lemon/network_simplex.h
author Peter Kovacs <kpeter@inf.elte.hu>
Sun, 15 Nov 2009 19:57:02 +0100
changeset 787 c2230649a493
parent 730 4a45c8808b33
child 788 c92296660262
permissions -rw-r--r--
Various doc improvements (#331)

- Add notes to the graph classes about the time of
item counting.
- Clarify the doc for run() in BFS and DFS.
- Other improvements.
     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 specialized version of the linear programming
    47   /// simplex method directly for the minimum cost flow problem.
    48   /// It is one of the most efficient solution methods.
    49   ///
    50   /// In general this class is the fastest implementation available
    51   /// in LEMON for the minimum cost flow 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 value type used for flow amounts, capacity bounds
    62   /// and supply values in the algorithm. By default it is \c int.
    63   /// \tparam C The value type used for costs and potentials in the
    64   /// algorithm. By default it is the same as \c V.
    65   ///
    66   /// \warning Both value 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 according to our benchmark tests.
   130     /// However another pivot rule can be selected using the \ref run()
   131     /// function with the proper parameter.
   132     enum PivotRule {
   133 
   134       /// The First Eligible pivot rule.
   135       /// The next eligible arc is selected in a wraparound fashion
   136       /// in every iteration.
   137       FIRST_ELIGIBLE,
   138 
   139       /// The Best Eligible pivot rule.
   140       /// The best eligible arc is selected in every iteration.
   141       BEST_ELIGIBLE,
   142 
   143       /// The Block 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 Candidate 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 Altering Candidate 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<bool> BoolVector;
   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 
   198     // Node and arc data
   199     ValueVector _lower;
   200     ValueVector _upper;
   201     ValueVector _cap;
   202     CostVector _cost;
   203     ValueVector _supply;
   204     ValueVector _flow;
   205     CostVector _pi;
   206 
   207     // Data for storing the spanning tree structure
   208     IntVector _parent;
   209     IntVector _pred;
   210     IntVector _thread;
   211     IntVector _rev_thread;
   212     IntVector _succ_num;
   213     IntVector _last_succ;
   214     IntVector _dirty_revs;
   215     BoolVector _forward;
   216     IntVector _state;
   217     int _root;
   218 
   219     // Temporary data used in the current pivot iteration
   220     int in_arc, join, u_in, v_in, u_out, v_out;
   221     int first, second, right, last;
   222     int stem, par_stem, new_stem;
   223     Value delta;
   224 
   225   public:
   226   
   227     /// \brief Constant for infinite upper bounds (capacities).
   228     ///
   229     /// Constant for infinite upper bounds (capacities).
   230     /// It is \c std::numeric_limits<Value>::infinity() if available,
   231     /// \c std::numeric_limits<Value>::max() otherwise.
   232     const Value INF;
   233 
   234   private:
   235 
   236     // Implementation of the First Eligible pivot rule
   237     class FirstEligiblePivotRule
   238     {
   239     private:
   240 
   241       // References to the NetworkSimplex class
   242       const IntVector  &_source;
   243       const IntVector  &_target;
   244       const CostVector &_cost;
   245       const IntVector  &_state;
   246       const CostVector &_pi;
   247       int &_in_arc;
   248       int _search_arc_num;
   249 
   250       // Pivot rule data
   251       int _next_arc;
   252 
   253     public:
   254 
   255       // Constructor
   256       FirstEligiblePivotRule(NetworkSimplex &ns) :
   257         _source(ns._source), _target(ns._target),
   258         _cost(ns._cost), _state(ns._state), _pi(ns._pi),
   259         _in_arc(ns.in_arc), _search_arc_num(ns._search_arc_num),
   260         _next_arc(0)
   261       {}
   262 
   263       // Find next entering arc
   264       bool findEnteringArc() {
   265         Cost c;
   266         for (int e = _next_arc; e < _search_arc_num; ++e) {
   267           c = _state[e] * (_cost[e] + _pi[_source[e]] - _pi[_target[e]]);
   268           if (c < 0) {
   269             _in_arc = e;
   270             _next_arc = e + 1;
   271             return true;
   272           }
   273         }
   274         for (int e = 0; e < _next_arc; ++e) {
   275           c = _state[e] * (_cost[e] + _pi[_source[e]] - _pi[_target[e]]);
   276           if (c < 0) {
   277             _in_arc = e;
   278             _next_arc = e + 1;
   279             return true;
   280           }
   281         }
   282         return false;
   283       }
   284 
   285     }; //class FirstEligiblePivotRule
   286 
   287 
   288     // Implementation of the Best Eligible pivot rule
   289     class BestEligiblePivotRule
   290     {
   291     private:
   292 
   293       // References to the NetworkSimplex class
   294       const IntVector  &_source;
   295       const IntVector  &_target;
   296       const CostVector &_cost;
   297       const IntVector  &_state;
   298       const CostVector &_pi;
   299       int &_in_arc;
   300       int _search_arc_num;
   301 
   302     public:
   303 
   304       // Constructor
   305       BestEligiblePivotRule(NetworkSimplex &ns) :
   306         _source(ns._source), _target(ns._target),
   307         _cost(ns._cost), _state(ns._state), _pi(ns._pi),
   308         _in_arc(ns.in_arc), _search_arc_num(ns._search_arc_num)
   309       {}
   310 
   311       // Find next entering arc
   312       bool findEnteringArc() {
   313         Cost c, min = 0;
   314         for (int e = 0; e < _search_arc_num; ++e) {
   315           c = _state[e] * (_cost[e] + _pi[_source[e]] - _pi[_target[e]]);
   316           if (c < min) {
   317             min = c;
   318             _in_arc = e;
   319           }
   320         }
   321         return min < 0;
   322       }
   323 
   324     }; //class BestEligiblePivotRule
   325 
   326 
   327     // Implementation of the Block Search pivot rule
   328     class BlockSearchPivotRule
   329     {
   330     private:
   331 
   332       // References to the NetworkSimplex class
   333       const IntVector  &_source;
   334       const IntVector  &_target;
   335       const CostVector &_cost;
   336       const IntVector  &_state;
   337       const CostVector &_pi;
   338       int &_in_arc;
   339       int _search_arc_num;
   340 
   341       // Pivot rule data
   342       int _block_size;
   343       int _next_arc;
   344 
   345     public:
   346 
   347       // Constructor
   348       BlockSearchPivotRule(NetworkSimplex &ns) :
   349         _source(ns._source), _target(ns._target),
   350         _cost(ns._cost), _state(ns._state), _pi(ns._pi),
   351         _in_arc(ns.in_arc), _search_arc_num(ns._search_arc_num),
   352         _next_arc(0)
   353       {
   354         // The main parameters of the pivot rule
   355         const double BLOCK_SIZE_FACTOR = 0.5;
   356         const int MIN_BLOCK_SIZE = 10;
   357 
   358         _block_size = std::max( int(BLOCK_SIZE_FACTOR *
   359                                     std::sqrt(double(_search_arc_num))),
   360                                 MIN_BLOCK_SIZE );
   361       }
   362 
   363       // Find next entering arc
   364       bool findEnteringArc() {
   365         Cost c, min = 0;
   366         int cnt = _block_size;
   367         int e;
   368         for (e = _next_arc; e < _search_arc_num; ++e) {
   369           c = _state[e] * (_cost[e] + _pi[_source[e]] - _pi[_target[e]]);
   370           if (c < min) {
   371             min = c;
   372             _in_arc = e;
   373           }
   374           if (--cnt == 0) {
   375             if (min < 0) goto search_end;
   376             cnt = _block_size;
   377           }
   378         }
   379         for (e = 0; e < _next_arc; ++e) {
   380           c = _state[e] * (_cost[e] + _pi[_source[e]] - _pi[_target[e]]);
   381           if (c < min) {
   382             min = c;
   383             _in_arc = e;
   384           }
   385           if (--cnt == 0) {
   386             if (min < 0) goto search_end;
   387             cnt = _block_size;
   388           }
   389         }
   390         if (min >= 0) return false;
   391 
   392       search_end:
   393         _next_arc = e;
   394         return true;
   395       }
   396 
   397     }; //class BlockSearchPivotRule
   398 
   399 
   400     // Implementation of the Candidate List pivot rule
   401     class CandidateListPivotRule
   402     {
   403     private:
   404 
   405       // References to the NetworkSimplex class
   406       const IntVector  &_source;
   407       const IntVector  &_target;
   408       const CostVector &_cost;
   409       const IntVector  &_state;
   410       const CostVector &_pi;
   411       int &_in_arc;
   412       int _search_arc_num;
   413 
   414       // Pivot rule data
   415       IntVector _candidates;
   416       int _list_length, _minor_limit;
   417       int _curr_length, _minor_count;
   418       int _next_arc;
   419 
   420     public:
   421 
   422       /// Constructor
   423       CandidateListPivotRule(NetworkSimplex &ns) :
   424         _source(ns._source), _target(ns._target),
   425         _cost(ns._cost), _state(ns._state), _pi(ns._pi),
   426         _in_arc(ns.in_arc), _search_arc_num(ns._search_arc_num),
   427         _next_arc(0)
   428       {
   429         // The main parameters of the pivot rule
   430         const double LIST_LENGTH_FACTOR = 0.25;
   431         const int MIN_LIST_LENGTH = 10;
   432         const double MINOR_LIMIT_FACTOR = 0.1;
   433         const int MIN_MINOR_LIMIT = 3;
   434 
   435         _list_length = std::max( int(LIST_LENGTH_FACTOR *
   436                                      std::sqrt(double(_search_arc_num))),
   437                                  MIN_LIST_LENGTH );
   438         _minor_limit = std::max( int(MINOR_LIMIT_FACTOR * _list_length),
   439                                  MIN_MINOR_LIMIT );
   440         _curr_length = _minor_count = 0;
   441         _candidates.resize(_list_length);
   442       }
   443 
   444       /// Find next entering arc
   445       bool findEnteringArc() {
   446         Cost min, c;
   447         int e;
   448         if (_curr_length > 0 && _minor_count < _minor_limit) {
   449           // Minor iteration: select the best eligible arc from the
   450           // current candidate list
   451           ++_minor_count;
   452           min = 0;
   453           for (int i = 0; i < _curr_length; ++i) {
   454             e = _candidates[i];
   455             c = _state[e] * (_cost[e] + _pi[_source[e]] - _pi[_target[e]]);
   456             if (c < min) {
   457               min = c;
   458               _in_arc = e;
   459             }
   460             else if (c >= 0) {
   461               _candidates[i--] = _candidates[--_curr_length];
   462             }
   463           }
   464           if (min < 0) return true;
   465         }
   466 
   467         // Major iteration: build a new candidate list
   468         min = 0;
   469         _curr_length = 0;
   470         for (e = _next_arc; e < _search_arc_num; ++e) {
   471           c = _state[e] * (_cost[e] + _pi[_source[e]] - _pi[_target[e]]);
   472           if (c < 0) {
   473             _candidates[_curr_length++] = e;
   474             if (c < min) {
   475               min = c;
   476               _in_arc = e;
   477             }
   478             if (_curr_length == _list_length) goto search_end;
   479           }
   480         }
   481         for (e = 0; e < _next_arc; ++e) {
   482           c = _state[e] * (_cost[e] + _pi[_source[e]] - _pi[_target[e]]);
   483           if (c < 0) {
   484             _candidates[_curr_length++] = e;
   485             if (c < min) {
   486               min = c;
   487               _in_arc = e;
   488             }
   489             if (_curr_length == _list_length) goto search_end;
   490           }
   491         }
   492         if (_curr_length == 0) return false;
   493       
   494       search_end:        
   495         _minor_count = 1;
   496         _next_arc = e;
   497         return true;
   498       }
   499 
   500     }; //class CandidateListPivotRule
   501 
   502 
   503     // Implementation of the Altering Candidate List pivot rule
   504     class AlteringListPivotRule
   505     {
   506     private:
   507 
   508       // References to the NetworkSimplex class
   509       const IntVector  &_source;
   510       const IntVector  &_target;
   511       const CostVector &_cost;
   512       const IntVector  &_state;
   513       const CostVector &_pi;
   514       int &_in_arc;
   515       int _search_arc_num;
   516 
   517       // Pivot rule data
   518       int _block_size, _head_length, _curr_length;
   519       int _next_arc;
   520       IntVector _candidates;
   521       CostVector _cand_cost;
   522 
   523       // Functor class to compare arcs during sort of the candidate list
   524       class SortFunc
   525       {
   526       private:
   527         const CostVector &_map;
   528       public:
   529         SortFunc(const CostVector &map) : _map(map) {}
   530         bool operator()(int left, int right) {
   531           return _map[left] > _map[right];
   532         }
   533       };
   534 
   535       SortFunc _sort_func;
   536 
   537     public:
   538 
   539       // Constructor
   540       AlteringListPivotRule(NetworkSimplex &ns) :
   541         _source(ns._source), _target(ns._target),
   542         _cost(ns._cost), _state(ns._state), _pi(ns._pi),
   543         _in_arc(ns.in_arc), _search_arc_num(ns._search_arc_num),
   544         _next_arc(0), _cand_cost(ns._search_arc_num), _sort_func(_cand_cost)
   545       {
   546         // The main parameters of the pivot rule
   547         const double BLOCK_SIZE_FACTOR = 1.0;
   548         const int MIN_BLOCK_SIZE = 10;
   549         const double HEAD_LENGTH_FACTOR = 0.1;
   550         const int MIN_HEAD_LENGTH = 3;
   551 
   552         _block_size = std::max( int(BLOCK_SIZE_FACTOR *
   553                                     std::sqrt(double(_search_arc_num))),
   554                                 MIN_BLOCK_SIZE );
   555         _head_length = std::max( int(HEAD_LENGTH_FACTOR * _block_size),
   556                                  MIN_HEAD_LENGTH );
   557         _candidates.resize(_head_length + _block_size);
   558         _curr_length = 0;
   559       }
   560 
   561       // Find next entering arc
   562       bool findEnteringArc() {
   563         // Check the current candidate list
   564         int e;
   565         for (int i = 0; i < _curr_length; ++i) {
   566           e = _candidates[i];
   567           _cand_cost[e] = _state[e] *
   568             (_cost[e] + _pi[_source[e]] - _pi[_target[e]]);
   569           if (_cand_cost[e] >= 0) {
   570             _candidates[i--] = _candidates[--_curr_length];
   571           }
   572         }
   573 
   574         // Extend the list
   575         int cnt = _block_size;
   576         int limit = _head_length;
   577 
   578         for (e = _next_arc; e < _search_arc_num; ++e) {
   579           _cand_cost[e] = _state[e] *
   580             (_cost[e] + _pi[_source[e]] - _pi[_target[e]]);
   581           if (_cand_cost[e] < 0) {
   582             _candidates[_curr_length++] = e;
   583           }
   584           if (--cnt == 0) {
   585             if (_curr_length > limit) goto search_end;
   586             limit = 0;
   587             cnt = _block_size;
   588           }
   589         }
   590         for (e = 0; e < _next_arc; ++e) {
   591           _cand_cost[e] = _state[e] *
   592             (_cost[e] + _pi[_source[e]] - _pi[_target[e]]);
   593           if (_cand_cost[e] < 0) {
   594             _candidates[_curr_length++] = e;
   595           }
   596           if (--cnt == 0) {
   597             if (_curr_length > limit) goto search_end;
   598             limit = 0;
   599             cnt = _block_size;
   600           }
   601         }
   602         if (_curr_length == 0) return false;
   603         
   604       search_end:
   605 
   606         // Make heap of the candidate list (approximating a partial sort)
   607         make_heap( _candidates.begin(), _candidates.begin() + _curr_length,
   608                    _sort_func );
   609 
   610         // Pop the first element of the heap
   611         _in_arc = _candidates[0];
   612         _next_arc = e;
   613         pop_heap( _candidates.begin(), _candidates.begin() + _curr_length,
   614                   _sort_func );
   615         _curr_length = std::min(_head_length, _curr_length - 1);
   616         return true;
   617       }
   618 
   619     }; //class AlteringListPivotRule
   620 
   621   public:
   622 
   623     /// \brief Constructor.
   624     ///
   625     /// The constructor of the class.
   626     ///
   627     /// \param graph The digraph the algorithm runs on.
   628     /// \param arc_mixing Indicate if the arcs have to be stored in a
   629     /// mixed order in the internal data structure. 
   630     /// In special cases, it could lead to better overall performance,
   631     /// but it is usually slower. Therefore it is disabled by default.
   632     NetworkSimplex(const GR& graph, bool arc_mixing = false) :
   633       _graph(graph), _node_id(graph), _arc_id(graph),
   634       INF(std::numeric_limits<Value>::has_infinity ?
   635           std::numeric_limits<Value>::infinity() :
   636           std::numeric_limits<Value>::max())
   637     {
   638       // Check the value types
   639       LEMON_ASSERT(std::numeric_limits<Value>::is_signed,
   640         "The flow type of NetworkSimplex must be signed");
   641       LEMON_ASSERT(std::numeric_limits<Cost>::is_signed,
   642         "The cost type of NetworkSimplex must be signed");
   643         
   644       // Resize vectors
   645       _node_num = countNodes(_graph);
   646       _arc_num = countArcs(_graph);
   647       int all_node_num = _node_num + 1;
   648       int max_arc_num = _arc_num + 2 * _node_num;
   649 
   650       _source.resize(max_arc_num);
   651       _target.resize(max_arc_num);
   652 
   653       _lower.resize(_arc_num);
   654       _upper.resize(_arc_num);
   655       _cap.resize(max_arc_num);
   656       _cost.resize(max_arc_num);
   657       _supply.resize(all_node_num);
   658       _flow.resize(max_arc_num);
   659       _pi.resize(all_node_num);
   660 
   661       _parent.resize(all_node_num);
   662       _pred.resize(all_node_num);
   663       _forward.resize(all_node_num);
   664       _thread.resize(all_node_num);
   665       _rev_thread.resize(all_node_num);
   666       _succ_num.resize(all_node_num);
   667       _last_succ.resize(all_node_num);
   668       _state.resize(max_arc_num);
   669 
   670       // Copy the graph
   671       int i = 0;
   672       for (NodeIt n(_graph); n != INVALID; ++n, ++i) {
   673         _node_id[n] = i;
   674       }
   675       if (arc_mixing) {
   676         // Store the arcs in a mixed order
   677         int k = std::max(int(std::sqrt(double(_arc_num))), 10);
   678         int i = 0, j = 0;
   679         for (ArcIt a(_graph); a != INVALID; ++a) {
   680           _arc_id[a] = i;
   681           _source[i] = _node_id[_graph.source(a)];
   682           _target[i] = _node_id[_graph.target(a)];
   683           if ((i += k) >= _arc_num) i = ++j;
   684         }
   685       } else {
   686         // Store the arcs in the original order
   687         int i = 0;
   688         for (ArcIt a(_graph); a != INVALID; ++a, ++i) {
   689           _arc_id[a] = i;
   690           _source[i] = _node_id[_graph.source(a)];
   691           _target[i] = _node_id[_graph.target(a)];
   692         }
   693       }
   694       
   695       // Reset parameters
   696       reset();
   697     }
   698 
   699     /// \name Parameters
   700     /// The parameters of the algorithm can be specified using these
   701     /// functions.
   702 
   703     /// @{
   704 
   705     /// \brief Set the lower bounds on the arcs.
   706     ///
   707     /// This function sets the lower bounds on the arcs.
   708     /// If it is not used before calling \ref run(), the lower bounds
   709     /// will be set to zero on all arcs.
   710     ///
   711     /// \param map An arc map storing the lower bounds.
   712     /// Its \c Value type must be convertible to the \c Value type
   713     /// of the algorithm.
   714     ///
   715     /// \return <tt>(*this)</tt>
   716     template <typename LowerMap>
   717     NetworkSimplex& lowerMap(const LowerMap& map) {
   718       _have_lower = true;
   719       for (ArcIt a(_graph); a != INVALID; ++a) {
   720         _lower[_arc_id[a]] = map[a];
   721       }
   722       return *this;
   723     }
   724 
   725     /// \brief Set the upper bounds (capacities) on the arcs.
   726     ///
   727     /// This function sets the upper bounds (capacities) on the arcs.
   728     /// If it is not used before calling \ref run(), the upper bounds
   729     /// will be set to \ref INF on all arcs (i.e. the flow value will be
   730     /// unbounded from above on each arc).
   731     ///
   732     /// \param map An arc map storing the upper bounds.
   733     /// Its \c Value type must be convertible to the \c Value type
   734     /// of the algorithm.
   735     ///
   736     /// \return <tt>(*this)</tt>
   737     template<typename UpperMap>
   738     NetworkSimplex& upperMap(const UpperMap& map) {
   739       for (ArcIt a(_graph); a != INVALID; ++a) {
   740         _upper[_arc_id[a]] = map[a];
   741       }
   742       return *this;
   743     }
   744 
   745     /// \brief Set the costs of the arcs.
   746     ///
   747     /// This function sets the costs of the arcs.
   748     /// If it is not used before calling \ref run(), the costs
   749     /// will be set to \c 1 on all arcs.
   750     ///
   751     /// \param map An arc map storing the costs.
   752     /// Its \c Value type must be convertible to the \c Cost type
   753     /// of the algorithm.
   754     ///
   755     /// \return <tt>(*this)</tt>
   756     template<typename CostMap>
   757     NetworkSimplex& costMap(const CostMap& map) {
   758       for (ArcIt a(_graph); a != INVALID; ++a) {
   759         _cost[_arc_id[a]] = map[a];
   760       }
   761       return *this;
   762     }
   763 
   764     /// \brief Set the supply values of the nodes.
   765     ///
   766     /// This function sets the supply values of the nodes.
   767     /// If neither this function nor \ref stSupply() is used before
   768     /// calling \ref run(), the supply of each node will be set to zero.
   769     ///
   770     /// \param map A node map storing the supply values.
   771     /// Its \c Value type must be convertible to the \c Value type
   772     /// of the algorithm.
   773     ///
   774     /// \return <tt>(*this)</tt>
   775     template<typename SupplyMap>
   776     NetworkSimplex& supplyMap(const SupplyMap& map) {
   777       for (NodeIt n(_graph); n != INVALID; ++n) {
   778         _supply[_node_id[n]] = map[n];
   779       }
   780       return *this;
   781     }
   782 
   783     /// \brief Set single source and target nodes and a supply value.
   784     ///
   785     /// This function sets a single source node and a single target node
   786     /// and the required flow value.
   787     /// If neither this function nor \ref supplyMap() is used before
   788     /// calling \ref run(), the supply of each node will be set to zero.
   789     ///
   790     /// Using this function has the same effect as using \ref supplyMap()
   791     /// with such a map in which \c k is assigned to \c s, \c -k is
   792     /// assigned to \c t and all other nodes have zero supply value.
   793     ///
   794     /// \param s The source node.
   795     /// \param t The target node.
   796     /// \param k The required amount of flow from node \c s to node \c t
   797     /// (i.e. the supply of \c s and the demand of \c t).
   798     ///
   799     /// \return <tt>(*this)</tt>
   800     NetworkSimplex& stSupply(const Node& s, const Node& t, Value k) {
   801       for (int i = 0; i != _node_num; ++i) {
   802         _supply[i] = 0;
   803       }
   804       _supply[_node_id[s]] =  k;
   805       _supply[_node_id[t]] = -k;
   806       return *this;
   807     }
   808     
   809     /// \brief Set the type of the supply constraints.
   810     ///
   811     /// This function sets the type of the supply/demand constraints.
   812     /// If it is not used before calling \ref run(), the \ref GEQ supply
   813     /// type will be used.
   814     ///
   815     /// For more information see \ref SupplyType.
   816     ///
   817     /// \return <tt>(*this)</tt>
   818     NetworkSimplex& supplyType(SupplyType supply_type) {
   819       _stype = supply_type;
   820       return *this;
   821     }
   822 
   823     /// @}
   824 
   825     /// \name Execution Control
   826     /// The algorithm can be executed using \ref run().
   827 
   828     /// @{
   829 
   830     /// \brief Run the algorithm.
   831     ///
   832     /// This function runs the algorithm.
   833     /// The paramters can be specified using functions \ref lowerMap(),
   834     /// \ref upperMap(), \ref costMap(), \ref supplyMap(), \ref stSupply(), 
   835     /// \ref supplyType().
   836     /// For example,
   837     /// \code
   838     ///   NetworkSimplex<ListDigraph> ns(graph);
   839     ///   ns.lowerMap(lower).upperMap(upper).costMap(cost)
   840     ///     .supplyMap(sup).run();
   841     /// \endcode
   842     ///
   843     /// This function can be called more than once. All the parameters
   844     /// that have been given are kept for the next call, unless
   845     /// \ref reset() is called, thus only the modified parameters
   846     /// have to be set again. See \ref reset() for examples.
   847     /// However the underlying digraph must not be modified after this
   848     /// class have been constructed, since it copies and extends the graph.
   849     ///
   850     /// \param pivot_rule The pivot rule that will be used during the
   851     /// algorithm. For more information see \ref PivotRule.
   852     ///
   853     /// \return \c INFEASIBLE if no feasible flow exists,
   854     /// \n \c OPTIMAL if the problem has optimal solution
   855     /// (i.e. it is feasible and bounded), and the algorithm has found
   856     /// optimal flow and node potentials (primal and dual solutions),
   857     /// \n \c UNBOUNDED if the objective function of the problem is
   858     /// unbounded, i.e. there is a directed cycle having negative total
   859     /// cost and infinite upper bound.
   860     ///
   861     /// \see ProblemType, PivotRule
   862     ProblemType run(PivotRule pivot_rule = BLOCK_SEARCH) {
   863       if (!init()) return INFEASIBLE;
   864       return start(pivot_rule);
   865     }
   866 
   867     /// \brief Reset all the parameters that have been given before.
   868     ///
   869     /// This function resets all the paramaters that have been given
   870     /// before using functions \ref lowerMap(), \ref upperMap(),
   871     /// \ref costMap(), \ref supplyMap(), \ref stSupply(), \ref supplyType().
   872     ///
   873     /// It is useful for multiple run() calls. If this function is not
   874     /// used, all the parameters given before are kept for the next
   875     /// \ref run() call.
   876     /// However the underlying digraph must not be modified after this
   877     /// class have been constructed, since it copies and extends the graph.
   878     ///
   879     /// For example,
   880     /// \code
   881     ///   NetworkSimplex<ListDigraph> ns(graph);
   882     ///
   883     ///   // First run
   884     ///   ns.lowerMap(lower).upperMap(upper).costMap(cost)
   885     ///     .supplyMap(sup).run();
   886     ///
   887     ///   // Run again with modified cost map (reset() is not called,
   888     ///   // so only the cost map have to be set again)
   889     ///   cost[e] += 100;
   890     ///   ns.costMap(cost).run();
   891     ///
   892     ///   // Run again from scratch using reset()
   893     ///   // (the lower bounds will be set to zero on all arcs)
   894     ///   ns.reset();
   895     ///   ns.upperMap(capacity).costMap(cost)
   896     ///     .supplyMap(sup).run();
   897     /// \endcode
   898     ///
   899     /// \return <tt>(*this)</tt>
   900     NetworkSimplex& reset() {
   901       for (int i = 0; i != _node_num; ++i) {
   902         _supply[i] = 0;
   903       }
   904       for (int i = 0; i != _arc_num; ++i) {
   905         _lower[i] = 0;
   906         _upper[i] = INF;
   907         _cost[i] = 1;
   908       }
   909       _have_lower = false;
   910       _stype = GEQ;
   911       return *this;
   912     }
   913 
   914     /// @}
   915 
   916     /// \name Query Functions
   917     /// The results of the algorithm can be obtained using these
   918     /// functions.\n
   919     /// The \ref run() function must be called before using them.
   920 
   921     /// @{
   922 
   923     /// \brief Return the total cost of the found flow.
   924     ///
   925     /// This function returns the total cost of the found flow.
   926     /// Its complexity is O(e).
   927     ///
   928     /// \note The return type of the function can be specified as a
   929     /// template parameter. For example,
   930     /// \code
   931     ///   ns.totalCost<double>();
   932     /// \endcode
   933     /// It is useful if the total cost cannot be stored in the \c Cost
   934     /// type of the algorithm, which is the default return type of the
   935     /// function.
   936     ///
   937     /// \pre \ref run() must be called before using this function.
   938     template <typename Number>
   939     Number totalCost() const {
   940       Number c = 0;
   941       for (ArcIt a(_graph); a != INVALID; ++a) {
   942         int i = _arc_id[a];
   943         c += Number(_flow[i]) * Number(_cost[i]);
   944       }
   945       return c;
   946     }
   947 
   948 #ifndef DOXYGEN
   949     Cost totalCost() const {
   950       return totalCost<Cost>();
   951     }
   952 #endif
   953 
   954     /// \brief Return the flow on the given arc.
   955     ///
   956     /// This function returns the flow on the given arc.
   957     ///
   958     /// \pre \ref run() must be called before using this function.
   959     Value flow(const Arc& a) const {
   960       return _flow[_arc_id[a]];
   961     }
   962 
   963     /// \brief Return the flow map (the primal solution).
   964     ///
   965     /// This function copies the flow value on each arc into the given
   966     /// map. The \c Value type of the algorithm must be convertible to
   967     /// the \c Value type of the map.
   968     ///
   969     /// \pre \ref run() must be called before using this function.
   970     template <typename FlowMap>
   971     void flowMap(FlowMap &map) const {
   972       for (ArcIt a(_graph); a != INVALID; ++a) {
   973         map.set(a, _flow[_arc_id[a]]);
   974       }
   975     }
   976 
   977     /// \brief Return the potential (dual value) of the given node.
   978     ///
   979     /// This function returns the potential (dual value) of the
   980     /// given node.
   981     ///
   982     /// \pre \ref run() must be called before using this function.
   983     Cost potential(const Node& n) const {
   984       return _pi[_node_id[n]];
   985     }
   986 
   987     /// \brief Return the potential map (the dual solution).
   988     ///
   989     /// This function copies the potential (dual value) of each node
   990     /// into the given map.
   991     /// The \c Cost type of the algorithm must be convertible to the
   992     /// \c Value type of the map.
   993     ///
   994     /// \pre \ref run() must be called before using this function.
   995     template <typename PotentialMap>
   996     void potentialMap(PotentialMap &map) const {
   997       for (NodeIt n(_graph); n != INVALID; ++n) {
   998         map.set(n, _pi[_node_id[n]]);
   999       }
  1000     }
  1001 
  1002     /// @}
  1003 
  1004   private:
  1005 
  1006     // Initialize internal data structures
  1007     bool init() {
  1008       if (_node_num == 0) return false;
  1009 
  1010       // Check the sum of supply values
  1011       _sum_supply = 0;
  1012       for (int i = 0; i != _node_num; ++i) {
  1013         _sum_supply += _supply[i];
  1014       }
  1015       if ( !((_stype == GEQ && _sum_supply <= 0) ||
  1016              (_stype == LEQ && _sum_supply >= 0)) ) return false;
  1017 
  1018       // Remove non-zero lower bounds
  1019       if (_have_lower) {
  1020         for (int i = 0; i != _arc_num; ++i) {
  1021           Value c = _lower[i];
  1022           if (c >= 0) {
  1023             _cap[i] = _upper[i] < INF ? _upper[i] - c : INF;
  1024           } else {
  1025             _cap[i] = _upper[i] < INF + c ? _upper[i] - c : INF;
  1026           }
  1027           _supply[_source[i]] -= c;
  1028           _supply[_target[i]] += c;
  1029         }
  1030       } else {
  1031         for (int i = 0; i != _arc_num; ++i) {
  1032           _cap[i] = _upper[i];
  1033         }
  1034       }
  1035 
  1036       // Initialize artifical cost
  1037       Cost ART_COST;
  1038       if (std::numeric_limits<Cost>::is_exact) {
  1039         ART_COST = std::numeric_limits<Cost>::max() / 2 + 1;
  1040       } else {
  1041         ART_COST = std::numeric_limits<Cost>::min();
  1042         for (int i = 0; i != _arc_num; ++i) {
  1043           if (_cost[i] > ART_COST) ART_COST = _cost[i];
  1044         }
  1045         ART_COST = (ART_COST + 1) * _node_num;
  1046       }
  1047 
  1048       // Initialize arc maps
  1049       for (int i = 0; i != _arc_num; ++i) {
  1050         _flow[i] = 0;
  1051         _state[i] = STATE_LOWER;
  1052       }
  1053       
  1054       // Set data for the artificial root node
  1055       _root = _node_num;
  1056       _parent[_root] = -1;
  1057       _pred[_root] = -1;
  1058       _thread[_root] = 0;
  1059       _rev_thread[0] = _root;
  1060       _succ_num[_root] = _node_num + 1;
  1061       _last_succ[_root] = _root - 1;
  1062       _supply[_root] = -_sum_supply;
  1063       _pi[_root] = 0;
  1064 
  1065       // Add artificial arcs and initialize the spanning tree data structure
  1066       if (_sum_supply == 0) {
  1067         // EQ supply constraints
  1068         _search_arc_num = _arc_num;
  1069         _all_arc_num = _arc_num + _node_num;
  1070         for (int u = 0, e = _arc_num; u != _node_num; ++u, ++e) {
  1071           _parent[u] = _root;
  1072           _pred[u] = e;
  1073           _thread[u] = u + 1;
  1074           _rev_thread[u + 1] = u;
  1075           _succ_num[u] = 1;
  1076           _last_succ[u] = u;
  1077           _cap[e] = INF;
  1078           _state[e] = STATE_TREE;
  1079           if (_supply[u] >= 0) {
  1080             _forward[u] = true;
  1081             _pi[u] = 0;
  1082             _source[e] = u;
  1083             _target[e] = _root;
  1084             _flow[e] = _supply[u];
  1085             _cost[e] = 0;
  1086           } else {
  1087             _forward[u] = false;
  1088             _pi[u] = ART_COST;
  1089             _source[e] = _root;
  1090             _target[e] = u;
  1091             _flow[e] = -_supply[u];
  1092             _cost[e] = ART_COST;
  1093           }
  1094         }
  1095       }
  1096       else if (_sum_supply > 0) {
  1097         // LEQ supply constraints
  1098         _search_arc_num = _arc_num + _node_num;
  1099         int f = _arc_num + _node_num;
  1100         for (int u = 0, e = _arc_num; u != _node_num; ++u, ++e) {
  1101           _parent[u] = _root;
  1102           _thread[u] = u + 1;
  1103           _rev_thread[u + 1] = u;
  1104           _succ_num[u] = 1;
  1105           _last_succ[u] = u;
  1106           if (_supply[u] >= 0) {
  1107             _forward[u] = true;
  1108             _pi[u] = 0;
  1109             _pred[u] = e;
  1110             _source[e] = u;
  1111             _target[e] = _root;
  1112             _cap[e] = INF;
  1113             _flow[e] = _supply[u];
  1114             _cost[e] = 0;
  1115             _state[e] = STATE_TREE;
  1116           } else {
  1117             _forward[u] = false;
  1118             _pi[u] = ART_COST;
  1119             _pred[u] = f;
  1120             _source[f] = _root;
  1121             _target[f] = u;
  1122             _cap[f] = INF;
  1123             _flow[f] = -_supply[u];
  1124             _cost[f] = ART_COST;
  1125             _state[f] = STATE_TREE;
  1126             _source[e] = u;
  1127             _target[e] = _root;
  1128             _cap[e] = INF;
  1129             _flow[e] = 0;
  1130             _cost[e] = 0;
  1131             _state[e] = STATE_LOWER;
  1132             ++f;
  1133           }
  1134         }
  1135         _all_arc_num = f;
  1136       }
  1137       else {
  1138         // GEQ supply constraints
  1139         _search_arc_num = _arc_num + _node_num;
  1140         int f = _arc_num + _node_num;
  1141         for (int u = 0, e = _arc_num; u != _node_num; ++u, ++e) {
  1142           _parent[u] = _root;
  1143           _thread[u] = u + 1;
  1144           _rev_thread[u + 1] = u;
  1145           _succ_num[u] = 1;
  1146           _last_succ[u] = u;
  1147           if (_supply[u] <= 0) {
  1148             _forward[u] = false;
  1149             _pi[u] = 0;
  1150             _pred[u] = e;
  1151             _source[e] = _root;
  1152             _target[e] = u;
  1153             _cap[e] = INF;
  1154             _flow[e] = -_supply[u];
  1155             _cost[e] = 0;
  1156             _state[e] = STATE_TREE;
  1157           } else {
  1158             _forward[u] = true;
  1159             _pi[u] = -ART_COST;
  1160             _pred[u] = f;
  1161             _source[f] = u;
  1162             _target[f] = _root;
  1163             _cap[f] = INF;
  1164             _flow[f] = _supply[u];
  1165             _state[f] = STATE_TREE;
  1166             _cost[f] = ART_COST;
  1167             _source[e] = _root;
  1168             _target[e] = u;
  1169             _cap[e] = INF;
  1170             _flow[e] = 0;
  1171             _cost[e] = 0;
  1172             _state[e] = STATE_LOWER;
  1173             ++f;
  1174           }
  1175         }
  1176         _all_arc_num = f;
  1177       }
  1178 
  1179       return true;
  1180     }
  1181 
  1182     // Find the join node
  1183     void findJoinNode() {
  1184       int u = _source[in_arc];
  1185       int v = _target[in_arc];
  1186       while (u != v) {
  1187         if (_succ_num[u] < _succ_num[v]) {
  1188           u = _parent[u];
  1189         } else {
  1190           v = _parent[v];
  1191         }
  1192       }
  1193       join = u;
  1194     }
  1195 
  1196     // Find the leaving arc of the cycle and returns true if the
  1197     // leaving arc is not the same as the entering arc
  1198     bool findLeavingArc() {
  1199       // Initialize first and second nodes according to the direction
  1200       // of the cycle
  1201       if (_state[in_arc] == STATE_LOWER) {
  1202         first  = _source[in_arc];
  1203         second = _target[in_arc];
  1204       } else {
  1205         first  = _target[in_arc];
  1206         second = _source[in_arc];
  1207       }
  1208       delta = _cap[in_arc];
  1209       int result = 0;
  1210       Value d;
  1211       int e;
  1212 
  1213       // Search the cycle along the path form the first node to the root
  1214       for (int u = first; u != join; u = _parent[u]) {
  1215         e = _pred[u];
  1216         d = _forward[u] ?
  1217           _flow[e] : (_cap[e] == INF ? INF : _cap[e] - _flow[e]);
  1218         if (d < delta) {
  1219           delta = d;
  1220           u_out = u;
  1221           result = 1;
  1222         }
  1223       }
  1224       // Search the cycle along the path form the second node to the root
  1225       for (int u = second; u != join; u = _parent[u]) {
  1226         e = _pred[u];
  1227         d = _forward[u] ? 
  1228           (_cap[e] == INF ? INF : _cap[e] - _flow[e]) : _flow[e];
  1229         if (d <= delta) {
  1230           delta = d;
  1231           u_out = u;
  1232           result = 2;
  1233         }
  1234       }
  1235 
  1236       if (result == 1) {
  1237         u_in = first;
  1238         v_in = second;
  1239       } else {
  1240         u_in = second;
  1241         v_in = first;
  1242       }
  1243       return result != 0;
  1244     }
  1245 
  1246     // Change _flow and _state vectors
  1247     void changeFlow(bool change) {
  1248       // Augment along the cycle
  1249       if (delta > 0) {
  1250         Value val = _state[in_arc] * delta;
  1251         _flow[in_arc] += val;
  1252         for (int u = _source[in_arc]; u != join; u = _parent[u]) {
  1253           _flow[_pred[u]] += _forward[u] ? -val : val;
  1254         }
  1255         for (int u = _target[in_arc]; u != join; u = _parent[u]) {
  1256           _flow[_pred[u]] += _forward[u] ? val : -val;
  1257         }
  1258       }
  1259       // Update the state of the entering and leaving arcs
  1260       if (change) {
  1261         _state[in_arc] = STATE_TREE;
  1262         _state[_pred[u_out]] =
  1263           (_flow[_pred[u_out]] == 0) ? STATE_LOWER : STATE_UPPER;
  1264       } else {
  1265         _state[in_arc] = -_state[in_arc];
  1266       }
  1267     }
  1268 
  1269     // Update the tree structure
  1270     void updateTreeStructure() {
  1271       int u, w;
  1272       int old_rev_thread = _rev_thread[u_out];
  1273       int old_succ_num = _succ_num[u_out];
  1274       int old_last_succ = _last_succ[u_out];
  1275       v_out = _parent[u_out];
  1276 
  1277       u = _last_succ[u_in];  // the last successor of u_in
  1278       right = _thread[u];    // the node after it
  1279 
  1280       // Handle the case when old_rev_thread equals to v_in
  1281       // (it also means that join and v_out coincide)
  1282       if (old_rev_thread == v_in) {
  1283         last = _thread[_last_succ[u_out]];
  1284       } else {
  1285         last = _thread[v_in];
  1286       }
  1287 
  1288       // Update _thread and _parent along the stem nodes (i.e. the nodes
  1289       // between u_in and u_out, whose parent have to be changed)
  1290       _thread[v_in] = stem = u_in;
  1291       _dirty_revs.clear();
  1292       _dirty_revs.push_back(v_in);
  1293       par_stem = v_in;
  1294       while (stem != u_out) {
  1295         // Insert the next stem node into the thread list
  1296         new_stem = _parent[stem];
  1297         _thread[u] = new_stem;
  1298         _dirty_revs.push_back(u);
  1299 
  1300         // Remove the subtree of stem from the thread list
  1301         w = _rev_thread[stem];
  1302         _thread[w] = right;
  1303         _rev_thread[right] = w;
  1304 
  1305         // Change the parent node and shift stem nodes
  1306         _parent[stem] = par_stem;
  1307         par_stem = stem;
  1308         stem = new_stem;
  1309 
  1310         // Update u and right
  1311         u = _last_succ[stem] == _last_succ[par_stem] ?
  1312           _rev_thread[par_stem] : _last_succ[stem];
  1313         right = _thread[u];
  1314       }
  1315       _parent[u_out] = par_stem;
  1316       _thread[u] = last;
  1317       _rev_thread[last] = u;
  1318       _last_succ[u_out] = u;
  1319 
  1320       // Remove the subtree of u_out from the thread list except for
  1321       // the case when old_rev_thread equals to v_in
  1322       // (it also means that join and v_out coincide)
  1323       if (old_rev_thread != v_in) {
  1324         _thread[old_rev_thread] = right;
  1325         _rev_thread[right] = old_rev_thread;
  1326       }
  1327 
  1328       // Update _rev_thread using the new _thread values
  1329       for (int i = 0; i < int(_dirty_revs.size()); ++i) {
  1330         u = _dirty_revs[i];
  1331         _rev_thread[_thread[u]] = u;
  1332       }
  1333 
  1334       // Update _pred, _forward, _last_succ and _succ_num for the
  1335       // stem nodes from u_out to u_in
  1336       int tmp_sc = 0, tmp_ls = _last_succ[u_out];
  1337       u = u_out;
  1338       while (u != u_in) {
  1339         w = _parent[u];
  1340         _pred[u] = _pred[w];
  1341         _forward[u] = !_forward[w];
  1342         tmp_sc += _succ_num[u] - _succ_num[w];
  1343         _succ_num[u] = tmp_sc;
  1344         _last_succ[w] = tmp_ls;
  1345         u = w;
  1346       }
  1347       _pred[u_in] = in_arc;
  1348       _forward[u_in] = (u_in == _source[in_arc]);
  1349       _succ_num[u_in] = old_succ_num;
  1350 
  1351       // Set limits for updating _last_succ form v_in and v_out
  1352       // towards the root
  1353       int up_limit_in = -1;
  1354       int up_limit_out = -1;
  1355       if (_last_succ[join] == v_in) {
  1356         up_limit_out = join;
  1357       } else {
  1358         up_limit_in = join;
  1359       }
  1360 
  1361       // Update _last_succ from v_in towards the root
  1362       for (u = v_in; u != up_limit_in && _last_succ[u] == v_in;
  1363            u = _parent[u]) {
  1364         _last_succ[u] = _last_succ[u_out];
  1365       }
  1366       // Update _last_succ from v_out towards the root
  1367       if (join != old_rev_thread && v_in != old_rev_thread) {
  1368         for (u = v_out; u != up_limit_out && _last_succ[u] == old_last_succ;
  1369              u = _parent[u]) {
  1370           _last_succ[u] = old_rev_thread;
  1371         }
  1372       } else {
  1373         for (u = v_out; u != up_limit_out && _last_succ[u] == old_last_succ;
  1374              u = _parent[u]) {
  1375           _last_succ[u] = _last_succ[u_out];
  1376         }
  1377       }
  1378 
  1379       // Update _succ_num from v_in to join
  1380       for (u = v_in; u != join; u = _parent[u]) {
  1381         _succ_num[u] += old_succ_num;
  1382       }
  1383       // Update _succ_num from v_out to join
  1384       for (u = v_out; u != join; u = _parent[u]) {
  1385         _succ_num[u] -= old_succ_num;
  1386       }
  1387     }
  1388 
  1389     // Update potentials
  1390     void updatePotential() {
  1391       Cost sigma = _forward[u_in] ?
  1392         _pi[v_in] - _pi[u_in] - _cost[_pred[u_in]] :
  1393         _pi[v_in] - _pi[u_in] + _cost[_pred[u_in]];
  1394       // Update potentials in the subtree, which has been moved
  1395       int end = _thread[_last_succ[u_in]];
  1396       for (int u = u_in; u != end; u = _thread[u]) {
  1397         _pi[u] += sigma;
  1398       }
  1399     }
  1400 
  1401     // Execute the algorithm
  1402     ProblemType start(PivotRule pivot_rule) {
  1403       // Select the pivot rule implementation
  1404       switch (pivot_rule) {
  1405         case FIRST_ELIGIBLE:
  1406           return start<FirstEligiblePivotRule>();
  1407         case BEST_ELIGIBLE:
  1408           return start<BestEligiblePivotRule>();
  1409         case BLOCK_SEARCH:
  1410           return start<BlockSearchPivotRule>();
  1411         case CANDIDATE_LIST:
  1412           return start<CandidateListPivotRule>();
  1413         case ALTERING_LIST:
  1414           return start<AlteringListPivotRule>();
  1415       }
  1416       return INFEASIBLE; // avoid warning
  1417     }
  1418 
  1419     template <typename PivotRuleImpl>
  1420     ProblemType start() {
  1421       PivotRuleImpl pivot(*this);
  1422 
  1423       // Execute the Network Simplex algorithm
  1424       while (pivot.findEnteringArc()) {
  1425         findJoinNode();
  1426         bool change = findLeavingArc();
  1427         if (delta >= INF) return UNBOUNDED;
  1428         changeFlow(change);
  1429         if (change) {
  1430           updateTreeStructure();
  1431           updatePotential();
  1432         }
  1433       }
  1434       
  1435       // Check feasibility
  1436       for (int e = _search_arc_num; e != _all_arc_num; ++e) {
  1437         if (_flow[e] != 0) return INFEASIBLE;
  1438       }
  1439 
  1440       // Transform the solution and the supply map to the original form
  1441       if (_have_lower) {
  1442         for (int i = 0; i != _arc_num; ++i) {
  1443           Value c = _lower[i];
  1444           if (c != 0) {
  1445             _flow[i] += c;
  1446             _supply[_source[i]] += c;
  1447             _supply[_target[i]] -= c;
  1448           }
  1449         }
  1450       }
  1451       
  1452       // Shift potentials to meet the requirements of the GEQ/LEQ type
  1453       // optimality conditions
  1454       if (_sum_supply == 0) {
  1455         if (_stype == GEQ) {
  1456           Cost max_pot = std::numeric_limits<Cost>::min();
  1457           for (int i = 0; i != _node_num; ++i) {
  1458             if (_pi[i] > max_pot) max_pot = _pi[i];
  1459           }
  1460           if (max_pot > 0) {
  1461             for (int i = 0; i != _node_num; ++i)
  1462               _pi[i] -= max_pot;
  1463           }
  1464         } else {
  1465           Cost min_pot = std::numeric_limits<Cost>::max();
  1466           for (int i = 0; i != _node_num; ++i) {
  1467             if (_pi[i] < min_pot) min_pot = _pi[i];
  1468           }
  1469           if (min_pot < 0) {
  1470             for (int i = 0; i != _node_num; ++i)
  1471               _pi[i] -= min_pot;
  1472           }
  1473         }
  1474       }
  1475 
  1476       return OPTIMAL;
  1477     }
  1478 
  1479   }; //class NetworkSimplex
  1480 
  1481   ///@}
  1482 
  1483 } //namespace lemon
  1484 
  1485 #endif //LEMON_NETWORK_SIMPLEX_H