lemon/network_simplex.h
author Peter Kovacs <kpeter@inf.elte.hu>
Sat, 20 Feb 2010 18:39:03 +0100
changeset 839 f3bc4e9b5f3a
parent 812 4b1b378823dc
child 840 2914b6f0fde0
permissions -rw-r--r--
New heuristics for MCF algorithms (#340)
and some implementation improvements.

- A useful heuristic is added to NetworkSimplex to make the
initial pivots faster.
- A powerful global update heuristic is added to CostScaling
and the implementation is reworked with various improvements.
- Better relabeling in CostScaling to improve numerical stability
and make the code faster.
- A small improvement is made in CapacityScaling for better
delta computation.
- Add notes to the classes about the usage of vector<char> instead
of vector<bool> for efficiency reasons.
     1 /* -*- mode: C++; indent-tabs-mode: nil; -*-
     2  *
     3  * This file is a part of LEMON, a generic C++ optimization library.
     4  *
     5  * Copyright (C) 2003-2009
     6  * Egervary Jeno Kombinatorikus Optimalizalasi Kutatocsoport
     7  * (Egervary Research Group on Combinatorial Optimization, EGRES).
     8  *
     9  * Permission to use, modify and distribute this software is granted
    10  * provided that this copyright notice appears in all copies. For
    11  * precise terms see the accompanying LICENSE file.
    12  *
    13  * This software is provided "AS IS" with no warranty of any kind,
    14  * express or implied, and with no claim as to its suitability for any
    15  * purpose.
    16  *
    17  */
    18 
    19 #ifndef LEMON_NETWORK_SIMPLEX_H
    20 #define LEMON_NETWORK_SIMPLEX_H
    21 
    22 /// \ingroup min_cost_flow_algs
    23 ///
    24 /// \file
    25 /// \brief Network Simplex algorithm for finding a minimum cost flow.
    26 
    27 #include <vector>
    28 #include <limits>
    29 #include <algorithm>
    30 
    31 #include <lemon/core.h>
    32 #include <lemon/math.h>
    33 
    34 namespace lemon {
    35 
    36   /// \addtogroup min_cost_flow_algs
    37   /// @{
    38 
    39   /// \brief Implementation of the primal Network Simplex algorithm
    40   /// for finding a \ref min_cost_flow "minimum cost flow".
    41   ///
    42   /// \ref NetworkSimplex implements the primal Network Simplex algorithm
    43   /// for finding a \ref min_cost_flow "minimum cost flow"
    44   /// \ref amo93networkflows, \ref dantzig63linearprog,
    45   /// \ref kellyoneill91netsimplex.
    46   /// This algorithm is a highly efficient specialized version of the
    47   /// linear programming simplex method directly for the minimum cost
    48   /// flow problem.
    49   ///
    50   /// In general, %NetworkSimplex is the fastest implementation available
    51   /// in LEMON for this problem.
    52   /// Moreover, it supports both directions of the supply/demand inequality
    53   /// constraints. For more information, see \ref SupplyType.
    54   ///
    55   /// Most of the parameters of the problem (except for the digraph)
    56   /// can be given using separate functions, and the algorithm can be
    57   /// executed using the \ref run() function. If some parameters are not
    58   /// specified, then default values will be used.
    59   ///
    60   /// \tparam GR The digraph type the algorithm runs on.
    61   /// \tparam V The number type used for flow amounts, capacity bounds
    62   /// and supply values in the algorithm. By default, it is \c int.
    63   /// \tparam C The number type used for costs and potentials in the
    64   /// algorithm. By default, it is the same as \c V.
    65   ///
    66   /// \warning Both number types must be signed and all input data must
    67   /// be integer.
    68   ///
    69   /// \note %NetworkSimplex provides five different pivot rule
    70   /// implementations, from which the most efficient one is used
    71   /// by default. For more information, see \ref PivotRule.
    72   template <typename GR, typename V = int, typename C = V>
    73   class NetworkSimplex
    74   {
    75   public:
    76 
    77     /// The type of the flow amounts, capacity bounds and supply values
    78     typedef V Value;
    79     /// The type of the arc costs
    80     typedef C Cost;
    81 
    82   public:
    83 
    84     /// \brief Problem type constants for the \c run() function.
    85     ///
    86     /// Enum type containing the problem type constants that can be
    87     /// returned by the \ref run() function of the algorithm.
    88     enum ProblemType {
    89       /// The problem has no feasible solution (flow).
    90       INFEASIBLE,
    91       /// The problem has optimal solution (i.e. it is feasible and
    92       /// bounded), and the algorithm has found optimal flow and node
    93       /// potentials (primal and dual solutions).
    94       OPTIMAL,
    95       /// The objective function of the problem is unbounded, i.e.
    96       /// there is a directed cycle having negative total cost and
    97       /// infinite upper bound.
    98       UNBOUNDED
    99     };
   100     
   101     /// \brief Constants for selecting the type of the supply constraints.
   102     ///
   103     /// Enum type containing constants for selecting the supply type,
   104     /// i.e. the direction of the inequalities in the supply/demand
   105     /// constraints of the \ref min_cost_flow "minimum cost flow problem".
   106     ///
   107     /// The default supply type is \c GEQ, the \c LEQ type can be
   108     /// selected using \ref supplyType().
   109     /// The equality form is a special case of both supply types.
   110     enum SupplyType {
   111       /// This option means that there are <em>"greater or equal"</em>
   112       /// supply/demand constraints in the definition of the problem.
   113       GEQ,
   114       /// This option means that there are <em>"less or equal"</em>
   115       /// supply/demand constraints in the definition of the problem.
   116       LEQ
   117     };
   118     
   119     /// \brief Constants for selecting the pivot rule.
   120     ///
   121     /// Enum type containing constants for selecting the pivot rule for
   122     /// the \ref run() function.
   123     ///
   124     /// \ref NetworkSimplex provides five different pivot rule
   125     /// implementations that significantly affect the running time
   126     /// of the algorithm.
   127     /// By default, \ref BLOCK_SEARCH "Block Search" is used, which
   128     /// proved to be the most efficient and the most robust on various
   129     /// test inputs.
   130     /// However, another pivot rule can be selected using the \ref run()
   131     /// function with the proper parameter.
   132     enum PivotRule {
   133 
   134       /// The \e First \e Eligible pivot rule.
   135       /// The next eligible arc is selected in a wraparound fashion
   136       /// in every iteration.
   137       FIRST_ELIGIBLE,
   138 
   139       /// The \e Best \e Eligible pivot rule.
   140       /// The best eligible arc is selected in every iteration.
   141       BEST_ELIGIBLE,
   142 
   143       /// The \e Block \e Search pivot rule.
   144       /// A specified number of arcs are examined in every iteration
   145       /// in a wraparound fashion and the best eligible arc is selected
   146       /// from this block.
   147       BLOCK_SEARCH,
   148 
   149       /// The \e Candidate \e List pivot rule.
   150       /// In a major iteration a candidate list is built from eligible arcs
   151       /// in a wraparound fashion and in the following minor iterations
   152       /// the best eligible arc is selected from this list.
   153       CANDIDATE_LIST,
   154 
   155       /// The \e Altering \e Candidate \e List pivot rule.
   156       /// It is a modified version of the Candidate List method.
   157       /// It keeps only the several best eligible arcs from the former
   158       /// candidate list and extends this list in every iteration.
   159       ALTERING_LIST
   160     };
   161     
   162   private:
   163 
   164     TEMPLATE_DIGRAPH_TYPEDEFS(GR);
   165 
   166     typedef std::vector<int> IntVector;
   167     typedef std::vector<Value> ValueVector;
   168     typedef std::vector<Cost> CostVector;
   169     typedef std::vector<char> BoolVector;
   170     // Note: vector<char> is used instead of vector<bool> for efficiency reasons
   171 
   172     // State constants for arcs
   173     enum ArcStateEnum {
   174       STATE_UPPER = -1,
   175       STATE_TREE  =  0,
   176       STATE_LOWER =  1
   177     };
   178 
   179   private:
   180 
   181     // Data related to the underlying digraph
   182     const GR &_graph;
   183     int _node_num;
   184     int _arc_num;
   185     int _all_arc_num;
   186     int _search_arc_num;
   187 
   188     // Parameters of the problem
   189     bool _have_lower;
   190     SupplyType _stype;
   191     Value _sum_supply;
   192 
   193     // Data structures for storing the digraph
   194     IntNodeMap _node_id;
   195     IntArcMap _arc_id;
   196     IntVector _source;
   197     IntVector _target;
   198 
   199     // Node and arc data
   200     ValueVector _lower;
   201     ValueVector _upper;
   202     ValueVector _cap;
   203     CostVector _cost;
   204     ValueVector _supply;
   205     ValueVector _flow;
   206     CostVector _pi;
   207 
   208     // Data for storing the spanning tree structure
   209     IntVector _parent;
   210     IntVector _pred;
   211     IntVector _thread;
   212     IntVector _rev_thread;
   213     IntVector _succ_num;
   214     IntVector _last_succ;
   215     IntVector _dirty_revs;
   216     BoolVector _forward;
   217     BoolVector _state;
   218     int _root;
   219 
   220     // Temporary data used in the current pivot iteration
   221     int in_arc, join, u_in, v_in, u_out, v_out;
   222     int first, second, right, last;
   223     int stem, par_stem, new_stem;
   224     Value delta;
   225     
   226     const Value MAX;
   227 
   228   public:
   229   
   230     /// \brief Constant for infinite upper bounds (capacities).
   231     ///
   232     /// Constant for infinite upper bounds (capacities).
   233     /// It is \c std::numeric_limits<Value>::infinity() if available,
   234     /// \c std::numeric_limits<Value>::max() otherwise.
   235     const Value INF;
   236 
   237   private:
   238 
   239     // Implementation of the First Eligible pivot rule
   240     class FirstEligiblePivotRule
   241     {
   242     private:
   243 
   244       // References to the NetworkSimplex class
   245       const IntVector  &_source;
   246       const IntVector  &_target;
   247       const CostVector &_cost;
   248       const BoolVector &_state;
   249       const CostVector &_pi;
   250       int &_in_arc;
   251       int _search_arc_num;
   252 
   253       // Pivot rule data
   254       int _next_arc;
   255 
   256     public:
   257 
   258       // Constructor
   259       FirstEligiblePivotRule(NetworkSimplex &ns) :
   260         _source(ns._source), _target(ns._target),
   261         _cost(ns._cost), _state(ns._state), _pi(ns._pi),
   262         _in_arc(ns.in_arc), _search_arc_num(ns._search_arc_num),
   263         _next_arc(0)
   264       {}
   265 
   266       // Find next entering arc
   267       bool findEnteringArc() {
   268         Cost c;
   269         for (int e = _next_arc; e != _search_arc_num; ++e) {
   270           c = _state[e] * (_cost[e] + _pi[_source[e]] - _pi[_target[e]]);
   271           if (c < 0) {
   272             _in_arc = e;
   273             _next_arc = e + 1;
   274             return true;
   275           }
   276         }
   277         for (int e = 0; e != _next_arc; ++e) {
   278           c = _state[e] * (_cost[e] + _pi[_source[e]] - _pi[_target[e]]);
   279           if (c < 0) {
   280             _in_arc = e;
   281             _next_arc = e + 1;
   282             return true;
   283           }
   284         }
   285         return false;
   286       }
   287 
   288     }; //class FirstEligiblePivotRule
   289 
   290 
   291     // Implementation of the Best Eligible pivot rule
   292     class BestEligiblePivotRule
   293     {
   294     private:
   295 
   296       // References to the NetworkSimplex class
   297       const IntVector  &_source;
   298       const IntVector  &_target;
   299       const CostVector &_cost;
   300       const BoolVector &_state;
   301       const CostVector &_pi;
   302       int &_in_arc;
   303       int _search_arc_num;
   304 
   305     public:
   306 
   307       // Constructor
   308       BestEligiblePivotRule(NetworkSimplex &ns) :
   309         _source(ns._source), _target(ns._target),
   310         _cost(ns._cost), _state(ns._state), _pi(ns._pi),
   311         _in_arc(ns.in_arc), _search_arc_num(ns._search_arc_num)
   312       {}
   313 
   314       // Find next entering arc
   315       bool findEnteringArc() {
   316         Cost c, min = 0;
   317         for (int e = 0; e != _search_arc_num; ++e) {
   318           c = _state[e] * (_cost[e] + _pi[_source[e]] - _pi[_target[e]]);
   319           if (c < min) {
   320             min = c;
   321             _in_arc = e;
   322           }
   323         }
   324         return min < 0;
   325       }
   326 
   327     }; //class BestEligiblePivotRule
   328 
   329 
   330     // Implementation of the Block Search pivot rule
   331     class BlockSearchPivotRule
   332     {
   333     private:
   334 
   335       // References to the NetworkSimplex class
   336       const IntVector  &_source;
   337       const IntVector  &_target;
   338       const CostVector &_cost;
   339       const BoolVector &_state;
   340       const CostVector &_pi;
   341       int &_in_arc;
   342       int _search_arc_num;
   343 
   344       // Pivot rule data
   345       int _block_size;
   346       int _next_arc;
   347 
   348     public:
   349 
   350       // Constructor
   351       BlockSearchPivotRule(NetworkSimplex &ns) :
   352         _source(ns._source), _target(ns._target),
   353         _cost(ns._cost), _state(ns._state), _pi(ns._pi),
   354         _in_arc(ns.in_arc), _search_arc_num(ns._search_arc_num),
   355         _next_arc(0)
   356       {
   357         // The main parameters of the pivot rule
   358         const double BLOCK_SIZE_FACTOR = 1.0;
   359         const int MIN_BLOCK_SIZE = 10;
   360 
   361         _block_size = std::max( int(BLOCK_SIZE_FACTOR *
   362                                     std::sqrt(double(_search_arc_num))),
   363                                 MIN_BLOCK_SIZE );
   364       }
   365 
   366       // Find next entering arc
   367       bool findEnteringArc() {
   368         Cost c, min = 0;
   369         int cnt = _block_size;
   370         int e;
   371         for (e = _next_arc; e != _search_arc_num; ++e) {
   372           c = _state[e] * (_cost[e] + _pi[_source[e]] - _pi[_target[e]]);
   373           if (c < min) {
   374             min = c;
   375             _in_arc = e;
   376           }
   377           if (--cnt == 0) {
   378             if (min < 0) goto search_end;
   379             cnt = _block_size;
   380           }
   381         }
   382         for (e = 0; e != _next_arc; ++e) {
   383           c = _state[e] * (_cost[e] + _pi[_source[e]] - _pi[_target[e]]);
   384           if (c < min) {
   385             min = c;
   386             _in_arc = e;
   387           }
   388           if (--cnt == 0) {
   389             if (min < 0) goto search_end;
   390             cnt = _block_size;
   391           }
   392         }
   393         if (min >= 0) return false;
   394 
   395       search_end:
   396         _next_arc = e;
   397         return true;
   398       }
   399 
   400     }; //class BlockSearchPivotRule
   401 
   402 
   403     // Implementation of the Candidate List pivot rule
   404     class CandidateListPivotRule
   405     {
   406     private:
   407 
   408       // References to the NetworkSimplex class
   409       const IntVector  &_source;
   410       const IntVector  &_target;
   411       const CostVector &_cost;
   412       const BoolVector &_state;
   413       const CostVector &_pi;
   414       int &_in_arc;
   415       int _search_arc_num;
   416 
   417       // Pivot rule data
   418       IntVector _candidates;
   419       int _list_length, _minor_limit;
   420       int _curr_length, _minor_count;
   421       int _next_arc;
   422 
   423     public:
   424 
   425       /// Constructor
   426       CandidateListPivotRule(NetworkSimplex &ns) :
   427         _source(ns._source), _target(ns._target),
   428         _cost(ns._cost), _state(ns._state), _pi(ns._pi),
   429         _in_arc(ns.in_arc), _search_arc_num(ns._search_arc_num),
   430         _next_arc(0)
   431       {
   432         // The main parameters of the pivot rule
   433         const double LIST_LENGTH_FACTOR = 0.25;
   434         const int MIN_LIST_LENGTH = 10;
   435         const double MINOR_LIMIT_FACTOR = 0.1;
   436         const int MIN_MINOR_LIMIT = 3;
   437 
   438         _list_length = std::max( int(LIST_LENGTH_FACTOR *
   439                                      std::sqrt(double(_search_arc_num))),
   440                                  MIN_LIST_LENGTH );
   441         _minor_limit = std::max( int(MINOR_LIMIT_FACTOR * _list_length),
   442                                  MIN_MINOR_LIMIT );
   443         _curr_length = _minor_count = 0;
   444         _candidates.resize(_list_length);
   445       }
   446 
   447       /// Find next entering arc
   448       bool findEnteringArc() {
   449         Cost min, c;
   450         int e;
   451         if (_curr_length > 0 && _minor_count < _minor_limit) {
   452           // Minor iteration: select the best eligible arc from the
   453           // current candidate list
   454           ++_minor_count;
   455           min = 0;
   456           for (int i = 0; i < _curr_length; ++i) {
   457             e = _candidates[i];
   458             c = _state[e] * (_cost[e] + _pi[_source[e]] - _pi[_target[e]]);
   459             if (c < min) {
   460               min = c;
   461               _in_arc = e;
   462             }
   463             else if (c >= 0) {
   464               _candidates[i--] = _candidates[--_curr_length];
   465             }
   466           }
   467           if (min < 0) return true;
   468         }
   469 
   470         // Major iteration: build a new candidate list
   471         min = 0;
   472         _curr_length = 0;
   473         for (e = _next_arc; e != _search_arc_num; ++e) {
   474           c = _state[e] * (_cost[e] + _pi[_source[e]] - _pi[_target[e]]);
   475           if (c < 0) {
   476             _candidates[_curr_length++] = e;
   477             if (c < min) {
   478               min = c;
   479               _in_arc = e;
   480             }
   481             if (_curr_length == _list_length) goto search_end;
   482           }
   483         }
   484         for (e = 0; e != _next_arc; ++e) {
   485           c = _state[e] * (_cost[e] + _pi[_source[e]] - _pi[_target[e]]);
   486           if (c < 0) {
   487             _candidates[_curr_length++] = e;
   488             if (c < min) {
   489               min = c;
   490               _in_arc = e;
   491             }
   492             if (_curr_length == _list_length) goto search_end;
   493           }
   494         }
   495         if (_curr_length == 0) return false;
   496       
   497       search_end:        
   498         _minor_count = 1;
   499         _next_arc = e;
   500         return true;
   501       }
   502 
   503     }; //class CandidateListPivotRule
   504 
   505 
   506     // Implementation of the Altering Candidate List pivot rule
   507     class AlteringListPivotRule
   508     {
   509     private:
   510 
   511       // References to the NetworkSimplex class
   512       const IntVector  &_source;
   513       const IntVector  &_target;
   514       const CostVector &_cost;
   515       const BoolVector &_state;
   516       const CostVector &_pi;
   517       int &_in_arc;
   518       int _search_arc_num;
   519 
   520       // Pivot rule data
   521       int _block_size, _head_length, _curr_length;
   522       int _next_arc;
   523       IntVector _candidates;
   524       CostVector _cand_cost;
   525 
   526       // Functor class to compare arcs during sort of the candidate list
   527       class SortFunc
   528       {
   529       private:
   530         const CostVector &_map;
   531       public:
   532         SortFunc(const CostVector &map) : _map(map) {}
   533         bool operator()(int left, int right) {
   534           return _map[left] > _map[right];
   535         }
   536       };
   537 
   538       SortFunc _sort_func;
   539 
   540     public:
   541 
   542       // Constructor
   543       AlteringListPivotRule(NetworkSimplex &ns) :
   544         _source(ns._source), _target(ns._target),
   545         _cost(ns._cost), _state(ns._state), _pi(ns._pi),
   546         _in_arc(ns.in_arc), _search_arc_num(ns._search_arc_num),
   547         _next_arc(0), _cand_cost(ns._search_arc_num), _sort_func(_cand_cost)
   548       {
   549         // The main parameters of the pivot rule
   550         const double BLOCK_SIZE_FACTOR = 1.0;
   551         const int MIN_BLOCK_SIZE = 10;
   552         const double HEAD_LENGTH_FACTOR = 0.1;
   553         const int MIN_HEAD_LENGTH = 3;
   554 
   555         _block_size = std::max( int(BLOCK_SIZE_FACTOR *
   556                                     std::sqrt(double(_search_arc_num))),
   557                                 MIN_BLOCK_SIZE );
   558         _head_length = std::max( int(HEAD_LENGTH_FACTOR * _block_size),
   559                                  MIN_HEAD_LENGTH );
   560         _candidates.resize(_head_length + _block_size);
   561         _curr_length = 0;
   562       }
   563 
   564       // Find next entering arc
   565       bool findEnteringArc() {
   566         // Check the current candidate list
   567         int e;
   568         for (int i = 0; i != _curr_length; ++i) {
   569           e = _candidates[i];
   570           _cand_cost[e] = _state[e] *
   571             (_cost[e] + _pi[_source[e]] - _pi[_target[e]]);
   572           if (_cand_cost[e] >= 0) {
   573             _candidates[i--] = _candidates[--_curr_length];
   574           }
   575         }
   576 
   577         // Extend the list
   578         int cnt = _block_size;
   579         int limit = _head_length;
   580 
   581         for (e = _next_arc; e != _search_arc_num; ++e) {
   582           _cand_cost[e] = _state[e] *
   583             (_cost[e] + _pi[_source[e]] - _pi[_target[e]]);
   584           if (_cand_cost[e] < 0) {
   585             _candidates[_curr_length++] = e;
   586           }
   587           if (--cnt == 0) {
   588             if (_curr_length > limit) goto search_end;
   589             limit = 0;
   590             cnt = _block_size;
   591           }
   592         }
   593         for (e = 0; e != _next_arc; ++e) {
   594           _cand_cost[e] = _state[e] *
   595             (_cost[e] + _pi[_source[e]] - _pi[_target[e]]);
   596           if (_cand_cost[e] < 0) {
   597             _candidates[_curr_length++] = e;
   598           }
   599           if (--cnt == 0) {
   600             if (_curr_length > limit) goto search_end;
   601             limit = 0;
   602             cnt = _block_size;
   603           }
   604         }
   605         if (_curr_length == 0) return false;
   606         
   607       search_end:
   608 
   609         // Make heap of the candidate list (approximating a partial sort)
   610         make_heap( _candidates.begin(), _candidates.begin() + _curr_length,
   611                    _sort_func );
   612 
   613         // Pop the first element of the heap
   614         _in_arc = _candidates[0];
   615         _next_arc = e;
   616         pop_heap( _candidates.begin(), _candidates.begin() + _curr_length,
   617                   _sort_func );
   618         _curr_length = std::min(_head_length, _curr_length - 1);
   619         return true;
   620       }
   621 
   622     }; //class AlteringListPivotRule
   623 
   624   public:
   625 
   626     /// \brief Constructor.
   627     ///
   628     /// The constructor of the class.
   629     ///
   630     /// \param graph The digraph the algorithm runs on.
   631     /// \param arc_mixing Indicate if the arcs have to be stored in a
   632     /// mixed order in the internal data structure. 
   633     /// In special cases, it could lead to better overall performance,
   634     /// but it is usually slower. Therefore it is disabled by default.
   635     NetworkSimplex(const GR& graph, bool arc_mixing = false) :
   636       _graph(graph), _node_id(graph), _arc_id(graph),
   637       MAX(std::numeric_limits<Value>::max()),
   638       INF(std::numeric_limits<Value>::has_infinity ?
   639           std::numeric_limits<Value>::infinity() : MAX)
   640     {
   641       // Check the number types
   642       LEMON_ASSERT(std::numeric_limits<Value>::is_signed,
   643         "The flow type of NetworkSimplex must be signed");
   644       LEMON_ASSERT(std::numeric_limits<Cost>::is_signed,
   645         "The cost type of NetworkSimplex must be signed");
   646         
   647       // Resize vectors
   648       _node_num = countNodes(_graph);
   649       _arc_num = countArcs(_graph);
   650       int all_node_num = _node_num + 1;
   651       int max_arc_num = _arc_num + 2 * _node_num;
   652 
   653       _source.resize(max_arc_num);
   654       _target.resize(max_arc_num);
   655 
   656       _lower.resize(_arc_num);
   657       _upper.resize(_arc_num);
   658       _cap.resize(max_arc_num);
   659       _cost.resize(max_arc_num);
   660       _supply.resize(all_node_num);
   661       _flow.resize(max_arc_num);
   662       _pi.resize(all_node_num);
   663 
   664       _parent.resize(all_node_num);
   665       _pred.resize(all_node_num);
   666       _forward.resize(all_node_num);
   667       _thread.resize(all_node_num);
   668       _rev_thread.resize(all_node_num);
   669       _succ_num.resize(all_node_num);
   670       _last_succ.resize(all_node_num);
   671       _state.resize(max_arc_num);
   672 
   673       // Copy the graph
   674       int i = 0;
   675       for (NodeIt n(_graph); n != INVALID; ++n, ++i) {
   676         _node_id[n] = i;
   677       }
   678       if (arc_mixing) {
   679         // Store the arcs in a mixed order
   680         int k = std::max(int(std::sqrt(double(_arc_num))), 10);
   681         int i = 0, j = 0;
   682         for (ArcIt a(_graph); a != INVALID; ++a) {
   683           _arc_id[a] = i;
   684           _source[i] = _node_id[_graph.source(a)];
   685           _target[i] = _node_id[_graph.target(a)];
   686           if ((i += k) >= _arc_num) i = ++j;
   687         }
   688       } else {
   689         // Store the arcs in the original order
   690         int i = 0;
   691         for (ArcIt a(_graph); a != INVALID; ++a, ++i) {
   692           _arc_id[a] = i;
   693           _source[i] = _node_id[_graph.source(a)];
   694           _target[i] = _node_id[_graph.target(a)];
   695         }
   696       }
   697       
   698       // Reset parameters
   699       reset();
   700     }
   701 
   702     /// \name Parameters
   703     /// The parameters of the algorithm can be specified using these
   704     /// functions.
   705 
   706     /// @{
   707 
   708     /// \brief Set the lower bounds on the arcs.
   709     ///
   710     /// This function sets the lower bounds on the arcs.
   711     /// If it is not used before calling \ref run(), the lower bounds
   712     /// will be set to zero on all arcs.
   713     ///
   714     /// \param map An arc map storing the lower bounds.
   715     /// Its \c Value type must be convertible to the \c Value type
   716     /// of the algorithm.
   717     ///
   718     /// \return <tt>(*this)</tt>
   719     template <typename LowerMap>
   720     NetworkSimplex& lowerMap(const LowerMap& map) {
   721       _have_lower = true;
   722       for (ArcIt a(_graph); a != INVALID; ++a) {
   723         _lower[_arc_id[a]] = map[a];
   724       }
   725       return *this;
   726     }
   727 
   728     /// \brief Set the upper bounds (capacities) on the arcs.
   729     ///
   730     /// This function sets the upper bounds (capacities) on the arcs.
   731     /// If it is not used before calling \ref run(), the upper bounds
   732     /// will be set to \ref INF on all arcs (i.e. the flow value will be
   733     /// unbounded from above).
   734     ///
   735     /// \param map An arc map storing the upper bounds.
   736     /// Its \c Value type must be convertible to the \c Value type
   737     /// of the algorithm.
   738     ///
   739     /// \return <tt>(*this)</tt>
   740     template<typename UpperMap>
   741     NetworkSimplex& upperMap(const UpperMap& map) {
   742       for (ArcIt a(_graph); a != INVALID; ++a) {
   743         _upper[_arc_id[a]] = map[a];
   744       }
   745       return *this;
   746     }
   747 
   748     /// \brief Set the costs of the arcs.
   749     ///
   750     /// This function sets the costs of the arcs.
   751     /// If it is not used before calling \ref run(), the costs
   752     /// will be set to \c 1 on all arcs.
   753     ///
   754     /// \param map An arc map storing the costs.
   755     /// Its \c Value type must be convertible to the \c Cost type
   756     /// of the algorithm.
   757     ///
   758     /// \return <tt>(*this)</tt>
   759     template<typename CostMap>
   760     NetworkSimplex& costMap(const CostMap& map) {
   761       for (ArcIt a(_graph); a != INVALID; ++a) {
   762         _cost[_arc_id[a]] = map[a];
   763       }
   764       return *this;
   765     }
   766 
   767     /// \brief Set the supply values of the nodes.
   768     ///
   769     /// This function sets the supply values of the nodes.
   770     /// If neither this function nor \ref stSupply() is used before
   771     /// calling \ref run(), the supply of each node will be set to zero.
   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     ///
   793     /// Using this function has the same effect as using \ref supplyMap()
   794     /// with such a map in which \c k is assigned to \c s, \c -k is
   795     /// assigned to \c t and all other nodes have zero supply value.
   796     ///
   797     /// \param s The source node.
   798     /// \param t The target node.
   799     /// \param k The required amount of flow from node \c s to node \c t
   800     /// (i.e. the supply of \c s and the demand of \c t).
   801     ///
   802     /// \return <tt>(*this)</tt>
   803     NetworkSimplex& stSupply(const Node& s, const Node& t, Value k) {
   804       for (int i = 0; i != _node_num; ++i) {
   805         _supply[i] = 0;
   806       }
   807       _supply[_node_id[s]] =  k;
   808       _supply[_node_id[t]] = -k;
   809       return *this;
   810     }
   811     
   812     /// \brief Set the type of the supply constraints.
   813     ///
   814     /// This function sets the type of the supply/demand constraints.
   815     /// If it is not used before calling \ref run(), the \ref GEQ supply
   816     /// type will be used.
   817     ///
   818     /// For more information, see \ref SupplyType.
   819     ///
   820     /// \return <tt>(*this)</tt>
   821     NetworkSimplex& supplyType(SupplyType supply_type) {
   822       _stype = supply_type;
   823       return *this;
   824     }
   825 
   826     /// @}
   827 
   828     /// \name Execution Control
   829     /// The algorithm can be executed using \ref run().
   830 
   831     /// @{
   832 
   833     /// \brief Run the algorithm.
   834     ///
   835     /// This function runs the algorithm.
   836     /// The paramters can be specified using functions \ref lowerMap(),
   837     /// \ref upperMap(), \ref costMap(), \ref supplyMap(), \ref stSupply(), 
   838     /// \ref supplyType().
   839     /// For example,
   840     /// \code
   841     ///   NetworkSimplex<ListDigraph> ns(graph);
   842     ///   ns.lowerMap(lower).upperMap(upper).costMap(cost)
   843     ///     .supplyMap(sup).run();
   844     /// \endcode
   845     ///
   846     /// This function can be called more than once. All the parameters
   847     /// that have been given are kept for the next call, unless
   848     /// \ref reset() is called, thus only the modified parameters
   849     /// have to be set again. See \ref reset() for examples.
   850     /// However, the underlying digraph must not be modified after this
   851     /// class have been constructed, since it copies and extends the graph.
   852     ///
   853     /// \param pivot_rule The pivot rule that will be used during the
   854     /// algorithm. For more information, see \ref PivotRule.
   855     ///
   856     /// \return \c INFEASIBLE if no feasible flow exists,
   857     /// \n \c OPTIMAL if the problem has optimal solution
   858     /// (i.e. it is feasible and bounded), and the algorithm has found
   859     /// optimal flow and node potentials (primal and dual solutions),
   860     /// \n \c UNBOUNDED if the objective function of the problem is
   861     /// unbounded, i.e. there is a directed cycle having negative total
   862     /// cost and infinite upper bound.
   863     ///
   864     /// \see ProblemType, PivotRule
   865     ProblemType run(PivotRule pivot_rule = BLOCK_SEARCH) {
   866       if (!init()) return INFEASIBLE;
   867       return start(pivot_rule);
   868     }
   869 
   870     /// \brief Reset all the parameters that have been given before.
   871     ///
   872     /// This function resets all the paramaters that have been given
   873     /// before using functions \ref lowerMap(), \ref upperMap(),
   874     /// \ref costMap(), \ref supplyMap(), \ref stSupply(), \ref supplyType().
   875     ///
   876     /// It is useful for multiple run() calls. If this function is not
   877     /// used, all the parameters given before are kept for the next
   878     /// \ref run() call.
   879     /// However, the underlying digraph must not be modified after this
   880     /// class have been constructed, since it copies and extends the graph.
   881     ///
   882     /// For example,
   883     /// \code
   884     ///   NetworkSimplex<ListDigraph> ns(graph);
   885     ///
   886     ///   // First run
   887     ///   ns.lowerMap(lower).upperMap(upper).costMap(cost)
   888     ///     .supplyMap(sup).run();
   889     ///
   890     ///   // Run again with modified cost map (reset() is not called,
   891     ///   // so only the cost map have to be set again)
   892     ///   cost[e] += 100;
   893     ///   ns.costMap(cost).run();
   894     ///
   895     ///   // Run again from scratch using reset()
   896     ///   // (the lower bounds will be set to zero on all arcs)
   897     ///   ns.reset();
   898     ///   ns.upperMap(capacity).costMap(cost)
   899     ///     .supplyMap(sup).run();
   900     /// \endcode
   901     ///
   902     /// \return <tt>(*this)</tt>
   903     NetworkSimplex& reset() {
   904       for (int i = 0; i != _node_num; ++i) {
   905         _supply[i] = 0;
   906       }
   907       for (int i = 0; i != _arc_num; ++i) {
   908         _lower[i] = 0;
   909         _upper[i] = INF;
   910         _cost[i] = 1;
   911       }
   912       _have_lower = false;
   913       _stype = GEQ;
   914       return *this;
   915     }
   916 
   917     /// @}
   918 
   919     /// \name Query Functions
   920     /// The results of the algorithm can be obtained using these
   921     /// functions.\n
   922     /// The \ref run() function must be called before using them.
   923 
   924     /// @{
   925 
   926     /// \brief Return the total cost of the found flow.
   927     ///
   928     /// This function returns the total cost of the found flow.
   929     /// Its complexity is O(e).
   930     ///
   931     /// \note The return type of the function can be specified as a
   932     /// template parameter. For example,
   933     /// \code
   934     ///   ns.totalCost<double>();
   935     /// \endcode
   936     /// It is useful if the total cost cannot be stored in the \c Cost
   937     /// type of the algorithm, which is the default return type of the
   938     /// function.
   939     ///
   940     /// \pre \ref run() must be called before using this function.
   941     template <typename Number>
   942     Number totalCost() const {
   943       Number c = 0;
   944       for (ArcIt a(_graph); a != INVALID; ++a) {
   945         int i = _arc_id[a];
   946         c += Number(_flow[i]) * Number(_cost[i]);
   947       }
   948       return c;
   949     }
   950 
   951 #ifndef DOXYGEN
   952     Cost totalCost() const {
   953       return totalCost<Cost>();
   954     }
   955 #endif
   956 
   957     /// \brief Return the flow on the given arc.
   958     ///
   959     /// This function returns the flow on the given arc.
   960     ///
   961     /// \pre \ref run() must be called before using this function.
   962     Value flow(const Arc& a) const {
   963       return _flow[_arc_id[a]];
   964     }
   965 
   966     /// \brief Return the flow map (the primal solution).
   967     ///
   968     /// This function copies the flow value on each arc into the given
   969     /// map. The \c Value type of the algorithm must be convertible to
   970     /// the \c Value type of the map.
   971     ///
   972     /// \pre \ref run() must be called before using this function.
   973     template <typename FlowMap>
   974     void flowMap(FlowMap &map) const {
   975       for (ArcIt a(_graph); a != INVALID; ++a) {
   976         map.set(a, _flow[_arc_id[a]]);
   977       }
   978     }
   979 
   980     /// \brief Return the potential (dual value) of the given node.
   981     ///
   982     /// This function returns the potential (dual value) of the
   983     /// given node.
   984     ///
   985     /// \pre \ref run() must be called before using this function.
   986     Cost potential(const Node& n) const {
   987       return _pi[_node_id[n]];
   988     }
   989 
   990     /// \brief Return the potential map (the dual solution).
   991     ///
   992     /// This function copies the potential (dual value) of each node
   993     /// into the given map.
   994     /// The \c Cost type of the algorithm must be convertible to the
   995     /// \c Value type of the map.
   996     ///
   997     /// \pre \ref run() must be called before using this function.
   998     template <typename PotentialMap>
   999     void potentialMap(PotentialMap &map) const {
  1000       for (NodeIt n(_graph); n != INVALID; ++n) {
  1001         map.set(n, _pi[_node_id[n]]);
  1002       }
  1003     }
  1004 
  1005     /// @}
  1006 
  1007   private:
  1008 
  1009     // Initialize internal data structures
  1010     bool init() {
  1011       if (_node_num == 0) return false;
  1012 
  1013       // Check the sum of supply values
  1014       _sum_supply = 0;
  1015       for (int i = 0; i != _node_num; ++i) {
  1016         _sum_supply += _supply[i];
  1017       }
  1018       if ( !((_stype == GEQ && _sum_supply <= 0) ||
  1019              (_stype == LEQ && _sum_supply >= 0)) ) return false;
  1020 
  1021       // Remove non-zero lower bounds
  1022       if (_have_lower) {
  1023         for (int i = 0; i != _arc_num; ++i) {
  1024           Value c = _lower[i];
  1025           if (c >= 0) {
  1026             _cap[i] = _upper[i] < MAX ? _upper[i] - c : INF;
  1027           } else {
  1028             _cap[i] = _upper[i] < MAX + c ? _upper[i] - c : INF;
  1029           }
  1030           _supply[_source[i]] -= c;
  1031           _supply[_target[i]] += c;
  1032         }
  1033       } else {
  1034         for (int i = 0; i != _arc_num; ++i) {
  1035           _cap[i] = _upper[i];
  1036         }
  1037       }
  1038 
  1039       // Initialize artifical cost
  1040       Cost ART_COST;
  1041       if (std::numeric_limits<Cost>::is_exact) {
  1042         ART_COST = std::numeric_limits<Cost>::max() / 2 + 1;
  1043       } else {
  1044         ART_COST = std::numeric_limits<Cost>::min();
  1045         for (int i = 0; i != _arc_num; ++i) {
  1046           if (_cost[i] > ART_COST) ART_COST = _cost[i];
  1047         }
  1048         ART_COST = (ART_COST + 1) * _node_num;
  1049       }
  1050 
  1051       // Initialize arc maps
  1052       for (int i = 0; i != _arc_num; ++i) {
  1053         _flow[i] = 0;
  1054         _state[i] = STATE_LOWER;
  1055       }
  1056       
  1057       // Set data for the artificial root node
  1058       _root = _node_num;
  1059       _parent[_root] = -1;
  1060       _pred[_root] = -1;
  1061       _thread[_root] = 0;
  1062       _rev_thread[0] = _root;
  1063       _succ_num[_root] = _node_num + 1;
  1064       _last_succ[_root] = _root - 1;
  1065       _supply[_root] = -_sum_supply;
  1066       _pi[_root] = 0;
  1067 
  1068       // Add artificial arcs and initialize the spanning tree data structure
  1069       if (_sum_supply == 0) {
  1070         // EQ supply constraints
  1071         _search_arc_num = _arc_num;
  1072         _all_arc_num = _arc_num + _node_num;
  1073         for (int u = 0, e = _arc_num; u != _node_num; ++u, ++e) {
  1074           _parent[u] = _root;
  1075           _pred[u] = e;
  1076           _thread[u] = u + 1;
  1077           _rev_thread[u + 1] = u;
  1078           _succ_num[u] = 1;
  1079           _last_succ[u] = u;
  1080           _cap[e] = INF;
  1081           _state[e] = STATE_TREE;
  1082           if (_supply[u] >= 0) {
  1083             _forward[u] = true;
  1084             _pi[u] = 0;
  1085             _source[e] = u;
  1086             _target[e] = _root;
  1087             _flow[e] = _supply[u];
  1088             _cost[e] = 0;
  1089           } else {
  1090             _forward[u] = false;
  1091             _pi[u] = ART_COST;
  1092             _source[e] = _root;
  1093             _target[e] = u;
  1094             _flow[e] = -_supply[u];
  1095             _cost[e] = ART_COST;
  1096           }
  1097         }
  1098       }
  1099       else if (_sum_supply > 0) {
  1100         // LEQ supply constraints
  1101         _search_arc_num = _arc_num + _node_num;
  1102         int f = _arc_num + _node_num;
  1103         for (int u = 0, e = _arc_num; u != _node_num; ++u, ++e) {
  1104           _parent[u] = _root;
  1105           _thread[u] = u + 1;
  1106           _rev_thread[u + 1] = u;
  1107           _succ_num[u] = 1;
  1108           _last_succ[u] = u;
  1109           if (_supply[u] >= 0) {
  1110             _forward[u] = true;
  1111             _pi[u] = 0;
  1112             _pred[u] = e;
  1113             _source[e] = u;
  1114             _target[e] = _root;
  1115             _cap[e] = INF;
  1116             _flow[e] = _supply[u];
  1117             _cost[e] = 0;
  1118             _state[e] = STATE_TREE;
  1119           } else {
  1120             _forward[u] = false;
  1121             _pi[u] = ART_COST;
  1122             _pred[u] = f;
  1123             _source[f] = _root;
  1124             _target[f] = u;
  1125             _cap[f] = INF;
  1126             _flow[f] = -_supply[u];
  1127             _cost[f] = ART_COST;
  1128             _state[f] = STATE_TREE;
  1129             _source[e] = u;
  1130             _target[e] = _root;
  1131             _cap[e] = INF;
  1132             _flow[e] = 0;
  1133             _cost[e] = 0;
  1134             _state[e] = STATE_LOWER;
  1135             ++f;
  1136           }
  1137         }
  1138         _all_arc_num = f;
  1139       }
  1140       else {
  1141         // GEQ supply constraints
  1142         _search_arc_num = _arc_num + _node_num;
  1143         int f = _arc_num + _node_num;
  1144         for (int u = 0, e = _arc_num; u != _node_num; ++u, ++e) {
  1145           _parent[u] = _root;
  1146           _thread[u] = u + 1;
  1147           _rev_thread[u + 1] = u;
  1148           _succ_num[u] = 1;
  1149           _last_succ[u] = u;
  1150           if (_supply[u] <= 0) {
  1151             _forward[u] = false;
  1152             _pi[u] = 0;
  1153             _pred[u] = e;
  1154             _source[e] = _root;
  1155             _target[e] = u;
  1156             _cap[e] = INF;
  1157             _flow[e] = -_supply[u];
  1158             _cost[e] = 0;
  1159             _state[e] = STATE_TREE;
  1160           } else {
  1161             _forward[u] = true;
  1162             _pi[u] = -ART_COST;
  1163             _pred[u] = f;
  1164             _source[f] = u;
  1165             _target[f] = _root;
  1166             _cap[f] = INF;
  1167             _flow[f] = _supply[u];
  1168             _state[f] = STATE_TREE;
  1169             _cost[f] = ART_COST;
  1170             _source[e] = _root;
  1171             _target[e] = u;
  1172             _cap[e] = INF;
  1173             _flow[e] = 0;
  1174             _cost[e] = 0;
  1175             _state[e] = STATE_LOWER;
  1176             ++f;
  1177           }
  1178         }
  1179         _all_arc_num = f;
  1180       }
  1181 
  1182       return true;
  1183     }
  1184 
  1185     // Find the join node
  1186     void findJoinNode() {
  1187       int u = _source[in_arc];
  1188       int v = _target[in_arc];
  1189       while (u != v) {
  1190         if (_succ_num[u] < _succ_num[v]) {
  1191           u = _parent[u];
  1192         } else {
  1193           v = _parent[v];
  1194         }
  1195       }
  1196       join = u;
  1197     }
  1198 
  1199     // Find the leaving arc of the cycle and returns true if the
  1200     // leaving arc is not the same as the entering arc
  1201     bool findLeavingArc() {
  1202       // Initialize first and second nodes according to the direction
  1203       // of the cycle
  1204       if (_state[in_arc] == STATE_LOWER) {
  1205         first  = _source[in_arc];
  1206         second = _target[in_arc];
  1207       } else {
  1208         first  = _target[in_arc];
  1209         second = _source[in_arc];
  1210       }
  1211       delta = _cap[in_arc];
  1212       int result = 0;
  1213       Value d;
  1214       int e;
  1215 
  1216       // Search the cycle along the path form the first node to the root
  1217       for (int u = first; u != join; u = _parent[u]) {
  1218         e = _pred[u];
  1219         d = _forward[u] ?
  1220           _flow[e] : (_cap[e] >= MAX ? INF : _cap[e] - _flow[e]);
  1221         if (d < delta) {
  1222           delta = d;
  1223           u_out = u;
  1224           result = 1;
  1225         }
  1226       }
  1227       // Search the cycle along the path form the second node to the root
  1228       for (int u = second; u != join; u = _parent[u]) {
  1229         e = _pred[u];
  1230         d = _forward[u] ? 
  1231           (_cap[e] >= MAX ? INF : _cap[e] - _flow[e]) : _flow[e];
  1232         if (d <= delta) {
  1233           delta = d;
  1234           u_out = u;
  1235           result = 2;
  1236         }
  1237       }
  1238 
  1239       if (result == 1) {
  1240         u_in = first;
  1241         v_in = second;
  1242       } else {
  1243         u_in = second;
  1244         v_in = first;
  1245       }
  1246       return result != 0;
  1247     }
  1248 
  1249     // Change _flow and _state vectors
  1250     void changeFlow(bool change) {
  1251       // Augment along the cycle
  1252       if (delta > 0) {
  1253         Value val = _state[in_arc] * delta;
  1254         _flow[in_arc] += val;
  1255         for (int u = _source[in_arc]; u != join; u = _parent[u]) {
  1256           _flow[_pred[u]] += _forward[u] ? -val : val;
  1257         }
  1258         for (int u = _target[in_arc]; u != join; u = _parent[u]) {
  1259           _flow[_pred[u]] += _forward[u] ? val : -val;
  1260         }
  1261       }
  1262       // Update the state of the entering and leaving arcs
  1263       if (change) {
  1264         _state[in_arc] = STATE_TREE;
  1265         _state[_pred[u_out]] =
  1266           (_flow[_pred[u_out]] == 0) ? STATE_LOWER : STATE_UPPER;
  1267       } else {
  1268         _state[in_arc] = -_state[in_arc];
  1269       }
  1270     }
  1271 
  1272     // Update the tree structure
  1273     void updateTreeStructure() {
  1274       int u, w;
  1275       int old_rev_thread = _rev_thread[u_out];
  1276       int old_succ_num = _succ_num[u_out];
  1277       int old_last_succ = _last_succ[u_out];
  1278       v_out = _parent[u_out];
  1279 
  1280       u = _last_succ[u_in];  // the last successor of u_in
  1281       right = _thread[u];    // the node after it
  1282 
  1283       // Handle the case when old_rev_thread equals to v_in
  1284       // (it also means that join and v_out coincide)
  1285       if (old_rev_thread == v_in) {
  1286         last = _thread[_last_succ[u_out]];
  1287       } else {
  1288         last = _thread[v_in];
  1289       }
  1290 
  1291       // Update _thread and _parent along the stem nodes (i.e. the nodes
  1292       // between u_in and u_out, whose parent have to be changed)
  1293       _thread[v_in] = stem = u_in;
  1294       _dirty_revs.clear();
  1295       _dirty_revs.push_back(v_in);
  1296       par_stem = v_in;
  1297       while (stem != u_out) {
  1298         // Insert the next stem node into the thread list
  1299         new_stem = _parent[stem];
  1300         _thread[u] = new_stem;
  1301         _dirty_revs.push_back(u);
  1302 
  1303         // Remove the subtree of stem from the thread list
  1304         w = _rev_thread[stem];
  1305         _thread[w] = right;
  1306         _rev_thread[right] = w;
  1307 
  1308         // Change the parent node and shift stem nodes
  1309         _parent[stem] = par_stem;
  1310         par_stem = stem;
  1311         stem = new_stem;
  1312 
  1313         // Update u and right
  1314         u = _last_succ[stem] == _last_succ[par_stem] ?
  1315           _rev_thread[par_stem] : _last_succ[stem];
  1316         right = _thread[u];
  1317       }
  1318       _parent[u_out] = par_stem;
  1319       _thread[u] = last;
  1320       _rev_thread[last] = u;
  1321       _last_succ[u_out] = u;
  1322 
  1323       // Remove the subtree of u_out from the thread list except for
  1324       // the case when old_rev_thread equals to v_in
  1325       // (it also means that join and v_out coincide)
  1326       if (old_rev_thread != v_in) {
  1327         _thread[old_rev_thread] = right;
  1328         _rev_thread[right] = old_rev_thread;
  1329       }
  1330 
  1331       // Update _rev_thread using the new _thread values
  1332       for (int i = 0; i != int(_dirty_revs.size()); ++i) {
  1333         u = _dirty_revs[i];
  1334         _rev_thread[_thread[u]] = u;
  1335       }
  1336 
  1337       // Update _pred, _forward, _last_succ and _succ_num for the
  1338       // stem nodes from u_out to u_in
  1339       int tmp_sc = 0, tmp_ls = _last_succ[u_out];
  1340       u = u_out;
  1341       while (u != u_in) {
  1342         w = _parent[u];
  1343         _pred[u] = _pred[w];
  1344         _forward[u] = !_forward[w];
  1345         tmp_sc += _succ_num[u] - _succ_num[w];
  1346         _succ_num[u] = tmp_sc;
  1347         _last_succ[w] = tmp_ls;
  1348         u = w;
  1349       }
  1350       _pred[u_in] = in_arc;
  1351       _forward[u_in] = (u_in == _source[in_arc]);
  1352       _succ_num[u_in] = old_succ_num;
  1353 
  1354       // Set limits for updating _last_succ form v_in and v_out
  1355       // towards the root
  1356       int up_limit_in = -1;
  1357       int up_limit_out = -1;
  1358       if (_last_succ[join] == v_in) {
  1359         up_limit_out = join;
  1360       } else {
  1361         up_limit_in = join;
  1362       }
  1363 
  1364       // Update _last_succ from v_in towards the root
  1365       for (u = v_in; u != up_limit_in && _last_succ[u] == v_in;
  1366            u = _parent[u]) {
  1367         _last_succ[u] = _last_succ[u_out];
  1368       }
  1369       // Update _last_succ from v_out towards the root
  1370       if (join != old_rev_thread && v_in != old_rev_thread) {
  1371         for (u = v_out; u != up_limit_out && _last_succ[u] == old_last_succ;
  1372              u = _parent[u]) {
  1373           _last_succ[u] = old_rev_thread;
  1374         }
  1375       } else {
  1376         for (u = v_out; u != up_limit_out && _last_succ[u] == old_last_succ;
  1377              u = _parent[u]) {
  1378           _last_succ[u] = _last_succ[u_out];
  1379         }
  1380       }
  1381 
  1382       // Update _succ_num from v_in to join
  1383       for (u = v_in; u != join; u = _parent[u]) {
  1384         _succ_num[u] += old_succ_num;
  1385       }
  1386       // Update _succ_num from v_out to join
  1387       for (u = v_out; u != join; u = _parent[u]) {
  1388         _succ_num[u] -= old_succ_num;
  1389       }
  1390     }
  1391 
  1392     // Update potentials
  1393     void updatePotential() {
  1394       Cost sigma = _forward[u_in] ?
  1395         _pi[v_in] - _pi[u_in] - _cost[_pred[u_in]] :
  1396         _pi[v_in] - _pi[u_in] + _cost[_pred[u_in]];
  1397       // Update potentials in the subtree, which has been moved
  1398       int end = _thread[_last_succ[u_in]];
  1399       for (int u = u_in; u != end; u = _thread[u]) {
  1400         _pi[u] += sigma;
  1401       }
  1402     }
  1403 
  1404     // Heuristic initial pivots
  1405     bool initialPivots() {
  1406       Value curr, total = 0;
  1407       std::vector<Node> supply_nodes, demand_nodes;
  1408       for (NodeIt u(_graph); u != INVALID; ++u) {
  1409         curr = _supply[_node_id[u]];
  1410         if (curr > 0) {
  1411           total += curr;
  1412           supply_nodes.push_back(u);
  1413         }
  1414         else if (curr < 0) {
  1415           demand_nodes.push_back(u);
  1416         }
  1417       }
  1418       if (_sum_supply > 0) total -= _sum_supply;
  1419       if (total <= 0) return true;
  1420 
  1421       IntVector arc_vector;
  1422       if (_sum_supply >= 0) {
  1423         if (supply_nodes.size() == 1 && demand_nodes.size() == 1) {
  1424           // Perform a reverse graph search from the sink to the source
  1425           typename GR::template NodeMap<bool> reached(_graph, false);
  1426           Node s = supply_nodes[0], t = demand_nodes[0];
  1427           std::vector<Node> stack;
  1428           reached[t] = true;
  1429           stack.push_back(t);
  1430           while (!stack.empty()) {
  1431             Node u, v = stack.back();
  1432             stack.pop_back();
  1433             if (v == s) break;
  1434             for (InArcIt a(_graph, v); a != INVALID; ++a) {
  1435               if (reached[u = _graph.source(a)]) continue;
  1436               int j = _arc_id[a];
  1437               if (_cap[j] >= total) {
  1438                 arc_vector.push_back(j);
  1439                 reached[u] = true;
  1440                 stack.push_back(u);
  1441               }
  1442             }
  1443           }
  1444         } else {
  1445           // Find the min. cost incomming arc for each demand node
  1446           for (int i = 0; i != int(demand_nodes.size()); ++i) {
  1447             Node v = demand_nodes[i];
  1448             Cost c, min_cost = std::numeric_limits<Cost>::max();
  1449             Arc min_arc = INVALID;
  1450             for (InArcIt a(_graph, v); a != INVALID; ++a) {
  1451               c = _cost[_arc_id[a]];
  1452               if (c < min_cost) {
  1453                 min_cost = c;
  1454                 min_arc = a;
  1455               }
  1456             }
  1457             if (min_arc != INVALID) {
  1458               arc_vector.push_back(_arc_id[min_arc]);
  1459             }
  1460           }
  1461         }
  1462       } else {
  1463         // Find the min. cost outgoing arc for each supply node
  1464         for (int i = 0; i != int(supply_nodes.size()); ++i) {
  1465           Node u = supply_nodes[i];
  1466           Cost c, min_cost = std::numeric_limits<Cost>::max();
  1467           Arc min_arc = INVALID;
  1468           for (OutArcIt a(_graph, u); a != INVALID; ++a) {
  1469             c = _cost[_arc_id[a]];
  1470             if (c < min_cost) {
  1471               min_cost = c;
  1472               min_arc = a;
  1473             }
  1474           }
  1475           if (min_arc != INVALID) {
  1476             arc_vector.push_back(_arc_id[min_arc]);
  1477           }
  1478         }
  1479       }
  1480 
  1481       // Perform heuristic initial pivots
  1482       for (int i = 0; i != int(arc_vector.size()); ++i) {
  1483         in_arc = arc_vector[i];
  1484         if (_state[in_arc] * (_cost[in_arc] + _pi[_source[in_arc]] -
  1485             _pi[_target[in_arc]]) >= 0) continue;
  1486         findJoinNode();
  1487         bool change = findLeavingArc();
  1488         if (delta >= MAX) return false;
  1489         changeFlow(change);
  1490         if (change) {
  1491           updateTreeStructure();
  1492           updatePotential();
  1493         }
  1494       }
  1495       return true;
  1496     }
  1497 
  1498     // Execute the algorithm
  1499     ProblemType start(PivotRule pivot_rule) {
  1500       // Select the pivot rule implementation
  1501       switch (pivot_rule) {
  1502         case FIRST_ELIGIBLE:
  1503           return start<FirstEligiblePivotRule>();
  1504         case BEST_ELIGIBLE:
  1505           return start<BestEligiblePivotRule>();
  1506         case BLOCK_SEARCH:
  1507           return start<BlockSearchPivotRule>();
  1508         case CANDIDATE_LIST:
  1509           return start<CandidateListPivotRule>();
  1510         case ALTERING_LIST:
  1511           return start<AlteringListPivotRule>();
  1512       }
  1513       return INFEASIBLE; // avoid warning
  1514     }
  1515 
  1516     template <typename PivotRuleImpl>
  1517     ProblemType start() {
  1518       PivotRuleImpl pivot(*this);
  1519 
  1520       // Perform heuristic initial pivots
  1521       if (!initialPivots()) return UNBOUNDED;
  1522 
  1523       // Execute the Network Simplex algorithm
  1524       while (pivot.findEnteringArc()) {
  1525         findJoinNode();
  1526         bool change = findLeavingArc();
  1527         if (delta >= MAX) return UNBOUNDED;
  1528         changeFlow(change);
  1529         if (change) {
  1530           updateTreeStructure();
  1531           updatePotential();
  1532         }
  1533       }
  1534       
  1535       // Check feasibility
  1536       for (int e = _search_arc_num; e != _all_arc_num; ++e) {
  1537         if (_flow[e] != 0) return INFEASIBLE;
  1538       }
  1539 
  1540       // Transform the solution and the supply map to the original form
  1541       if (_have_lower) {
  1542         for (int i = 0; i != _arc_num; ++i) {
  1543           Value c = _lower[i];
  1544           if (c != 0) {
  1545             _flow[i] += c;
  1546             _supply[_source[i]] += c;
  1547             _supply[_target[i]] -= c;
  1548           }
  1549         }
  1550       }
  1551       
  1552       // Shift potentials to meet the requirements of the GEQ/LEQ type
  1553       // optimality conditions
  1554       if (_sum_supply == 0) {
  1555         if (_stype == GEQ) {
  1556           Cost max_pot = std::numeric_limits<Cost>::min();
  1557           for (int i = 0; i != _node_num; ++i) {
  1558             if (_pi[i] > max_pot) max_pot = _pi[i];
  1559           }
  1560           if (max_pot > 0) {
  1561             for (int i = 0; i != _node_num; ++i)
  1562               _pi[i] -= max_pot;
  1563           }
  1564         } else {
  1565           Cost min_pot = std::numeric_limits<Cost>::max();
  1566           for (int i = 0; i != _node_num; ++i) {
  1567             if (_pi[i] < min_pot) min_pot = _pi[i];
  1568           }
  1569           if (min_pot < 0) {
  1570             for (int i = 0; i != _node_num; ++i)
  1571               _pi[i] -= min_pot;
  1572           }
  1573         }
  1574       }
  1575 
  1576       return OPTIMAL;
  1577     }
  1578 
  1579   }; //class NetworkSimplex
  1580 
  1581   ///@}
  1582 
  1583 } //namespace lemon
  1584 
  1585 #endif //LEMON_NETWORK_SIMPLEX_H