COIN-OR::LEMON - Graph Library

source: lemon/lemon/network_simplex.h @ 991:fb932bcfd803

Last change on this file since 991:fb932bcfd803 was 991:fb932bcfd803, checked in by Peter Kovacs <kpeter@…>, 9 years ago

Improve arc mixing in NS and enable it by default (#391)

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 will be stored in a
640    /// mixed order in the internal data structure.
641    /// In general, it leads to similar performance as using the original
642    /// arc order, but it makes the algorithm more robust and in special
643    /// cases, even significantly faster. Therefore, it is enabled by default.
644    NetworkSimplex(const GR& graph, bool arc_mixing = true) :
645      _graph(graph), _node_id(graph), _arc_id(graph),
646      _arc_mixing(arc_mixing),
647      MAX(std::numeric_limits<Value>::max()),
648      INF(std::numeric_limits<Value>::has_infinity ?
649          std::numeric_limits<Value>::infinity() : MAX)
650    {
651      // Check the number types
652      LEMON_ASSERT(std::numeric_limits<Value>::is_signed,
653        "The flow type of NetworkSimplex must be signed");
654      LEMON_ASSERT(std::numeric_limits<Cost>::is_signed,
655        "The cost type of NetworkSimplex must be signed");
656
657      // Reset data structures
658      reset();
659    }
660
661    /// \name Parameters
662    /// The parameters of the algorithm can be specified using these
663    /// functions.
664
665    /// @{
666
667    /// \brief Set the lower bounds on the arcs.
668    ///
669    /// This function sets the lower bounds on the arcs.
670    /// If it is not used before calling \ref run(), the lower bounds
671    /// will be set to zero on all arcs.
672    ///
673    /// \param map An arc map storing the lower bounds.
674    /// Its \c Value type must be convertible to the \c Value type
675    /// of the algorithm.
676    ///
677    /// \return <tt>(*this)</tt>
678    template <typename LowerMap>
679    NetworkSimplex& lowerMap(const LowerMap& map) {
680      _have_lower = true;
681      for (ArcIt a(_graph); a != INVALID; ++a) {
682        _lower[_arc_id[a]] = map[a];
683      }
684      return *this;
685    }
686
687    /// \brief Set the upper bounds (capacities) on the arcs.
688    ///
689    /// This function sets the upper bounds (capacities) on the arcs.
690    /// If it is not used before calling \ref run(), the upper bounds
691    /// will be set to \ref INF on all arcs (i.e. the flow value will be
692    /// unbounded from above).
693    ///
694    /// \param map An arc map storing the upper bounds.
695    /// Its \c Value type must be convertible to the \c Value type
696    /// of the algorithm.
697    ///
698    /// \return <tt>(*this)</tt>
699    template<typename UpperMap>
700    NetworkSimplex& upperMap(const UpperMap& map) {
701      for (ArcIt a(_graph); a != INVALID; ++a) {
702        _upper[_arc_id[a]] = map[a];
703      }
704      return *this;
705    }
706
707    /// \brief Set the costs of the arcs.
708    ///
709    /// This function sets the costs of the arcs.
710    /// If it is not used before calling \ref run(), the costs
711    /// will be set to \c 1 on all arcs.
712    ///
713    /// \param map An arc map storing the costs.
714    /// Its \c Value type must be convertible to the \c Cost type
715    /// of the algorithm.
716    ///
717    /// \return <tt>(*this)</tt>
718    template<typename CostMap>
719    NetworkSimplex& costMap(const CostMap& map) {
720      for (ArcIt a(_graph); a != INVALID; ++a) {
721        _cost[_arc_id[a]] = map[a];
722      }
723      return *this;
724    }
725
726    /// \brief Set the supply values of the nodes.
727    ///
728    /// This function sets the supply values of the nodes.
729    /// If neither this function nor \ref stSupply() is used before
730    /// calling \ref run(), the supply of each node will be set to zero.
731    ///
732    /// \param map A node map storing the supply values.
733    /// Its \c Value type must be convertible to the \c Value type
734    /// of the algorithm.
735    ///
736    /// \return <tt>(*this)</tt>
737    template<typename SupplyMap>
738    NetworkSimplex& supplyMap(const SupplyMap& map) {
739      for (NodeIt n(_graph); n != INVALID; ++n) {
740        _supply[_node_id[n]] = map[n];
741      }
742      return *this;
743    }
744
745    /// \brief Set single source and target nodes and a supply value.
746    ///
747    /// This function sets a single source node and a single target node
748    /// and the required flow value.
749    /// If neither this function nor \ref supplyMap() is used before
750    /// calling \ref run(), the supply of each node will be set to zero.
751    ///
752    /// Using this function has the same effect as using \ref supplyMap()
753    /// with such a map in which \c k is assigned to \c s, \c -k is
754    /// assigned to \c t and all other nodes have zero supply value.
755    ///
756    /// \param s The source node.
757    /// \param t The target node.
758    /// \param k The required amount of flow from node \c s to node \c t
759    /// (i.e. the supply of \c s and the demand of \c t).
760    ///
761    /// \return <tt>(*this)</tt>
762    NetworkSimplex& stSupply(const Node& s, const Node& t, Value k) {
763      for (int i = 0; i != _node_num; ++i) {
764        _supply[i] = 0;
765      }
766      _supply[_node_id[s]] =  k;
767      _supply[_node_id[t]] = -k;
768      return *this;
769    }
770
771    /// \brief Set the type of the supply constraints.
772    ///
773    /// This function sets the type of the supply/demand constraints.
774    /// If it is not used before calling \ref run(), the \ref GEQ supply
775    /// type will be used.
776    ///
777    /// For more information, see \ref SupplyType.
778    ///
779    /// \return <tt>(*this)</tt>
780    NetworkSimplex& supplyType(SupplyType supply_type) {
781      _stype = supply_type;
782      return *this;
783    }
784
785    /// @}
786
787    /// \name Execution Control
788    /// The algorithm can be executed using \ref run().
789
790    /// @{
791
792    /// \brief Run the algorithm.
793    ///
794    /// This function runs the algorithm.
795    /// The paramters can be specified using functions \ref lowerMap(),
796    /// \ref upperMap(), \ref costMap(), \ref supplyMap(), \ref stSupply(),
797    /// \ref supplyType().
798    /// For example,
799    /// \code
800    ///   NetworkSimplex<ListDigraph> ns(graph);
801    ///   ns.lowerMap(lower).upperMap(upper).costMap(cost)
802    ///     .supplyMap(sup).run();
803    /// \endcode
804    ///
805    /// This function can be called more than once. All the given parameters
806    /// are kept for the next call, unless \ref resetParams() or \ref reset()
807    /// is used, thus only the modified parameters have to be set again.
808    /// If the underlying digraph was also modified after the construction
809    /// of the class (or the last \ref reset() call), then the \ref reset()
810    /// function must be called.
811    ///
812    /// \param pivot_rule The pivot rule that will be used during the
813    /// algorithm. For more information, see \ref PivotRule.
814    ///
815    /// \return \c INFEASIBLE if no feasible flow exists,
816    /// \n \c OPTIMAL if the problem has optimal solution
817    /// (i.e. it is feasible and bounded), and the algorithm has found
818    /// optimal flow and node potentials (primal and dual solutions),
819    /// \n \c UNBOUNDED if the objective function of the problem is
820    /// unbounded, i.e. there is a directed cycle having negative total
821    /// cost and infinite upper bound.
822    ///
823    /// \see ProblemType, PivotRule
824    /// \see resetParams(), reset()
825    ProblemType run(PivotRule pivot_rule = BLOCK_SEARCH) {
826      if (!init()) return INFEASIBLE;
827      return start(pivot_rule);
828    }
829
830    /// \brief Reset all the parameters that have been given before.
831    ///
832    /// This function resets all the paramaters that have been given
833    /// before using functions \ref lowerMap(), \ref upperMap(),
834    /// \ref costMap(), \ref supplyMap(), \ref stSupply(), \ref supplyType().
835    ///
836    /// It is useful for multiple \ref run() calls. Basically, all the given
837    /// parameters are kept for the next \ref run() call, unless
838    /// \ref resetParams() or \ref reset() is used.
839    /// If the underlying digraph was also modified after the construction
840    /// of the class or the last \ref reset() call, then the \ref reset()
841    /// function must be used, otherwise \ref resetParams() is sufficient.
842    ///
843    /// For example,
844    /// \code
845    ///   NetworkSimplex<ListDigraph> ns(graph);
846    ///
847    ///   // First run
848    ///   ns.lowerMap(lower).upperMap(upper).costMap(cost)
849    ///     .supplyMap(sup).run();
850    ///
851    ///   // Run again with modified cost map (resetParams() is not called,
852    ///   // so only the cost map have to be set again)
853    ///   cost[e] += 100;
854    ///   ns.costMap(cost).run();
855    ///
856    ///   // Run again from scratch using resetParams()
857    ///   // (the lower bounds will be set to zero on all arcs)
858    ///   ns.resetParams();
859    ///   ns.upperMap(capacity).costMap(cost)
860    ///     .supplyMap(sup).run();
861    /// \endcode
862    ///
863    /// \return <tt>(*this)</tt>
864    ///
865    /// \see reset(), run()
866    NetworkSimplex& resetParams() {
867      for (int i = 0; i != _node_num; ++i) {
868        _supply[i] = 0;
869      }
870      for (int i = 0; i != _arc_num; ++i) {
871        _lower[i] = 0;
872        _upper[i] = INF;
873        _cost[i] = 1;
874      }
875      _have_lower = false;
876      _stype = GEQ;
877      return *this;
878    }
879
880    /// \brief Reset the internal data structures and all the parameters
881    /// that have been given before.
882    ///
883    /// This function resets the internal data structures and all the
884    /// paramaters that have been given before using functions \ref lowerMap(),
885    /// \ref upperMap(), \ref costMap(), \ref supplyMap(), \ref stSupply(),
886    /// \ref supplyType().
887    ///
888    /// It is useful for multiple \ref run() calls. Basically, all the given
889    /// parameters are kept for the next \ref run() call, unless
890    /// \ref resetParams() or \ref reset() is used.
891    /// If the underlying digraph was also modified after the construction
892    /// of the class or the last \ref reset() call, then the \ref reset()
893    /// function must be used, otherwise \ref resetParams() is sufficient.
894    ///
895    /// See \ref resetParams() for examples.
896    ///
897    /// \return <tt>(*this)</tt>
898    ///
899    /// \see resetParams(), run()
900    NetworkSimplex& reset() {
901      // Resize vectors
902      _node_num = countNodes(_graph);
903      _arc_num = countArcs(_graph);
904      int all_node_num = _node_num + 1;
905      int max_arc_num = _arc_num + 2 * _node_num;
906
907      _source.resize(max_arc_num);
908      _target.resize(max_arc_num);
909
910      _lower.resize(_arc_num);
911      _upper.resize(_arc_num);
912      _cap.resize(max_arc_num);
913      _cost.resize(max_arc_num);
914      _supply.resize(all_node_num);
915      _flow.resize(max_arc_num);
916      _pi.resize(all_node_num);
917
918      _parent.resize(all_node_num);
919      _pred.resize(all_node_num);
920      _pred_dir.resize(all_node_num);
921      _thread.resize(all_node_num);
922      _rev_thread.resize(all_node_num);
923      _succ_num.resize(all_node_num);
924      _last_succ.resize(all_node_num);
925      _state.resize(max_arc_num);
926
927      // Copy the graph
928      int i = 0;
929      for (NodeIt n(_graph); n != INVALID; ++n, ++i) {
930        _node_id[n] = i;
931      }
932      if (_arc_mixing) {
933        // Store the arcs in a mixed order
934        const int skip = std::max(_arc_num / _node_num, 3);
935        int i = 0, j = 0;
936        for (ArcIt a(_graph); a != INVALID; ++a) {
937          _arc_id[a] = i;
938          _source[i] = _node_id[_graph.source(a)];
939          _target[i] = _node_id[_graph.target(a)];
940          if ((i += skip) >= _arc_num) i = ++j;
941        }
942      } else {
943        // Store the arcs in the original order
944        int i = 0;
945        for (ArcIt a(_graph); a != INVALID; ++a, ++i) {
946          _arc_id[a] = i;
947          _source[i] = _node_id[_graph.source(a)];
948          _target[i] = _node_id[_graph.target(a)];
949        }
950      }
951
952      // Reset parameters
953      resetParams();
954      return *this;
955    }
956
957    /// @}
958
959    /// \name Query Functions
960    /// The results of the algorithm can be obtained using these
961    /// functions.\n
962    /// The \ref run() function must be called before using them.
963
964    /// @{
965
966    /// \brief Return the total cost of the found flow.
967    ///
968    /// This function returns the total cost of the found flow.
969    /// Its complexity is O(e).
970    ///
971    /// \note The return type of the function can be specified as a
972    /// template parameter. For example,
973    /// \code
974    ///   ns.totalCost<double>();
975    /// \endcode
976    /// It is useful if the total cost cannot be stored in the \c Cost
977    /// type of the algorithm, which is the default return type of the
978    /// function.
979    ///
980    /// \pre \ref run() must be called before using this function.
981    template <typename Number>
982    Number totalCost() const {
983      Number c = 0;
984      for (ArcIt a(_graph); a != INVALID; ++a) {
985        int i = _arc_id[a];
986        c += Number(_flow[i]) * Number(_cost[i]);
987      }
988      return c;
989    }
990
991#ifndef DOXYGEN
992    Cost totalCost() const {
993      return totalCost<Cost>();
994    }
995#endif
996
997    /// \brief Return the flow on the given arc.
998    ///
999    /// This function returns the flow on the given arc.
1000    ///
1001    /// \pre \ref run() must be called before using this function.
1002    Value flow(const Arc& a) const {
1003      return _flow[_arc_id[a]];
1004    }
1005
1006    /// \brief Return the flow map (the primal solution).
1007    ///
1008    /// This function copies the flow value on each arc into the given
1009    /// map. The \c Value type of the algorithm must be convertible to
1010    /// the \c Value type of the map.
1011    ///
1012    /// \pre \ref run() must be called before using this function.
1013    template <typename FlowMap>
1014    void flowMap(FlowMap &map) const {
1015      for (ArcIt a(_graph); a != INVALID; ++a) {
1016        map.set(a, _flow[_arc_id[a]]);
1017      }
1018    }
1019
1020    /// \brief Return the potential (dual value) of the given node.
1021    ///
1022    /// This function returns the potential (dual value) of the
1023    /// given node.
1024    ///
1025    /// \pre \ref run() must be called before using this function.
1026    Cost potential(const Node& n) const {
1027      return _pi[_node_id[n]];
1028    }
1029
1030    /// \brief Return the potential map (the dual solution).
1031    ///
1032    /// This function copies the potential (dual value) of each node
1033    /// into the given map.
1034    /// The \c Cost type of the algorithm must be convertible to the
1035    /// \c Value type of the map.
1036    ///
1037    /// \pre \ref run() must be called before using this function.
1038    template <typename PotentialMap>
1039    void potentialMap(PotentialMap &map) const {
1040      for (NodeIt n(_graph); n != INVALID; ++n) {
1041        map.set(n, _pi[_node_id[n]]);
1042      }
1043    }
1044
1045    /// @}
1046
1047  private:
1048
1049    // Initialize internal data structures
1050    bool init() {
1051      if (_node_num == 0) return false;
1052
1053      // Check the sum of supply values
1054      _sum_supply = 0;
1055      for (int i = 0; i != _node_num; ++i) {
1056        _sum_supply += _supply[i];
1057      }
1058      if ( !((_stype == GEQ && _sum_supply <= 0) ||
1059             (_stype == LEQ && _sum_supply >= 0)) ) return false;
1060
1061      // Remove non-zero lower bounds
1062      if (_have_lower) {
1063        for (int i = 0; i != _arc_num; ++i) {
1064          Value c = _lower[i];
1065          if (c >= 0) {
1066            _cap[i] = _upper[i] < MAX ? _upper[i] - c : INF;
1067          } else {
1068            _cap[i] = _upper[i] < MAX + c ? _upper[i] - c : INF;
1069          }
1070          _supply[_source[i]] -= c;
1071          _supply[_target[i]] += c;
1072        }
1073      } else {
1074        for (int i = 0; i != _arc_num; ++i) {
1075          _cap[i] = _upper[i];
1076        }
1077      }
1078
1079      // Initialize artifical cost
1080      Cost ART_COST;
1081      if (std::numeric_limits<Cost>::is_exact) {
1082        ART_COST = std::numeric_limits<Cost>::max() / 2 + 1;
1083      } else {
1084        ART_COST = 0;
1085        for (int i = 0; i != _arc_num; ++i) {
1086          if (_cost[i] > ART_COST) ART_COST = _cost[i];
1087        }
1088        ART_COST = (ART_COST + 1) * _node_num;
1089      }
1090
1091      // Initialize arc maps
1092      for (int i = 0; i != _arc_num; ++i) {
1093        _flow[i] = 0;
1094        _state[i] = STATE_LOWER;
1095      }
1096
1097      // Set data for the artificial root node
1098      _root = _node_num;
1099      _parent[_root] = -1;
1100      _pred[_root] = -1;
1101      _thread[_root] = 0;
1102      _rev_thread[0] = _root;
1103      _succ_num[_root] = _node_num + 1;
1104      _last_succ[_root] = _root - 1;
1105      _supply[_root] = -_sum_supply;
1106      _pi[_root] = 0;
1107
1108      // Add artificial arcs and initialize the spanning tree data structure
1109      if (_sum_supply == 0) {
1110        // EQ supply constraints
1111        _search_arc_num = _arc_num;
1112        _all_arc_num = _arc_num + _node_num;
1113        for (int u = 0, e = _arc_num; u != _node_num; ++u, ++e) {
1114          _parent[u] = _root;
1115          _pred[u] = e;
1116          _thread[u] = u + 1;
1117          _rev_thread[u + 1] = u;
1118          _succ_num[u] = 1;
1119          _last_succ[u] = u;
1120          _cap[e] = INF;
1121          _state[e] = STATE_TREE;
1122          if (_supply[u] >= 0) {
1123            _pred_dir[u] = DIR_UP;
1124            _pi[u] = 0;
1125            _source[e] = u;
1126            _target[e] = _root;
1127            _flow[e] = _supply[u];
1128            _cost[e] = 0;
1129          } else {
1130            _pred_dir[u] = DIR_DOWN;
1131            _pi[u] = ART_COST;
1132            _source[e] = _root;
1133            _target[e] = u;
1134            _flow[e] = -_supply[u];
1135            _cost[e] = ART_COST;
1136          }
1137        }
1138      }
1139      else if (_sum_supply > 0) {
1140        // LEQ supply constraints
1141        _search_arc_num = _arc_num + _node_num;
1142        int f = _arc_num + _node_num;
1143        for (int u = 0, e = _arc_num; u != _node_num; ++u, ++e) {
1144          _parent[u] = _root;
1145          _thread[u] = u + 1;
1146          _rev_thread[u + 1] = u;
1147          _succ_num[u] = 1;
1148          _last_succ[u] = u;
1149          if (_supply[u] >= 0) {
1150            _pred_dir[u] = DIR_UP;
1151            _pi[u] = 0;
1152            _pred[u] = e;
1153            _source[e] = u;
1154            _target[e] = _root;
1155            _cap[e] = INF;
1156            _flow[e] = _supply[u];
1157            _cost[e] = 0;
1158            _state[e] = STATE_TREE;
1159          } else {
1160            _pred_dir[u] = DIR_DOWN;
1161            _pi[u] = ART_COST;
1162            _pred[u] = f;
1163            _source[f] = _root;
1164            _target[f] = u;
1165            _cap[f] = INF;
1166            _flow[f] = -_supply[u];
1167            _cost[f] = ART_COST;
1168            _state[f] = STATE_TREE;
1169            _source[e] = u;
1170            _target[e] = _root;
1171            _cap[e] = INF;
1172            _flow[e] = 0;
1173            _cost[e] = 0;
1174            _state[e] = STATE_LOWER;
1175            ++f;
1176          }
1177        }
1178        _all_arc_num = f;
1179      }
1180      else {
1181        // GEQ supply constraints
1182        _search_arc_num = _arc_num + _node_num;
1183        int f = _arc_num + _node_num;
1184        for (int u = 0, e = _arc_num; u != _node_num; ++u, ++e) {
1185          _parent[u] = _root;
1186          _thread[u] = u + 1;
1187          _rev_thread[u + 1] = u;
1188          _succ_num[u] = 1;
1189          _last_succ[u] = u;
1190          if (_supply[u] <= 0) {
1191            _pred_dir[u] = DIR_DOWN;
1192            _pi[u] = 0;
1193            _pred[u] = e;
1194            _source[e] = _root;
1195            _target[e] = u;
1196            _cap[e] = INF;
1197            _flow[e] = -_supply[u];
1198            _cost[e] = 0;
1199            _state[e] = STATE_TREE;
1200          } else {
1201            _pred_dir[u] = DIR_UP;
1202            _pi[u] = -ART_COST;
1203            _pred[u] = f;
1204            _source[f] = u;
1205            _target[f] = _root;
1206            _cap[f] = INF;
1207            _flow[f] = _supply[u];
1208            _state[f] = STATE_TREE;
1209            _cost[f] = ART_COST;
1210            _source[e] = _root;
1211            _target[e] = u;
1212            _cap[e] = INF;
1213            _flow[e] = 0;
1214            _cost[e] = 0;
1215            _state[e] = STATE_LOWER;
1216            ++f;
1217          }
1218        }
1219        _all_arc_num = f;
1220      }
1221
1222      return true;
1223    }
1224
1225    // Find the join node
1226    void findJoinNode() {
1227      int u = _source[in_arc];
1228      int v = _target[in_arc];
1229      while (u != v) {
1230        if (_succ_num[u] < _succ_num[v]) {
1231          u = _parent[u];
1232        } else {
1233          v = _parent[v];
1234        }
1235      }
1236      join = u;
1237    }
1238
1239    // Find the leaving arc of the cycle and returns true if the
1240    // leaving arc is not the same as the entering arc
1241    bool findLeavingArc() {
1242      // Initialize first and second nodes according to the direction
1243      // of the cycle
1244      int first, second;
1245      if (_state[in_arc] == STATE_LOWER) {
1246        first  = _source[in_arc];
1247        second = _target[in_arc];
1248      } else {
1249        first  = _target[in_arc];
1250        second = _source[in_arc];
1251      }
1252      delta = _cap[in_arc];
1253      int result = 0;
1254      Value c, d;
1255      int e;
1256
1257      // Search the cycle form the first node to the join node
1258      for (int u = first; u != join; u = _parent[u]) {
1259        e = _pred[u];
1260        d = _flow[e];
1261        if (_pred_dir[u] == DIR_DOWN) {
1262          c = _cap[e];
1263          d = c >= MAX ? INF : c - d;
1264        }
1265        if (d < delta) {
1266          delta = d;
1267          u_out = u;
1268          result = 1;
1269        }
1270      }
1271
1272      // Search the cycle form the second node to the join node
1273      for (int u = second; u != join; u = _parent[u]) {
1274        e = _pred[u];
1275        d = _flow[e];
1276        if (_pred_dir[u] == DIR_UP) {
1277          c = _cap[e];
1278          d = c >= MAX ? INF : c - d;
1279        }
1280        if (d <= delta) {
1281          delta = d;
1282          u_out = u;
1283          result = 2;
1284        }
1285      }
1286
1287      if (result == 1) {
1288        u_in = first;
1289        v_in = second;
1290      } else {
1291        u_in = second;
1292        v_in = first;
1293      }
1294      return result != 0;
1295    }
1296
1297    // Change _flow and _state vectors
1298    void changeFlow(bool change) {
1299      // Augment along the cycle
1300      if (delta > 0) {
1301        Value val = _state[in_arc] * delta;
1302        _flow[in_arc] += val;
1303        for (int u = _source[in_arc]; u != join; u = _parent[u]) {
1304          _flow[_pred[u]] -= _pred_dir[u] * val;
1305        }
1306        for (int u = _target[in_arc]; u != join; u = _parent[u]) {
1307          _flow[_pred[u]] += _pred_dir[u] * val;
1308        }
1309      }
1310      // Update the state of the entering and leaving arcs
1311      if (change) {
1312        _state[in_arc] = STATE_TREE;
1313        _state[_pred[u_out]] =
1314          (_flow[_pred[u_out]] == 0) ? STATE_LOWER : STATE_UPPER;
1315      } else {
1316        _state[in_arc] = -_state[in_arc];
1317      }
1318    }
1319
1320    // Update the tree structure
1321    void updateTreeStructure() {
1322      int old_rev_thread = _rev_thread[u_out];
1323      int old_succ_num = _succ_num[u_out];
1324      int old_last_succ = _last_succ[u_out];
1325      v_out = _parent[u_out];
1326
1327      // Check if u_in and u_out coincide
1328      if (u_in == u_out) {
1329        // Update _parent, _pred, _pred_dir
1330        _parent[u_in] = v_in;
1331        _pred[u_in] = in_arc;
1332        _pred_dir[u_in] = u_in == _source[in_arc] ? DIR_UP : DIR_DOWN;
1333
1334        // Update _thread and _rev_thread
1335        if (_thread[v_in] != u_out) {
1336          int after = _thread[old_last_succ];
1337          _thread[old_rev_thread] = after;
1338          _rev_thread[after] = old_rev_thread;
1339          after = _thread[v_in];
1340          _thread[v_in] = u_out;
1341          _rev_thread[u_out] = v_in;
1342          _thread[old_last_succ] = after;
1343          _rev_thread[after] = old_last_succ;
1344        }
1345      } else {
1346        // Handle the case when old_rev_thread equals to v_in
1347        // (it also means that join and v_out coincide)
1348        int thread_continue = old_rev_thread == v_in ?
1349          _thread[old_last_succ] : _thread[v_in];
1350
1351        // Update _thread and _parent along the stem nodes (i.e. the nodes
1352        // between u_in and u_out, whose parent have to be changed)
1353        int stem = u_in;              // the current stem node
1354        int par_stem = v_in;          // the new parent of stem
1355        int next_stem;                // the next stem node
1356        int last = _last_succ[u_in];  // the last successor of stem
1357        int before, after = _thread[last];
1358        _thread[v_in] = u_in;
1359        _dirty_revs.clear();
1360        _dirty_revs.push_back(v_in);
1361        while (stem != u_out) {
1362          // Insert the next stem node into the thread list
1363          next_stem = _parent[stem];
1364          _thread[last] = next_stem;
1365          _dirty_revs.push_back(last);
1366
1367          // Remove the subtree of stem from the thread list
1368          before = _rev_thread[stem];
1369          _thread[before] = after;
1370          _rev_thread[after] = before;
1371
1372          // Change the parent node and shift stem nodes
1373          _parent[stem] = par_stem;
1374          par_stem = stem;
1375          stem = next_stem;
1376
1377          // Update last and after
1378          last = _last_succ[stem] == _last_succ[par_stem] ?
1379            _rev_thread[par_stem] : _last_succ[stem];
1380          after = _thread[last];
1381        }
1382        _parent[u_out] = par_stem;
1383        _thread[last] = thread_continue;
1384        _rev_thread[thread_continue] = last;
1385        _last_succ[u_out] = last;
1386
1387        // Remove the subtree of u_out from the thread list except for
1388        // the case when old_rev_thread equals to v_in
1389        if (old_rev_thread != v_in) {
1390          _thread[old_rev_thread] = after;
1391          _rev_thread[after] = old_rev_thread;
1392        }
1393
1394        // Update _rev_thread using the new _thread values
1395        for (int i = 0; i != int(_dirty_revs.size()); ++i) {
1396          int u = _dirty_revs[i];
1397          _rev_thread[_thread[u]] = u;
1398        }
1399
1400        // Update _pred, _pred_dir, _last_succ and _succ_num for the
1401        // stem nodes from u_out to u_in
1402        int tmp_sc = 0, tmp_ls = _last_succ[u_out];
1403        for (int u = u_out, p = _parent[u]; u != u_in; u = p, p = _parent[u]) {
1404          _pred[u] = _pred[p];
1405          _pred_dir[u] = -_pred_dir[p];
1406          tmp_sc += _succ_num[u] - _succ_num[p];
1407          _succ_num[u] = tmp_sc;
1408          _last_succ[p] = tmp_ls;
1409        }
1410        _pred[u_in] = in_arc;
1411        _pred_dir[u_in] = u_in == _source[in_arc] ? DIR_UP : DIR_DOWN;
1412        _succ_num[u_in] = old_succ_num;
1413      }
1414
1415      // Update _last_succ from v_in towards the root
1416      int up_limit_out = _last_succ[join] == v_in ? join : -1;
1417      int last_succ_out = _last_succ[u_out];
1418      for (int u = v_in; u != -1 && _last_succ[u] == v_in; u = _parent[u]) {
1419        _last_succ[u] = last_succ_out;
1420      }
1421
1422      // Update _last_succ from v_out towards the root
1423      if (join != old_rev_thread && v_in != old_rev_thread) {
1424        for (int u = v_out; u != up_limit_out && _last_succ[u] == old_last_succ;
1425             u = _parent[u]) {
1426          _last_succ[u] = old_rev_thread;
1427        }
1428      }
1429      else if (last_succ_out != old_last_succ) {
1430        for (int u = v_out; u != up_limit_out && _last_succ[u] == old_last_succ;
1431             u = _parent[u]) {
1432          _last_succ[u] = last_succ_out;
1433        }
1434      }
1435
1436      // Update _succ_num from v_in to join
1437      for (int u = v_in; u != join; u = _parent[u]) {
1438        _succ_num[u] += old_succ_num;
1439      }
1440      // Update _succ_num from v_out to join
1441      for (int u = v_out; u != join; u = _parent[u]) {
1442        _succ_num[u] -= old_succ_num;
1443      }
1444    }
1445
1446    // Update potentials in the subtree that has been moved
1447    void updatePotential() {
1448      Cost sigma = _pi[v_in] - _pi[u_in] -
1449                   _pred_dir[u_in] * _cost[in_arc];
1450      int end = _thread[_last_succ[u_in]];
1451      for (int u = u_in; u != end; u = _thread[u]) {
1452        _pi[u] += sigma;
1453      }
1454    }
1455
1456    // Heuristic initial pivots
1457    bool initialPivots() {
1458      Value curr, total = 0;
1459      std::vector<Node> supply_nodes, demand_nodes;
1460      for (NodeIt u(_graph); u != INVALID; ++u) {
1461        curr = _supply[_node_id[u]];
1462        if (curr > 0) {
1463          total += curr;
1464          supply_nodes.push_back(u);
1465        }
1466        else if (curr < 0) {
1467          demand_nodes.push_back(u);
1468        }
1469      }
1470      if (_sum_supply > 0) total -= _sum_supply;
1471      if (total <= 0) return true;
1472
1473      IntVector arc_vector;
1474      if (_sum_supply >= 0) {
1475        if (supply_nodes.size() == 1 && demand_nodes.size() == 1) {
1476          // Perform a reverse graph search from the sink to the source
1477          typename GR::template NodeMap<bool> reached(_graph, false);
1478          Node s = supply_nodes[0], t = demand_nodes[0];
1479          std::vector<Node> stack;
1480          reached[t] = true;
1481          stack.push_back(t);
1482          while (!stack.empty()) {
1483            Node u, v = stack.back();
1484            stack.pop_back();
1485            if (v == s) break;
1486            for (InArcIt a(_graph, v); a != INVALID; ++a) {
1487              if (reached[u = _graph.source(a)]) continue;
1488              int j = _arc_id[a];
1489              if (_cap[j] >= total) {
1490                arc_vector.push_back(j);
1491                reached[u] = true;
1492                stack.push_back(u);
1493              }
1494            }
1495          }
1496        } else {
1497          // Find the min. cost incomming arc for each demand node
1498          for (int i = 0; i != int(demand_nodes.size()); ++i) {
1499            Node v = demand_nodes[i];
1500            Cost c, min_cost = std::numeric_limits<Cost>::max();
1501            Arc min_arc = INVALID;
1502            for (InArcIt a(_graph, v); a != INVALID; ++a) {
1503              c = _cost[_arc_id[a]];
1504              if (c < min_cost) {
1505                min_cost = c;
1506                min_arc = a;
1507              }
1508            }
1509            if (min_arc != INVALID) {
1510              arc_vector.push_back(_arc_id[min_arc]);
1511            }
1512          }
1513        }
1514      } else {
1515        // Find the min. cost outgoing arc for each supply node
1516        for (int i = 0; i != int(supply_nodes.size()); ++i) {
1517          Node u = supply_nodes[i];
1518          Cost c, min_cost = std::numeric_limits<Cost>::max();
1519          Arc min_arc = INVALID;
1520          for (OutArcIt a(_graph, u); a != INVALID; ++a) {
1521            c = _cost[_arc_id[a]];
1522            if (c < min_cost) {
1523              min_cost = c;
1524              min_arc = a;
1525            }
1526          }
1527          if (min_arc != INVALID) {
1528            arc_vector.push_back(_arc_id[min_arc]);
1529          }
1530        }
1531      }
1532
1533      // Perform heuristic initial pivots
1534      for (int i = 0; i != int(arc_vector.size()); ++i) {
1535        in_arc = arc_vector[i];
1536        if (_state[in_arc] * (_cost[in_arc] + _pi[_source[in_arc]] -
1537            _pi[_target[in_arc]]) >= 0) continue;
1538        findJoinNode();
1539        bool change = findLeavingArc();
1540        if (delta >= MAX) return false;
1541        changeFlow(change);
1542        if (change) {
1543          updateTreeStructure();
1544          updatePotential();
1545        }
1546      }
1547      return true;
1548    }
1549
1550    // Execute the algorithm
1551    ProblemType start(PivotRule pivot_rule) {
1552      // Select the pivot rule implementation
1553      switch (pivot_rule) {
1554        case FIRST_ELIGIBLE:
1555          return start<FirstEligiblePivotRule>();
1556        case BEST_ELIGIBLE:
1557          return start<BestEligiblePivotRule>();
1558        case BLOCK_SEARCH:
1559          return start<BlockSearchPivotRule>();
1560        case CANDIDATE_LIST:
1561          return start<CandidateListPivotRule>();
1562        case ALTERING_LIST:
1563          return start<AlteringListPivotRule>();
1564      }
1565      return INFEASIBLE; // avoid warning
1566    }
1567
1568    template <typename PivotRuleImpl>
1569    ProblemType start() {
1570      PivotRuleImpl pivot(*this);
1571
1572      // Perform heuristic initial pivots
1573      if (!initialPivots()) return UNBOUNDED;
1574
1575      // Execute the Network Simplex algorithm
1576      while (pivot.findEnteringArc()) {
1577        findJoinNode();
1578        bool change = findLeavingArc();
1579        if (delta >= MAX) return UNBOUNDED;
1580        changeFlow(change);
1581        if (change) {
1582          updateTreeStructure();
1583          updatePotential();
1584        }
1585      }
1586
1587      // Check feasibility
1588      for (int e = _search_arc_num; e != _all_arc_num; ++e) {
1589        if (_flow[e] != 0) return INFEASIBLE;
1590      }
1591
1592      // Transform the solution and the supply map to the original form
1593      if (_have_lower) {
1594        for (int i = 0; i != _arc_num; ++i) {
1595          Value c = _lower[i];
1596          if (c != 0) {
1597            _flow[i] += c;
1598            _supply[_source[i]] += c;
1599            _supply[_target[i]] -= c;
1600          }
1601        }
1602      }
1603
1604      // Shift potentials to meet the requirements of the GEQ/LEQ type
1605      // optimality conditions
1606      if (_sum_supply == 0) {
1607        if (_stype == GEQ) {
1608          Cost max_pot = -std::numeric_limits<Cost>::max();
1609          for (int i = 0; i != _node_num; ++i) {
1610            if (_pi[i] > max_pot) max_pot = _pi[i];
1611          }
1612          if (max_pot > 0) {
1613            for (int i = 0; i != _node_num; ++i)
1614              _pi[i] -= max_pot;
1615          }
1616        } else {
1617          Cost min_pot = std::numeric_limits<Cost>::max();
1618          for (int i = 0; i != _node_num; ++i) {
1619            if (_pi[i] < min_pot) min_pot = _pi[i];
1620          }
1621          if (min_pot < 0) {
1622            for (int i = 0; i != _node_num; ++i)
1623              _pi[i] -= min_pot;
1624          }
1625        }
1626      }
1627
1628      return OPTIMAL;
1629    }
1630
1631  }; //class NetworkSimplex
1632
1633  ///@}
1634
1635} //namespace lemon
1636
1637#endif //LEMON_NETWORK_SIMPLEX_H
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