COIN-OR::LEMON - Graph Library

source: lemon/lemon/network_simplex.h @ 990:dca9eed2c375

Last change on this file since 990:dca9eed2c375 was 990:dca9eed2c375, checked in by Peter Kovacs <kpeter@…>, 9 years ago

Improve the tree update process and a pivot rule (#391)
and make some parts of the code clearer using better names

File size: 50.7 KB
Line 
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) 2003-2010
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
34namespace 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<signed char> CharVector;
170    // Note: vector<signed char> is used instead of vector<ArcState> and
171    // vector<ArcDirection> for efficiency reasons
172
173    // State constants for arcs
174    enum ArcState {
175      STATE_UPPER = -1,
176      STATE_TREE  =  0,
177      STATE_LOWER =  1
178    };
179
180    // Direction constants for tree arcs
181    enum ArcDirection {
182      DIR_DOWN = -1,
183      DIR_UP   =  1
184    };
185
186  private:
187
188    // Data related to the underlying digraph
189    const GR &_graph;
190    int _node_num;
191    int _arc_num;
192    int _all_arc_num;
193    int _search_arc_num;
194
195    // Parameters of the problem
196    bool _have_lower;
197    SupplyType _stype;
198    Value _sum_supply;
199
200    // Data structures for storing the digraph
201    IntNodeMap _node_id;
202    IntArcMap _arc_id;
203    IntVector _source;
204    IntVector _target;
205    bool _arc_mixing;
206
207    // Node and arc data
208    ValueVector _lower;
209    ValueVector _upper;
210    ValueVector _cap;
211    CostVector _cost;
212    ValueVector _supply;
213    ValueVector _flow;
214    CostVector _pi;
215
216    // Data for storing the spanning tree structure
217    IntVector _parent;
218    IntVector _pred;
219    IntVector _thread;
220    IntVector _rev_thread;
221    IntVector _succ_num;
222    IntVector _last_succ;
223    CharVector _pred_dir;
224    CharVector _state;
225    IntVector _dirty_revs;
226    int _root;
227
228    // Temporary data used in the current pivot iteration
229    int in_arc, join, u_in, v_in, u_out, v_out;
230    Value delta;
231
232    const Value MAX;
233
234  public:
235
236    /// \brief Constant for infinite upper bounds (capacities).
237    ///
238    /// Constant for infinite upper bounds (capacities).
239    /// It is \c std::numeric_limits<Value>::infinity() if available,
240    /// \c std::numeric_limits<Value>::max() otherwise.
241    const Value INF;
242
243  private:
244
245    // Implementation of the First Eligible pivot rule
246    class FirstEligiblePivotRule
247    {
248    private:
249
250      // References to the NetworkSimplex class
251      const IntVector  &_source;
252      const IntVector  &_target;
253      const CostVector &_cost;
254      const CharVector &_state;
255      const CostVector &_pi;
256      int &_in_arc;
257      int _search_arc_num;
258
259      // Pivot rule data
260      int _next_arc;
261
262    public:
263
264      // Constructor
265      FirstEligiblePivotRule(NetworkSimplex &ns) :
266        _source(ns._source), _target(ns._target),
267        _cost(ns._cost), _state(ns._state), _pi(ns._pi),
268        _in_arc(ns.in_arc), _search_arc_num(ns._search_arc_num),
269        _next_arc(0)
270      {}
271
272      // Find next entering arc
273      bool findEnteringArc() {
274        Cost c;
275        for (int e = _next_arc; e != _search_arc_num; ++e) {
276          c = _state[e] * (_cost[e] + _pi[_source[e]] - _pi[_target[e]]);
277          if (c < 0) {
278            _in_arc = e;
279            _next_arc = e + 1;
280            return true;
281          }
282        }
283        for (int e = 0; e != _next_arc; ++e) {
284          c = _state[e] * (_cost[e] + _pi[_source[e]] - _pi[_target[e]]);
285          if (c < 0) {
286            _in_arc = e;
287            _next_arc = e + 1;
288            return true;
289          }
290        }
291        return false;
292      }
293
294    }; //class FirstEligiblePivotRule
295
296
297    // Implementation of the Best Eligible pivot rule
298    class BestEligiblePivotRule
299    {
300    private:
301
302      // References to the NetworkSimplex class
303      const IntVector  &_source;
304      const IntVector  &_target;
305      const CostVector &_cost;
306      const CharVector &_state;
307      const CostVector &_pi;
308      int &_in_arc;
309      int _search_arc_num;
310
311    public:
312
313      // Constructor
314      BestEligiblePivotRule(NetworkSimplex &ns) :
315        _source(ns._source), _target(ns._target),
316        _cost(ns._cost), _state(ns._state), _pi(ns._pi),
317        _in_arc(ns.in_arc), _search_arc_num(ns._search_arc_num)
318      {}
319
320      // Find next entering arc
321      bool findEnteringArc() {
322        Cost c, min = 0;
323        for (int e = 0; e != _search_arc_num; ++e) {
324          c = _state[e] * (_cost[e] + _pi[_source[e]] - _pi[_target[e]]);
325          if (c < min) {
326            min = c;
327            _in_arc = e;
328          }
329        }
330        return min < 0;
331      }
332
333    }; //class BestEligiblePivotRule
334
335
336    // Implementation of the Block Search pivot rule
337    class BlockSearchPivotRule
338    {
339    private:
340
341      // References to the NetworkSimplex class
342      const IntVector  &_source;
343      const IntVector  &_target;
344      const CostVector &_cost;
345      const CharVector &_state;
346      const CostVector &_pi;
347      int &_in_arc;
348      int _search_arc_num;
349
350      // Pivot rule data
351      int _block_size;
352      int _next_arc;
353
354    public:
355
356      // Constructor
357      BlockSearchPivotRule(NetworkSimplex &ns) :
358        _source(ns._source), _target(ns._target),
359        _cost(ns._cost), _state(ns._state), _pi(ns._pi),
360        _in_arc(ns.in_arc), _search_arc_num(ns._search_arc_num),
361        _next_arc(0)
362      {
363        // The main parameters of the pivot rule
364        const double BLOCK_SIZE_FACTOR = 1.0;
365        const int MIN_BLOCK_SIZE = 10;
366
367        _block_size = std::max( int(BLOCK_SIZE_FACTOR *
368                                    std::sqrt(double(_search_arc_num))),
369                                MIN_BLOCK_SIZE );
370      }
371
372      // Find next entering arc
373      bool findEnteringArc() {
374        Cost c, min = 0;
375        int cnt = _block_size;
376        int e;
377        for (e = _next_arc; e != _search_arc_num; ++e) {
378          c = _state[e] * (_cost[e] + _pi[_source[e]] - _pi[_target[e]]);
379          if (c < min) {
380            min = c;
381            _in_arc = e;
382          }
383          if (--cnt == 0) {
384            if (min < 0) goto search_end;
385            cnt = _block_size;
386          }
387        }
388        for (e = 0; e != _next_arc; ++e) {
389          c = _state[e] * (_cost[e] + _pi[_source[e]] - _pi[_target[e]]);
390          if (c < min) {
391            min = c;
392            _in_arc = e;
393          }
394          if (--cnt == 0) {
395            if (min < 0) goto search_end;
396            cnt = _block_size;
397          }
398        }
399        if (min >= 0) return false;
400
401      search_end:
402        _next_arc = e;
403        return true;
404      }
405
406    }; //class BlockSearchPivotRule
407
408
409    // Implementation of the Candidate List pivot rule
410    class CandidateListPivotRule
411    {
412    private:
413
414      // References to the NetworkSimplex class
415      const IntVector  &_source;
416      const IntVector  &_target;
417      const CostVector &_cost;
418      const CharVector &_state;
419      const CostVector &_pi;
420      int &_in_arc;
421      int _search_arc_num;
422
423      // Pivot rule data
424      IntVector _candidates;
425      int _list_length, _minor_limit;
426      int _curr_length, _minor_count;
427      int _next_arc;
428
429    public:
430
431      /// Constructor
432      CandidateListPivotRule(NetworkSimplex &ns) :
433        _source(ns._source), _target(ns._target),
434        _cost(ns._cost), _state(ns._state), _pi(ns._pi),
435        _in_arc(ns.in_arc), _search_arc_num(ns._search_arc_num),
436        _next_arc(0)
437      {
438        // The main parameters of the pivot rule
439        const double LIST_LENGTH_FACTOR = 0.25;
440        const int MIN_LIST_LENGTH = 10;
441        const double MINOR_LIMIT_FACTOR = 0.1;
442        const int MIN_MINOR_LIMIT = 3;
443
444        _list_length = std::max( int(LIST_LENGTH_FACTOR *
445                                     std::sqrt(double(_search_arc_num))),
446                                 MIN_LIST_LENGTH );
447        _minor_limit = std::max( int(MINOR_LIMIT_FACTOR * _list_length),
448                                 MIN_MINOR_LIMIT );
449        _curr_length = _minor_count = 0;
450        _candidates.resize(_list_length);
451      }
452
453      /// Find next entering arc
454      bool findEnteringArc() {
455        Cost min, c;
456        int e;
457        if (_curr_length > 0 && _minor_count < _minor_limit) {
458          // Minor iteration: select the best eligible arc from the
459          // current candidate list
460          ++_minor_count;
461          min = 0;
462          for (int i = 0; i < _curr_length; ++i) {
463            e = _candidates[i];
464            c = _state[e] * (_cost[e] + _pi[_source[e]] - _pi[_target[e]]);
465            if (c < min) {
466              min = c;
467              _in_arc = e;
468            }
469            else if (c >= 0) {
470              _candidates[i--] = _candidates[--_curr_length];
471            }
472          }
473          if (min < 0) return true;
474        }
475
476        // Major iteration: build a new candidate list
477        min = 0;
478        _curr_length = 0;
479        for (e = _next_arc; e != _search_arc_num; ++e) {
480          c = _state[e] * (_cost[e] + _pi[_source[e]] - _pi[_target[e]]);
481          if (c < 0) {
482            _candidates[_curr_length++] = e;
483            if (c < min) {
484              min = c;
485              _in_arc = e;
486            }
487            if (_curr_length == _list_length) goto search_end;
488          }
489        }
490        for (e = 0; e != _next_arc; ++e) {
491          c = _state[e] * (_cost[e] + _pi[_source[e]] - _pi[_target[e]]);
492          if (c < 0) {
493            _candidates[_curr_length++] = e;
494            if (c < min) {
495              min = c;
496              _in_arc = e;
497            }
498            if (_curr_length == _list_length) goto search_end;
499          }
500        }
501        if (_curr_length == 0) return false;
502
503      search_end:
504        _minor_count = 1;
505        _next_arc = e;
506        return true;
507      }
508
509    }; //class CandidateListPivotRule
510
511
512    // Implementation of the Altering Candidate List pivot rule
513    class AlteringListPivotRule
514    {
515    private:
516
517      // References to the NetworkSimplex class
518      const IntVector  &_source;
519      const IntVector  &_target;
520      const CostVector &_cost;
521      const CharVector &_state;
522      const CostVector &_pi;
523      int &_in_arc;
524      int _search_arc_num;
525
526      // Pivot rule data
527      int _block_size, _head_length, _curr_length;
528      int _next_arc;
529      IntVector _candidates;
530      CostVector _cand_cost;
531
532      // Functor class to compare arcs during sort of the candidate list
533      class SortFunc
534      {
535      private:
536        const CostVector &_map;
537      public:
538        SortFunc(const CostVector &map) : _map(map) {}
539        bool operator()(int left, int right) {
540          return _map[left] > _map[right];
541        }
542      };
543
544      SortFunc _sort_func;
545
546    public:
547
548      // Constructor
549      AlteringListPivotRule(NetworkSimplex &ns) :
550        _source(ns._source), _target(ns._target),
551        _cost(ns._cost), _state(ns._state), _pi(ns._pi),
552        _in_arc(ns.in_arc), _search_arc_num(ns._search_arc_num),
553        _next_arc(0), _cand_cost(ns._search_arc_num), _sort_func(_cand_cost)
554      {
555        // The main parameters of the pivot rule
556        const double BLOCK_SIZE_FACTOR = 1.0;
557        const int MIN_BLOCK_SIZE = 10;
558        const double HEAD_LENGTH_FACTOR = 0.1;
559        const int MIN_HEAD_LENGTH = 3;
560
561        _block_size = std::max( int(BLOCK_SIZE_FACTOR *
562                                    std::sqrt(double(_search_arc_num))),
563                                MIN_BLOCK_SIZE );
564        _head_length = std::max( int(HEAD_LENGTH_FACTOR * _block_size),
565                                 MIN_HEAD_LENGTH );
566        _candidates.resize(_head_length + _block_size);
567        _curr_length = 0;
568      }
569
570      // Find next entering arc
571      bool findEnteringArc() {
572        // Check the current candidate list
573        int e;
574        Cost c;
575        for (int i = 0; i != _curr_length; ++i) {
576          e = _candidates[i];
577          c = _state[e] * (_cost[e] + _pi[_source[e]] - _pi[_target[e]]);
578          if (c < 0) {
579            _cand_cost[e] = c;
580          } else {
581            _candidates[i--] = _candidates[--_curr_length];
582          }
583        }
584
585        // Extend the list
586        int cnt = _block_size;
587        int limit = _head_length;
588
589        for (e = _next_arc; e != _search_arc_num; ++e) {
590          c = _state[e] * (_cost[e] + _pi[_source[e]] - _pi[_target[e]]);
591          if (c < 0) {
592            _cand_cost[e] = c;
593            _candidates[_curr_length++] = e;
594          }
595          if (--cnt == 0) {
596            if (_curr_length > limit) goto search_end;
597            limit = 0;
598            cnt = _block_size;
599          }
600        }
601        for (e = 0; e != _next_arc; ++e) {
602          _cand_cost[e] = _state[e] *
603            (_cost[e] + _pi[_source[e]] - _pi[_target[e]]);
604          if (_cand_cost[e] < 0) {
605            _candidates[_curr_length++] = e;
606          }
607          if (--cnt == 0) {
608            if (_curr_length > limit) goto search_end;
609            limit = 0;
610            cnt = _block_size;
611          }
612        }
613        if (_curr_length == 0) return false;
614
615      search_end:
616
617        // Make heap of the candidate list (approximating a partial sort)
618        make_heap( _candidates.begin(), _candidates.begin() + _curr_length,
619                   _sort_func );
620
621        // Pop the first element of the heap
622        _in_arc = _candidates[0];
623        _next_arc = e;
624        pop_heap( _candidates.begin(), _candidates.begin() + _curr_length,
625                  _sort_func );
626        _curr_length = std::min(_head_length, _curr_length - 1);
627        return true;
628      }
629
630    }; //class AlteringListPivotRule
631
632  public:
633
634    /// \brief Constructor.
635    ///
636    /// The constructor of the class.
637    ///
638    /// \param graph The digraph the algorithm runs on.
639    /// \param arc_mixing Indicate if the arcs have to be stored in a
640    /// mixed order in the internal data structure.
641    /// In special cases, it could lead to better overall performance,
642    /// but it is usually slower. Therefore it is disabled by default.
643    NetworkSimplex(const GR& graph, bool arc_mixing = false) :
644      _graph(graph), _node_id(graph), _arc_id(graph),
645      _arc_mixing(arc_mixing),
646      MAX(std::numeric_limits<Value>::max()),
647      INF(std::numeric_limits<Value>::has_infinity ?
648          std::numeric_limits<Value>::infinity() : MAX)
649    {
650      // Check the number types
651      LEMON_ASSERT(std::numeric_limits<Value>::is_signed,
652        "The flow type of NetworkSimplex must be signed");
653      LEMON_ASSERT(std::numeric_limits<Cost>::is_signed,
654        "The cost type of NetworkSimplex must be signed");
655
656      // Reset data structures
657      reset();
658    }
659
660    /// \name Parameters
661    /// The parameters of the algorithm can be specified using these
662    /// functions.
663
664    /// @{
665
666    /// \brief Set the lower bounds on the arcs.
667    ///
668    /// This function sets the lower bounds on the arcs.
669    /// If it is not used before calling \ref run(), the lower bounds
670    /// will be set to zero on all arcs.
671    ///
672    /// \param map An arc map storing the lower bounds.
673    /// Its \c Value type must be convertible to the \c Value type
674    /// of the algorithm.
675    ///
676    /// \return <tt>(*this)</tt>
677    template <typename LowerMap>
678    NetworkSimplex& lowerMap(const LowerMap& map) {
679      _have_lower = true;
680      for (ArcIt a(_graph); a != INVALID; ++a) {
681        _lower[_arc_id[a]] = map[a];
682      }
683      return *this;
684    }
685
686    /// \brief Set the upper bounds (capacities) on the arcs.
687    ///
688    /// This function sets the upper bounds (capacities) on the arcs.
689    /// If it is not used before calling \ref run(), the upper bounds
690    /// will be set to \ref INF on all arcs (i.e. the flow value will be
691    /// unbounded from above).
692    ///
693    /// \param map An arc map storing the upper bounds.
694    /// Its \c Value type must be convertible to the \c Value type
695    /// of the algorithm.
696    ///
697    /// \return <tt>(*this)</tt>
698    template<typename UpperMap>
699    NetworkSimplex& upperMap(const UpperMap& map) {
700      for (ArcIt a(_graph); a != INVALID; ++a) {
701        _upper[_arc_id[a]] = map[a];
702      }
703      return *this;
704    }
705
706    /// \brief Set the costs of the arcs.
707    ///
708    /// This function sets the costs of the arcs.
709    /// If it is not used before calling \ref run(), the costs
710    /// will be set to \c 1 on all arcs.
711    ///
712    /// \param map An arc map storing the costs.
713    /// Its \c Value type must be convertible to the \c Cost type
714    /// of the algorithm.
715    ///
716    /// \return <tt>(*this)</tt>
717    template<typename CostMap>
718    NetworkSimplex& costMap(const CostMap& map) {
719      for (ArcIt a(_graph); a != INVALID; ++a) {
720        _cost[_arc_id[a]] = map[a];
721      }
722      return *this;
723    }
724
725    /// \brief Set the supply values of the nodes.
726    ///
727    /// This function sets the supply values of the nodes.
728    /// If neither this function nor \ref stSupply() is used before
729    /// calling \ref run(), the supply of each node will be set to zero.
730    ///
731    /// \param map A node map storing the supply values.
732    /// Its \c Value type must be convertible to the \c Value type
733    /// of the algorithm.
734    ///
735    /// \return <tt>(*this)</tt>
736    template<typename SupplyMap>
737    NetworkSimplex& supplyMap(const SupplyMap& map) {
738      for (NodeIt n(_graph); n != INVALID; ++n) {
739        _supply[_node_id[n]] = map[n];
740      }
741      return *this;
742    }
743
744    /// \brief Set single source and target nodes and a supply value.
745    ///
746    /// This function sets a single source node and a single target node
747    /// and the required flow value.
748    /// If neither this function nor \ref supplyMap() is used before
749    /// calling \ref run(), the supply of each node will be set to zero.
750    ///
751    /// Using this function has the same effect as using \ref supplyMap()
752    /// with such a map in which \c k is assigned to \c s, \c -k is
753    /// assigned to \c t and all other nodes have zero supply value.
754    ///
755    /// \param s The source node.
756    /// \param t The target node.
757    /// \param k The required amount of flow from node \c s to node \c t
758    /// (i.e. the supply of \c s and the demand of \c t).
759    ///
760    /// \return <tt>(*this)</tt>
761    NetworkSimplex& stSupply(const Node& s, const Node& t, Value k) {
762      for (int i = 0; i != _node_num; ++i) {
763        _supply[i] = 0;
764      }
765      _supply[_node_id[s]] =  k;
766      _supply[_node_id[t]] = -k;
767      return *this;
768    }
769
770    /// \brief Set the type of the supply constraints.
771    ///
772    /// This function sets the type of the supply/demand constraints.
773    /// If it is not used before calling \ref run(), the \ref GEQ supply
774    /// type will be used.
775    ///
776    /// For more information, see \ref SupplyType.
777    ///
778    /// \return <tt>(*this)</tt>
779    NetworkSimplex& supplyType(SupplyType supply_type) {
780      _stype = supply_type;
781      return *this;
782    }
783
784    /// @}
785
786    /// \name Execution Control
787    /// The algorithm can be executed using \ref run().
788
789    /// @{
790
791    /// \brief Run the algorithm.
792    ///
793    /// This function runs the algorithm.
794    /// The paramters can be specified using functions \ref lowerMap(),
795    /// \ref upperMap(), \ref costMap(), \ref supplyMap(), \ref stSupply(),
796    /// \ref supplyType().
797    /// For example,
798    /// \code
799    ///   NetworkSimplex<ListDigraph> ns(graph);
800    ///   ns.lowerMap(lower).upperMap(upper).costMap(cost)
801    ///     .supplyMap(sup).run();
802    /// \endcode
803    ///
804    /// This function can be called more than once. All the given parameters
805    /// are kept for the next call, unless \ref resetParams() or \ref reset()
806    /// is used, thus only the modified parameters have to be set again.
807    /// If the underlying digraph was also modified after the construction
808    /// of the class (or the last \ref reset() call), then the \ref reset()
809    /// function must be called.
810    ///
811    /// \param pivot_rule The pivot rule that will be used during the
812    /// algorithm. For more information, see \ref PivotRule.
813    ///
814    /// \return \c INFEASIBLE if no feasible flow exists,
815    /// \n \c OPTIMAL if the problem has optimal solution
816    /// (i.e. it is feasible and bounded), and the algorithm has found
817    /// optimal flow and node potentials (primal and dual solutions),
818    /// \n \c UNBOUNDED if the objective function of the problem is
819    /// unbounded, i.e. there is a directed cycle having negative total
820    /// cost and infinite upper bound.
821    ///
822    /// \see ProblemType, PivotRule
823    /// \see resetParams(), reset()
824    ProblemType run(PivotRule pivot_rule = BLOCK_SEARCH) {
825      if (!init()) return INFEASIBLE;
826      return start(pivot_rule);
827    }
828
829    /// \brief Reset all the parameters that have been given before.
830    ///
831    /// This function resets all the paramaters that have been given
832    /// before using functions \ref lowerMap(), \ref upperMap(),
833    /// \ref costMap(), \ref supplyMap(), \ref stSupply(), \ref supplyType().
834    ///
835    /// It is useful for multiple \ref run() calls. Basically, all the given
836    /// parameters are kept for the next \ref run() call, unless
837    /// \ref resetParams() or \ref reset() is used.
838    /// If the underlying digraph was also modified after the construction
839    /// of the class or the last \ref reset() call, then the \ref reset()
840    /// function must be used, otherwise \ref resetParams() is sufficient.
841    ///
842    /// For example,
843    /// \code
844    ///   NetworkSimplex<ListDigraph> ns(graph);
845    ///
846    ///   // First run
847    ///   ns.lowerMap(lower).upperMap(upper).costMap(cost)
848    ///     .supplyMap(sup).run();
849    ///
850    ///   // Run again with modified cost map (resetParams() is not called,
851    ///   // so only the cost map have to be set again)
852    ///   cost[e] += 100;
853    ///   ns.costMap(cost).run();
854    ///
855    ///   // Run again from scratch using resetParams()
856    ///   // (the lower bounds will be set to zero on all arcs)
857    ///   ns.resetParams();
858    ///   ns.upperMap(capacity).costMap(cost)
859    ///     .supplyMap(sup).run();
860    /// \endcode
861    ///
862    /// \return <tt>(*this)</tt>
863    ///
864    /// \see reset(), run()
865    NetworkSimplex& resetParams() {
866      for (int i = 0; i != _node_num; ++i) {
867        _supply[i] = 0;
868      }
869      for (int i = 0; i != _arc_num; ++i) {
870        _lower[i] = 0;
871        _upper[i] = INF;
872        _cost[i] = 1;
873      }
874      _have_lower = false;
875      _stype = GEQ;
876      return *this;
877    }
878
879    /// \brief Reset the internal data structures and all the parameters
880    /// that have been given before.
881    ///
882    /// This function resets the internal data structures and all the
883    /// paramaters that have been given before using functions \ref lowerMap(),
884    /// \ref upperMap(), \ref costMap(), \ref supplyMap(), \ref stSupply(),
885    /// \ref supplyType().
886    ///
887    /// It is useful for multiple \ref run() calls. Basically, all the given
888    /// parameters are kept for the next \ref run() call, unless
889    /// \ref resetParams() or \ref reset() is used.
890    /// If the underlying digraph was also modified after the construction
891    /// of the class or the last \ref reset() call, then the \ref reset()
892    /// function must be used, otherwise \ref resetParams() is sufficient.
893    ///
894    /// See \ref resetParams() for examples.
895    ///
896    /// \return <tt>(*this)</tt>
897    ///
898    /// \see resetParams(), run()
899    NetworkSimplex& reset() {
900      // Resize vectors
901      _node_num = countNodes(_graph);
902      _arc_num = countArcs(_graph);
903      int all_node_num = _node_num + 1;
904      int max_arc_num = _arc_num + 2 * _node_num;
905
906      _source.resize(max_arc_num);
907      _target.resize(max_arc_num);
908
909      _lower.resize(_arc_num);
910      _upper.resize(_arc_num);
911      _cap.resize(max_arc_num);
912      _cost.resize(max_arc_num);
913      _supply.resize(all_node_num);
914      _flow.resize(max_arc_num);
915      _pi.resize(all_node_num);
916
917      _parent.resize(all_node_num);
918      _pred.resize(all_node_num);
919      _pred_dir.resize(all_node_num);
920      _thread.resize(all_node_num);
921      _rev_thread.resize(all_node_num);
922      _succ_num.resize(all_node_num);
923      _last_succ.resize(all_node_num);
924      _state.resize(max_arc_num);
925
926      // Copy the graph
927      int i = 0;
928      for (NodeIt n(_graph); n != INVALID; ++n, ++i) {
929        _node_id[n] = i;
930      }
931      if (_arc_mixing) {
932        // Store the arcs in a mixed order
933        int k = std::max(int(std::sqrt(double(_arc_num))), 10);
934        int i = 0, j = 0;
935        for (ArcIt a(_graph); a != INVALID; ++a) {
936          _arc_id[a] = i;
937          _source[i] = _node_id[_graph.source(a)];
938          _target[i] = _node_id[_graph.target(a)];
939          if ((i += k) >= _arc_num) i = ++j;
940        }
941      } else {
942        // Store the arcs in the original order
943        int i = 0;
944        for (ArcIt a(_graph); a != INVALID; ++a, ++i) {
945          _arc_id[a] = i;
946          _source[i] = _node_id[_graph.source(a)];
947          _target[i] = _node_id[_graph.target(a)];
948        }
949      }
950
951      // Reset parameters
952      resetParams();
953      return *this;
954    }
955
956    /// @}
957
958    /// \name Query Functions
959    /// The results of the algorithm can be obtained using these
960    /// functions.\n
961    /// The \ref run() function must be called before using them.
962
963    /// @{
964
965    /// \brief Return the total cost of the found flow.
966    ///
967    /// This function returns the total cost of the found flow.
968    /// Its complexity is O(e).
969    ///
970    /// \note The return type of the function can be specified as a
971    /// template parameter. For example,
972    /// \code
973    ///   ns.totalCost<double>();
974    /// \endcode
975    /// It is useful if the total cost cannot be stored in the \c Cost
976    /// type of the algorithm, which is the default return type of the
977    /// function.
978    ///
979    /// \pre \ref run() must be called before using this function.
980    template <typename Number>
981    Number totalCost() const {
982      Number c = 0;
983      for (ArcIt a(_graph); a != INVALID; ++a) {
984        int i = _arc_id[a];
985        c += Number(_flow[i]) * Number(_cost[i]);
986      }
987      return c;
988    }
989
990#ifndef DOXYGEN
991    Cost totalCost() const {
992      return totalCost<Cost>();
993    }
994#endif
995
996    /// \brief Return the flow on the given arc.
997    ///
998    /// This function returns the flow on the given arc.
999    ///
1000    /// \pre \ref run() must be called before using this function.
1001    Value flow(const Arc& a) const {
1002      return _flow[_arc_id[a]];
1003    }
1004
1005    /// \brief Return the flow map (the primal solution).
1006    ///
1007    /// This function copies the flow value on each arc into the given
1008    /// map. The \c Value type of the algorithm must be convertible to
1009    /// the \c Value type of the map.
1010    ///
1011    /// \pre \ref run() must be called before using this function.
1012    template <typename FlowMap>
1013    void flowMap(FlowMap &map) const {
1014      for (ArcIt a(_graph); a != INVALID; ++a) {
1015        map.set(a, _flow[_arc_id[a]]);
1016      }
1017    }
1018
1019    /// \brief Return the potential (dual value) of the given node.
1020    ///
1021    /// This function returns the potential (dual value) of the
1022    /// given node.
1023    ///
1024    /// \pre \ref run() must be called before using this function.
1025    Cost potential(const Node& n) const {
1026      return _pi[_node_id[n]];
1027    }
1028
1029    /// \brief Return the potential map (the dual solution).
1030    ///
1031    /// This function copies the potential (dual value) of each node
1032    /// into the given map.
1033    /// The \c Cost type of the algorithm must be convertible to the
1034    /// \c Value type of the map.
1035    ///
1036    /// \pre \ref run() must be called before using this function.
1037    template <typename PotentialMap>
1038    void potentialMap(PotentialMap &map) const {
1039      for (NodeIt n(_graph); n != INVALID; ++n) {
1040        map.set(n, _pi[_node_id[n]]);
1041      }
1042    }
1043
1044    /// @}
1045
1046  private:
1047
1048    // Initialize internal data structures
1049    bool init() {
1050      if (_node_num == 0) return false;
1051
1052      // Check the sum of supply values
1053      _sum_supply = 0;
1054      for (int i = 0; i != _node_num; ++i) {
1055        _sum_supply += _supply[i];
1056      }
1057      if ( !((_stype == GEQ && _sum_supply <= 0) ||
1058             (_stype == LEQ && _sum_supply >= 0)) ) return false;
1059
1060      // Remove non-zero lower bounds
1061      if (_have_lower) {
1062        for (int i = 0; i != _arc_num; ++i) {
1063          Value c = _lower[i];
1064          if (c >= 0) {
1065            _cap[i] = _upper[i] < MAX ? _upper[i] - c : INF;
1066          } else {
1067            _cap[i] = _upper[i] < MAX + c ? _upper[i] - c : INF;
1068          }
1069          _supply[_source[i]] -= c;
1070          _supply[_target[i]] += c;
1071        }
1072      } else {
1073        for (int i = 0; i != _arc_num; ++i) {
1074          _cap[i] = _upper[i];
1075        }
1076      }
1077
1078      // Initialize artifical cost
1079      Cost ART_COST;
1080      if (std::numeric_limits<Cost>::is_exact) {
1081        ART_COST = std::numeric_limits<Cost>::max() / 2 + 1;
1082      } else {
1083        ART_COST = 0;
1084        for (int i = 0; i != _arc_num; ++i) {
1085          if (_cost[i] > ART_COST) ART_COST = _cost[i];
1086        }
1087        ART_COST = (ART_COST + 1) * _node_num;
1088      }
1089
1090      // Initialize arc maps
1091      for (int i = 0; i != _arc_num; ++i) {
1092        _flow[i] = 0;
1093        _state[i] = STATE_LOWER;
1094      }
1095
1096      // Set data for the artificial root node
1097      _root = _node_num;
1098      _parent[_root] = -1;
1099      _pred[_root] = -1;
1100      _thread[_root] = 0;
1101      _rev_thread[0] = _root;
1102      _succ_num[_root] = _node_num + 1;
1103      _last_succ[_root] = _root - 1;
1104      _supply[_root] = -_sum_supply;
1105      _pi[_root] = 0;
1106
1107      // Add artificial arcs and initialize the spanning tree data structure
1108      if (_sum_supply == 0) {
1109        // EQ supply constraints
1110        _search_arc_num = _arc_num;
1111        _all_arc_num = _arc_num + _node_num;
1112        for (int u = 0, e = _arc_num; u != _node_num; ++u, ++e) {
1113          _parent[u] = _root;
1114          _pred[u] = e;
1115          _thread[u] = u + 1;
1116          _rev_thread[u + 1] = u;
1117          _succ_num[u] = 1;
1118          _last_succ[u] = u;
1119          _cap[e] = INF;
1120          _state[e] = STATE_TREE;
1121          if (_supply[u] >= 0) {
1122            _pred_dir[u] = DIR_UP;
1123            _pi[u] = 0;
1124            _source[e] = u;
1125            _target[e] = _root;
1126            _flow[e] = _supply[u];
1127            _cost[e] = 0;
1128          } else {
1129            _pred_dir[u] = DIR_DOWN;
1130            _pi[u] = ART_COST;
1131            _source[e] = _root;
1132            _target[e] = u;
1133            _flow[e] = -_supply[u];
1134            _cost[e] = ART_COST;
1135          }
1136        }
1137      }
1138      else if (_sum_supply > 0) {
1139        // LEQ supply constraints
1140        _search_arc_num = _arc_num + _node_num;
1141        int f = _arc_num + _node_num;
1142        for (int u = 0, e = _arc_num; u != _node_num; ++u, ++e) {
1143          _parent[u] = _root;
1144          _thread[u] = u + 1;
1145          _rev_thread[u + 1] = u;
1146          _succ_num[u] = 1;
1147          _last_succ[u] = u;
1148          if (_supply[u] >= 0) {
1149            _pred_dir[u] = DIR_UP;
1150            _pi[u] = 0;
1151            _pred[u] = e;
1152            _source[e] = u;
1153            _target[e] = _root;
1154            _cap[e] = INF;
1155            _flow[e] = _supply[u];
1156            _cost[e] = 0;
1157            _state[e] = STATE_TREE;
1158          } else {
1159            _pred_dir[u] = DIR_DOWN;
1160            _pi[u] = ART_COST;
1161            _pred[u] = f;
1162            _source[f] = _root;
1163            _target[f] = u;
1164            _cap[f] = INF;
1165            _flow[f] = -_supply[u];
1166            _cost[f] = ART_COST;
1167            _state[f] = STATE_TREE;
1168            _source[e] = u;
1169            _target[e] = _root;
1170            _cap[e] = INF;
1171            _flow[e] = 0;
1172            _cost[e] = 0;
1173            _state[e] = STATE_LOWER;
1174            ++f;
1175          }
1176        }
1177        _all_arc_num = f;
1178      }
1179      else {
1180        // GEQ supply constraints
1181        _search_arc_num = _arc_num + _node_num;
1182        int f = _arc_num + _node_num;
1183        for (int u = 0, e = _arc_num; u != _node_num; ++u, ++e) {
1184          _parent[u] = _root;
1185          _thread[u] = u + 1;
1186          _rev_thread[u + 1] = u;
1187          _succ_num[u] = 1;
1188          _last_succ[u] = u;
1189          if (_supply[u] <= 0) {
1190            _pred_dir[u] = DIR_DOWN;
1191            _pi[u] = 0;
1192            _pred[u] = e;
1193            _source[e] = _root;
1194            _target[e] = u;
1195            _cap[e] = INF;
1196            _flow[e] = -_supply[u];
1197            _cost[e] = 0;
1198            _state[e] = STATE_TREE;
1199          } else {
1200            _pred_dir[u] = DIR_UP;
1201            _pi[u] = -ART_COST;
1202            _pred[u] = f;
1203            _source[f] = u;
1204            _target[f] = _root;
1205            _cap[f] = INF;
1206            _flow[f] = _supply[u];
1207            _state[f] = STATE_TREE;
1208            _cost[f] = ART_COST;
1209            _source[e] = _root;
1210            _target[e] = u;
1211            _cap[e] = INF;
1212            _flow[e] = 0;
1213            _cost[e] = 0;
1214            _state[e] = STATE_LOWER;
1215            ++f;
1216          }
1217        }
1218        _all_arc_num = f;
1219      }
1220
1221      return true;
1222    }
1223
1224    // Find the join node
1225    void findJoinNode() {
1226      int u = _source[in_arc];
1227      int v = _target[in_arc];
1228      while (u != v) {
1229        if (_succ_num[u] < _succ_num[v]) {
1230          u = _parent[u];
1231        } else {
1232          v = _parent[v];
1233        }
1234      }
1235      join = u;
1236    }
1237
1238    // Find the leaving arc of the cycle and returns true if the
1239    // leaving arc is not the same as the entering arc
1240    bool findLeavingArc() {
1241      // Initialize first and second nodes according to the direction
1242      // of the cycle
1243      int first, second;
1244      if (_state[in_arc] == STATE_LOWER) {
1245        first  = _source[in_arc];
1246        second = _target[in_arc];
1247      } else {
1248        first  = _target[in_arc];
1249        second = _source[in_arc];
1250      }
1251      delta = _cap[in_arc];
1252      int result = 0;
1253      Value c, d;
1254      int e;
1255
1256      // Search the cycle form the first node to the join node
1257      for (int u = first; u != join; u = _parent[u]) {
1258        e = _pred[u];
1259        d = _flow[e];
1260        if (_pred_dir[u] == DIR_DOWN) {
1261          c = _cap[e];
1262          d = c >= MAX ? INF : c - d;
1263        }
1264        if (d < delta) {
1265          delta = d;
1266          u_out = u;
1267          result = 1;
1268        }
1269      }
1270
1271      // Search the cycle form the second node to the join node
1272      for (int u = second; u != join; u = _parent[u]) {
1273        e = _pred[u];
1274        d = _flow[e];
1275        if (_pred_dir[u] == DIR_UP) {
1276          c = _cap[e];
1277          d = c >= MAX ? INF : c - d;
1278        }
1279        if (d <= delta) {
1280          delta = d;
1281          u_out = u;
1282          result = 2;
1283        }
1284      }
1285
1286      if (result == 1) {
1287        u_in = first;
1288        v_in = second;
1289      } else {
1290        u_in = second;
1291        v_in = first;
1292      }
1293      return result != 0;
1294    }
1295
1296    // Change _flow and _state vectors
1297    void changeFlow(bool change) {
1298      // Augment along the cycle
1299      if (delta > 0) {
1300        Value val = _state[in_arc] * delta;
1301        _flow[in_arc] += val;
1302        for (int u = _source[in_arc]; u != join; u = _parent[u]) {
1303          _flow[_pred[u]] -= _pred_dir[u] * val;
1304        }
1305        for (int u = _target[in_arc]; u != join; u = _parent[u]) {
1306          _flow[_pred[u]] += _pred_dir[u] * val;
1307        }
1308      }
1309      // Update the state of the entering and leaving arcs
1310      if (change) {
1311        _state[in_arc] = STATE_TREE;
1312        _state[_pred[u_out]] =
1313          (_flow[_pred[u_out]] == 0) ? STATE_LOWER : STATE_UPPER;
1314      } else {
1315        _state[in_arc] = -_state[in_arc];
1316      }
1317    }
1318
1319    // Update the tree structure
1320    void updateTreeStructure() {
1321      int old_rev_thread = _rev_thread[u_out];
1322      int old_succ_num = _succ_num[u_out];
1323      int old_last_succ = _last_succ[u_out];
1324      v_out = _parent[u_out];
1325
1326      // Check if u_in and u_out coincide
1327      if (u_in == u_out) {
1328        // Update _parent, _pred, _pred_dir
1329        _parent[u_in] = v_in;
1330        _pred[u_in] = in_arc;
1331        _pred_dir[u_in] = u_in == _source[in_arc] ? DIR_UP : DIR_DOWN;
1332
1333        // Update _thread and _rev_thread
1334        if (_thread[v_in] != u_out) {
1335          int after = _thread[old_last_succ];
1336          _thread[old_rev_thread] = after;
1337          _rev_thread[after] = old_rev_thread;
1338          after = _thread[v_in];
1339          _thread[v_in] = u_out;
1340          _rev_thread[u_out] = v_in;
1341          _thread[old_last_succ] = after;
1342          _rev_thread[after] = old_last_succ;
1343        }
1344      } else {
1345        // Handle the case when old_rev_thread equals to v_in
1346        // (it also means that join and v_out coincide)
1347        int thread_continue = old_rev_thread == v_in ?
1348          _thread[old_last_succ] : _thread[v_in];
1349
1350        // Update _thread and _parent along the stem nodes (i.e. the nodes
1351        // between u_in and u_out, whose parent have to be changed)
1352        int stem = u_in;              // the current stem node
1353        int par_stem = v_in;          // the new parent of stem
1354        int next_stem;                // the next stem node
1355        int last = _last_succ[u_in];  // the last successor of stem
1356        int before, after = _thread[last];
1357        _thread[v_in] = u_in;
1358        _dirty_revs.clear();
1359        _dirty_revs.push_back(v_in);
1360        while (stem != u_out) {
1361          // Insert the next stem node into the thread list
1362          next_stem = _parent[stem];
1363          _thread[last] = next_stem;
1364          _dirty_revs.push_back(last);
1365
1366          // Remove the subtree of stem from the thread list
1367          before = _rev_thread[stem];
1368          _thread[before] = after;
1369          _rev_thread[after] = before;
1370
1371          // Change the parent node and shift stem nodes
1372          _parent[stem] = par_stem;
1373          par_stem = stem;
1374          stem = next_stem;
1375
1376          // Update last and after
1377          last = _last_succ[stem] == _last_succ[par_stem] ?
1378            _rev_thread[par_stem] : _last_succ[stem];
1379          after = _thread[last];
1380        }
1381        _parent[u_out] = par_stem;
1382        _thread[last] = thread_continue;
1383        _rev_thread[thread_continue] = last;
1384        _last_succ[u_out] = last;
1385
1386        // Remove the subtree of u_out from the thread list except for
1387        // the case when old_rev_thread equals to v_in
1388        if (old_rev_thread != v_in) {
1389          _thread[old_rev_thread] = after;
1390          _rev_thread[after] = old_rev_thread;
1391        }
1392
1393        // Update _rev_thread using the new _thread values
1394        for (int i = 0; i != int(_dirty_revs.size()); ++i) {
1395          int u = _dirty_revs[i];
1396          _rev_thread[_thread[u]] = u;
1397        }
1398
1399        // Update _pred, _pred_dir, _last_succ and _succ_num for the
1400        // stem nodes from u_out to u_in
1401        int tmp_sc = 0, tmp_ls = _last_succ[u_out];
1402        for (int u = u_out, p = _parent[u]; u != u_in; u = p, p = _parent[u]) {
1403          _pred[u] = _pred[p];
1404          _pred_dir[u] = -_pred_dir[p];
1405          tmp_sc += _succ_num[u] - _succ_num[p];
1406          _succ_num[u] = tmp_sc;
1407          _last_succ[p] = tmp_ls;
1408        }
1409        _pred[u_in] = in_arc;
1410        _pred_dir[u_in] = u_in == _source[in_arc] ? DIR_UP : DIR_DOWN;
1411        _succ_num[u_in] = old_succ_num;
1412      }
1413
1414      // Update _last_succ from v_in towards the root
1415      int up_limit_out = _last_succ[join] == v_in ? join : -1;
1416      int last_succ_out = _last_succ[u_out];
1417      for (int u = v_in; u != -1 && _last_succ[u] == v_in; u = _parent[u]) {
1418        _last_succ[u] = last_succ_out;
1419      }
1420
1421      // Update _last_succ from v_out towards the root
1422      if (join != old_rev_thread && v_in != old_rev_thread) {
1423        for (int u = v_out; u != up_limit_out && _last_succ[u] == old_last_succ;
1424             u = _parent[u]) {
1425          _last_succ[u] = old_rev_thread;
1426        }
1427      }
1428      else if (last_succ_out != old_last_succ) {
1429        for (int u = v_out; u != up_limit_out && _last_succ[u] == old_last_succ;
1430             u = _parent[u]) {
1431          _last_succ[u] = last_succ_out;
1432        }
1433      }
1434
1435      // Update _succ_num from v_in to join
1436      for (int u = v_in; u != join; u = _parent[u]) {
1437        _succ_num[u] += old_succ_num;
1438      }
1439      // Update _succ_num from v_out to join
1440      for (int u = v_out; u != join; u = _parent[u]) {
1441        _succ_num[u] -= old_succ_num;
1442      }
1443    }
1444
1445    // Update potentials in the subtree that has been moved
1446    void updatePotential() {
1447      Cost sigma = _pi[v_in] - _pi[u_in] -
1448                   _pred_dir[u_in] * _cost[in_arc];
1449      int end = _thread[_last_succ[u_in]];
1450      for (int u = u_in; u != end; u = _thread[u]) {
1451        _pi[u] += sigma;
1452      }
1453    }
1454
1455    // Heuristic initial pivots
1456    bool initialPivots() {
1457      Value curr, total = 0;
1458      std::vector<Node> supply_nodes, demand_nodes;
1459      for (NodeIt u(_graph); u != INVALID; ++u) {
1460        curr = _supply[_node_id[u]];
1461        if (curr > 0) {
1462          total += curr;
1463          supply_nodes.push_back(u);
1464        }
1465        else if (curr < 0) {
1466          demand_nodes.push_back(u);
1467        }
1468      }
1469      if (_sum_supply > 0) total -= _sum_supply;
1470      if (total <= 0) return true;
1471
1472      IntVector arc_vector;
1473      if (_sum_supply >= 0) {
1474        if (supply_nodes.size() == 1 && demand_nodes.size() == 1) {
1475          // Perform a reverse graph search from the sink to the source
1476          typename GR::template NodeMap<bool> reached(_graph, false);
1477          Node s = supply_nodes[0], t = demand_nodes[0];
1478          std::vector<Node> stack;
1479          reached[t] = true;
1480          stack.push_back(t);
1481          while (!stack.empty()) {
1482            Node u, v = stack.back();
1483            stack.pop_back();
1484            if (v == s) break;
1485            for (InArcIt a(_graph, v); a != INVALID; ++a) {
1486              if (reached[u = _graph.source(a)]) continue;
1487              int j = _arc_id[a];
1488              if (_cap[j] >= total) {
1489                arc_vector.push_back(j);
1490                reached[u] = true;
1491                stack.push_back(u);
1492              }
1493            }
1494          }
1495        } else {
1496          // Find the min. cost incomming arc for each demand node
1497          for (int i = 0; i != int(demand_nodes.size()); ++i) {
1498            Node v = demand_nodes[i];
1499            Cost c, min_cost = std::numeric_limits<Cost>::max();
1500            Arc min_arc = INVALID;
1501            for (InArcIt a(_graph, v); a != INVALID; ++a) {
1502              c = _cost[_arc_id[a]];
1503              if (c < min_cost) {
1504                min_cost = c;
1505                min_arc = a;
1506              }
1507            }
1508            if (min_arc != INVALID) {
1509              arc_vector.push_back(_arc_id[min_arc]);
1510            }
1511          }
1512        }
1513      } else {
1514        // Find the min. cost outgoing arc for each supply node
1515        for (int i = 0; i != int(supply_nodes.size()); ++i) {
1516          Node u = supply_nodes[i];
1517          Cost c, min_cost = std::numeric_limits<Cost>::max();
1518          Arc min_arc = INVALID;
1519          for (OutArcIt a(_graph, u); a != INVALID; ++a) {
1520            c = _cost[_arc_id[a]];
1521            if (c < min_cost) {
1522              min_cost = c;
1523              min_arc = a;
1524            }
1525          }
1526          if (min_arc != INVALID) {
1527            arc_vector.push_back(_arc_id[min_arc]);
1528          }
1529        }
1530      }
1531
1532      // Perform heuristic initial pivots
1533      for (int i = 0; i != int(arc_vector.size()); ++i) {
1534        in_arc = arc_vector[i];
1535        if (_state[in_arc] * (_cost[in_arc] + _pi[_source[in_arc]] -
1536            _pi[_target[in_arc]]) >= 0) continue;
1537        findJoinNode();
1538        bool change = findLeavingArc();
1539        if (delta >= MAX) return false;
1540        changeFlow(change);
1541        if (change) {
1542          updateTreeStructure();
1543          updatePotential();
1544        }
1545      }
1546      return true;
1547    }
1548
1549    // Execute the algorithm
1550    ProblemType start(PivotRule pivot_rule) {
1551      // Select the pivot rule implementation
1552      switch (pivot_rule) {
1553        case FIRST_ELIGIBLE:
1554          return start<FirstEligiblePivotRule>();
1555        case BEST_ELIGIBLE:
1556          return start<BestEligiblePivotRule>();
1557        case BLOCK_SEARCH:
1558          return start<BlockSearchPivotRule>();
1559        case CANDIDATE_LIST:
1560          return start<CandidateListPivotRule>();
1561        case ALTERING_LIST:
1562          return start<AlteringListPivotRule>();
1563      }
1564      return INFEASIBLE; // avoid warning
1565    }
1566
1567    template <typename PivotRuleImpl>
1568    ProblemType start() {
1569      PivotRuleImpl pivot(*this);
1570
1571      // Perform heuristic initial pivots
1572      if (!initialPivots()) return UNBOUNDED;
1573
1574      // Execute the Network Simplex algorithm
1575      while (pivot.findEnteringArc()) {
1576        findJoinNode();
1577        bool change = findLeavingArc();
1578        if (delta >= MAX) return UNBOUNDED;
1579        changeFlow(change);
1580        if (change) {
1581          updateTreeStructure();
1582          updatePotential();
1583        }
1584      }
1585
1586      // Check feasibility
1587      for (int e = _search_arc_num; e != _all_arc_num; ++e) {
1588        if (_flow[e] != 0) return INFEASIBLE;
1589      }
1590
1591      // Transform the solution and the supply map to the original form
1592      if (_have_lower) {
1593        for (int i = 0; i != _arc_num; ++i) {
1594          Value c = _lower[i];
1595          if (c != 0) {
1596            _flow[i] += c;
1597            _supply[_source[i]] += c;
1598            _supply[_target[i]] -= c;
1599          }
1600        }
1601      }
1602
1603      // Shift potentials to meet the requirements of the GEQ/LEQ type
1604      // optimality conditions
1605      if (_sum_supply == 0) {
1606        if (_stype == GEQ) {
1607          Cost max_pot = -std::numeric_limits<Cost>::max();
1608          for (int i = 0; i != _node_num; ++i) {
1609            if (_pi[i] > max_pot) max_pot = _pi[i];
1610          }
1611          if (max_pot > 0) {
1612            for (int i = 0; i != _node_num; ++i)
1613              _pi[i] -= max_pot;
1614          }
1615        } else {
1616          Cost min_pot = std::numeric_limits<Cost>::max();
1617          for (int i = 0; i != _node_num; ++i) {
1618            if (_pi[i] < min_pot) min_pot = _pi[i];
1619          }
1620          if (min_pot < 0) {
1621            for (int i = 0; i != _node_num; ++i)
1622              _pi[i] -= min_pot;
1623          }
1624        }
1625      }
1626
1627      return OPTIMAL;
1628    }
1629
1630  }; //class NetworkSimplex
1631
1632  ///@}
1633
1634} //namespace lemon
1635
1636#endif //LEMON_NETWORK_SIMPLEX_H
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