lemon/network_simplex.h
author Peter Kovacs <kpeter@inf.elte.hu>
Mon, 11 May 2009 16:42:42 +0200
changeset 658 ebdcc68fe79e
parent 642 111698359429
child 663 8b0df68370a4
permissions -rw-r--r--
Change the explanation of the acronym LEMON (#289)
     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
    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
    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   /// This algorithm is a specialized version of the linear programming
    45   /// simplex method directly for the minimum cost flow problem.
    46   /// It is one of the most efficient solution methods.
    47   ///
    48   /// In general this class is the fastest implementation available
    49   /// in LEMON for the minimum cost flow problem.
    50   /// Moreover it supports both directions of the supply/demand inequality
    51   /// constraints. For more information see \ref SupplyType.
    52   ///
    53   /// Most of the parameters of the problem (except for the digraph)
    54   /// can be given using separate functions, and the algorithm can be
    55   /// executed using the \ref run() function. If some parameters are not
    56   /// specified, then default values will be used.
    57   ///
    58   /// \tparam GR The digraph type the algorithm runs on.
    59   /// \tparam V The value type used for flow amounts, capacity bounds
    60   /// and supply values in the algorithm. By default it is \c int.
    61   /// \tparam C The value type used for costs and potentials in the
    62   /// algorithm. By default it is the same as \c V.
    63   ///
    64   /// \warning Both value types must be signed and all input data must
    65   /// be integer.
    66   ///
    67   /// \note %NetworkSimplex provides five different pivot rule
    68   /// implementations, from which the most efficient one is used
    69   /// by default. For more information see \ref PivotRule.
    70   template <typename GR, typename V = int, typename C = V>
    71   class NetworkSimplex
    72   {
    73   public:
    74 
    75     /// The type of the flow amounts, capacity bounds and supply values
    76     typedef V Value;
    77     /// The type of the arc costs
    78     typedef C Cost;
    79 
    80   public:
    81 
    82     /// \brief Problem type constants for the \c run() function.
    83     ///
    84     /// Enum type containing the problem type constants that can be
    85     /// returned by the \ref run() function of the algorithm.
    86     enum ProblemType {
    87       /// The problem has no feasible solution (flow).
    88       INFEASIBLE,
    89       /// The problem has optimal solution (i.e. it is feasible and
    90       /// bounded), and the algorithm has found optimal flow and node
    91       /// potentials (primal and dual solutions).
    92       OPTIMAL,
    93       /// The objective function of the problem is unbounded, i.e.
    94       /// there is a directed cycle having negative total cost and
    95       /// infinite upper bound.
    96       UNBOUNDED
    97     };
    98     
    99     /// \brief Constants for selecting the type of the supply constraints.
   100     ///
   101     /// Enum type containing constants for selecting the supply type,
   102     /// i.e. the direction of the inequalities in the supply/demand
   103     /// constraints of the \ref min_cost_flow "minimum cost flow problem".
   104     ///
   105     /// The default supply type is \c GEQ, since this form is supported
   106     /// by other minimum cost flow algorithms and the \ref Circulation
   107     /// algorithm, as well.
   108     /// The \c LEQ problem type can be selected using the \ref supplyType()
   109     /// function.
   110     ///
   111     /// Note that the equality form is a special case of both supply types.
   112     enum SupplyType {
   113 
   114       /// This option means that there are <em>"greater or equal"</em>
   115       /// supply/demand constraints in the definition, i.e. the exact
   116       /// formulation of the problem is the following.
   117       /**
   118           \f[ \min\sum_{uv\in A} f(uv) \cdot cost(uv) \f]
   119           \f[ \sum_{uv\in A} f(uv) - \sum_{vu\in A} f(vu) \geq
   120               sup(u) \quad \forall u\in V \f]
   121           \f[ lower(uv) \leq f(uv) \leq upper(uv) \quad \forall uv\in A \f]
   122       */
   123       /// It means that the total demand must be greater or equal to the 
   124       /// total supply (i.e. \f$\sum_{u\in V} sup(u)\f$ must be zero or
   125       /// negative) and all the supplies have to be carried out from 
   126       /// the supply nodes, but there could be demands that are not 
   127       /// satisfied.
   128       GEQ,
   129       /// It is just an alias for the \c GEQ option.
   130       CARRY_SUPPLIES = GEQ,
   131 
   132       /// This option means that there are <em>"less or equal"</em>
   133       /// supply/demand constraints in the definition, i.e. the exact
   134       /// formulation of the problem is the following.
   135       /**
   136           \f[ \min\sum_{uv\in A} f(uv) \cdot cost(uv) \f]
   137           \f[ \sum_{uv\in A} f(uv) - \sum_{vu\in A} f(vu) \leq
   138               sup(u) \quad \forall u\in V \f]
   139           \f[ lower(uv) \leq f(uv) \leq upper(uv) \quad \forall uv\in A \f]
   140       */
   141       /// It means that the total demand must be less or equal to the 
   142       /// total supply (i.e. \f$\sum_{u\in V} sup(u)\f$ must be zero or
   143       /// positive) and all the demands have to be satisfied, but there
   144       /// could be supplies that are not carried out from the supply
   145       /// nodes.
   146       LEQ,
   147       /// It is just an alias for the \c LEQ option.
   148       SATISFY_DEMANDS = LEQ
   149     };
   150     
   151     /// \brief Constants for selecting the pivot rule.
   152     ///
   153     /// Enum type containing constants for selecting the pivot rule for
   154     /// the \ref run() function.
   155     ///
   156     /// \ref NetworkSimplex provides five different pivot rule
   157     /// implementations that significantly affect the running time
   158     /// of the algorithm.
   159     /// By default \ref BLOCK_SEARCH "Block Search" is used, which
   160     /// proved to be the most efficient and the most robust on various
   161     /// test inputs according to our benchmark tests.
   162     /// However another pivot rule can be selected using the \ref run()
   163     /// function with the proper parameter.
   164     enum PivotRule {
   165 
   166       /// The First Eligible pivot rule.
   167       /// The next eligible arc is selected in a wraparound fashion
   168       /// in every iteration.
   169       FIRST_ELIGIBLE,
   170 
   171       /// The Best Eligible pivot rule.
   172       /// The best eligible arc is selected in every iteration.
   173       BEST_ELIGIBLE,
   174 
   175       /// The Block Search pivot rule.
   176       /// A specified number of arcs are examined in every iteration
   177       /// in a wraparound fashion and the best eligible arc is selected
   178       /// from this block.
   179       BLOCK_SEARCH,
   180 
   181       /// The Candidate List pivot rule.
   182       /// In a major iteration a candidate list is built from eligible arcs
   183       /// in a wraparound fashion and in the following minor iterations
   184       /// the best eligible arc is selected from this list.
   185       CANDIDATE_LIST,
   186 
   187       /// The Altering Candidate List pivot rule.
   188       /// It is a modified version of the Candidate List method.
   189       /// It keeps only the several best eligible arcs from the former
   190       /// candidate list and extends this list in every iteration.
   191       ALTERING_LIST
   192     };
   193     
   194   private:
   195 
   196     TEMPLATE_DIGRAPH_TYPEDEFS(GR);
   197 
   198     typedef std::vector<Arc> ArcVector;
   199     typedef std::vector<Node> NodeVector;
   200     typedef std::vector<int> IntVector;
   201     typedef std::vector<bool> BoolVector;
   202     typedef std::vector<Value> ValueVector;
   203     typedef std::vector<Cost> CostVector;
   204 
   205     // State constants for arcs
   206     enum ArcStateEnum {
   207       STATE_UPPER = -1,
   208       STATE_TREE  =  0,
   209       STATE_LOWER =  1
   210     };
   211 
   212   private:
   213 
   214     // Data related to the underlying digraph
   215     const GR &_graph;
   216     int _node_num;
   217     int _arc_num;
   218 
   219     // Parameters of the problem
   220     bool _have_lower;
   221     SupplyType _stype;
   222     Value _sum_supply;
   223 
   224     // Data structures for storing the digraph
   225     IntNodeMap _node_id;
   226     IntArcMap _arc_id;
   227     IntVector _source;
   228     IntVector _target;
   229 
   230     // Node and arc data
   231     ValueVector _lower;
   232     ValueVector _upper;
   233     ValueVector _cap;
   234     CostVector _cost;
   235     ValueVector _supply;
   236     ValueVector _flow;
   237     CostVector _pi;
   238 
   239     // Data for storing the spanning tree structure
   240     IntVector _parent;
   241     IntVector _pred;
   242     IntVector _thread;
   243     IntVector _rev_thread;
   244     IntVector _succ_num;
   245     IntVector _last_succ;
   246     IntVector _dirty_revs;
   247     BoolVector _forward;
   248     IntVector _state;
   249     int _root;
   250 
   251     // Temporary data used in the current pivot iteration
   252     int in_arc, join, u_in, v_in, u_out, v_out;
   253     int first, second, right, last;
   254     int stem, par_stem, new_stem;
   255     Value delta;
   256 
   257   public:
   258   
   259     /// \brief Constant for infinite upper bounds (capacities).
   260     ///
   261     /// Constant for infinite upper bounds (capacities).
   262     /// It is \c std::numeric_limits<Value>::infinity() if available,
   263     /// \c std::numeric_limits<Value>::max() otherwise.
   264     const Value INF;
   265 
   266   private:
   267 
   268     // Implementation of the First Eligible pivot rule
   269     class FirstEligiblePivotRule
   270     {
   271     private:
   272 
   273       // References to the NetworkSimplex class
   274       const IntVector  &_source;
   275       const IntVector  &_target;
   276       const CostVector &_cost;
   277       const IntVector  &_state;
   278       const CostVector &_pi;
   279       int &_in_arc;
   280       int _arc_num;
   281 
   282       // Pivot rule data
   283       int _next_arc;
   284 
   285     public:
   286 
   287       // Constructor
   288       FirstEligiblePivotRule(NetworkSimplex &ns) :
   289         _source(ns._source), _target(ns._target),
   290         _cost(ns._cost), _state(ns._state), _pi(ns._pi),
   291         _in_arc(ns.in_arc), _arc_num(ns._arc_num), _next_arc(0)
   292       {}
   293 
   294       // Find next entering arc
   295       bool findEnteringArc() {
   296         Cost c;
   297         for (int e = _next_arc; e < _arc_num; ++e) {
   298           c = _state[e] * (_cost[e] + _pi[_source[e]] - _pi[_target[e]]);
   299           if (c < 0) {
   300             _in_arc = e;
   301             _next_arc = e + 1;
   302             return true;
   303           }
   304         }
   305         for (int e = 0; e < _next_arc; ++e) {
   306           c = _state[e] * (_cost[e] + _pi[_source[e]] - _pi[_target[e]]);
   307           if (c < 0) {
   308             _in_arc = e;
   309             _next_arc = e + 1;
   310             return true;
   311           }
   312         }
   313         return false;
   314       }
   315 
   316     }; //class FirstEligiblePivotRule
   317 
   318 
   319     // Implementation of the Best Eligible pivot rule
   320     class BestEligiblePivotRule
   321     {
   322     private:
   323 
   324       // References to the NetworkSimplex class
   325       const IntVector  &_source;
   326       const IntVector  &_target;
   327       const CostVector &_cost;
   328       const IntVector  &_state;
   329       const CostVector &_pi;
   330       int &_in_arc;
   331       int _arc_num;
   332 
   333     public:
   334 
   335       // Constructor
   336       BestEligiblePivotRule(NetworkSimplex &ns) :
   337         _source(ns._source), _target(ns._target),
   338         _cost(ns._cost), _state(ns._state), _pi(ns._pi),
   339         _in_arc(ns.in_arc), _arc_num(ns._arc_num)
   340       {}
   341 
   342       // Find next entering arc
   343       bool findEnteringArc() {
   344         Cost c, min = 0;
   345         for (int e = 0; e < _arc_num; ++e) {
   346           c = _state[e] * (_cost[e] + _pi[_source[e]] - _pi[_target[e]]);
   347           if (c < min) {
   348             min = c;
   349             _in_arc = e;
   350           }
   351         }
   352         return min < 0;
   353       }
   354 
   355     }; //class BestEligiblePivotRule
   356 
   357 
   358     // Implementation of the Block Search pivot rule
   359     class BlockSearchPivotRule
   360     {
   361     private:
   362 
   363       // References to the NetworkSimplex class
   364       const IntVector  &_source;
   365       const IntVector  &_target;
   366       const CostVector &_cost;
   367       const IntVector  &_state;
   368       const CostVector &_pi;
   369       int &_in_arc;
   370       int _arc_num;
   371 
   372       // Pivot rule data
   373       int _block_size;
   374       int _next_arc;
   375 
   376     public:
   377 
   378       // Constructor
   379       BlockSearchPivotRule(NetworkSimplex &ns) :
   380         _source(ns._source), _target(ns._target),
   381         _cost(ns._cost), _state(ns._state), _pi(ns._pi),
   382         _in_arc(ns.in_arc), _arc_num(ns._arc_num), _next_arc(0)
   383       {
   384         // The main parameters of the pivot rule
   385         const double BLOCK_SIZE_FACTOR = 2.0;
   386         const int MIN_BLOCK_SIZE = 10;
   387 
   388         _block_size = std::max( int(BLOCK_SIZE_FACTOR *
   389                                     std::sqrt(double(_arc_num))),
   390                                 MIN_BLOCK_SIZE );
   391       }
   392 
   393       // Find next entering arc
   394       bool findEnteringArc() {
   395         Cost c, min = 0;
   396         int cnt = _block_size;
   397         int e, min_arc = _next_arc;
   398         for (e = _next_arc; e < _arc_num; ++e) {
   399           c = _state[e] * (_cost[e] + _pi[_source[e]] - _pi[_target[e]]);
   400           if (c < min) {
   401             min = c;
   402             min_arc = e;
   403           }
   404           if (--cnt == 0) {
   405             if (min < 0) break;
   406             cnt = _block_size;
   407           }
   408         }
   409         if (min == 0 || cnt > 0) {
   410           for (e = 0; e < _next_arc; ++e) {
   411             c = _state[e] * (_cost[e] + _pi[_source[e]] - _pi[_target[e]]);
   412             if (c < min) {
   413               min = c;
   414               min_arc = e;
   415             }
   416             if (--cnt == 0) {
   417               if (min < 0) break;
   418               cnt = _block_size;
   419             }
   420           }
   421         }
   422         if (min >= 0) return false;
   423         _in_arc = min_arc;
   424         _next_arc = e;
   425         return true;
   426       }
   427 
   428     }; //class BlockSearchPivotRule
   429 
   430 
   431     // Implementation of the Candidate List pivot rule
   432     class CandidateListPivotRule
   433     {
   434     private:
   435 
   436       // References to the NetworkSimplex class
   437       const IntVector  &_source;
   438       const IntVector  &_target;
   439       const CostVector &_cost;
   440       const IntVector  &_state;
   441       const CostVector &_pi;
   442       int &_in_arc;
   443       int _arc_num;
   444 
   445       // Pivot rule data
   446       IntVector _candidates;
   447       int _list_length, _minor_limit;
   448       int _curr_length, _minor_count;
   449       int _next_arc;
   450 
   451     public:
   452 
   453       /// Constructor
   454       CandidateListPivotRule(NetworkSimplex &ns) :
   455         _source(ns._source), _target(ns._target),
   456         _cost(ns._cost), _state(ns._state), _pi(ns._pi),
   457         _in_arc(ns.in_arc), _arc_num(ns._arc_num), _next_arc(0)
   458       {
   459         // The main parameters of the pivot rule
   460         const double LIST_LENGTH_FACTOR = 1.0;
   461         const int MIN_LIST_LENGTH = 10;
   462         const double MINOR_LIMIT_FACTOR = 0.1;
   463         const int MIN_MINOR_LIMIT = 3;
   464 
   465         _list_length = std::max( int(LIST_LENGTH_FACTOR *
   466                                      std::sqrt(double(_arc_num))),
   467                                  MIN_LIST_LENGTH );
   468         _minor_limit = std::max( int(MINOR_LIMIT_FACTOR * _list_length),
   469                                  MIN_MINOR_LIMIT );
   470         _curr_length = _minor_count = 0;
   471         _candidates.resize(_list_length);
   472       }
   473 
   474       /// Find next entering arc
   475       bool findEnteringArc() {
   476         Cost min, c;
   477         int e, min_arc = _next_arc;
   478         if (_curr_length > 0 && _minor_count < _minor_limit) {
   479           // Minor iteration: select the best eligible arc from the
   480           // current candidate list
   481           ++_minor_count;
   482           min = 0;
   483           for (int i = 0; i < _curr_length; ++i) {
   484             e = _candidates[i];
   485             c = _state[e] * (_cost[e] + _pi[_source[e]] - _pi[_target[e]]);
   486             if (c < min) {
   487               min = c;
   488               min_arc = e;
   489             }
   490             if (c >= 0) {
   491               _candidates[i--] = _candidates[--_curr_length];
   492             }
   493           }
   494           if (min < 0) {
   495             _in_arc = min_arc;
   496             return true;
   497           }
   498         }
   499 
   500         // Major iteration: build a new candidate list
   501         min = 0;
   502         _curr_length = 0;
   503         for (e = _next_arc; e < _arc_num; ++e) {
   504           c = _state[e] * (_cost[e] + _pi[_source[e]] - _pi[_target[e]]);
   505           if (c < 0) {
   506             _candidates[_curr_length++] = e;
   507             if (c < min) {
   508               min = c;
   509               min_arc = e;
   510             }
   511             if (_curr_length == _list_length) break;
   512           }
   513         }
   514         if (_curr_length < _list_length) {
   515           for (e = 0; e < _next_arc; ++e) {
   516             c = _state[e] * (_cost[e] + _pi[_source[e]] - _pi[_target[e]]);
   517             if (c < 0) {
   518               _candidates[_curr_length++] = e;
   519               if (c < min) {
   520                 min = c;
   521                 min_arc = e;
   522               }
   523               if (_curr_length == _list_length) break;
   524             }
   525           }
   526         }
   527         if (_curr_length == 0) return false;
   528         _minor_count = 1;
   529         _in_arc = min_arc;
   530         _next_arc = e;
   531         return true;
   532       }
   533 
   534     }; //class CandidateListPivotRule
   535 
   536 
   537     // Implementation of the Altering Candidate List pivot rule
   538     class AlteringListPivotRule
   539     {
   540     private:
   541 
   542       // References to the NetworkSimplex class
   543       const IntVector  &_source;
   544       const IntVector  &_target;
   545       const CostVector &_cost;
   546       const IntVector  &_state;
   547       const CostVector &_pi;
   548       int &_in_arc;
   549       int _arc_num;
   550 
   551       // Pivot rule data
   552       int _block_size, _head_length, _curr_length;
   553       int _next_arc;
   554       IntVector _candidates;
   555       CostVector _cand_cost;
   556 
   557       // Functor class to compare arcs during sort of the candidate list
   558       class SortFunc
   559       {
   560       private:
   561         const CostVector &_map;
   562       public:
   563         SortFunc(const CostVector &map) : _map(map) {}
   564         bool operator()(int left, int right) {
   565           return _map[left] > _map[right];
   566         }
   567       };
   568 
   569       SortFunc _sort_func;
   570 
   571     public:
   572 
   573       // Constructor
   574       AlteringListPivotRule(NetworkSimplex &ns) :
   575         _source(ns._source), _target(ns._target),
   576         _cost(ns._cost), _state(ns._state), _pi(ns._pi),
   577         _in_arc(ns.in_arc), _arc_num(ns._arc_num),
   578         _next_arc(0), _cand_cost(ns._arc_num), _sort_func(_cand_cost)
   579       {
   580         // The main parameters of the pivot rule
   581         const double BLOCK_SIZE_FACTOR = 1.5;
   582         const int MIN_BLOCK_SIZE = 10;
   583         const double HEAD_LENGTH_FACTOR = 0.1;
   584         const int MIN_HEAD_LENGTH = 3;
   585 
   586         _block_size = std::max( int(BLOCK_SIZE_FACTOR *
   587                                     std::sqrt(double(_arc_num))),
   588                                 MIN_BLOCK_SIZE );
   589         _head_length = std::max( int(HEAD_LENGTH_FACTOR * _block_size),
   590                                  MIN_HEAD_LENGTH );
   591         _candidates.resize(_head_length + _block_size);
   592         _curr_length = 0;
   593       }
   594 
   595       // Find next entering arc
   596       bool findEnteringArc() {
   597         // Check the current candidate list
   598         int e;
   599         for (int i = 0; i < _curr_length; ++i) {
   600           e = _candidates[i];
   601           _cand_cost[e] = _state[e] *
   602             (_cost[e] + _pi[_source[e]] - _pi[_target[e]]);
   603           if (_cand_cost[e] >= 0) {
   604             _candidates[i--] = _candidates[--_curr_length];
   605           }
   606         }
   607 
   608         // Extend the list
   609         int cnt = _block_size;
   610         int last_arc = 0;
   611         int limit = _head_length;
   612 
   613         for (int e = _next_arc; e < _arc_num; ++e) {
   614           _cand_cost[e] = _state[e] *
   615             (_cost[e] + _pi[_source[e]] - _pi[_target[e]]);
   616           if (_cand_cost[e] < 0) {
   617             _candidates[_curr_length++] = e;
   618             last_arc = e;
   619           }
   620           if (--cnt == 0) {
   621             if (_curr_length > limit) break;
   622             limit = 0;
   623             cnt = _block_size;
   624           }
   625         }
   626         if (_curr_length <= limit) {
   627           for (int e = 0; e < _next_arc; ++e) {
   628             _cand_cost[e] = _state[e] *
   629               (_cost[e] + _pi[_source[e]] - _pi[_target[e]]);
   630             if (_cand_cost[e] < 0) {
   631               _candidates[_curr_length++] = e;
   632               last_arc = e;
   633             }
   634             if (--cnt == 0) {
   635               if (_curr_length > limit) break;
   636               limit = 0;
   637               cnt = _block_size;
   638             }
   639           }
   640         }
   641         if (_curr_length == 0) return false;
   642         _next_arc = last_arc + 1;
   643 
   644         // Make heap of the candidate list (approximating a partial sort)
   645         make_heap( _candidates.begin(), _candidates.begin() + _curr_length,
   646                    _sort_func );
   647 
   648         // Pop the first element of the heap
   649         _in_arc = _candidates[0];
   650         pop_heap( _candidates.begin(), _candidates.begin() + _curr_length,
   651                   _sort_func );
   652         _curr_length = std::min(_head_length, _curr_length - 1);
   653         return true;
   654       }
   655 
   656     }; //class AlteringListPivotRule
   657 
   658   public:
   659 
   660     /// \brief Constructor.
   661     ///
   662     /// The constructor of the class.
   663     ///
   664     /// \param graph The digraph the algorithm runs on.
   665     NetworkSimplex(const GR& graph) :
   666       _graph(graph), _node_id(graph), _arc_id(graph),
   667       INF(std::numeric_limits<Value>::has_infinity ?
   668           std::numeric_limits<Value>::infinity() :
   669           std::numeric_limits<Value>::max())
   670     {
   671       // Check the value types
   672       LEMON_ASSERT(std::numeric_limits<Value>::is_signed,
   673         "The flow type of NetworkSimplex must be signed");
   674       LEMON_ASSERT(std::numeric_limits<Cost>::is_signed,
   675         "The cost type of NetworkSimplex must be signed");
   676         
   677       // Resize vectors
   678       _node_num = countNodes(_graph);
   679       _arc_num = countArcs(_graph);
   680       int all_node_num = _node_num + 1;
   681       int all_arc_num = _arc_num + _node_num;
   682 
   683       _source.resize(all_arc_num);
   684       _target.resize(all_arc_num);
   685 
   686       _lower.resize(all_arc_num);
   687       _upper.resize(all_arc_num);
   688       _cap.resize(all_arc_num);
   689       _cost.resize(all_arc_num);
   690       _supply.resize(all_node_num);
   691       _flow.resize(all_arc_num);
   692       _pi.resize(all_node_num);
   693 
   694       _parent.resize(all_node_num);
   695       _pred.resize(all_node_num);
   696       _forward.resize(all_node_num);
   697       _thread.resize(all_node_num);
   698       _rev_thread.resize(all_node_num);
   699       _succ_num.resize(all_node_num);
   700       _last_succ.resize(all_node_num);
   701       _state.resize(all_arc_num);
   702 
   703       // Copy the graph (store the arcs in a mixed order)
   704       int i = 0;
   705       for (NodeIt n(_graph); n != INVALID; ++n, ++i) {
   706         _node_id[n] = i;
   707       }
   708       int k = std::max(int(std::sqrt(double(_arc_num))), 10);
   709       i = 0;
   710       for (ArcIt a(_graph); a != INVALID; ++a) {
   711         _arc_id[a] = i;
   712         _source[i] = _node_id[_graph.source(a)];
   713         _target[i] = _node_id[_graph.target(a)];
   714         if ((i += k) >= _arc_num) i = (i % k) + 1;
   715       }
   716       
   717       // Initialize maps
   718       for (int i = 0; i != _node_num; ++i) {
   719         _supply[i] = 0;
   720       }
   721       for (int i = 0; i != _arc_num; ++i) {
   722         _lower[i] = 0;
   723         _upper[i] = INF;
   724         _cost[i] = 1;
   725       }
   726       _have_lower = false;
   727       _stype = GEQ;
   728     }
   729 
   730     /// \name Parameters
   731     /// The parameters of the algorithm can be specified using these
   732     /// functions.
   733 
   734     /// @{
   735 
   736     /// \brief Set the lower bounds on the arcs.
   737     ///
   738     /// This function sets the lower bounds on the arcs.
   739     /// If it is not used before calling \ref run(), the lower bounds
   740     /// will be set to zero on all arcs.
   741     ///
   742     /// \param map An arc map storing the lower bounds.
   743     /// Its \c Value type must be convertible to the \c Value type
   744     /// of the algorithm.
   745     ///
   746     /// \return <tt>(*this)</tt>
   747     template <typename LowerMap>
   748     NetworkSimplex& lowerMap(const LowerMap& map) {
   749       _have_lower = true;
   750       for (ArcIt a(_graph); a != INVALID; ++a) {
   751         _lower[_arc_id[a]] = map[a];
   752       }
   753       return *this;
   754     }
   755 
   756     /// \brief Set the upper bounds (capacities) on the arcs.
   757     ///
   758     /// This function sets the upper bounds (capacities) on the arcs.
   759     /// If it is not used before calling \ref run(), the upper bounds
   760     /// will be set to \ref INF on all arcs (i.e. the flow value will be
   761     /// unbounded from above on each arc).
   762     ///
   763     /// \param map An arc map storing the upper bounds.
   764     /// Its \c Value type must be convertible to the \c Value type
   765     /// of the algorithm.
   766     ///
   767     /// \return <tt>(*this)</tt>
   768     template<typename UpperMap>
   769     NetworkSimplex& upperMap(const UpperMap& map) {
   770       for (ArcIt a(_graph); a != INVALID; ++a) {
   771         _upper[_arc_id[a]] = map[a];
   772       }
   773       return *this;
   774     }
   775 
   776     /// \brief Set the costs of the arcs.
   777     ///
   778     /// This function sets the costs of the arcs.
   779     /// If it is not used before calling \ref run(), the costs
   780     /// will be set to \c 1 on all arcs.
   781     ///
   782     /// \param map An arc map storing the costs.
   783     /// Its \c Value type must be convertible to the \c Cost type
   784     /// of the algorithm.
   785     ///
   786     /// \return <tt>(*this)</tt>
   787     template<typename CostMap>
   788     NetworkSimplex& costMap(const CostMap& map) {
   789       for (ArcIt a(_graph); a != INVALID; ++a) {
   790         _cost[_arc_id[a]] = map[a];
   791       }
   792       return *this;
   793     }
   794 
   795     /// \brief Set the supply values of the nodes.
   796     ///
   797     /// This function sets the supply values of the nodes.
   798     /// If neither this function nor \ref stSupply() is used before
   799     /// calling \ref run(), the supply of each node will be set to zero.
   800     /// (It makes sense only if non-zero lower bounds are given.)
   801     ///
   802     /// \param map A node map storing the supply values.
   803     /// Its \c Value type must be convertible to the \c Value type
   804     /// of the algorithm.
   805     ///
   806     /// \return <tt>(*this)</tt>
   807     template<typename SupplyMap>
   808     NetworkSimplex& supplyMap(const SupplyMap& map) {
   809       for (NodeIt n(_graph); n != INVALID; ++n) {
   810         _supply[_node_id[n]] = map[n];
   811       }
   812       return *this;
   813     }
   814 
   815     /// \brief Set single source and target nodes and a supply value.
   816     ///
   817     /// This function sets a single source node and a single target node
   818     /// and the required flow value.
   819     /// If neither this function nor \ref supplyMap() is used before
   820     /// calling \ref run(), the supply of each node will be set to zero.
   821     /// (It makes sense only if non-zero lower bounds are given.)
   822     ///
   823     /// Using this function has the same effect as using \ref supplyMap()
   824     /// with such a map in which \c k is assigned to \c s, \c -k is
   825     /// assigned to \c t and all other nodes have zero supply value.
   826     ///
   827     /// \param s The source node.
   828     /// \param t The target node.
   829     /// \param k The required amount of flow from node \c s to node \c t
   830     /// (i.e. the supply of \c s and the demand of \c t).
   831     ///
   832     /// \return <tt>(*this)</tt>
   833     NetworkSimplex& stSupply(const Node& s, const Node& t, Value k) {
   834       for (int i = 0; i != _node_num; ++i) {
   835         _supply[i] = 0;
   836       }
   837       _supply[_node_id[s]] =  k;
   838       _supply[_node_id[t]] = -k;
   839       return *this;
   840     }
   841     
   842     /// \brief Set the type of the supply constraints.
   843     ///
   844     /// This function sets the type of the supply/demand constraints.
   845     /// If it is not used before calling \ref run(), the \ref GEQ supply
   846     /// type will be used.
   847     ///
   848     /// For more information see \ref SupplyType.
   849     ///
   850     /// \return <tt>(*this)</tt>
   851     NetworkSimplex& supplyType(SupplyType supply_type) {
   852       _stype = supply_type;
   853       return *this;
   854     }
   855 
   856     /// @}
   857 
   858     /// \name Execution Control
   859     /// The algorithm can be executed using \ref run().
   860 
   861     /// @{
   862 
   863     /// \brief Run the algorithm.
   864     ///
   865     /// This function runs the algorithm.
   866     /// The paramters can be specified using functions \ref lowerMap(),
   867     /// \ref upperMap(), \ref costMap(), \ref supplyMap(), \ref stSupply(), 
   868     /// \ref supplyType().
   869     /// For example,
   870     /// \code
   871     ///   NetworkSimplex<ListDigraph> ns(graph);
   872     ///   ns.lowerMap(lower).upperMap(upper).costMap(cost)
   873     ///     .supplyMap(sup).run();
   874     /// \endcode
   875     ///
   876     /// This function can be called more than once. All the parameters
   877     /// that have been given are kept for the next call, unless
   878     /// \ref reset() is called, thus only the modified parameters
   879     /// have to be set again. See \ref reset() for examples.
   880     /// However the underlying digraph must not be modified after this
   881     /// class have been constructed, since it copies and extends the graph.
   882     ///
   883     /// \param pivot_rule The pivot rule that will be used during the
   884     /// algorithm. For more information see \ref PivotRule.
   885     ///
   886     /// \return \c INFEASIBLE if no feasible flow exists,
   887     /// \n \c OPTIMAL if the problem has optimal solution
   888     /// (i.e. it is feasible and bounded), and the algorithm has found
   889     /// optimal flow and node potentials (primal and dual solutions),
   890     /// \n \c UNBOUNDED if the objective function of the problem is
   891     /// unbounded, i.e. there is a directed cycle having negative total
   892     /// cost and infinite upper bound.
   893     ///
   894     /// \see ProblemType, PivotRule
   895     ProblemType run(PivotRule pivot_rule = BLOCK_SEARCH) {
   896       if (!init()) return INFEASIBLE;
   897       return start(pivot_rule);
   898     }
   899 
   900     /// \brief Reset all the parameters that have been given before.
   901     ///
   902     /// This function resets all the paramaters that have been given
   903     /// before using functions \ref lowerMap(), \ref upperMap(),
   904     /// \ref costMap(), \ref supplyMap(), \ref stSupply(), \ref supplyType().
   905     ///
   906     /// It is useful for multiple run() calls. If this function is not
   907     /// used, all the parameters given before are kept for the next
   908     /// \ref run() call.
   909     /// However the underlying digraph must not be modified after this
   910     /// class have been constructed, since it copies and extends the graph.
   911     ///
   912     /// For example,
   913     /// \code
   914     ///   NetworkSimplex<ListDigraph> ns(graph);
   915     ///
   916     ///   // First run
   917     ///   ns.lowerMap(lower).upperMap(upper).costMap(cost)
   918     ///     .supplyMap(sup).run();
   919     ///
   920     ///   // Run again with modified cost map (reset() is not called,
   921     ///   // so only the cost map have to be set again)
   922     ///   cost[e] += 100;
   923     ///   ns.costMap(cost).run();
   924     ///
   925     ///   // Run again from scratch using reset()
   926     ///   // (the lower bounds will be set to zero on all arcs)
   927     ///   ns.reset();
   928     ///   ns.upperMap(capacity).costMap(cost)
   929     ///     .supplyMap(sup).run();
   930     /// \endcode
   931     ///
   932     /// \return <tt>(*this)</tt>
   933     NetworkSimplex& reset() {
   934       for (int i = 0; i != _node_num; ++i) {
   935         _supply[i] = 0;
   936       }
   937       for (int i = 0; i != _arc_num; ++i) {
   938         _lower[i] = 0;
   939         _upper[i] = INF;
   940         _cost[i] = 1;
   941       }
   942       _have_lower = false;
   943       _stype = GEQ;
   944       return *this;
   945     }
   946 
   947     /// @}
   948 
   949     /// \name Query Functions
   950     /// The results of the algorithm can be obtained using these
   951     /// functions.\n
   952     /// The \ref run() function must be called before using them.
   953 
   954     /// @{
   955 
   956     /// \brief Return the total cost of the found flow.
   957     ///
   958     /// This function returns the total cost of the found flow.
   959     /// Its complexity is O(e).
   960     ///
   961     /// \note The return type of the function can be specified as a
   962     /// template parameter. For example,
   963     /// \code
   964     ///   ns.totalCost<double>();
   965     /// \endcode
   966     /// It is useful if the total cost cannot be stored in the \c Cost
   967     /// type of the algorithm, which is the default return type of the
   968     /// function.
   969     ///
   970     /// \pre \ref run() must be called before using this function.
   971     template <typename Number>
   972     Number totalCost() const {
   973       Number c = 0;
   974       for (ArcIt a(_graph); a != INVALID; ++a) {
   975         int i = _arc_id[a];
   976         c += Number(_flow[i]) * Number(_cost[i]);
   977       }
   978       return c;
   979     }
   980 
   981 #ifndef DOXYGEN
   982     Cost totalCost() const {
   983       return totalCost<Cost>();
   984     }
   985 #endif
   986 
   987     /// \brief Return the flow on the given arc.
   988     ///
   989     /// This function returns the flow on the given arc.
   990     ///
   991     /// \pre \ref run() must be called before using this function.
   992     Value flow(const Arc& a) const {
   993       return _flow[_arc_id[a]];
   994     }
   995 
   996     /// \brief Return the flow map (the primal solution).
   997     ///
   998     /// This function copies the flow value on each arc into the given
   999     /// map. The \c Value type of the algorithm must be convertible to
  1000     /// the \c Value type of the map.
  1001     ///
  1002     /// \pre \ref run() must be called before using this function.
  1003     template <typename FlowMap>
  1004     void flowMap(FlowMap &map) const {
  1005       for (ArcIt a(_graph); a != INVALID; ++a) {
  1006         map.set(a, _flow[_arc_id[a]]);
  1007       }
  1008     }
  1009 
  1010     /// \brief Return the potential (dual value) of the given node.
  1011     ///
  1012     /// This function returns the potential (dual value) of the
  1013     /// given node.
  1014     ///
  1015     /// \pre \ref run() must be called before using this function.
  1016     Cost potential(const Node& n) const {
  1017       return _pi[_node_id[n]];
  1018     }
  1019 
  1020     /// \brief Return the potential map (the dual solution).
  1021     ///
  1022     /// This function copies the potential (dual value) of each node
  1023     /// into the given map.
  1024     /// The \c Cost type of the algorithm must be convertible to the
  1025     /// \c Value type of the map.
  1026     ///
  1027     /// \pre \ref run() must be called before using this function.
  1028     template <typename PotentialMap>
  1029     void potentialMap(PotentialMap &map) const {
  1030       for (NodeIt n(_graph); n != INVALID; ++n) {
  1031         map.set(n, _pi[_node_id[n]]);
  1032       }
  1033     }
  1034 
  1035     /// @}
  1036 
  1037   private:
  1038 
  1039     // Initialize internal data structures
  1040     bool init() {
  1041       if (_node_num == 0) return false;
  1042 
  1043       // Check the sum of supply values
  1044       _sum_supply = 0;
  1045       for (int i = 0; i != _node_num; ++i) {
  1046         _sum_supply += _supply[i];
  1047       }
  1048       if ( !((_stype == GEQ && _sum_supply <= 0) ||
  1049              (_stype == LEQ && _sum_supply >= 0)) ) return false;
  1050 
  1051       // Remove non-zero lower bounds
  1052       if (_have_lower) {
  1053         for (int i = 0; i != _arc_num; ++i) {
  1054           Value c = _lower[i];
  1055           if (c >= 0) {
  1056             _cap[i] = _upper[i] < INF ? _upper[i] - c : INF;
  1057           } else {
  1058             _cap[i] = _upper[i] < INF + c ? _upper[i] - c : INF;
  1059           }
  1060           _supply[_source[i]] -= c;
  1061           _supply[_target[i]] += c;
  1062         }
  1063       } else {
  1064         for (int i = 0; i != _arc_num; ++i) {
  1065           _cap[i] = _upper[i];
  1066         }
  1067       }
  1068 
  1069       // Initialize artifical cost
  1070       Cost ART_COST;
  1071       if (std::numeric_limits<Cost>::is_exact) {
  1072         ART_COST = std::numeric_limits<Cost>::max() / 4 + 1;
  1073       } else {
  1074         ART_COST = std::numeric_limits<Cost>::min();
  1075         for (int i = 0; i != _arc_num; ++i) {
  1076           if (_cost[i] > ART_COST) ART_COST = _cost[i];
  1077         }
  1078         ART_COST = (ART_COST + 1) * _node_num;
  1079       }
  1080 
  1081       // Initialize arc maps
  1082       for (int i = 0; i != _arc_num; ++i) {
  1083         _flow[i] = 0;
  1084         _state[i] = STATE_LOWER;
  1085       }
  1086       
  1087       // Set data for the artificial root node
  1088       _root = _node_num;
  1089       _parent[_root] = -1;
  1090       _pred[_root] = -1;
  1091       _thread[_root] = 0;
  1092       _rev_thread[0] = _root;
  1093       _succ_num[_root] = _node_num + 1;
  1094       _last_succ[_root] = _root - 1;
  1095       _supply[_root] = -_sum_supply;
  1096       _pi[_root] = _sum_supply < 0 ? -ART_COST : ART_COST;
  1097 
  1098       // Add artificial arcs and initialize the spanning tree data structure
  1099       for (int u = 0, e = _arc_num; u != _node_num; ++u, ++e) {
  1100         _parent[u] = _root;
  1101         _pred[u] = e;
  1102         _thread[u] = u + 1;
  1103         _rev_thread[u + 1] = u;
  1104         _succ_num[u] = 1;
  1105         _last_succ[u] = u;
  1106         _cost[e] = ART_COST;
  1107         _cap[e] = INF;
  1108         _state[e] = STATE_TREE;
  1109         if (_supply[u] > 0 || (_supply[u] == 0 && _sum_supply <= 0)) {
  1110           _flow[e] = _supply[u];
  1111           _forward[u] = true;
  1112           _pi[u] = -ART_COST + _pi[_root];
  1113         } else {
  1114           _flow[e] = -_supply[u];
  1115           _forward[u] = false;
  1116           _pi[u] = ART_COST + _pi[_root];
  1117         }
  1118       }
  1119 
  1120       return true;
  1121     }
  1122 
  1123     // Find the join node
  1124     void findJoinNode() {
  1125       int u = _source[in_arc];
  1126       int v = _target[in_arc];
  1127       while (u != v) {
  1128         if (_succ_num[u] < _succ_num[v]) {
  1129           u = _parent[u];
  1130         } else {
  1131           v = _parent[v];
  1132         }
  1133       }
  1134       join = u;
  1135     }
  1136 
  1137     // Find the leaving arc of the cycle and returns true if the
  1138     // leaving arc is not the same as the entering arc
  1139     bool findLeavingArc() {
  1140       // Initialize first and second nodes according to the direction
  1141       // of the cycle
  1142       if (_state[in_arc] == STATE_LOWER) {
  1143         first  = _source[in_arc];
  1144         second = _target[in_arc];
  1145       } else {
  1146         first  = _target[in_arc];
  1147         second = _source[in_arc];
  1148       }
  1149       delta = _cap[in_arc];
  1150       int result = 0;
  1151       Value d;
  1152       int e;
  1153 
  1154       // Search the cycle along the path form the first node to the root
  1155       for (int u = first; u != join; u = _parent[u]) {
  1156         e = _pred[u];
  1157         d = _forward[u] ?
  1158           _flow[e] : (_cap[e] == INF ? INF : _cap[e] - _flow[e]);
  1159         if (d < delta) {
  1160           delta = d;
  1161           u_out = u;
  1162           result = 1;
  1163         }
  1164       }
  1165       // Search the cycle along the path form the second node to the root
  1166       for (int u = second; u != join; u = _parent[u]) {
  1167         e = _pred[u];
  1168         d = _forward[u] ? 
  1169           (_cap[e] == INF ? INF : _cap[e] - _flow[e]) : _flow[e];
  1170         if (d <= delta) {
  1171           delta = d;
  1172           u_out = u;
  1173           result = 2;
  1174         }
  1175       }
  1176 
  1177       if (result == 1) {
  1178         u_in = first;
  1179         v_in = second;
  1180       } else {
  1181         u_in = second;
  1182         v_in = first;
  1183       }
  1184       return result != 0;
  1185     }
  1186 
  1187     // Change _flow and _state vectors
  1188     void changeFlow(bool change) {
  1189       // Augment along the cycle
  1190       if (delta > 0) {
  1191         Value val = _state[in_arc] * delta;
  1192         _flow[in_arc] += val;
  1193         for (int u = _source[in_arc]; u != join; u = _parent[u]) {
  1194           _flow[_pred[u]] += _forward[u] ? -val : val;
  1195         }
  1196         for (int u = _target[in_arc]; u != join; u = _parent[u]) {
  1197           _flow[_pred[u]] += _forward[u] ? val : -val;
  1198         }
  1199       }
  1200       // Update the state of the entering and leaving arcs
  1201       if (change) {
  1202         _state[in_arc] = STATE_TREE;
  1203         _state[_pred[u_out]] =
  1204           (_flow[_pred[u_out]] == 0) ? STATE_LOWER : STATE_UPPER;
  1205       } else {
  1206         _state[in_arc] = -_state[in_arc];
  1207       }
  1208     }
  1209 
  1210     // Update the tree structure
  1211     void updateTreeStructure() {
  1212       int u, w;
  1213       int old_rev_thread = _rev_thread[u_out];
  1214       int old_succ_num = _succ_num[u_out];
  1215       int old_last_succ = _last_succ[u_out];
  1216       v_out = _parent[u_out];
  1217 
  1218       u = _last_succ[u_in];  // the last successor of u_in
  1219       right = _thread[u];    // the node after it
  1220 
  1221       // Handle the case when old_rev_thread equals to v_in
  1222       // (it also means that join and v_out coincide)
  1223       if (old_rev_thread == v_in) {
  1224         last = _thread[_last_succ[u_out]];
  1225       } else {
  1226         last = _thread[v_in];
  1227       }
  1228 
  1229       // Update _thread and _parent along the stem nodes (i.e. the nodes
  1230       // between u_in and u_out, whose parent have to be changed)
  1231       _thread[v_in] = stem = u_in;
  1232       _dirty_revs.clear();
  1233       _dirty_revs.push_back(v_in);
  1234       par_stem = v_in;
  1235       while (stem != u_out) {
  1236         // Insert the next stem node into the thread list
  1237         new_stem = _parent[stem];
  1238         _thread[u] = new_stem;
  1239         _dirty_revs.push_back(u);
  1240 
  1241         // Remove the subtree of stem from the thread list
  1242         w = _rev_thread[stem];
  1243         _thread[w] = right;
  1244         _rev_thread[right] = w;
  1245 
  1246         // Change the parent node and shift stem nodes
  1247         _parent[stem] = par_stem;
  1248         par_stem = stem;
  1249         stem = new_stem;
  1250 
  1251         // Update u and right
  1252         u = _last_succ[stem] == _last_succ[par_stem] ?
  1253           _rev_thread[par_stem] : _last_succ[stem];
  1254         right = _thread[u];
  1255       }
  1256       _parent[u_out] = par_stem;
  1257       _thread[u] = last;
  1258       _rev_thread[last] = u;
  1259       _last_succ[u_out] = u;
  1260 
  1261       // Remove the subtree of u_out from the thread list except for
  1262       // the case when old_rev_thread equals to v_in
  1263       // (it also means that join and v_out coincide)
  1264       if (old_rev_thread != v_in) {
  1265         _thread[old_rev_thread] = right;
  1266         _rev_thread[right] = old_rev_thread;
  1267       }
  1268 
  1269       // Update _rev_thread using the new _thread values
  1270       for (int i = 0; i < int(_dirty_revs.size()); ++i) {
  1271         u = _dirty_revs[i];
  1272         _rev_thread[_thread[u]] = u;
  1273       }
  1274 
  1275       // Update _pred, _forward, _last_succ and _succ_num for the
  1276       // stem nodes from u_out to u_in
  1277       int tmp_sc = 0, tmp_ls = _last_succ[u_out];
  1278       u = u_out;
  1279       while (u != u_in) {
  1280         w = _parent[u];
  1281         _pred[u] = _pred[w];
  1282         _forward[u] = !_forward[w];
  1283         tmp_sc += _succ_num[u] - _succ_num[w];
  1284         _succ_num[u] = tmp_sc;
  1285         _last_succ[w] = tmp_ls;
  1286         u = w;
  1287       }
  1288       _pred[u_in] = in_arc;
  1289       _forward[u_in] = (u_in == _source[in_arc]);
  1290       _succ_num[u_in] = old_succ_num;
  1291 
  1292       // Set limits for updating _last_succ form v_in and v_out
  1293       // towards the root
  1294       int up_limit_in = -1;
  1295       int up_limit_out = -1;
  1296       if (_last_succ[join] == v_in) {
  1297         up_limit_out = join;
  1298       } else {
  1299         up_limit_in = join;
  1300       }
  1301 
  1302       // Update _last_succ from v_in towards the root
  1303       for (u = v_in; u != up_limit_in && _last_succ[u] == v_in;
  1304            u = _parent[u]) {
  1305         _last_succ[u] = _last_succ[u_out];
  1306       }
  1307       // Update _last_succ from v_out towards the root
  1308       if (join != old_rev_thread && v_in != old_rev_thread) {
  1309         for (u = v_out; u != up_limit_out && _last_succ[u] == old_last_succ;
  1310              u = _parent[u]) {
  1311           _last_succ[u] = old_rev_thread;
  1312         }
  1313       } else {
  1314         for (u = v_out; u != up_limit_out && _last_succ[u] == old_last_succ;
  1315              u = _parent[u]) {
  1316           _last_succ[u] = _last_succ[u_out];
  1317         }
  1318       }
  1319 
  1320       // Update _succ_num from v_in to join
  1321       for (u = v_in; u != join; u = _parent[u]) {
  1322         _succ_num[u] += old_succ_num;
  1323       }
  1324       // Update _succ_num from v_out to join
  1325       for (u = v_out; u != join; u = _parent[u]) {
  1326         _succ_num[u] -= old_succ_num;
  1327       }
  1328     }
  1329 
  1330     // Update potentials
  1331     void updatePotential() {
  1332       Cost sigma = _forward[u_in] ?
  1333         _pi[v_in] - _pi[u_in] - _cost[_pred[u_in]] :
  1334         _pi[v_in] - _pi[u_in] + _cost[_pred[u_in]];
  1335       // Update potentials in the subtree, which has been moved
  1336       int end = _thread[_last_succ[u_in]];
  1337       for (int u = u_in; u != end; u = _thread[u]) {
  1338         _pi[u] += sigma;
  1339       }
  1340     }
  1341 
  1342     // Execute the algorithm
  1343     ProblemType start(PivotRule pivot_rule) {
  1344       // Select the pivot rule implementation
  1345       switch (pivot_rule) {
  1346         case FIRST_ELIGIBLE:
  1347           return start<FirstEligiblePivotRule>();
  1348         case BEST_ELIGIBLE:
  1349           return start<BestEligiblePivotRule>();
  1350         case BLOCK_SEARCH:
  1351           return start<BlockSearchPivotRule>();
  1352         case CANDIDATE_LIST:
  1353           return start<CandidateListPivotRule>();
  1354         case ALTERING_LIST:
  1355           return start<AlteringListPivotRule>();
  1356       }
  1357       return INFEASIBLE; // avoid warning
  1358     }
  1359 
  1360     template <typename PivotRuleImpl>
  1361     ProblemType start() {
  1362       PivotRuleImpl pivot(*this);
  1363 
  1364       // Execute the Network Simplex algorithm
  1365       while (pivot.findEnteringArc()) {
  1366         findJoinNode();
  1367         bool change = findLeavingArc();
  1368         if (delta >= INF) return UNBOUNDED;
  1369         changeFlow(change);
  1370         if (change) {
  1371           updateTreeStructure();
  1372           updatePotential();
  1373         }
  1374       }
  1375       
  1376       // Check feasibility
  1377       if (_sum_supply < 0) {
  1378         for (int u = 0, e = _arc_num; u != _node_num; ++u, ++e) {
  1379           if (_supply[u] >= 0 && _flow[e] != 0) return INFEASIBLE;
  1380         }
  1381       }
  1382       else if (_sum_supply > 0) {
  1383         for (int u = 0, e = _arc_num; u != _node_num; ++u, ++e) {
  1384           if (_supply[u] <= 0 && _flow[e] != 0) return INFEASIBLE;
  1385         }
  1386       }
  1387       else {
  1388         for (int u = 0, e = _arc_num; u != _node_num; ++u, ++e) {
  1389           if (_flow[e] != 0) return INFEASIBLE;
  1390         }
  1391       }
  1392 
  1393       // Transform the solution and the supply map to the original form
  1394       if (_have_lower) {
  1395         for (int i = 0; i != _arc_num; ++i) {
  1396           Value c = _lower[i];
  1397           if (c != 0) {
  1398             _flow[i] += c;
  1399             _supply[_source[i]] += c;
  1400             _supply[_target[i]] -= c;
  1401           }
  1402         }
  1403       }
  1404 
  1405       return OPTIMAL;
  1406     }
  1407 
  1408   }; //class NetworkSimplex
  1409 
  1410   ///@}
  1411 
  1412 } //namespace lemon
  1413 
  1414 #endif //LEMON_NETWORK_SIMPLEX_H