   259       FirstEligiblePivotRule(NetworkSimplex &ns) :
   260         _source(ns._source), _target(ns._target),
   261         _cost(ns._cost), _state(ns._state), _pi(ns._pi),
   262         _in_arc(ns.in_arc), _search_arc_num(ns._search_arc_num),
   263         _next_arc(0)
   264       {}
   265 
   266       // Find next entering arc
   267       bool findEnteringArc() {
   268         Cost c;
   269         for (int e = _next_arc; e < _search_arc_num; ++e) {
   270           c = _state[e] * (_cost[e] + _pi[_source[e]] - _pi[_target[e]]);
   271           if (c < 0) {
   272             _in_arc = e;
   273             _next_arc = e + 1;
   274             return true;
   275           }
   276         }
   277         for (int e = 0; e < _next_arc; ++e) {
   278           c = _state[e] * (_cost[e] + _pi[_source[e]] - _pi[_target[e]]);
   279           if (c < 0) {
   280             _in_arc = e;
   281             _next_arc = e + 1;
   282             return true;
   283           }
   284         }
   285         return false;
   286       }
   287 
   288     }; //class FirstEligiblePivotRule
   289 
   290 
   291     // Implementation of the Best Eligible pivot rule
   292     class BestEligiblePivotRule
   293     {
   294     private:
   295 
   296       // References to the NetworkSimplex class
   297       const IntVector  &_source;
   298       const IntVector  &_target;
   299       const CostVector &_cost;
   300       const CharVector &_state;
   301       const CostVector &_pi;
   302       int &_in_arc;
   303       int _search_arc_num;
   304 
   305     public:
   306 
   307       // Constructor
   308       BestEligiblePivotRule(NetworkSimplex &ns) :
   309         _source(ns._source), _target(ns._target),
   310         _cost(ns._cost), _state(ns._state), _pi(ns._pi),
   311         _in_arc(ns.in_arc), _search_arc_num(ns._search_arc_num)
   312       {}
   313 
   314       // Find next entering arc
   315       bool findEnteringArc() {
   316         Cost c, min = 0;
   317         for (int e = 0; e < _search_arc_num; ++e) {
   318           c = _state[e] * (_cost[e] + _pi[_source[e]] - _pi[_target[e]]);
   319           if (c < min) {
   320             min = c;
   321             _in_arc = e;
   322           }
   323         }
   324         return min < 0;
   325       }
   326 
   327     }; //class BestEligiblePivotRule
   328 
   329 
   330     // Implementation of the Block Search pivot rule
   331     class BlockSearchPivotRule
   332     {
   333     private:
   334 
   335       // References to the NetworkSimplex class
   336       const IntVector  &_source;
   337       const IntVector  &_target;
   338       const CostVector &_cost;
   339       const CharVector &_state;
   340       const CostVector &_pi;
   341       int &_in_arc;
   342       int _search_arc_num;
   343 
   344       // Pivot rule data
   345       int _block_size;
   346       int _next_arc;
   347 
   348     public:
   349 
   350       // Constructor
   351       BlockSearchPivotRule(NetworkSimplex &ns) :
   352         _source(ns._source), _target(ns._target),
   353         _cost(ns._cost), _state(ns._state), _pi(ns._pi),
   354         _in_arc(ns.in_arc), _search_arc_num(ns._search_arc_num),
   355         _next_arc(0)
   356       {
   357         // The main parameters of the pivot rule
   358         const double BLOCK_SIZE_FACTOR = 0.5;
   359         const int MIN_BLOCK_SIZE = 10;
   360 
   361         _block_size = std::max( int(BLOCK_SIZE_FACTOR *
   362                                     std::sqrt(double(_search_arc_num))),
   363                                 MIN_BLOCK_SIZE );
   364       }
   365 
   366       // Find next entering arc
   367       bool findEnteringArc() {
   368         Cost c, min = 0;
   369         int cnt = _block_size;
   370         int e;
   371         for (e = _next_arc; e < _search_arc_num; ++e) {
   372           c = _state[e] * (_cost[e] + _pi[_source[e]] - _pi[_target[e]]);
   373           if (c < min) {
   374             min = c;
   375             _in_arc = e;
   376           }
   377           if (--cnt == 0) {
   378             if (min < 0) goto search_end;
   379             cnt = _block_size;
   380           }
   381         }
   382         for (e = 0; e < _next_arc; ++e) {
   383           c = _state[e] * (_cost[e] + _pi[_source[e]] - _pi[_target[e]]);
   384           if (c < min) {
   385             min = c;
   386             _in_arc = e;
   387           }
   388           if (--cnt == 0) {
   389             if (min < 0) goto search_end;
   390             cnt = _block_size;
   391           }
   392         }
   393         if (min >= 0) return false;
   394 
   395       search_end:
   396         _next_arc = e;
   397         return true;
   398       }
   399 
   400     }; //class BlockSearchPivotRule
   401 
   402 
   403     // Implementation of the Candidate List pivot rule
   404     class CandidateListPivotRule
   405     {
   406     private:
   407 
   408       // References to the NetworkSimplex class
   409       const IntVector  &_source;
   410       const IntVector  &_target;
   411       const CostVector &_cost;
   412       const CharVector &_state;
   413       const CostVector &_pi;
   414       int &_in_arc;
   415       int _search_arc_num;
   416 
   417       // Pivot rule data
   418       IntVector _candidates;
   419       int _list_length, _minor_limit;
   420       int _curr_length, _minor_count;
   421       int _next_arc;
   422 
   423     public:
   424 
   425       /// Constructor
   426       CandidateListPivotRule(NetworkSimplex &ns) :
   427         _source(ns._source), _target(ns._target),
   428         _cost(ns._cost), _state(ns._state), _pi(ns._pi),
   429         _in_arc(ns.in_arc), _search_arc_num(ns._search_arc_num),
   430         _next_arc(0)
   431       {
   432         // The main parameters of the pivot rule
   433         const double LIST_LENGTH_FACTOR = 0.25;
   434         const int MIN_LIST_LENGTH = 10;
   435         const double MINOR_LIMIT_FACTOR = 0.1;
   436         const int MIN_MINOR_LIMIT = 3;
   437 
   438         _list_length = std::max( int(LIST_LENGTH_FACTOR *
   439                                      std::sqrt(double(_search_arc_num))),
   440                                  MIN_LIST_LENGTH );
   441         _minor_limit = std::max( int(MINOR_LIMIT_FACTOR * _list_length),
   442                                  MIN_MINOR_LIMIT );
   443         _curr_length = _minor_count = 0;
   444         _candidates.resize(_list_length);
   445       }
   446 
   447       /// Find next entering arc
   448       bool findEnteringArc() {
   449         Cost min, c;
   450         int e;
   451         if (_curr_length > 0 && _minor_count < _minor_limit) {
   452           // Minor iteration: select the best eligible arc from the
   453           // current candidate list
   454           ++_minor_count;
   455           min = 0;
   456           for (int i = 0; i < _curr_length; ++i) {
   457             e = _candidates[i];
   458             c = _state[e] * (_cost[e] + _pi[_source[e]] - _pi[_target[e]]);
   459             if (c < min) {
   460               min = c;
   461               _in_arc = e;
   462             }
   463             else if (c >= 0) {
   464               _candidates[i--] = _candidates[--_curr_length];
   465             }
   466           }
   467           if (min < 0) return true;
   468         }
   469 
   470         // Major iteration: build a new candidate list
   471         min = 0;
   472         _curr_length = 0;
   473         for (e = _next_arc; e < _search_arc_num; ++e) {
   474           c = _state[e] * (_cost[e] + _pi[_source[e]] - _pi[_target[e]]);
   475           if (c < 0) {
   476             _candidates[_curr_length++] = e;
   477             if (c < min) {
   478               min = c;
   479               _in_arc = e;
   480             }
   481             if (_curr_length == _list_length) goto search_end;
   482           }
   483         }
   484         for (e = 0; e < _next_arc; ++e) {
   485           c = _state[e] * (_cost[e] + _pi[_source[e]] - _pi[_target[e]]);
   486           if (c < 0) {
   487             _candidates[_curr_length++] = e;
   488             if (c < min) {
   489               min = c;
   490               _in_arc = e;
   491             }
   492             if (_curr_length == _list_length) goto search_end;
   493           }
   494         }
   495         if (_curr_length == 0) return false;
   496       
   497       search_end:        
   498         _minor_count = 1;
   499         _next_arc = e;
   500         return true;
   501       }
   502 
   503     }; //class CandidateListPivotRule
   504 
   505 
   506     // Implementation of the Altering Candidate List pivot rule
   507     class AlteringListPivotRule
   508     {
   509     private:
   510 
   511       // References to the NetworkSimplex class
   512       const IntVector  &_source;
   513       const IntVector  &_target;
   514       const CostVector &_cost;
   515       const CharVector &_state;
   516       const CostVector &_pi;
   517       int &_in_arc;
   518       int _search_arc_num;
   519 
   520       // Pivot rule data
   521       int _block_size, _head_length, _curr_length;
   522       int _next_arc;
   523       IntVector _candidates;
   524       CostVector _cand_cost;
   525 
   526       // Functor class to compare arcs during sort of the candidate list
   527       class SortFunc
   528       {
   529       private:
   530         const CostVector &_map;
   531       public:
   532         SortFunc(const CostVector &map) : _map(map) {}
   533         bool operator()(int left, int right) {
   534           return _map[left] > _map[right];
   535         }
   536       };
   537 
   538       SortFunc _sort_func;
   539 
   540     public:
   541 
   542       // Constructor
   543       AlteringListPivotRule(NetworkSimplex &ns) :
   544         _source(ns._source), _target(ns._target),
   545         _cost(ns._cost), _state(ns._state), _pi(ns._pi),
   546         _in_arc(ns.in_arc), _search_arc_num(ns._search_arc_num),
   547         _next_arc(0), _cand_cost(ns._search_arc_num), _sort_func(_cand_cost)
   548       {
   549         // The main parameters of the pivot rule
   550         const double BLOCK_SIZE_FACTOR = 1.0;
   551         const int MIN_BLOCK_SIZE = 10;
   552         const double HEAD_LENGTH_FACTOR = 0.1;
   553         const int MIN_HEAD_LENGTH = 3;
   554 
   555         _block_size = std::max( int(BLOCK_SIZE_FACTOR *
   556                                     std::sqrt(double(_search_arc_num))),
   557                                 MIN_BLOCK_SIZE );
   558         _head_length = std::max( int(HEAD_LENGTH_FACTOR * _block_size),
   559                                  MIN_HEAD_LENGTH );
   560         _candidates.resize(_head_length + _block_size);
   561         _curr_length = 0;
   562       }
   563 
   564       // Find next entering arc
   565       bool findEnteringArc() {
   566         // Check the current candidate list
   567         int e;
   568         for (int i = 0; i < _curr_length; ++i) {
   569           e = _candidates[i];
   570           _cand_cost[e] = _state[e] *
   571             (_cost[e] + _pi[_source[e]] - _pi[_target[e]]);
   572           if (_cand_cost[e] >= 0) {
   573             _candidates[i--] = _candidates[--_curr_length];
   574           }
   575         }
   576 
   577         // Extend the list
   578         int cnt = _block_size;
   579         int limit = _head_length;
   580 
   581         for (e = _next_arc; e < _search_arc_num; ++e) {
   582           _cand_cost[e] = _state[e] *
   583             (_cost[e] + _pi[_source[e]] - _pi[_target[e]]);
   584           if (_cand_cost[e] < 0) {
   585             _candidates[_curr_length++] = e;
   586           }
   587           if (--cnt == 0) {
   588             if (_curr_length > limit) goto search_end;
   589             limit = 0;
   590             cnt = _block_size;
   591           }
   592         }
   593         for (e = 0; e < _next_arc; ++e) {
   594           _cand_cost[e] = _state[e] *
   595             (_cost[e] + _pi[_source[e]] - _pi[_target[e]]);
   596           if (_cand_cost[e] < 0) {
   597             _candidates[_curr_length++] = e;
   598           }
   599           if (--cnt == 0) {
   600             if (_curr_length > limit) goto search_end;
   601             limit = 0;
   602             cnt = _block_size;
   603           }
   604         }
   605         if (_curr_length == 0) return false;
   606         
   607       search_end:
   608 
   609         // Make heap of the candidate list (approximating a partial sort)
   610         make_heap( _candidates.begin(), _candidates.begin() + _curr_length,
   611                    _sort_func );
   612 
   613         // Pop the first element of the heap
   614         _in_arc = _candidates[0];
   615         _next_arc = e;
   616         pop_heap( _candidates.begin(), _candidates.begin() + _curr_length,
   617                   _sort_func );
   618         _curr_length = std::min(_head_length, _curr_length - 1);
   619         return true;
   620       }
   621 
   622     }; //class AlteringListPivotRule
   623 
   624   public:
   625 
   626     /// \brief Constructor.
   627     ///
   628     /// The constructor of the class.
   629     ///
   630     /// \param graph The digraph the algorithm runs on.
   631     /// \param arc_mixing Indicate if the arcs have to be stored in a
   632     /// mixed order in the internal data structure. 
   633     /// In special cases, it could lead to better overall performance,
   634     /// but it is usually slower. Therefore it is disabled by default.
   635     NetworkSimplex(const GR& graph, bool arc_mixing = false) :
   636       _graph(graph), _node_id(graph), _arc_id(graph),
   637       _arc_mixing(arc_mixing),
   638       MAX(std::numeric_limits<Value>::max()),
   639       INF(std::numeric_limits<Value>::has_infinity ?
   640           std::numeric_limits<Value>::infinity() : MAX)
   641     {
   642       // Check the number types
   643       LEMON_ASSERT(std::numeric_limits<Value>::is_signed,
   644         "The flow type of NetworkSimplex must be signed");
   645       LEMON_ASSERT(std::numeric_limits<Cost>::is_signed,
   646         "The cost type of NetworkSimplex must be signed");
   647         
   648       // Reset data structures
   649       reset();
   650     }
   651 
   652     /// \name Parameters
   653     /// The parameters of the algorithm can be specified using these
   654     /// functions.
   655 
   656     /// @{
   657 
   658     /// \brief Set the lower bounds on the arcs.
   659     ///
   660     /// This function sets the lower bounds on the arcs.
   661     /// If it is not used before calling \ref run(), the lower bounds
   662     /// will be set to zero on all arcs.
   663     ///
   664     /// \param map An arc map storing the lower bounds.
   665     /// Its \c Value type must be convertible to the \c Value type
   666     /// of the algorithm.
   667     ///
   668     /// \return <tt>(*this)</tt>
   669     template <typename LowerMap>
   670     NetworkSimplex& lowerMap(const LowerMap& map) {
   671       _have_lower = true;
   672       for (ArcIt a(_graph); a != INVALID; ++a) {
   673         _lower[_arc_id[a]] = map[a];
   674       }
   675       return *this;
   676     }
   677 
   678     /// \brief Set the upper bounds (capacities) on the arcs.
   679     ///
   680     /// This function sets the upper bounds (capacities) on the arcs.
   681     /// If it is not used before calling \ref run(), the upper bounds
   682     /// will be set to \ref INF on all arcs (i.e. the flow value will be
   683     /// unbounded from above).
   684     ///
   685     /// \param map An arc map storing the upper bounds.
   686     /// Its \c Value type must be convertible to the \c Value type
   687     /// of the algorithm.
   688     ///
   689     /// \return <tt>(*this)</tt>
   690     template<typename UpperMap>
   691     NetworkSimplex& upperMap(const UpperMap& map) {
   692       for (ArcIt a(_graph); a != INVALID; ++a) {
   693         _upper[_arc_id[a]] = map[a];
   694       }
   695       return *this;
   696     }
   697 
   698     /// \brief Set the costs of the arcs.
   699     ///
   700     /// This function sets the costs of the arcs.
   701     /// If it is not used before calling \ref run(), the costs
   702     /// will be set to \c 1 on all arcs.
   703     ///
   704     /// \param map An arc map storing the costs.
   705     /// Its \c Value type must be convertible to the \c Cost type
   706     /// of the algorithm.
   707     ///
   708     /// \return <tt>(*this)</tt>
   709     template<typename CostMap>
   710     NetworkSimplex& costMap(const CostMap& map) {
   711       for (ArcIt a(_graph); a != INVALID; ++a) {
   712         _cost[_arc_id[a]] = map[a];
   713       }
   714       return *this;
   715     }
   716 
   717     /// \brief Set the supply values of the nodes.
   718     ///
   719     /// This function sets the supply values of the nodes.
   720     /// If neither this function nor \ref stSupply() is used before
   721     /// calling \ref run(), the supply of each node will be set to zero.
   722     ///
   723     /// \param map A node map storing the supply values.
   724     /// Its \c Value type must be convertible to the \c Value type
   725     /// of the algorithm.
   726     ///
   727     /// \return <tt>(*this)</tt>
   728     template<typename SupplyMap>
   729     NetworkSimplex& supplyMap(const SupplyMap& map) {
   730       for (NodeIt n(_graph); n != INVALID; ++n) {
   731         _supply[_node_id[n]] = map[n];
   732       }
   733       return *this;
   734     }
   735 
   736     /// \brief Set single source and target nodes and a supply value.
   737     ///
   738     /// This function sets a single source node and a single target node
   739     /// and the required flow value.
   740     /// If neither this function nor \ref supplyMap() is used before
   741     /// calling \ref run(), the supply of each node will be set to zero.
   742     ///
   743     /// Using this function has the same effect as using \ref supplyMap()
   744     /// with such a map in which \c k is assigned to \c s, \c -k is
   745     /// assigned to \c t and all other nodes have zero supply value.
   746     ///
   747     /// \param s The source node.
   748     /// \param t The target node.
   749     /// \param k The required amount of flow from node \c s to node \c t
   750     /// (i.e. the supply of \c s and the demand of \c t).
   751     ///
   752     /// \return <tt>(*this)</tt>
   753     NetworkSimplex& stSupply(const Node& s, const Node& t, Value k) {
   754       for (int i = 0; i != _node_num; ++i) {
   755         _supply[i] = 0;
   756       }
   757       _supply[_node_id[s]] =  k;
   758       _supply[_node_id[t]] = -k;
   759       return *this;
   760     }
   761     
   762     /// \brief Set the type of the supply constraints.
   763     ///
   764     /// This function sets the type of the supply/demand constraints.
   765     /// If it is not used before calling \ref run(), the \ref GEQ supply
   766     /// type will be used.
   767     ///
   768     /// For more information, see \ref SupplyType.
   769     ///
   770     /// \return <tt>(*this)</tt>
   771     NetworkSimplex& supplyType(SupplyType supply_type) {
   772       _stype = supply_type;
   773       return *this;
   774     }
   775 
   776     /// @}
   777 
   778     /// \name Execution Control
   779     /// The algorithm can be executed using \ref run().
   780 
   781     /// @{
   782 
   783     /// \brief Run the algorithm.
   784     ///
   785     /// This function runs the algorithm.
   786     /// The paramters can be specified using functions \ref lowerMap(),
   787     /// \ref upperMap(), \ref costMap(), \ref supplyMap(), \ref stSupply(), 
   788     /// \ref supplyType().
   789     /// For example,
   790     /// \code
   791     ///   NetworkSimplex<ListDigraph> ns(graph);
   792     ///   ns.lowerMap(lower).upperMap(upper).costMap(cost)
   793     ///     .supplyMap(sup).run();
   794     /// \endcode
   795     ///
   796     /// This function can be called more than once. All the given parameters
   797     /// are kept for the next call, unless \ref resetParams() or \ref reset()
   798     /// is used, thus only the modified parameters have to be set again.
   799     /// If the underlying digraph was also modified after the construction
   800     /// of the class (or the last \ref reset() call), then the \ref reset()
   801     /// function must be called.
   802     ///
   803     /// \param pivot_rule The pivot rule that will be used during the
   804     /// algorithm. For more information, see \ref PivotRule.
   805     ///
   806     /// \return \c INFEASIBLE if no feasible flow exists,
   807     /// \n \c OPTIMAL if the problem has optimal solution
   808     /// (i.e. it is feasible and bounded), and the algorithm has found
   809     /// optimal flow and node potentials (primal and dual solutions),
   810     /// \n \c UNBOUNDED if the objective function of the problem is
   811     /// unbounded, i.e. there is a directed cycle having negative total
   812     /// cost and infinite upper bound.
   813     ///
   814     /// \see ProblemType, PivotRule
   815     /// \see resetParams(), reset()
   816     ProblemType run(PivotRule pivot_rule = BLOCK_SEARCH) {
   817       if (!init()) return INFEASIBLE;
   818       return start(pivot_rule);
   819     }
   820 
   821     /// \brief Reset all the parameters that have been given before.
   822     ///
   823     /// This function resets all the paramaters that have been given
   824     /// before using functions \ref lowerMap(), \ref upperMap(),
   825     /// \ref costMap(), \ref supplyMap(), \ref stSupply(), \ref supplyType().
   826     ///
   827     /// It is useful for multiple \ref run() calls. Basically, all the given
   828     /// parameters are kept for the next \ref run() call, unless
   829     /// \ref resetParams() or \ref reset() is used.
   830     /// If the underlying digraph was also modified after the construction
   831     /// of the class or the last \ref reset() call, then the \ref reset()
   832     /// function must be used, otherwise \ref resetParams() is sufficient.
   833     ///
   834     /// For example,
   835     /// \code
   836     ///   NetworkSimplex<ListDigraph> ns(graph);
   837     ///
   838     ///   // First run
   839     ///   ns.lowerMap(lower).upperMap(upper).costMap(cost)
   840     ///     .supplyMap(sup).run();
   841     ///
   842     ///   // Run again with modified cost map (resetParams() is not called,
   843     ///   // so only the cost map have to be set again)
   844     ///   cost[e] += 100;
   845     ///   ns.costMap(cost).run();
   846     ///
   847     ///   // Run again from scratch using resetParams()
   848     ///   // (the lower bounds will be set to zero on all arcs)
   849     ///   ns.resetParams();
   850     ///   ns.upperMap(capacity).costMap(cost)
   851     ///     .supplyMap(sup).run();
   852     /// \endcode
   853     ///
   854     /// \return <tt>(*this)</tt>
   855     ///
   856     /// \see reset(), run()
   857     NetworkSimplex& resetParams() {
   858       for (int i = 0; i != _node_num; ++i) {
   859         _supply[i] = 0;
   860       }
   861       for (int i = 0; i != _arc_num; ++i) {
   862         _lower[i] = 0;
   863         _upper[i] = INF;
   864         _cost[i] = 1;
   865       }
   866       _have_lower = false;
   867       _stype = GEQ;
   868       return *this;
   869     }
   870 
   871     /// \brief Reset the internal data structures and all the parameters
   872     /// that have been given before.
   873     ///
   874     /// This function resets the internal data structures and all the
   875     /// paramaters that have been given before using functions \ref lowerMap(),
   876     /// \ref upperMap(), \ref costMap(), \ref supplyMap(), \ref stSupply(),
   877     /// \ref supplyType().
   878     ///
   879     /// It is useful for multiple \ref run() calls. Basically, all the given
   880     /// parameters are kept for the next \ref run() call, unless
   881     /// \ref resetParams() or \ref reset() is used.
   882     /// If the underlying digraph was also modified after the construction
   883     /// of the class or the last \ref reset() call, then the \ref reset()
   884     /// function must be used, otherwise \ref resetParams() is sufficient.
   885     ///
   886     /// See \ref resetParams() for examples.
   887     ///
   888     /// \return <tt>(*this)</tt>
   889     ///
   890     /// \see resetParams(), run()
   891     NetworkSimplex& reset() {
   892       // Resize vectors
   893       _node_num = countNodes(_graph);
   894       _arc_num = countArcs(_graph);
   895       int all_node_num = _node_num + 1;
   896       int max_arc_num = _arc_num + 2 * _node_num;
   897 
   898       _source.resize(max_arc_num);
   899       _target.resize(max_arc_num);
   900 
   901       _lower.resize(_arc_num);
   902       _upper.resize(_arc_num);
   903       _cap.resize(max_arc_num);
   904       _cost.resize(max_arc_num);
   905       _supply.resize(all_node_num);
   906       _flow.resize(max_arc_num);
   907       _pi.resize(all_node_num);
   908 
   909       _parent.resize(all_node_num);
   910       _pred.resize(all_node_num);
   911       _forward.resize(all_node_num);
   912       _thread.resize(all_node_num);
   913       _rev_thread.resize(all_node_num);
   914       _succ_num.resize(all_node_num);
   915       _last_succ.resize(all_node_num);
   916       _state.resize(max_arc_num);
   917 
   918       // Copy the graph
   919       int i = 0;
   920       for (NodeIt n(_graph); n != INVALID; ++n, ++i) {
   921         _node_id[n] = i;
   922       }
   923       if (_arc_mixing) {
   924         // Store the arcs in a mixed order
   925         int k = std::max(int(std::sqrt(double(_arc_num))), 10);
   926         int i = 0, j = 0;
   927         for (ArcIt a(_graph); a != INVALID; ++a) {
   928           _arc_id[a] = i;
   929           _source[i] = _node_id[_graph.source(a)];
   930           _target[i] = _node_id[_graph.target(a)];
   931           if ((i += k) >= _arc_num) i = ++j;
   932         }
   933       } else {
   934         // Store the arcs in the original order
   935         int i = 0;
   936         for (ArcIt a(_graph); a != INVALID; ++a, ++i) {
   937           _arc_id[a] = i;
   938           _source[i] = _node_id[_graph.source(a)];
   939           _target[i] = _node_id[_graph.target(a)];
   940         }
   941       }
   942       
   943       // Reset parameters
   944       resetParams();
   945       return *this;
   946     }
   947     
   948     /// @}
   949 
   950     /// \name Query Functions
   951     /// The results of the algorithm can be obtained using these
   952     /// functions.\n
   953     /// The \ref run() function must be called before using them.
   954 
   955     /// @{
   956 
   957     /// \brief Return the total cost of the found flow.
   958     ///
   959     /// This function returns the total cost of the found flow.
   960     /// Its complexity is O(e).
   961     ///
   962     /// \note The return type of the function can be specified as a
   963     /// template parameter. For example,
   964     /// \code
   965     ///   ns.totalCost<double>();
   966     /// \endcode
   967     /// It is useful if the total cost cannot be stored in the \c Cost
   968     /// type of the algorithm, which is the default return type of the
   969     /// function.
   970     ///
   971     /// \pre \ref run() must be called before using this function.
   972     template <typename Number>
   973     Number totalCost() const {
   974       Number c = 0;
   975       for (ArcIt a(_graph); a != INVALID; ++a) {
   976         int i = _arc_id[a];
   977         c += Number(_flow[i]) * Number(_cost[i]);
   978       }
   979       return c;
   980     }
   981 
   982 #ifndef DOXYGEN
   983     Cost totalCost() const {
   984       return totalCost<Cost>();
   985     }
   986 #endif
   987 
   988     /// \brief Return the flow on the given arc.
   989     ///
   990     /// This function returns the flow on the given arc.
   991     ///
   992     /// \pre \ref run() must be called before using this function.
   993     Value flow(const Arc& a) const {
   994       return _flow[_arc_id[a]];
   995     }
   996 
   997     /// \brief Return the flow map (the primal solution).
   998     ///
   999     /// This function copies the flow value on each arc into the given
  1000     /// map. The \c Value type of the algorithm must be convertible to
  1001     /// the \c Value type of the map.
  1002     ///
  1003     /// \pre \ref run() must be called before using this function.
  1004     template <typename FlowMap>
  1005     void flowMap(FlowMap &map) const {
  1006       for (ArcIt a(_graph); a != INVALID; ++a) {
  1007         map.set(a, _flow[_arc_id[a]]);
  1008       }
  1009     }
  1010 
  1011     /// \brief Return the potential (dual value) of the given node.
  1012     ///
  1013     /// This function returns the potential (dual value) of the
  1014     /// given node.
  1015     ///
  1016     /// \pre \ref run() must be called before using this function.
  1017     Cost potential(const Node& n) const {
  1018       return _pi[_node_id[n]];
  1019     }
  1020 
  1021     /// \brief Return the potential map (the dual solution).
  1022     ///
  1023     /// This function copies the potential (dual value) of each node
  1024     /// into the given map.
  1025     /// The \c Cost type of the algorithm must be convertible to the
  1026     /// \c Value type of the map.
  1027     ///
  1028     /// \pre \ref run() must be called before using this function.
  1029     template <typename PotentialMap>
  1030     void potentialMap(PotentialMap &map) const {
  1031       for (NodeIt n(_graph); n != INVALID; ++n) {
  1032         map.set(n, _pi[_node_id[n]]);
  1033       }
  1034     }
  1035 
  1036     /// @}
  1037 
  1038   private:
  1039 
  1040     // Initialize internal data structures
  1041     bool init() {
  1042       if (_node_num == 0) return false;
  1043 
  1044       // Check the sum of supply values
  1045       _sum_supply = 0;
  1046       for (int i = 0; i != _node_num; ++i) {
  1047         _sum_supply += _supply[i];
  1048       }
  1049       if ( !((_stype == GEQ && _sum_supply <= 0) ||
  1050              (_stype == LEQ && _sum_supply >= 0)) ) return false;
  1051 
  1052       // Remove non-zero lower bounds
  1053       if (_have_lower) {
  1054         for (int i = 0; i != _arc_num; ++i) {
  1055           Value c = _lower[i];
  1056           if (c >= 0) {
  1057             _cap[i] = _upper[i] < MAX ? _upper[i] - c : INF;
  1058           } else {
  1059             _cap[i] = _upper[i] < MAX + c ? _upper[i] - c : INF;
  1060           }
  1061           _supply[_source[i]] -= c;
  1062           _supply[_target[i]] += c;
  1063         }
  1064       } else {
  1065         for (int i = 0; i != _arc_num; ++i) {
  1066           _cap[i] = _upper[i];
  1067         }
  1068       }
  1069 
  1070       // Initialize artifical cost
  1071       Cost ART_COST;
  1072       if (std::numeric_limits<Cost>::is_exact) {
  1073         ART_COST = std::numeric_limits<Cost>::max() / 2 + 1;
  1074       } else {
  1075         ART_COST = std::numeric_limits<Cost>::min();
  1076         for (int i = 0; i != _arc_num; ++i) {
  1077           if (_cost[i] > ART_COST) ART_COST = _cost[i];
  1078         }
  1079         ART_COST = (ART_COST + 1) * _node_num;
  1080       }
  1081 
  1082       // Initialize arc maps
  1083       for (int i = 0; i != _arc_num; ++i) {
  1084         _flow[i] = 0;
  1085         _state[i] = STATE_LOWER;
  1086       }
  1087       
  1088       // Set data for the artificial root node
  1089       _root = _node_num;
  1090       _parent[_root] = -1;
  1091       _pred[_root] = -1;
  1092       _thread[_root] = 0;
  1093       _rev_thread[0] = _root;
  1094       _succ_num[_root] = _node_num + 1;
  1095       _last_succ[_root] = _root - 1;
  1096       _supply[_root] = -_sum_supply;
  1097       _pi[_root] = 0;
  1098 
  1099       // Add artificial arcs and initialize the spanning tree data structure
  1100       if (_sum_supply == 0) {
  1101         // EQ supply constraints
  1102         _search_arc_num = _arc_num;
  1103         _all_arc_num = _arc_num + _node_num;
  1104         for (int u = 0, e = _arc_num; u != _node_num; ++u, ++e) {
  1105           _parent[u] = _root;
  1106           _pred[u] = e;
  1107           _thread[u] = u + 1;
  1108           _rev_thread[u + 1] = u;
  1109           _succ_num[u] = 1;
  1110           _last_succ[u] = u;
  1111           _cap[e] = INF;
  1112           _state[e] = STATE_TREE;
  1113           if (_supply[u] >= 0) {
  1114             _forward[u] = true;
  1115             _pi[u] = 0;
  1116             _source[e] = u;
  1117             _target[e] = _root;
  1118             _flow[e] = _supply[u];
  1119             _cost[e] = 0;
  1120           } else {
  1121             _forward[u] = false;
  1122             _pi[u] = ART_COST;
  1123             _source[e] = _root;
  1124             _target[e] = u;
  1125             _flow[e] = -_supply[u];
  1126             _cost[e] = ART_COST;
  1127           }
  1128         }
  1129       }
  1130       else if (_sum_supply > 0) {
  1131         // LEQ supply constraints
  1132         _search_arc_num = _arc_num + _node_num;
  1133         int f = _arc_num + _node_num;
  1134         for (int u = 0, e = _arc_num; u != _node_num; ++u, ++e) {
  1135           _parent[u] = _root;
  1136           _thread[u] = u + 1;
  1137           _rev_thread[u + 1] = u;
  1138           _succ_num[u] = 1;
  1139           _last_succ[u] = u;
  1140           if (_supply[u] >= 0) {
  1141             _forward[u] = true;
  1142             _pi[u] = 0;
  1143             _pred[u] = e;
  1144             _source[e] = u;
  1145             _target[e] = _root;
  1146             _cap[e] = INF;
  1147             _flow[e] = _supply[u];
  1148             _cost[e] = 0;
  1149             _state[e] = STATE_TREE;
  1150           } else {
  1151             _forward[u] = false;
  1152             _pi[u] = ART_COST;
  1153             _pred[u] = f;
  1154             _source[f] = _root;
  1155             _target[f] = u;
  1156             _cap[f] = INF;
  1157             _flow[f] = -_supply[u];
  1158             _cost[f] = ART_COST;
  1159             _state[f] = STATE_TREE;
  1160             _source[e] = u;
  1161             _target[e] = _root;
  1162             _cap[e] = INF;
  1163             _flow[e] = 0;
  1164             _cost[e] = 0;
  1165             _state[e] = STATE_LOWER;
  1166             ++f;
  1167           }
  1168         }
  1169         _all_arc_num = f;
  1170       }
  1171       else {
  1172         // GEQ supply constraints
  1173         _search_arc_num = _arc_num + _node_num;
  1174         int f = _arc_num + _node_num;
  1175         for (int u = 0, e = _arc_num; u != _node_num; ++u, ++e) {
  1176           _parent[u] = _root;
  1177           _thread[u] = u + 1;
  1178           _rev_thread[u + 1] = u;
  1179           _succ_num[u] = 1;
  1180           _last_succ[u] = u;
  1181           if (_supply[u] <= 0) {
  1182             _forward[u] = false;
  1183             _pi[u] = 0;
  1184             _pred[u] = e;
  1185             _source[e] = _root;
  1186             _target[e] = u;
  1187             _cap[e] = INF;
  1188             _flow[e] = -_supply[u];
  1189             _cost[e] = 0;
  1190             _state[e] = STATE_TREE;
  1191           } else {
  1192             _forward[u] = true;
  1193             _pi[u] = -ART_COST;
  1194             _pred[u] = f;
  1195             _source[f] = u;
  1196             _target[f] = _root;
  1197             _cap[f] = INF;
  1198             _flow[f] = _supply[u];
  1199             _state[f] = STATE_TREE;
  1200             _cost[f] = ART_COST;
  1201             _source[e] = _root;
  1202             _target[e] = u;
  1203             _cap[e] = INF;
  1204             _flow[e] = 0;
  1205             _cost[e] = 0;
  1206             _state[e] = STATE_LOWER;
  1207             ++f;
  1208           }
  1209         }
  1210         _all_arc_num = f;
  1211       }
  1212 
  1213       return true;
  1214     }
  1215 
  1216     // Find the join node
  1217     void findJoinNode() {
  1218       int u = _source[in_arc];
  1219       int v = _target[in_arc];
  1220       while (u != v) {
  1221         if (_succ_num[u] < _succ_num[v]) {
  1222           u = _parent[u];
  1223         } else {
  1224           v = _parent[v];
  1225         }
  1226       }
  1227       join = u;
  1228     }
  1229 
  1230     // Find the leaving arc of the cycle and returns true if the
  1231     // leaving arc is not the same as the entering arc
  1232     bool findLeavingArc() {
  1233       // Initialize first and second nodes according to the direction
  1234       // of the cycle
  1235       if (_state[in_arc] == STATE_LOWER) {
  1236         first  = _source[in_arc];
  1237         second = _target[in_arc];
  1238       } else {
  1239         first  = _target[in_arc];
  1240         second = _source[in_arc];
  1241       }
  1242       delta = _cap[in_arc];
  1243       int result = 0;
  1244       Value d;
  1245       int e;
  1246 
  1247       // Search the cycle along the path form the first node to the root
  1248       for (int u = first; u != join; u = _parent[u]) {
  1249         e = _pred[u];
  1250         d = _forward[u] ?
  1251           _flow[e] : (_cap[e] >= MAX ? INF : _cap[e] - _flow[e]);
  1252         if (d < delta) {
  1253           delta = d;
  1254           u_out = u;
  1255           result = 1;
  1256         }
  1257       }
  1258       // Search the cycle along the path form the second node to the root
  1259       for (int u = second; u != join; u = _parent[u]) {
  1260         e = _pred[u];
  1261         d = _forward[u] ? 
  1262           (_cap[e] >= MAX ? INF : _cap[e] - _flow[e]) : _flow[e];
  1263         if (d <= delta) {
  1264           delta = d;
  1265           u_out = u;
  1266           result = 2;
  1267         }
  1268       }
  1269 
  1270       if (result == 1) {
  1271         u_in = first;
  1272         v_in = second;
  1273       } else {
  1274         u_in = second;
  1275         v_in = first;
  1276       }
  1277       return result != 0;
  1278     }
  1279 
  1280     // Change _flow and _state vectors
  1281     void changeFlow(bool change) {
  1282       // Augment along the cycle
  1283       if (delta > 0) {
  1284         Value val = _state[in_arc] * delta;
  1285         _flow[in_arc] += val;
  1286         for (int u = _source[in_arc]; u != join; u = _parent[u]) {
  1287           _flow[_pred[u]] += _forward[u] ? -val : val;
  1288         }
  1289         for (int u = _target[in_arc]; u != join; u = _parent[u]) {
  1290           _flow[_pred[u]] += _forward[u] ? val : -val;
  1291         }
  1292       }
  1293       // Update the state of the entering and leaving arcs
  1294       if (change) {
  1295         _state[in_arc] = STATE_TREE;
  1296         _state[_pred[u_out]] =
  1297           (_flow[_pred[u_out]] == 0) ? STATE_LOWER : STATE_UPPER;
  1298       } else {
  1299         _state[in_arc] = -_state[in_arc];
  1300       }
  1301     }
  1302 
  1303     // Update the tree structure
  1304     void updateTreeStructure() {
  1305       int u, w;
  1306       int old_rev_thread = _rev_thread[u_out];
  1307       int old_succ_num = _succ_num[u_out];
  1308       int old_last_succ = _last_succ[u_out];
  1309       v_out = _parent[u_out];
  1310 
  1311       u = _last_succ[u_in];  // the last successor of u_in
  1312       right = _thread[u];    // the node after it
  1313 
  1314       // Handle the case when old_rev_thread equals to v_in
  1315       // (it also means that join and v_out coincide)
  1316       if (old_rev_thread == v_in) {
  1317         last = _thread[_last_succ[u_out]];
  1318       } else {
  1319         last = _thread[v_in];
  1320       }
  1321 
  1322       // Update _thread and _parent along the stem nodes (i.e. the nodes
  1323       // between u_in and u_out, whose parent have to be changed)
  1324       _thread[v_in] = stem = u_in;
  1325       _dirty_revs.clear();
  1326       _dirty_revs.push_back(v_in);
  1327       par_stem = v_in;
  1328       while (stem != u_out) {
  1329         // Insert the next stem node into the thread list
  1330         new_stem = _parent[stem];
  1331         _thread[u] = new_stem;
  1332         _dirty_revs.push_back(u);
  1333 
  1334         // Remove the subtree of stem from the thread list
  1335         w = _rev_thread[stem];
  1336         _thread[w] = right;
  1337         _rev_thread[right] = w;
  1338 
  1339         // Change the parent node and shift stem nodes
  1340         _parent[stem] = par_stem;
  1341         par_stem = stem;
  1342         stem = new_stem;
  1343 
  1344         // Update u and right
  1345         u = _last_succ[stem] == _last_succ[par_stem] ?
  1346           _rev_thread[par_stem] : _last_succ[stem];
  1347         right = _thread[u];
  1348       }
  1349       _parent[u_out] = par_stem;
  1350       _thread[u] = last;
  1351       _rev_thread[last] = u;
  1352       _last_succ[u_out] = u;
  1353 
  1354       // Remove the subtree of u_out from the thread list except for
  1355       // the case when old_rev_thread equals to v_in
  1356       // (it also means that join and v_out coincide)
  1357       if (old_rev_thread != v_in) {
  1358         _thread[old_rev_thread] = right;
  1359         _rev_thread[right] = old_rev_thread;
  1360       }
  1361 
  1362       // Update _rev_thread using the new _thread values
  1363       for (int i = 0; i < int(_dirty_revs.size()); ++i) {
  1364         u = _dirty_revs[i];
  1365         _rev_thread[_thread[u]] = u;
  1366       }
  1367 
  1368       // Update _pred, _forward, _last_succ and _succ_num for the
  1369       // stem nodes from u_out to u_in
  1370       int tmp_sc = 0, tmp_ls = _last_succ[u_out];
  1371       u = u_out;
  1372       while (u != u_in) {
  1373         w = _parent[u];
  1374         _pred[u] = _pred[w];
  1375         _forward[u] = !_forward[w];
  1376         tmp_sc += _succ_num[u] - _succ_num[w];
  1377         _succ_num[u] = tmp_sc;
  1378         _last_succ[w] = tmp_ls;
  1379         u = w;
  1380       }
  1381       _pred[u_in] = in_arc;
  1382       _forward[u_in] = (u_in == _source[in_arc]);
  1383       _succ_num[u_in] = old_succ_num;
  1384 
  1385       // Set limits for updating _last_succ form v_in and v_out
  1386       // towards the root
  1387       int up_limit_in = -1;
  1388       int up_limit_out = -1;
  1389       if (_last_succ[join] == v_in) {
  1390         up_limit_out = join;
  1391       } else {
  1392         up_limit_in = join;
  1393       }
  1394 
  1395       // Update _last_succ from v_in towards the root
  1396       for (u = v_in; u != up_limit_in && _last_succ[u] == v_in;
  1397            u = _parent[u]) {
  1398         _last_succ[u] = _last_succ[u_out];
  1399       }
  1400       // Update _last_succ from v_out towards the root
  1401       if (join != old_rev_thread && v_in != old_rev_thread) {
  1402         for (u = v_out; u != up_limit_out && _last_succ[u] == old_last_succ;
  1403              u = _parent[u]) {
  1404           _last_succ[u] = old_rev_thread;
  1405         }
  1406       } else {
  1407         for (u = v_out; u != up_limit_out && _last_succ[u] == old_last_succ;
  1408              u = _parent[u]) {
  1409           _last_succ[u] = _last_succ[u_out];
  1410         }
  1411       }
  1412 
  1413       // Update _succ_num from v_in to join
  1414       for (u = v_in; u != join; u = _parent[u]) {
  1415         _succ_num[u] += old_succ_num;
  1416       }
  1417       // Update _succ_num from v_out to join
  1418       for (u = v_out; u != join; u = _parent[u]) {
  1419         _succ_num[u] -= old_succ_num;
  1420       }
  1421     }
  1422 
  1423     // Update potentials
  1424     void updatePotential() {
  1425       Cost sigma = _forward[u_in] ?
  1426         _pi[v_in] - _pi[u_in] - _cost[_pred[u_in]] :
  1427         _pi[v_in] - _pi[u_in] + _cost[_pred[u_in]];
  1428       // Update potentials in the subtree, which has been moved
  1429       int end = _thread[_last_succ[u_in]];
  1430       for (int u = u_in; u != end; u = _thread[u]) {
  1431         _pi[u] += sigma;
  1432       }
  1433     }
  1434 
  1435     // Execute the algorithm
  1436     ProblemType start(PivotRule pivot_rule) {
  1437       // Select the pivot rule implementation
  1438       switch (pivot_rule) {
  1439         case FIRST_ELIGIBLE:
  1440           return start<FirstEligiblePivotRule>();
  1441         case BEST_ELIGIBLE:
  1442           return start<BestEligiblePivotRule>();
  1443         case BLOCK_SEARCH:
  1444           return start<BlockSearchPivotRule>();
  1445         case CANDIDATE_LIST:
  1446           return start<CandidateListPivotRule>();
  1447         case ALTERING_LIST:
  1448           return start<AlteringListPivotRule>();
  1449       }
  1450       return INFEASIBLE; // avoid warning
  1451     }
  1452 
  1453     template <typename PivotRuleImpl>
  1454     ProblemType start() {
  1455       PivotRuleImpl pivot(*this);
  1456 
  1457       // Execute the Network Simplex algorithm
  1458       while (pivot.findEnteringArc()) {
  1459         findJoinNode();
  1460         bool change = findLeavingArc();
  1461         if (delta >= MAX) return UNBOUNDED;
  1462         changeFlow(change);
  1463         if (change) {
  1464           updateTreeStructure();
  1465           updatePotential();
  1466         }
  1467       }
  1468       
  1469       // Check feasibility
  1470       for (int e = _search_arc_num; e != _all_arc_num; ++e) {
  1471         if (_flow[e] != 0) return INFEASIBLE;
  1472       }
  1473 
  1474       // Transform the solution and the supply map to the original form
  1475       if (_have_lower) {
  1476         for (int i = 0; i != _arc_num; ++i) {
  1477           Value c = _lower[i];
  1478           if (c != 0) {
  1479             _flow[i] += c;
  1480             _supply[_source[i]] += c;
  1481             _supply[_target[i]] -= c;
  1482           }
  1483         }
  1484       }
  1485       
  1486       // Shift potentials to meet the requirements of the GEQ/LEQ type
  1487       // optimality conditions
  1488       if (_sum_supply == 0) {
  1489         if (_stype == GEQ) {
  1490           Cost max_pot = std::numeric_limits<Cost>::min();
  1491           for (int i = 0; i != _node_num; ++i) {
  1492             if (_pi[i] > max_pot) max_pot = _pi[i];
  1493           }
  1494           if (max_pot > 0) {
  1495             for (int i = 0; i != _node_num; ++i)
  1496               _pi[i] -= max_pot;
  1497           }
  1498         } else {
  1499           Cost min_pot = std::numeric_limits<Cost>::max();
  1500           for (int i = 0; i != _node_num; ++i) {
  1501             if (_pi[i] < min_pot) min_pot = _pi[i];
  1502           }
  1503           if (min_pot < 0) {
  1504             for (int i = 0; i != _node_num; ++i)
  1505               _pi[i] -= min_pot;
  1506           }
  1507         }
  1508       }
  1509 
  1510       return OPTIMAL;
  1511     }
  1512 
  1513   }; //class NetworkSimplex
  1514 
  1515   ///@}
  1516 
  1517 } //namespace lemon
  1518 
  1519 #endif //LEMON_NETWORK_SIMPLEX_H