lemon/network_simplex.h
author Balazs Dezso <deba@inf.elte.hu>
Thu, 04 Mar 2010 15:20:59 +0100
changeset 951 41d7ac528c3a
parent 690 f3792d5bb294
child 774 cab85bd7859b
child 776 be48a648d28f
child 976 5205145fabf6
permissions -rw-r--r--
Uniforming primal scale to 2 (#314)
     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   /// 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, the \c LEQ type can be
   106     /// selected using \ref supplyType().
   107     /// The equality form is a special case of both supply types.
   108     enum SupplyType {
   109       /// This option means that there are <em>"greater or equal"</em>
   110       /// supply/demand constraints in the definition of the problem.
   111       GEQ,
   112       /// This option means that there are <em>"less or equal"</em>
   113       /// supply/demand constraints in the definition of the problem.
   114       LEQ
   115     };
   116     
   117     /// \brief Constants for selecting the pivot rule.
   118     ///
   119     /// Enum type containing constants for selecting the pivot rule for
   120     /// the \ref run() function.
   121     ///
   122     /// \ref NetworkSimplex provides five different pivot rule
   123     /// implementations that significantly affect the running time
   124     /// of the algorithm.
   125     /// By default \ref BLOCK_SEARCH "Block Search" is used, which
   126     /// proved to be the most efficient and the most robust on various
   127     /// test inputs according to our benchmark tests.
   128     /// However another pivot rule can be selected using the \ref run()
   129     /// function with the proper parameter.
   130     enum PivotRule {
   131 
   132       /// The First Eligible pivot rule.
   133       /// The next eligible arc is selected in a wraparound fashion
   134       /// in every iteration.
   135       FIRST_ELIGIBLE,
   136 
   137       /// The Best Eligible pivot rule.
   138       /// The best eligible arc is selected in every iteration.
   139       BEST_ELIGIBLE,
   140 
   141       /// The Block Search pivot rule.
   142       /// A specified number of arcs are examined in every iteration
   143       /// in a wraparound fashion and the best eligible arc is selected
   144       /// from this block.
   145       BLOCK_SEARCH,
   146 
   147       /// The Candidate List pivot rule.
   148       /// In a major iteration a candidate list is built from eligible arcs
   149       /// in a wraparound fashion and in the following minor iterations
   150       /// the best eligible arc is selected from this list.
   151       CANDIDATE_LIST,
   152 
   153       /// The Altering Candidate List pivot rule.
   154       /// It is a modified version of the Candidate List method.
   155       /// It keeps only the several best eligible arcs from the former
   156       /// candidate list and extends this list in every iteration.
   157       ALTERING_LIST
   158     };
   159     
   160   private:
   161 
   162     TEMPLATE_DIGRAPH_TYPEDEFS(GR);
   163 
   164     typedef std::vector<Arc> ArcVector;
   165     typedef std::vector<Node> NodeVector;
   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, min_arc = _next_arc;
   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             min_arc = e;
   373           }
   374           if (--cnt == 0) {
   375             if (min < 0) break;
   376             cnt = _block_size;
   377           }
   378         }
   379         if (min == 0 || cnt > 0) {
   380           for (e = 0; e < _next_arc; ++e) {
   381             c = _state[e] * (_cost[e] + _pi[_source[e]] - _pi[_target[e]]);
   382             if (c < min) {
   383               min = c;
   384               min_arc = e;
   385             }
   386             if (--cnt == 0) {
   387               if (min < 0) break;
   388               cnt = _block_size;
   389             }
   390           }
   391         }
   392         if (min >= 0) return false;
   393         _in_arc = min_arc;
   394         _next_arc = e;
   395         return true;
   396       }
   397 
   398     }; //class BlockSearchPivotRule
   399 
   400 
   401     // Implementation of the Candidate List pivot rule
   402     class CandidateListPivotRule
   403     {
   404     private:
   405 
   406       // References to the NetworkSimplex class
   407       const IntVector  &_source;
   408       const IntVector  &_target;
   409       const CostVector &_cost;
   410       const IntVector  &_state;
   411       const CostVector &_pi;
   412       int &_in_arc;
   413       int _search_arc_num;
   414 
   415       // Pivot rule data
   416       IntVector _candidates;
   417       int _list_length, _minor_limit;
   418       int _curr_length, _minor_count;
   419       int _next_arc;
   420 
   421     public:
   422 
   423       /// Constructor
   424       CandidateListPivotRule(NetworkSimplex &ns) :
   425         _source(ns._source), _target(ns._target),
   426         _cost(ns._cost), _state(ns._state), _pi(ns._pi),
   427         _in_arc(ns.in_arc), _search_arc_num(ns._search_arc_num),
   428         _next_arc(0)
   429       {
   430         // The main parameters of the pivot rule
   431         const double LIST_LENGTH_FACTOR = 1.0;
   432         const int MIN_LIST_LENGTH = 10;
   433         const double MINOR_LIMIT_FACTOR = 0.1;
   434         const int MIN_MINOR_LIMIT = 3;
   435 
   436         _list_length = std::max( int(LIST_LENGTH_FACTOR *
   437                                      std::sqrt(double(_search_arc_num))),
   438                                  MIN_LIST_LENGTH );
   439         _minor_limit = std::max( int(MINOR_LIMIT_FACTOR * _list_length),
   440                                  MIN_MINOR_LIMIT );
   441         _curr_length = _minor_count = 0;
   442         _candidates.resize(_list_length);
   443       }
   444 
   445       /// Find next entering arc
   446       bool findEnteringArc() {
   447         Cost min, c;
   448         int e, min_arc = _next_arc;
   449         if (_curr_length > 0 && _minor_count < _minor_limit) {
   450           // Minor iteration: select the best eligible arc from the
   451           // current candidate list
   452           ++_minor_count;
   453           min = 0;
   454           for (int i = 0; i < _curr_length; ++i) {
   455             e = _candidates[i];
   456             c = _state[e] * (_cost[e] + _pi[_source[e]] - _pi[_target[e]]);
   457             if (c < min) {
   458               min = c;
   459               min_arc = e;
   460             }
   461             if (c >= 0) {
   462               _candidates[i--] = _candidates[--_curr_length];
   463             }
   464           }
   465           if (min < 0) {
   466             _in_arc = min_arc;
   467             return true;
   468           }
   469         }
   470 
   471         // Major iteration: build a new candidate list
   472         min = 0;
   473         _curr_length = 0;
   474         for (e = _next_arc; e < _search_arc_num; ++e) {
   475           c = _state[e] * (_cost[e] + _pi[_source[e]] - _pi[_target[e]]);
   476           if (c < 0) {
   477             _candidates[_curr_length++] = e;
   478             if (c < min) {
   479               min = c;
   480               min_arc = e;
   481             }
   482             if (_curr_length == _list_length) break;
   483           }
   484         }
   485         if (_curr_length < _list_length) {
   486           for (e = 0; e < _next_arc; ++e) {
   487             c = _state[e] * (_cost[e] + _pi[_source[e]] - _pi[_target[e]]);
   488             if (c < 0) {
   489               _candidates[_curr_length++] = e;
   490               if (c < min) {
   491                 min = c;
   492                 min_arc = e;
   493               }
   494               if (_curr_length == _list_length) break;
   495             }
   496           }
   497         }
   498         if (_curr_length == 0) return false;
   499         _minor_count = 1;
   500         _in_arc = min_arc;
   501         _next_arc = e;
   502         return true;
   503       }
   504 
   505     }; //class CandidateListPivotRule
   506 
   507 
   508     // Implementation of the Altering Candidate List pivot rule
   509     class AlteringListPivotRule
   510     {
   511     private:
   512 
   513       // References to the NetworkSimplex class
   514       const IntVector  &_source;
   515       const IntVector  &_target;
   516       const CostVector &_cost;
   517       const IntVector  &_state;
   518       const CostVector &_pi;
   519       int &_in_arc;
   520       int _search_arc_num;
   521 
   522       // Pivot rule data
   523       int _block_size, _head_length, _curr_length;
   524       int _next_arc;
   525       IntVector _candidates;
   526       CostVector _cand_cost;
   527 
   528       // Functor class to compare arcs during sort of the candidate list
   529       class SortFunc
   530       {
   531       private:
   532         const CostVector &_map;
   533       public:
   534         SortFunc(const CostVector &map) : _map(map) {}
   535         bool operator()(int left, int right) {
   536           return _map[left] > _map[right];
   537         }
   538       };
   539 
   540       SortFunc _sort_func;
   541 
   542     public:
   543 
   544       // Constructor
   545       AlteringListPivotRule(NetworkSimplex &ns) :
   546         _source(ns._source), _target(ns._target),
   547         _cost(ns._cost), _state(ns._state), _pi(ns._pi),
   548         _in_arc(ns.in_arc), _search_arc_num(ns._search_arc_num),
   549         _next_arc(0), _cand_cost(ns._search_arc_num), _sort_func(_cand_cost)
   550       {
   551         // The main parameters of the pivot rule
   552         const double BLOCK_SIZE_FACTOR = 1.5;
   553         const int MIN_BLOCK_SIZE = 10;
   554         const double HEAD_LENGTH_FACTOR = 0.1;
   555         const int MIN_HEAD_LENGTH = 3;
   556 
   557         _block_size = std::max( int(BLOCK_SIZE_FACTOR *
   558                                     std::sqrt(double(_search_arc_num))),
   559                                 MIN_BLOCK_SIZE );
   560         _head_length = std::max( int(HEAD_LENGTH_FACTOR * _block_size),
   561                                  MIN_HEAD_LENGTH );
   562         _candidates.resize(_head_length + _block_size);
   563         _curr_length = 0;
   564       }
   565 
   566       // Find next entering arc
   567       bool findEnteringArc() {
   568         // Check the current candidate list
   569         int e;
   570         for (int i = 0; i < _curr_length; ++i) {
   571           e = _candidates[i];
   572           _cand_cost[e] = _state[e] *
   573             (_cost[e] + _pi[_source[e]] - _pi[_target[e]]);
   574           if (_cand_cost[e] >= 0) {
   575             _candidates[i--] = _candidates[--_curr_length];
   576           }
   577         }
   578 
   579         // Extend the list
   580         int cnt = _block_size;
   581         int last_arc = 0;
   582         int limit = _head_length;
   583 
   584         for (int e = _next_arc; e < _search_arc_num; ++e) {
   585           _cand_cost[e] = _state[e] *
   586             (_cost[e] + _pi[_source[e]] - _pi[_target[e]]);
   587           if (_cand_cost[e] < 0) {
   588             _candidates[_curr_length++] = e;
   589             last_arc = e;
   590           }
   591           if (--cnt == 0) {
   592             if (_curr_length > limit) break;
   593             limit = 0;
   594             cnt = _block_size;
   595           }
   596         }
   597         if (_curr_length <= limit) {
   598           for (int e = 0; e < _next_arc; ++e) {
   599             _cand_cost[e] = _state[e] *
   600               (_cost[e] + _pi[_source[e]] - _pi[_target[e]]);
   601             if (_cand_cost[e] < 0) {
   602               _candidates[_curr_length++] = e;
   603               last_arc = e;
   604             }
   605             if (--cnt == 0) {
   606               if (_curr_length > limit) break;
   607               limit = 0;
   608               cnt = _block_size;
   609             }
   610           }
   611         }
   612         if (_curr_length == 0) return false;
   613         _next_arc = last_arc + 1;
   614 
   615         // Make heap of the candidate list (approximating a partial sort)
   616         make_heap( _candidates.begin(), _candidates.begin() + _curr_length,
   617                    _sort_func );
   618 
   619         // Pop the first element of the heap
   620         _in_arc = _candidates[0];
   621         pop_heap( _candidates.begin(), _candidates.begin() + _curr_length,
   622                   _sort_func );
   623         _curr_length = std::min(_head_length, _curr_length - 1);
   624         return true;
   625       }
   626 
   627     }; //class AlteringListPivotRule
   628 
   629   public:
   630 
   631     /// \brief Constructor.
   632     ///
   633     /// The constructor of the class.
   634     ///
   635     /// \param graph The digraph the algorithm runs on.
   636     NetworkSimplex(const GR& graph) :
   637       _graph(graph), _node_id(graph), _arc_id(graph),
   638       INF(std::numeric_limits<Value>::has_infinity ?
   639           std::numeric_limits<Value>::infinity() :
   640           std::numeric_limits<Value>::max())
   641     {
   642       // Check the value types
   643       LEMON_ASSERT(std::numeric_limits<Value>::is_signed,
   644         "The flow type of NetworkSimplex must be signed");
   645       LEMON_ASSERT(std::numeric_limits<Cost>::is_signed,
   646         "The cost type of NetworkSimplex must be signed");
   647         
   648       // Resize vectors
   649       _node_num = countNodes(_graph);
   650       _arc_num = countArcs(_graph);
   651       int all_node_num = _node_num + 1;
   652       int max_arc_num = _arc_num + 2 * _node_num;
   653 
   654       _source.resize(max_arc_num);
   655       _target.resize(max_arc_num);
   656 
   657       _lower.resize(_arc_num);
   658       _upper.resize(_arc_num);
   659       _cap.resize(max_arc_num);
   660       _cost.resize(max_arc_num);
   661       _supply.resize(all_node_num);
   662       _flow.resize(max_arc_num);
   663       _pi.resize(all_node_num);
   664 
   665       _parent.resize(all_node_num);
   666       _pred.resize(all_node_num);
   667       _forward.resize(all_node_num);
   668       _thread.resize(all_node_num);
   669       _rev_thread.resize(all_node_num);
   670       _succ_num.resize(all_node_num);
   671       _last_succ.resize(all_node_num);
   672       _state.resize(max_arc_num);
   673 
   674       // Copy the graph (store the arcs in a mixed order)
   675       int i = 0;
   676       for (NodeIt n(_graph); n != INVALID; ++n, ++i) {
   677         _node_id[n] = i;
   678       }
   679       int k = std::max(int(std::sqrt(double(_arc_num))), 10);
   680       i = 0;
   681       for (ArcIt a(_graph); a != INVALID; ++a) {
   682         _arc_id[a] = i;
   683         _source[i] = _node_id[_graph.source(a)];
   684         _target[i] = _node_id[_graph.target(a)];
   685         if ((i += k) >= _arc_num) i = (i % k) + 1;
   686       }
   687       
   688       // Initialize maps
   689       for (int i = 0; i != _node_num; ++i) {
   690         _supply[i] = 0;
   691       }
   692       for (int i = 0; i != _arc_num; ++i) {
   693         _lower[i] = 0;
   694         _upper[i] = INF;
   695         _cost[i] = 1;
   696       }
   697       _have_lower = false;
   698       _stype = GEQ;
   699     }
   700 
   701     /// \name Parameters
   702     /// The parameters of the algorithm can be specified using these
   703     /// functions.
   704 
   705     /// @{
   706 
   707     /// \brief Set the lower bounds on the arcs.
   708     ///
   709     /// This function sets the lower bounds on the arcs.
   710     /// If it is not used before calling \ref run(), the lower bounds
   711     /// will be set to zero on all arcs.
   712     ///
   713     /// \param map An arc map storing the lower bounds.
   714     /// Its \c Value type must be convertible to the \c Value type
   715     /// of the algorithm.
   716     ///
   717     /// \return <tt>(*this)</tt>
   718     template <typename LowerMap>
   719     NetworkSimplex& lowerMap(const LowerMap& map) {
   720       _have_lower = true;
   721       for (ArcIt a(_graph); a != INVALID; ++a) {
   722         _lower[_arc_id[a]] = map[a];
   723       }
   724       return *this;
   725     }
   726 
   727     /// \brief Set the upper bounds (capacities) on the arcs.
   728     ///
   729     /// This function sets the upper bounds (capacities) on the arcs.
   730     /// If it is not used before calling \ref run(), the upper bounds
   731     /// will be set to \ref INF on all arcs (i.e. the flow value will be
   732     /// unbounded from above on each arc).
   733     ///
   734     /// \param map An arc map storing the upper bounds.
   735     /// Its \c Value type must be convertible to the \c Value type
   736     /// of the algorithm.
   737     ///
   738     /// \return <tt>(*this)</tt>
   739     template<typename UpperMap>
   740     NetworkSimplex& upperMap(const UpperMap& map) {
   741       for (ArcIt a(_graph); a != INVALID; ++a) {
   742         _upper[_arc_id[a]] = map[a];
   743       }
   744       return *this;
   745     }
   746 
   747     /// \brief Set the costs of the arcs.
   748     ///
   749     /// This function sets the costs of the arcs.
   750     /// If it is not used before calling \ref run(), the costs
   751     /// will be set to \c 1 on all arcs.
   752     ///
   753     /// \param map An arc map storing the costs.
   754     /// Its \c Value type must be convertible to the \c Cost type
   755     /// of the algorithm.
   756     ///
   757     /// \return <tt>(*this)</tt>
   758     template<typename CostMap>
   759     NetworkSimplex& costMap(const CostMap& map) {
   760       for (ArcIt a(_graph); a != INVALID; ++a) {
   761         _cost[_arc_id[a]] = map[a];
   762       }
   763       return *this;
   764     }
   765 
   766     /// \brief Set the supply values of the nodes.
   767     ///
   768     /// This function sets the supply values of the nodes.
   769     /// If neither this function nor \ref stSupply() is used before
   770     /// calling \ref run(), the supply of each node will be set to zero.
   771     /// (It makes sense only if non-zero lower bounds are given.)
   772     ///
   773     /// \param map A node map storing the supply values.
   774     /// Its \c Value type must be convertible to the \c Value type
   775     /// of the algorithm.
   776     ///
   777     /// \return <tt>(*this)</tt>
   778     template<typename SupplyMap>
   779     NetworkSimplex& supplyMap(const SupplyMap& map) {
   780       for (NodeIt n(_graph); n != INVALID; ++n) {
   781         _supply[_node_id[n]] = map[n];
   782       }
   783       return *this;
   784     }
   785 
   786     /// \brief Set single source and target nodes and a supply value.
   787     ///
   788     /// This function sets a single source node and a single target node
   789     /// and the required flow value.
   790     /// If neither this function nor \ref supplyMap() is used before
   791     /// calling \ref run(), the supply of each node will be set to zero.
   792     /// (It makes sense only if non-zero lower bounds are given.)
   793     ///
   794     /// Using this function has the same effect as using \ref supplyMap()
   795     /// with such a map in which \c k is assigned to \c s, \c -k is
   796     /// assigned to \c t and all other nodes have zero supply value.
   797     ///
   798     /// \param s The source node.
   799     /// \param t The target node.
   800     /// \param k The required amount of flow from node \c s to node \c t
   801     /// (i.e. the supply of \c s and the demand of \c t).
   802     ///
   803     /// \return <tt>(*this)</tt>
   804     NetworkSimplex& stSupply(const Node& s, const Node& t, Value k) {
   805       for (int i = 0; i != _node_num; ++i) {
   806         _supply[i] = 0;
   807       }
   808       _supply[_node_id[s]] =  k;
   809       _supply[_node_id[t]] = -k;
   810       return *this;
   811     }
   812     
   813     /// \brief Set the type of the supply constraints.
   814     ///
   815     /// This function sets the type of the supply/demand constraints.
   816     /// If it is not used before calling \ref run(), the \ref GEQ supply
   817     /// type will be used.
   818     ///
   819     /// For more information see \ref SupplyType.
   820     ///
   821     /// \return <tt>(*this)</tt>
   822     NetworkSimplex& supplyType(SupplyType supply_type) {
   823       _stype = supply_type;
   824       return *this;
   825     }
   826 
   827     /// @}
   828 
   829     /// \name Execution Control
   830     /// The algorithm can be executed using \ref run().
   831 
   832     /// @{
   833 
   834     /// \brief Run the algorithm.
   835     ///
   836     /// This function runs the algorithm.
   837     /// The paramters can be specified using functions \ref lowerMap(),
   838     /// \ref upperMap(), \ref costMap(), \ref supplyMap(), \ref stSupply(), 
   839     /// \ref supplyType().
   840     /// For example,
   841     /// \code
   842     ///   NetworkSimplex<ListDigraph> ns(graph);
   843     ///   ns.lowerMap(lower).upperMap(upper).costMap(cost)
   844     ///     .supplyMap(sup).run();
   845     /// \endcode
   846     ///
   847     /// This function can be called more than once. All the parameters
   848     /// that have been given are kept for the next call, unless
   849     /// \ref reset() is called, thus only the modified parameters
   850     /// have to be set again. See \ref reset() for examples.
   851     /// However the underlying digraph must not be modified after this
   852     /// class have been constructed, since it copies and extends the graph.
   853     ///
   854     /// \param pivot_rule The pivot rule that will be used during the
   855     /// algorithm. For more information see \ref PivotRule.
   856     ///
   857     /// \return \c INFEASIBLE if no feasible flow exists,
   858     /// \n \c OPTIMAL if the problem has optimal solution
   859     /// (i.e. it is feasible and bounded), and the algorithm has found
   860     /// optimal flow and node potentials (primal and dual solutions),
   861     /// \n \c UNBOUNDED if the objective function of the problem is
   862     /// unbounded, i.e. there is a directed cycle having negative total
   863     /// cost and infinite upper bound.
   864     ///
   865     /// \see ProblemType, PivotRule
   866     ProblemType run(PivotRule pivot_rule = BLOCK_SEARCH) {
   867       if (!init()) return INFEASIBLE;
   868       return start(pivot_rule);
   869     }
   870 
   871     /// \brief Reset all the parameters that have been given before.
   872     ///
   873     /// This function resets all the paramaters that have been given
   874     /// before using functions \ref lowerMap(), \ref upperMap(),
   875     /// \ref costMap(), \ref supplyMap(), \ref stSupply(), \ref supplyType().
   876     ///
   877     /// It is useful for multiple run() calls. If this function is not
   878     /// used, all the parameters given before are kept for the next
   879     /// \ref run() call.
   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     /// For example,
   884     /// \code
   885     ///   NetworkSimplex<ListDigraph> ns(graph);
   886     ///
   887     ///   // First run
   888     ///   ns.lowerMap(lower).upperMap(upper).costMap(cost)
   889     ///     .supplyMap(sup).run();
   890     ///
   891     ///   // Run again with modified cost map (reset() is not called,
   892     ///   // so only the cost map have to be set again)
   893     ///   cost[e] += 100;
   894     ///   ns.costMap(cost).run();
   895     ///
   896     ///   // Run again from scratch using reset()
   897     ///   // (the lower bounds will be set to zero on all arcs)
   898     ///   ns.reset();
   899     ///   ns.upperMap(capacity).costMap(cost)
   900     ///     .supplyMap(sup).run();
   901     /// \endcode
   902     ///
   903     /// \return <tt>(*this)</tt>
   904     NetworkSimplex& reset() {
   905       for (int i = 0; i != _node_num; ++i) {
   906         _supply[i] = 0;
   907       }
   908       for (int i = 0; i != _arc_num; ++i) {
   909         _lower[i] = 0;
   910         _upper[i] = INF;
   911         _cost[i] = 1;
   912       }
   913       _have_lower = false;
   914       _stype = GEQ;
   915       return *this;
   916     }
   917 
   918     /// @}
   919 
   920     /// \name Query Functions
   921     /// The results of the algorithm can be obtained using these
   922     /// functions.\n
   923     /// The \ref run() function must be called before using them.
   924 
   925     /// @{
   926 
   927     /// \brief Return the total cost of the found flow.
   928     ///
   929     /// This function returns the total cost of the found flow.
   930     /// Its complexity is O(e).
   931     ///
   932     /// \note The return type of the function can be specified as a
   933     /// template parameter. For example,
   934     /// \code
   935     ///   ns.totalCost<double>();
   936     /// \endcode
   937     /// It is useful if the total cost cannot be stored in the \c Cost
   938     /// type of the algorithm, which is the default return type of the
   939     /// function.
   940     ///
   941     /// \pre \ref run() must be called before using this function.
   942     template <typename Number>
   943     Number totalCost() const {
   944       Number c = 0;
   945       for (ArcIt a(_graph); a != INVALID; ++a) {
   946         int i = _arc_id[a];
   947         c += Number(_flow[i]) * Number(_cost[i]);
   948       }
   949       return c;
   950     }
   951 
   952 #ifndef DOXYGEN
   953     Cost totalCost() const {
   954       return totalCost<Cost>();
   955     }
   956 #endif
   957 
   958     /// \brief Return the flow on the given arc.
   959     ///
   960     /// This function returns the flow on the given arc.
   961     ///
   962     /// \pre \ref run() must be called before using this function.
   963     Value flow(const Arc& a) const {
   964       return _flow[_arc_id[a]];
   965     }
   966 
   967     /// \brief Return the flow map (the primal solution).
   968     ///
   969     /// This function copies the flow value on each arc into the given
   970     /// map. The \c Value type of the algorithm must be convertible to
   971     /// the \c Value type of the map.
   972     ///
   973     /// \pre \ref run() must be called before using this function.
   974     template <typename FlowMap>
   975     void flowMap(FlowMap &map) const {
   976       for (ArcIt a(_graph); a != INVALID; ++a) {
   977         map.set(a, _flow[_arc_id[a]]);
   978       }
   979     }
   980 
   981     /// \brief Return the potential (dual value) of the given node.
   982     ///
   983     /// This function returns the potential (dual value) of the
   984     /// given node.
   985     ///
   986     /// \pre \ref run() must be called before using this function.
   987     Cost potential(const Node& n) const {
   988       return _pi[_node_id[n]];
   989     }
   990 
   991     /// \brief Return the potential map (the dual solution).
   992     ///
   993     /// This function copies the potential (dual value) of each node
   994     /// into the given map.
   995     /// The \c Cost type of the algorithm must be convertible to the
   996     /// \c Value type of the map.
   997     ///
   998     /// \pre \ref run() must be called before using this function.
   999     template <typename PotentialMap>
  1000     void potentialMap(PotentialMap &map) const {
  1001       for (NodeIt n(_graph); n != INVALID; ++n) {
  1002         map.set(n, _pi[_node_id[n]]);
  1003       }
  1004     }
  1005 
  1006     /// @}
  1007 
  1008   private:
  1009 
  1010     // Initialize internal data structures
  1011     bool init() {
  1012       if (_node_num == 0) return false;
  1013 
  1014       // Check the sum of supply values
  1015       _sum_supply = 0;
  1016       for (int i = 0; i != _node_num; ++i) {
  1017         _sum_supply += _supply[i];
  1018       }
  1019       if ( !((_stype == GEQ && _sum_supply <= 0) ||
  1020              (_stype == LEQ && _sum_supply >= 0)) ) return false;
  1021 
  1022       // Remove non-zero lower bounds
  1023       if (_have_lower) {
  1024         for (int i = 0; i != _arc_num; ++i) {
  1025           Value c = _lower[i];
  1026           if (c >= 0) {
  1027             _cap[i] = _upper[i] < INF ? _upper[i] - c : INF;
  1028           } else {
  1029             _cap[i] = _upper[i] < INF + c ? _upper[i] - c : INF;
  1030           }
  1031           _supply[_source[i]] -= c;
  1032           _supply[_target[i]] += c;
  1033         }
  1034       } else {
  1035         for (int i = 0; i != _arc_num; ++i) {
  1036           _cap[i] = _upper[i];
  1037         }
  1038       }
  1039 
  1040       // Initialize artifical cost
  1041       Cost ART_COST;
  1042       if (std::numeric_limits<Cost>::is_exact) {
  1043         ART_COST = std::numeric_limits<Cost>::max() / 2 + 1;
  1044       } else {
  1045         ART_COST = std::numeric_limits<Cost>::min();
  1046         for (int i = 0; i != _arc_num; ++i) {
  1047           if (_cost[i] > ART_COST) ART_COST = _cost[i];
  1048         }
  1049         ART_COST = (ART_COST + 1) * _node_num;
  1050       }
  1051 
  1052       // Initialize arc maps
  1053       for (int i = 0; i != _arc_num; ++i) {
  1054         _flow[i] = 0;
  1055         _state[i] = STATE_LOWER;
  1056       }
  1057       
  1058       // Set data for the artificial root node
  1059       _root = _node_num;
  1060       _parent[_root] = -1;
  1061       _pred[_root] = -1;
  1062       _thread[_root] = 0;
  1063       _rev_thread[0] = _root;
  1064       _succ_num[_root] = _node_num + 1;
  1065       _last_succ[_root] = _root - 1;
  1066       _supply[_root] = -_sum_supply;
  1067       _pi[_root] = 0;
  1068 
  1069       // Add artificial arcs and initialize the spanning tree data structure
  1070       if (_sum_supply == 0) {
  1071         // EQ supply constraints
  1072         _search_arc_num = _arc_num;
  1073         _all_arc_num = _arc_num + _node_num;
  1074         for (int u = 0, e = _arc_num; u != _node_num; ++u, ++e) {
  1075           _parent[u] = _root;
  1076           _pred[u] = e;
  1077           _thread[u] = u + 1;
  1078           _rev_thread[u + 1] = u;
  1079           _succ_num[u] = 1;
  1080           _last_succ[u] = u;
  1081           _cap[e] = INF;
  1082           _state[e] = STATE_TREE;
  1083           if (_supply[u] >= 0) {
  1084             _forward[u] = true;
  1085             _pi[u] = 0;
  1086             _source[e] = u;
  1087             _target[e] = _root;
  1088             _flow[e] = _supply[u];
  1089             _cost[e] = 0;
  1090           } else {
  1091             _forward[u] = false;
  1092             _pi[u] = ART_COST;
  1093             _source[e] = _root;
  1094             _target[e] = u;
  1095             _flow[e] = -_supply[u];
  1096             _cost[e] = ART_COST;
  1097           }
  1098         }
  1099       }
  1100       else if (_sum_supply > 0) {
  1101         // LEQ supply constraints
  1102         _search_arc_num = _arc_num + _node_num;
  1103         int f = _arc_num + _node_num;
  1104         for (int u = 0, e = _arc_num; u != _node_num; ++u, ++e) {
  1105           _parent[u] = _root;
  1106           _thread[u] = u + 1;
  1107           _rev_thread[u + 1] = u;
  1108           _succ_num[u] = 1;
  1109           _last_succ[u] = u;
  1110           if (_supply[u] >= 0) {
  1111             _forward[u] = true;
  1112             _pi[u] = 0;
  1113             _pred[u] = e;
  1114             _source[e] = u;
  1115             _target[e] = _root;
  1116             _cap[e] = INF;
  1117             _flow[e] = _supply[u];
  1118             _cost[e] = 0;
  1119             _state[e] = STATE_TREE;
  1120           } else {
  1121             _forward[u] = false;
  1122             _pi[u] = ART_COST;
  1123             _pred[u] = f;
  1124             _source[f] = _root;
  1125             _target[f] = u;
  1126             _cap[f] = INF;
  1127             _flow[f] = -_supply[u];
  1128             _cost[f] = ART_COST;
  1129             _state[f] = STATE_TREE;
  1130             _source[e] = u;
  1131             _target[e] = _root;
  1132             _cap[e] = INF;
  1133             _flow[e] = 0;
  1134             _cost[e] = 0;
  1135             _state[e] = STATE_LOWER;
  1136             ++f;
  1137           }
  1138         }
  1139         _all_arc_num = f;
  1140       }
  1141       else {
  1142         // GEQ supply constraints
  1143         _search_arc_num = _arc_num + _node_num;
  1144         int f = _arc_num + _node_num;
  1145         for (int u = 0, e = _arc_num; u != _node_num; ++u, ++e) {
  1146           _parent[u] = _root;
  1147           _thread[u] = u + 1;
  1148           _rev_thread[u + 1] = u;
  1149           _succ_num[u] = 1;
  1150           _last_succ[u] = u;
  1151           if (_supply[u] <= 0) {
  1152             _forward[u] = false;
  1153             _pi[u] = 0;
  1154             _pred[u] = e;
  1155             _source[e] = _root;
  1156             _target[e] = u;
  1157             _cap[e] = INF;
  1158             _flow[e] = -_supply[u];
  1159             _cost[e] = 0;
  1160             _state[e] = STATE_TREE;
  1161           } else {
  1162             _forward[u] = true;
  1163             _pi[u] = -ART_COST;
  1164             _pred[u] = f;
  1165             _source[f] = u;
  1166             _target[f] = _root;
  1167             _cap[f] = INF;
  1168             _flow[f] = _supply[u];
  1169             _state[f] = STATE_TREE;
  1170             _cost[f] = ART_COST;
  1171             _source[e] = _root;
  1172             _target[e] = u;
  1173             _cap[e] = INF;
  1174             _flow[e] = 0;
  1175             _cost[e] = 0;
  1176             _state[e] = STATE_LOWER;
  1177             ++f;
  1178           }
  1179         }
  1180         _all_arc_num = f;
  1181       }
  1182 
  1183       return true;
  1184     }
  1185 
  1186     // Find the join node
  1187     void findJoinNode() {
  1188       int u = _source[in_arc];
  1189       int v = _target[in_arc];
  1190       while (u != v) {
  1191         if (_succ_num[u] < _succ_num[v]) {
  1192           u = _parent[u];
  1193         } else {
  1194           v = _parent[v];
  1195         }
  1196       }
  1197       join = u;
  1198     }
  1199 
  1200     // Find the leaving arc of the cycle and returns true if the
  1201     // leaving arc is not the same as the entering arc
  1202     bool findLeavingArc() {
  1203       // Initialize first and second nodes according to the direction
  1204       // of the cycle
  1205       if (_state[in_arc] == STATE_LOWER) {
  1206         first  = _source[in_arc];
  1207         second = _target[in_arc];
  1208       } else {
  1209         first  = _target[in_arc];
  1210         second = _source[in_arc];
  1211       }
  1212       delta = _cap[in_arc];
  1213       int result = 0;
  1214       Value d;
  1215       int e;
  1216 
  1217       // Search the cycle along the path form the first node to the root
  1218       for (int u = first; u != join; u = _parent[u]) {
  1219         e = _pred[u];
  1220         d = _forward[u] ?
  1221           _flow[e] : (_cap[e] == INF ? INF : _cap[e] - _flow[e]);
  1222         if (d < delta) {
  1223           delta = d;
  1224           u_out = u;
  1225           result = 1;
  1226         }
  1227       }
  1228       // Search the cycle along the path form the second node to the root
  1229       for (int u = second; u != join; u = _parent[u]) {
  1230         e = _pred[u];
  1231         d = _forward[u] ? 
  1232           (_cap[e] == INF ? INF : _cap[e] - _flow[e]) : _flow[e];
  1233         if (d <= delta) {
  1234           delta = d;
  1235           u_out = u;
  1236           result = 2;
  1237         }
  1238       }
  1239 
  1240       if (result == 1) {
  1241         u_in = first;
  1242         v_in = second;
  1243       } else {
  1244         u_in = second;
  1245         v_in = first;
  1246       }
  1247       return result != 0;
  1248     }
  1249 
  1250     // Change _flow and _state vectors
  1251     void changeFlow(bool change) {
  1252       // Augment along the cycle
  1253       if (delta > 0) {
  1254         Value val = _state[in_arc] * delta;
  1255         _flow[in_arc] += val;
  1256         for (int u = _source[in_arc]; u != join; u = _parent[u]) {
  1257           _flow[_pred[u]] += _forward[u] ? -val : val;
  1258         }
  1259         for (int u = _target[in_arc]; u != join; u = _parent[u]) {
  1260           _flow[_pred[u]] += _forward[u] ? val : -val;
  1261         }
  1262       }
  1263       // Update the state of the entering and leaving arcs
  1264       if (change) {
  1265         _state[in_arc] = STATE_TREE;
  1266         _state[_pred[u_out]] =
  1267           (_flow[_pred[u_out]] == 0) ? STATE_LOWER : STATE_UPPER;
  1268       } else {
  1269         _state[in_arc] = -_state[in_arc];
  1270       }
  1271     }
  1272 
  1273     // Update the tree structure
  1274     void updateTreeStructure() {
  1275       int u, w;
  1276       int old_rev_thread = _rev_thread[u_out];
  1277       int old_succ_num = _succ_num[u_out];
  1278       int old_last_succ = _last_succ[u_out];
  1279       v_out = _parent[u_out];
  1280 
  1281       u = _last_succ[u_in];  // the last successor of u_in
  1282       right = _thread[u];    // the node after it
  1283 
  1284       // Handle the case when old_rev_thread equals to v_in
  1285       // (it also means that join and v_out coincide)
  1286       if (old_rev_thread == v_in) {
  1287         last = _thread[_last_succ[u_out]];
  1288       } else {
  1289         last = _thread[v_in];
  1290       }
  1291 
  1292       // Update _thread and _parent along the stem nodes (i.e. the nodes
  1293       // between u_in and u_out, whose parent have to be changed)
  1294       _thread[v_in] = stem = u_in;
  1295       _dirty_revs.clear();
  1296       _dirty_revs.push_back(v_in);
  1297       par_stem = v_in;
  1298       while (stem != u_out) {
  1299         // Insert the next stem node into the thread list
  1300         new_stem = _parent[stem];
  1301         _thread[u] = new_stem;
  1302         _dirty_revs.push_back(u);
  1303 
  1304         // Remove the subtree of stem from the thread list
  1305         w = _rev_thread[stem];
  1306         _thread[w] = right;
  1307         _rev_thread[right] = w;
  1308 
  1309         // Change the parent node and shift stem nodes
  1310         _parent[stem] = par_stem;
  1311         par_stem = stem;
  1312         stem = new_stem;
  1313 
  1314         // Update u and right
  1315         u = _last_succ[stem] == _last_succ[par_stem] ?
  1316           _rev_thread[par_stem] : _last_succ[stem];
  1317         right = _thread[u];
  1318       }
  1319       _parent[u_out] = par_stem;
  1320       _thread[u] = last;
  1321       _rev_thread[last] = u;
  1322       _last_succ[u_out] = u;
  1323 
  1324       // Remove the subtree of u_out from the thread list except for
  1325       // the case when old_rev_thread equals to v_in
  1326       // (it also means that join and v_out coincide)
  1327       if (old_rev_thread != v_in) {
  1328         _thread[old_rev_thread] = right;
  1329         _rev_thread[right] = old_rev_thread;
  1330       }
  1331 
  1332       // Update _rev_thread using the new _thread values
  1333       for (int i = 0; i < int(_dirty_revs.size()); ++i) {
  1334         u = _dirty_revs[i];
  1335         _rev_thread[_thread[u]] = u;
  1336       }
  1337 
  1338       // Update _pred, _forward, _last_succ and _succ_num for the
  1339       // stem nodes from u_out to u_in
  1340       int tmp_sc = 0, tmp_ls = _last_succ[u_out];
  1341       u = u_out;
  1342       while (u != u_in) {
  1343         w = _parent[u];
  1344         _pred[u] = _pred[w];
  1345         _forward[u] = !_forward[w];
  1346         tmp_sc += _succ_num[u] - _succ_num[w];
  1347         _succ_num[u] = tmp_sc;
  1348         _last_succ[w] = tmp_ls;
  1349         u = w;
  1350       }
  1351       _pred[u_in] = in_arc;
  1352       _forward[u_in] = (u_in == _source[in_arc]);
  1353       _succ_num[u_in] = old_succ_num;
  1354 
  1355       // Set limits for updating _last_succ form v_in and v_out
  1356       // towards the root
  1357       int up_limit_in = -1;
  1358       int up_limit_out = -1;
  1359       if (_last_succ[join] == v_in) {
  1360         up_limit_out = join;
  1361       } else {
  1362         up_limit_in = join;
  1363       }
  1364 
  1365       // Update _last_succ from v_in towards the root
  1366       for (u = v_in; u != up_limit_in && _last_succ[u] == v_in;
  1367            u = _parent[u]) {
  1368         _last_succ[u] = _last_succ[u_out];
  1369       }
  1370       // Update _last_succ from v_out towards the root
  1371       if (join != old_rev_thread && v_in != old_rev_thread) {
  1372         for (u = v_out; u != up_limit_out && _last_succ[u] == old_last_succ;
  1373              u = _parent[u]) {
  1374           _last_succ[u] = old_rev_thread;
  1375         }
  1376       } else {
  1377         for (u = v_out; u != up_limit_out && _last_succ[u] == old_last_succ;
  1378              u = _parent[u]) {
  1379           _last_succ[u] = _last_succ[u_out];
  1380         }
  1381       }
  1382 
  1383       // Update _succ_num from v_in to join
  1384       for (u = v_in; u != join; u = _parent[u]) {
  1385         _succ_num[u] += old_succ_num;
  1386       }
  1387       // Update _succ_num from v_out to join
  1388       for (u = v_out; u != join; u = _parent[u]) {
  1389         _succ_num[u] -= old_succ_num;
  1390       }
  1391     }
  1392 
  1393     // Update potentials
  1394     void updatePotential() {
  1395       Cost sigma = _forward[u_in] ?
  1396         _pi[v_in] - _pi[u_in] - _cost[_pred[u_in]] :
  1397         _pi[v_in] - _pi[u_in] + _cost[_pred[u_in]];
  1398       // Update potentials in the subtree, which has been moved
  1399       int end = _thread[_last_succ[u_in]];
  1400       for (int u = u_in; u != end; u = _thread[u]) {
  1401         _pi[u] += sigma;
  1402       }
  1403     }
  1404 
  1405     // Execute the algorithm
  1406     ProblemType start(PivotRule pivot_rule) {
  1407       // Select the pivot rule implementation
  1408       switch (pivot_rule) {
  1409         case FIRST_ELIGIBLE:
  1410           return start<FirstEligiblePivotRule>();
  1411         case BEST_ELIGIBLE:
  1412           return start<BestEligiblePivotRule>();
  1413         case BLOCK_SEARCH:
  1414           return start<BlockSearchPivotRule>();
  1415         case CANDIDATE_LIST:
  1416           return start<CandidateListPivotRule>();
  1417         case ALTERING_LIST:
  1418           return start<AlteringListPivotRule>();
  1419       }
  1420       return INFEASIBLE; // avoid warning
  1421     }
  1422 
  1423     template <typename PivotRuleImpl>
  1424     ProblemType start() {
  1425       PivotRuleImpl pivot(*this);
  1426 
  1427       // Execute the Network Simplex algorithm
  1428       while (pivot.findEnteringArc()) {
  1429         findJoinNode();
  1430         bool change = findLeavingArc();
  1431         if (delta >= INF) return UNBOUNDED;
  1432         changeFlow(change);
  1433         if (change) {
  1434           updateTreeStructure();
  1435           updatePotential();
  1436         }
  1437       }
  1438       
  1439       // Check feasibility
  1440       for (int e = _search_arc_num; e != _all_arc_num; ++e) {
  1441         if (_flow[e] != 0) return INFEASIBLE;
  1442       }
  1443 
  1444       // Transform the solution and the supply map to the original form
  1445       if (_have_lower) {
  1446         for (int i = 0; i != _arc_num; ++i) {
  1447           Value c = _lower[i];
  1448           if (c != 0) {
  1449             _flow[i] += c;
  1450             _supply[_source[i]] += c;
  1451             _supply[_target[i]] -= c;
  1452           }
  1453         }
  1454       }
  1455       
  1456       // Shift potentials to meet the requirements of the GEQ/LEQ type
  1457       // optimality conditions
  1458       if (_sum_supply == 0) {
  1459         if (_stype == GEQ) {
  1460           Cost max_pot = std::numeric_limits<Cost>::min();
  1461           for (int i = 0; i != _node_num; ++i) {
  1462             if (_pi[i] > max_pot) max_pot = _pi[i];
  1463           }
  1464           if (max_pot > 0) {
  1465             for (int i = 0; i != _node_num; ++i)
  1466               _pi[i] -= max_pot;
  1467           }
  1468         } else {
  1469           Cost min_pot = std::numeric_limits<Cost>::max();
  1470           for (int i = 0; i != _node_num; ++i) {
  1471             if (_pi[i] < min_pot) min_pot = _pi[i];
  1472           }
  1473           if (min_pot < 0) {
  1474             for (int i = 0; i != _node_num; ++i)
  1475               _pi[i] -= min_pot;
  1476           }
  1477         }
  1478       }
  1479 
  1480       return OPTIMAL;
  1481     }
  1482 
  1483   }; //class NetworkSimplex
  1484 
  1485   ///@}
  1486 
  1487 } //namespace lemon
  1488 
  1489 #endif //LEMON_NETWORK_SIMPLEX_H