COIN-OR::LEMON - Graph Library

source: lemon-1.2/lemon/network_simplex.h @ 830:75c97c3786d6

Last change on this file since 830:75c97c3786d6 was 830:75c97c3786d6, checked in by Peter Kovacs <kpeter@…>, 10 years ago

Handle graph changes in the MCF algorithms (#327)

The reset() functions are renamed to resetParams() and the new reset()
functions handle the graph chnages, as well.

File size: 46.6 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) 2003-2009
6 * Egervary Jeno Kombinatorikus Optimalizalasi Kutatocsoport
7 * (Egervary Research Group on Combinatorial Optimization, EGRES).
8 *
9 * Permission to use, modify and distribute this software is granted
10 * provided that this copyright notice appears in all copies. For
11 * precise terms see the accompanying LICENSE file.
12 *
13 * This software is provided "AS IS" with no warranty of any kind,
14 * express or implied, and with no claim as to its suitability for any
15 * purpose.
16 *
17 */
18
19#ifndef LEMON_NETWORK_SIMPLEX_H
20#define LEMON_NETWORK_SIMPLEX_H
21
22/// \ingroup min_cost_flow_algs
23///
24/// \file
25/// \brief Network Simplex algorithm for finding a minimum cost flow.
26
27#include <vector>
28#include <limits>
29#include <algorithm>
30
31#include <lemon/core.h>
32#include <lemon/math.h>
33
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<char> CharVector;
168    typedef std::vector<Value> ValueVector;
169    typedef std::vector<Cost> CostVector;
170
171    // State constants for arcs
172    enum ArcStateEnum {
173      STATE_UPPER = -1,
174      STATE_TREE  =  0,
175      STATE_LOWER =  1
176    };
177
178  private:
179
180    // Data related to the underlying digraph
181    const GR &_graph;
182    int _node_num;
183    int _arc_num;
184    int _all_arc_num;
185    int _search_arc_num;
186
187    // Parameters of the problem
188    bool _have_lower;
189    SupplyType _stype;
190    Value _sum_supply;
191
192    // Data structures for storing the digraph
193    IntNodeMap _node_id;
194    IntArcMap _arc_id;
195    IntVector _source;
196    IntVector _target;
197    bool _arc_mixing;
198
199    // Node and arc data
200    ValueVector _lower;
201    ValueVector _upper;
202    ValueVector _cap;
203    CostVector _cost;
204    ValueVector _supply;
205    ValueVector _flow;
206    CostVector _pi;
207
208    // Data for storing the spanning tree structure
209    IntVector _parent;
210    IntVector _pred;
211    IntVector _thread;
212    IntVector _rev_thread;
213    IntVector _succ_num;
214    IntVector _last_succ;
215    IntVector _dirty_revs;
216    CharVector _forward;
217    CharVector _state;
218    int _root;
219
220    // Temporary data used in the current pivot iteration
221    int in_arc, join, u_in, v_in, u_out, v_out;
222    int first, second, right, last;
223    int stem, par_stem, new_stem;
224    Value delta;
225   
226    const Value MAX;
227
228  public:
229 
230    /// \brief Constant for infinite upper bounds (capacities).
231    ///
232    /// Constant for infinite upper bounds (capacities).
233    /// It is \c std::numeric_limits<Value>::infinity() if available,
234    /// \c std::numeric_limits<Value>::max() otherwise.
235    const Value INF;
236
237  private:
238
239    // Implementation of the First Eligible pivot rule
240    class FirstEligiblePivotRule
241    {
242    private:
243
244      // References to the NetworkSimplex class
245      const IntVector  &_source;
246      const IntVector  &_target;
247      const CostVector &_cost;
248      const CharVector &_state;
249      const CostVector &_pi;
250      int &_in_arc;
251      int _search_arc_num;
252
253      // Pivot rule data
254      int _next_arc;
255
256    public:
257
258      // Constructor
259      FirstEligiblePivotRule(NetworkSimplex &ns) :
260        _source(ns._source), _target(ns._target),
261        _cost(ns._cost), _state(ns._state), _pi(ns._pi),
262        _in_arc(ns.in_arc), _search_arc_num(ns._search_arc_num),
263        _next_arc(0)
264      {}
265
266      // Find next entering arc
267      bool findEnteringArc() {
268        Cost c;
269        for (int e = _next_arc; e < _search_arc_num; ++e) {
270          c = _state[e] * (_cost[e] + _pi[_source[e]] - _pi[_target[e]]);
271          if (c < 0) {
272            _in_arc = e;
273            _next_arc = e + 1;
274            return true;
275          }
276        }
277        for (int e = 0; e < _next_arc; ++e) {
278          c = _state[e] * (_cost[e] + _pi[_source[e]] - _pi[_target[e]]);
279          if (c < 0) {
280            _in_arc = e;
281            _next_arc = e + 1;
282            return true;
283          }
284        }
285        return false;
286      }
287
288    }; //class FirstEligiblePivotRule
289
290
291    // Implementation of the Best Eligible pivot rule
292    class BestEligiblePivotRule
293    {
294    private:
295
296      // References to the NetworkSimplex class
297      const IntVector  &_source;
298      const IntVector  &_target;
299      const CostVector &_cost;
300      const CharVector &_state;
301      const CostVector &_pi;
302      int &_in_arc;
303      int _search_arc_num;
304
305    public:
306
307      // Constructor
308      BestEligiblePivotRule(NetworkSimplex &ns) :
309        _source(ns._source), _target(ns._target),
310        _cost(ns._cost), _state(ns._state), _pi(ns._pi),
311        _in_arc(ns.in_arc), _search_arc_num(ns._search_arc_num)
312      {}
313
314      // Find next entering arc
315      bool findEnteringArc() {
316        Cost c, min = 0;
317        for (int e = 0; e < _search_arc_num; ++e) {
318          c = _state[e] * (_cost[e] + _pi[_source[e]] - _pi[_target[e]]);
319          if (c < min) {
320            min = c;
321            _in_arc = e;
322          }
323        }
324        return min < 0;
325      }
326
327    }; //class BestEligiblePivotRule
328
329
330    // Implementation of the Block Search pivot rule
331    class BlockSearchPivotRule
332    {
333    private:
334
335      // References to the NetworkSimplex class
336      const IntVector  &_source;
337      const IntVector  &_target;
338      const CostVector &_cost;
339      const CharVector &_state;
340      const CostVector &_pi;
341      int &_in_arc;
342      int _search_arc_num;
343
344      // Pivot rule data
345      int _block_size;
346      int _next_arc;
347
348    public:
349
350      // Constructor
351      BlockSearchPivotRule(NetworkSimplex &ns) :
352        _source(ns._source), _target(ns._target),
353        _cost(ns._cost), _state(ns._state), _pi(ns._pi),
354        _in_arc(ns.in_arc), _search_arc_num(ns._search_arc_num),
355        _next_arc(0)
356      {
357        // The main parameters of the pivot rule
358        const double BLOCK_SIZE_FACTOR = 0.5;
359        const int MIN_BLOCK_SIZE = 10;
360
361        _block_size = std::max( int(BLOCK_SIZE_FACTOR *
362                                    std::sqrt(double(_search_arc_num))),
363                                MIN_BLOCK_SIZE );
364      }
365
366      // Find next entering arc
367      bool findEnteringArc() {
368        Cost c, min = 0;
369        int cnt = _block_size;
370        int e;
371        for (e = _next_arc; e < _search_arc_num; ++e) {
372          c = _state[e] * (_cost[e] + _pi[_source[e]] - _pi[_target[e]]);
373          if (c < min) {
374            min = c;
375            _in_arc = e;
376          }
377          if (--cnt == 0) {
378            if (min < 0) goto search_end;
379            cnt = _block_size;
380          }
381        }
382        for (e = 0; e < _next_arc; ++e) {
383          c = _state[e] * (_cost[e] + _pi[_source[e]] - _pi[_target[e]]);
384          if (c < min) {
385            min = c;
386            _in_arc = e;
387          }
388          if (--cnt == 0) {
389            if (min < 0) goto search_end;
390            cnt = _block_size;
391          }
392        }
393        if (min >= 0) return false;
394
395      search_end:
396        _next_arc = e;
397        return true;
398      }
399
400    }; //class BlockSearchPivotRule
401
402
403    // Implementation of the Candidate List pivot rule
404    class CandidateListPivotRule
405    {
406    private:
407
408      // References to the NetworkSimplex class
409      const IntVector  &_source;
410      const IntVector  &_target;
411      const CostVector &_cost;
412      const CharVector &_state;
413      const CostVector &_pi;
414      int &_in_arc;
415      int _search_arc_num;
416
417      // Pivot rule data
418      IntVector _candidates;
419      int _list_length, _minor_limit;
420      int _curr_length, _minor_count;
421      int _next_arc;
422
423    public:
424
425      /// Constructor
426      CandidateListPivotRule(NetworkSimplex &ns) :
427        _source(ns._source), _target(ns._target),
428        _cost(ns._cost), _state(ns._state), _pi(ns._pi),
429        _in_arc(ns.in_arc), _search_arc_num(ns._search_arc_num),
430        _next_arc(0)
431      {
432        // The main parameters of the pivot rule
433        const double LIST_LENGTH_FACTOR = 0.25;
434        const int MIN_LIST_LENGTH = 10;
435        const double MINOR_LIMIT_FACTOR = 0.1;
436        const int MIN_MINOR_LIMIT = 3;
437
438        _list_length = std::max( int(LIST_LENGTH_FACTOR *
439                                     std::sqrt(double(_search_arc_num))),
440                                 MIN_LIST_LENGTH );
441        _minor_limit = std::max( int(MINOR_LIMIT_FACTOR * _list_length),
442                                 MIN_MINOR_LIMIT );
443        _curr_length = _minor_count = 0;
444        _candidates.resize(_list_length);
445      }
446
447      /// Find next entering arc
448      bool findEnteringArc() {
449        Cost min, c;
450        int e;
451        if (_curr_length > 0 && _minor_count < _minor_limit) {
452          // Minor iteration: select the best eligible arc from the
453          // current candidate list
454          ++_minor_count;
455          min = 0;
456          for (int i = 0; i < _curr_length; ++i) {
457            e = _candidates[i];
458            c = _state[e] * (_cost[e] + _pi[_source[e]] - _pi[_target[e]]);
459            if (c < min) {
460              min = c;
461              _in_arc = e;
462            }
463            else if (c >= 0) {
464              _candidates[i--] = _candidates[--_curr_length];
465            }
466          }
467          if (min < 0) return true;
468        }
469
470        // Major iteration: build a new candidate list
471        min = 0;
472        _curr_length = 0;
473        for (e = _next_arc; e < _search_arc_num; ++e) {
474          c = _state[e] * (_cost[e] + _pi[_source[e]] - _pi[_target[e]]);
475          if (c < 0) {
476            _candidates[_curr_length++] = e;
477            if (c < min) {
478              min = c;
479              _in_arc = e;
480            }
481            if (_curr_length == _list_length) goto search_end;
482          }
483        }
484        for (e = 0; e < _next_arc; ++e) {
485          c = _state[e] * (_cost[e] + _pi[_source[e]] - _pi[_target[e]]);
486          if (c < 0) {
487            _candidates[_curr_length++] = e;
488            if (c < min) {
489              min = c;
490              _in_arc = e;
491            }
492            if (_curr_length == _list_length) goto search_end;
493          }
494        }
495        if (_curr_length == 0) return false;
496     
497      search_end:       
498        _minor_count = 1;
499        _next_arc = e;
500        return true;
501      }
502
503    }; //class CandidateListPivotRule
504
505
506    // Implementation of the Altering Candidate List pivot rule
507    class AlteringListPivotRule
508    {
509    private:
510
511      // References to the NetworkSimplex class
512      const IntVector  &_source;
513      const IntVector  &_target;
514      const CostVector &_cost;
515      const CharVector &_state;
516      const CostVector &_pi;
517      int &_in_arc;
518      int _search_arc_num;
519
520      // Pivot rule data
521      int _block_size, _head_length, _curr_length;
522      int _next_arc;
523      IntVector _candidates;
524      CostVector _cand_cost;
525
526      // Functor class to compare arcs during sort of the candidate list
527      class SortFunc
528      {
529      private:
530        const CostVector &_map;
531      public:
532        SortFunc(const CostVector &map) : _map(map) {}
533        bool operator()(int left, int right) {
534          return _map[left] > _map[right];
535        }
536      };
537
538      SortFunc _sort_func;
539
540    public:
541
542      // Constructor
543      AlteringListPivotRule(NetworkSimplex &ns) :
544        _source(ns._source), _target(ns._target),
545        _cost(ns._cost), _state(ns._state), _pi(ns._pi),
546        _in_arc(ns.in_arc), _search_arc_num(ns._search_arc_num),
547        _next_arc(0), _cand_cost(ns._search_arc_num), _sort_func(_cand_cost)
548      {
549        // The main parameters of the pivot rule
550        const double BLOCK_SIZE_FACTOR = 1.0;
551        const int MIN_BLOCK_SIZE = 10;
552        const double HEAD_LENGTH_FACTOR = 0.1;
553        const int MIN_HEAD_LENGTH = 3;
554
555        _block_size = std::max( int(BLOCK_SIZE_FACTOR *
556                                    std::sqrt(double(_search_arc_num))),
557                                MIN_BLOCK_SIZE );
558        _head_length = std::max( int(HEAD_LENGTH_FACTOR * _block_size),
559                                 MIN_HEAD_LENGTH );
560        _candidates.resize(_head_length + _block_size);
561        _curr_length = 0;
562      }
563
564      // Find next entering arc
565      bool findEnteringArc() {
566        // Check the current candidate list
567        int e;
568        for (int i = 0; i < _curr_length; ++i) {
569          e = _candidates[i];
570          _cand_cost[e] = _state[e] *
571            (_cost[e] + _pi[_source[e]] - _pi[_target[e]]);
572          if (_cand_cost[e] >= 0) {
573            _candidates[i--] = _candidates[--_curr_length];
574          }
575        }
576
577        // Extend the list
578        int cnt = _block_size;
579        int limit = _head_length;
580
581        for (e = _next_arc; e < _search_arc_num; ++e) {
582          _cand_cost[e] = _state[e] *
583            (_cost[e] + _pi[_source[e]] - _pi[_target[e]]);
584          if (_cand_cost[e] < 0) {
585            _candidates[_curr_length++] = e;
586          }
587          if (--cnt == 0) {
588            if (_curr_length > limit) goto search_end;
589            limit = 0;
590            cnt = _block_size;
591          }
592        }
593        for (e = 0; e < _next_arc; ++e) {
594          _cand_cost[e] = _state[e] *
595            (_cost[e] + _pi[_source[e]] - _pi[_target[e]]);
596          if (_cand_cost[e] < 0) {
597            _candidates[_curr_length++] = e;
598          }
599          if (--cnt == 0) {
600            if (_curr_length > limit) goto search_end;
601            limit = 0;
602            cnt = _block_size;
603          }
604        }
605        if (_curr_length == 0) return false;
606       
607      search_end:
608
609        // Make heap of the candidate list (approximating a partial sort)
610        make_heap( _candidates.begin(), _candidates.begin() + _curr_length,
611                   _sort_func );
612
613        // Pop the first element of the heap
614        _in_arc = _candidates[0];
615        _next_arc = e;
616        pop_heap( _candidates.begin(), _candidates.begin() + _curr_length,
617                  _sort_func );
618        _curr_length = std::min(_head_length, _curr_length - 1);
619        return true;
620      }
621
622    }; //class AlteringListPivotRule
623
624  public:
625
626    /// \brief Constructor.
627    ///
628    /// The constructor of the class.
629    ///
630    /// \param graph The digraph the algorithm runs on.
631    /// \param arc_mixing Indicate if the arcs have to be stored in a
632    /// mixed order in the internal data structure.
633    /// In special cases, it could lead to better overall performance,
634    /// but it is usually slower. Therefore it is disabled by default.
635    NetworkSimplex(const GR& graph, bool arc_mixing = false) :
636      _graph(graph), _node_id(graph), _arc_id(graph),
637      _arc_mixing(arc_mixing),
638      MAX(std::numeric_limits<Value>::max()),
639      INF(std::numeric_limits<Value>::has_infinity ?
640          std::numeric_limits<Value>::infinity() : MAX)
641    {
642      // Check the number types
643      LEMON_ASSERT(std::numeric_limits<Value>::is_signed,
644        "The flow type of NetworkSimplex must be signed");
645      LEMON_ASSERT(std::numeric_limits<Cost>::is_signed,
646        "The cost type of NetworkSimplex must be signed");
647       
648      // Reset data structures
649      reset();
650    }
651
652    /// \name Parameters
653    /// The parameters of the algorithm can be specified using these
654    /// functions.
655
656    /// @{
657
658    /// \brief Set the lower bounds on the arcs.
659    ///
660    /// This function sets the lower bounds on the arcs.
661    /// If it is not used before calling \ref run(), the lower bounds
662    /// will be set to zero on all arcs.
663    ///
664    /// \param map An arc map storing the lower bounds.
665    /// Its \c Value type must be convertible to the \c Value type
666    /// of the algorithm.
667    ///
668    /// \return <tt>(*this)</tt>
669    template <typename LowerMap>
670    NetworkSimplex& lowerMap(const LowerMap& map) {
671      _have_lower = true;
672      for (ArcIt a(_graph); a != INVALID; ++a) {
673        _lower[_arc_id[a]] = map[a];
674      }
675      return *this;
676    }
677
678    /// \brief Set the upper bounds (capacities) on the arcs.
679    ///
680    /// This function sets the upper bounds (capacities) on the arcs.
681    /// If it is not used before calling \ref run(), the upper bounds
682    /// will be set to \ref INF on all arcs (i.e. the flow value will be
683    /// unbounded from above).
684    ///
685    /// \param map An arc map storing the upper bounds.
686    /// Its \c Value type must be convertible to the \c Value type
687    /// of the algorithm.
688    ///
689    /// \return <tt>(*this)</tt>
690    template<typename UpperMap>
691    NetworkSimplex& upperMap(const UpperMap& map) {
692      for (ArcIt a(_graph); a != INVALID; ++a) {
693        _upper[_arc_id[a]] = map[a];
694      }
695      return *this;
696    }
697
698    /// \brief Set the costs of the arcs.
699    ///
700    /// This function sets the costs of the arcs.
701    /// If it is not used before calling \ref run(), the costs
702    /// will be set to \c 1 on all arcs.
703    ///
704    /// \param map An arc map storing the costs.
705    /// Its \c Value type must be convertible to the \c Cost type
706    /// of the algorithm.
707    ///
708    /// \return <tt>(*this)</tt>
709    template<typename CostMap>
710    NetworkSimplex& costMap(const CostMap& map) {
711      for (ArcIt a(_graph); a != INVALID; ++a) {
712        _cost[_arc_id[a]] = map[a];
713      }
714      return *this;
715    }
716
717    /// \brief Set the supply values of the nodes.
718    ///
719    /// This function sets the supply values of the nodes.
720    /// If neither this function nor \ref stSupply() is used before
721    /// calling \ref run(), the supply of each node will be set to zero.
722    ///
723    /// \param map A node map storing the supply values.
724    /// Its \c Value type must be convertible to the \c Value type
725    /// of the algorithm.
726    ///
727    /// \return <tt>(*this)</tt>
728    template<typename SupplyMap>
729    NetworkSimplex& supplyMap(const SupplyMap& map) {
730      for (NodeIt n(_graph); n != INVALID; ++n) {
731        _supply[_node_id[n]] = map[n];
732      }
733      return *this;
734    }
735
736    /// \brief Set single source and target nodes and a supply value.
737    ///
738    /// This function sets a single source node and a single target node
739    /// and the required flow value.
740    /// If neither this function nor \ref supplyMap() is used before
741    /// calling \ref run(), the supply of each node will be set to zero.
742    ///
743    /// Using this function has the same effect as using \ref supplyMap()
744    /// with such a map in which \c k is assigned to \c s, \c -k is
745    /// assigned to \c t and all other nodes have zero supply value.
746    ///
747    /// \param s The source node.
748    /// \param t The target node.
749    /// \param k The required amount of flow from node \c s to node \c t
750    /// (i.e. the supply of \c s and the demand of \c t).
751    ///
752    /// \return <tt>(*this)</tt>
753    NetworkSimplex& stSupply(const Node& s, const Node& t, Value k) {
754      for (int i = 0; i != _node_num; ++i) {
755        _supply[i] = 0;
756      }
757      _supply[_node_id[s]] =  k;
758      _supply[_node_id[t]] = -k;
759      return *this;
760    }
761   
762    /// \brief Set the type of the supply constraints.
763    ///
764    /// This function sets the type of the supply/demand constraints.
765    /// If it is not used before calling \ref run(), the \ref GEQ supply
766    /// type will be used.
767    ///
768    /// For more information, see \ref SupplyType.
769    ///
770    /// \return <tt>(*this)</tt>
771    NetworkSimplex& supplyType(SupplyType supply_type) {
772      _stype = supply_type;
773      return *this;
774    }
775
776    /// @}
777
778    /// \name Execution Control
779    /// The algorithm can be executed using \ref run().
780
781    /// @{
782
783    /// \brief Run the algorithm.
784    ///
785    /// This function runs the algorithm.
786    /// The paramters can be specified using functions \ref lowerMap(),
787    /// \ref upperMap(), \ref costMap(), \ref supplyMap(), \ref stSupply(),
788    /// \ref supplyType().
789    /// For example,
790    /// \code
791    ///   NetworkSimplex<ListDigraph> ns(graph);
792    ///   ns.lowerMap(lower).upperMap(upper).costMap(cost)
793    ///     .supplyMap(sup).run();
794    /// \endcode
795    ///
796    /// This function can be called more than once. All the given parameters
797    /// are kept for the next call, unless \ref resetParams() or \ref reset()
798    /// is used, thus only the modified parameters have to be set again.
799    /// If the underlying digraph was also modified after the construction
800    /// of the class (or the last \ref reset() call), then the \ref reset()
801    /// function must be called.
802    ///
803    /// \param pivot_rule The pivot rule that will be used during the
804    /// algorithm. For more information, see \ref PivotRule.
805    ///
806    /// \return \c INFEASIBLE if no feasible flow exists,
807    /// \n \c OPTIMAL if the problem has optimal solution
808    /// (i.e. it is feasible and bounded), and the algorithm has found
809    /// optimal flow and node potentials (primal and dual solutions),
810    /// \n \c UNBOUNDED if the objective function of the problem is
811    /// unbounded, i.e. there is a directed cycle having negative total
812    /// cost and infinite upper bound.
813    ///
814    /// \see ProblemType, PivotRule
815    /// \see resetParams(), reset()
816    ProblemType run(PivotRule pivot_rule = BLOCK_SEARCH) {
817      if (!init()) return INFEASIBLE;
818      return start(pivot_rule);
819    }
820
821    /// \brief Reset all the parameters that have been given before.
822    ///
823    /// This function resets all the paramaters that have been given
824    /// before using functions \ref lowerMap(), \ref upperMap(),
825    /// \ref costMap(), \ref supplyMap(), \ref stSupply(), \ref supplyType().
826    ///
827    /// It is useful for multiple \ref run() calls. Basically, all the given
828    /// parameters are kept for the next \ref run() call, unless
829    /// \ref resetParams() or \ref reset() is used.
830    /// If the underlying digraph was also modified after the construction
831    /// of the class or the last \ref reset() call, then the \ref reset()
832    /// function must be used, otherwise \ref resetParams() is sufficient.
833    ///
834    /// For example,
835    /// \code
836    ///   NetworkSimplex<ListDigraph> ns(graph);
837    ///
838    ///   // First run
839    ///   ns.lowerMap(lower).upperMap(upper).costMap(cost)
840    ///     .supplyMap(sup).run();
841    ///
842    ///   // Run again with modified cost map (resetParams() is not called,
843    ///   // so only the cost map have to be set again)
844    ///   cost[e] += 100;
845    ///   ns.costMap(cost).run();
846    ///
847    ///   // Run again from scratch using resetParams()
848    ///   // (the lower bounds will be set to zero on all arcs)
849    ///   ns.resetParams();
850    ///   ns.upperMap(capacity).costMap(cost)
851    ///     .supplyMap(sup).run();
852    /// \endcode
853    ///
854    /// \return <tt>(*this)</tt>
855    ///
856    /// \see reset(), run()
857    NetworkSimplex& resetParams() {
858      for (int i = 0; i != _node_num; ++i) {
859        _supply[i] = 0;
860      }
861      for (int i = 0; i != _arc_num; ++i) {
862        _lower[i] = 0;
863        _upper[i] = INF;
864        _cost[i] = 1;
865      }
866      _have_lower = false;
867      _stype = GEQ;
868      return *this;
869    }
870
871    /// \brief Reset the internal data structures and all the parameters
872    /// that have been given before.
873    ///
874    /// This function resets the internal data structures and all the
875    /// paramaters that have been given before using functions \ref lowerMap(),
876    /// \ref upperMap(), \ref costMap(), \ref supplyMap(), \ref stSupply(),
877    /// \ref supplyType().
878    ///
879    /// It is useful for multiple \ref run() calls. Basically, all the given
880    /// parameters are kept for the next \ref run() call, unless
881    /// \ref resetParams() or \ref reset() is used.
882    /// If the underlying digraph was also modified after the construction
883    /// of the class or the last \ref reset() call, then the \ref reset()
884    /// function must be used, otherwise \ref resetParams() is sufficient.
885    ///
886    /// See \ref resetParams() for examples.
887    ///
888    /// \return <tt>(*this)</tt>
889    ///
890    /// \see resetParams(), run()
891    NetworkSimplex& reset() {
892      // Resize vectors
893      _node_num = countNodes(_graph);
894      _arc_num = countArcs(_graph);
895      int all_node_num = _node_num + 1;
896      int max_arc_num = _arc_num + 2 * _node_num;
897
898      _source.resize(max_arc_num);
899      _target.resize(max_arc_num);
900
901      _lower.resize(_arc_num);
902      _upper.resize(_arc_num);
903      _cap.resize(max_arc_num);
904      _cost.resize(max_arc_num);
905      _supply.resize(all_node_num);
906      _flow.resize(max_arc_num);
907      _pi.resize(all_node_num);
908
909      _parent.resize(all_node_num);
910      _pred.resize(all_node_num);
911      _forward.resize(all_node_num);
912      _thread.resize(all_node_num);
913      _rev_thread.resize(all_node_num);
914      _succ_num.resize(all_node_num);
915      _last_succ.resize(all_node_num);
916      _state.resize(max_arc_num);
917
918      // Copy the graph
919      int i = 0;
920      for (NodeIt n(_graph); n != INVALID; ++n, ++i) {
921        _node_id[n] = i;
922      }
923      if (_arc_mixing) {
924        // Store the arcs in a mixed order
925        int k = std::max(int(std::sqrt(double(_arc_num))), 10);
926        int i = 0, j = 0;
927        for (ArcIt a(_graph); a != INVALID; ++a) {
928          _arc_id[a] = i;
929          _source[i] = _node_id[_graph.source(a)];
930          _target[i] = _node_id[_graph.target(a)];
931          if ((i += k) >= _arc_num) i = ++j;
932        }
933      } else {
934        // Store the arcs in the original order
935        int i = 0;
936        for (ArcIt a(_graph); a != INVALID; ++a, ++i) {
937          _arc_id[a] = i;
938          _source[i] = _node_id[_graph.source(a)];
939          _target[i] = _node_id[_graph.target(a)];
940        }
941      }
942     
943      // Reset parameters
944      resetParams();
945      return *this;
946    }
947   
948    /// @}
949
950    /// \name Query Functions
951    /// The results of the algorithm can be obtained using these
952    /// functions.\n
953    /// The \ref run() function must be called before using them.
954
955    /// @{
956
957    /// \brief Return the total cost of the found flow.
958    ///
959    /// This function returns the total cost of the found flow.
960    /// Its complexity is O(e).
961    ///
962    /// \note The return type of the function can be specified as a
963    /// template parameter. For example,
964    /// \code
965    ///   ns.totalCost<double>();
966    /// \endcode
967    /// It is useful if the total cost cannot be stored in the \c Cost
968    /// type of the algorithm, which is the default return type of the
969    /// function.
970    ///
971    /// \pre \ref run() must be called before using this function.
972    template <typename Number>
973    Number totalCost() const {
974      Number c = 0;
975      for (ArcIt a(_graph); a != INVALID; ++a) {
976        int i = _arc_id[a];
977        c += Number(_flow[i]) * Number(_cost[i]);
978      }
979      return c;
980    }
981
982#ifndef DOXYGEN
983    Cost totalCost() const {
984      return totalCost<Cost>();
985    }
986#endif
987
988    /// \brief Return the flow on the given arc.
989    ///
990    /// This function returns the flow on the given arc.
991    ///
992    /// \pre \ref run() must be called before using this function.
993    Value flow(const Arc& a) const {
994      return _flow[_arc_id[a]];
995    }
996
997    /// \brief Return the flow map (the primal solution).
998    ///
999    /// This function copies the flow value on each arc into the given
1000    /// map. The \c Value type of the algorithm must be convertible to
1001    /// the \c Value type of the map.
1002    ///
1003    /// \pre \ref run() must be called before using this function.
1004    template <typename FlowMap>
1005    void flowMap(FlowMap &map) const {
1006      for (ArcIt a(_graph); a != INVALID; ++a) {
1007        map.set(a, _flow[_arc_id[a]]);
1008      }
1009    }
1010
1011    /// \brief Return the potential (dual value) of the given node.
1012    ///
1013    /// This function returns the potential (dual value) of the
1014    /// given node.
1015    ///
1016    /// \pre \ref run() must be called before using this function.
1017    Cost potential(const Node& n) const {
1018      return _pi[_node_id[n]];
1019    }
1020
1021    /// \brief Return the potential map (the dual solution).
1022    ///
1023    /// This function copies the potential (dual value) of each node
1024    /// into the given map.
1025    /// The \c Cost type of the algorithm must be convertible to the
1026    /// \c Value type of the map.
1027    ///
1028    /// \pre \ref run() must be called before using this function.
1029    template <typename PotentialMap>
1030    void potentialMap(PotentialMap &map) const {
1031      for (NodeIt n(_graph); n != INVALID; ++n) {
1032        map.set(n, _pi[_node_id[n]]);
1033      }
1034    }
1035
1036    /// @}
1037
1038  private:
1039
1040    // Initialize internal data structures
1041    bool init() {
1042      if (_node_num == 0) return false;
1043
1044      // Check the sum of supply values
1045      _sum_supply = 0;
1046      for (int i = 0; i != _node_num; ++i) {
1047        _sum_supply += _supply[i];
1048      }
1049      if ( !((_stype == GEQ && _sum_supply <= 0) ||
1050             (_stype == LEQ && _sum_supply >= 0)) ) return false;
1051
1052      // Remove non-zero lower bounds
1053      if (_have_lower) {
1054        for (int i = 0; i != _arc_num; ++i) {
1055          Value c = _lower[i];
1056          if (c >= 0) {
1057            _cap[i] = _upper[i] < MAX ? _upper[i] - c : INF;
1058          } else {
1059            _cap[i] = _upper[i] < MAX + c ? _upper[i] - c : INF;
1060          }
1061          _supply[_source[i]] -= c;
1062          _supply[_target[i]] += c;
1063        }
1064      } else {
1065        for (int i = 0; i != _arc_num; ++i) {
1066          _cap[i] = _upper[i];
1067        }
1068      }
1069
1070      // Initialize artifical cost
1071      Cost ART_COST;
1072      if (std::numeric_limits<Cost>::is_exact) {
1073        ART_COST = std::numeric_limits<Cost>::max() / 2 + 1;
1074      } else {
1075        ART_COST = std::numeric_limits<Cost>::min();
1076        for (int i = 0; i != _arc_num; ++i) {
1077          if (_cost[i] > ART_COST) ART_COST = _cost[i];
1078        }
1079        ART_COST = (ART_COST + 1) * _node_num;
1080      }
1081
1082      // Initialize arc maps
1083      for (int i = 0; i != _arc_num; ++i) {
1084        _flow[i] = 0;
1085        _state[i] = STATE_LOWER;
1086      }
1087     
1088      // Set data for the artificial root node
1089      _root = _node_num;
1090      _parent[_root] = -1;
1091      _pred[_root] = -1;
1092      _thread[_root] = 0;
1093      _rev_thread[0] = _root;
1094      _succ_num[_root] = _node_num + 1;
1095      _last_succ[_root] = _root - 1;
1096      _supply[_root] = -_sum_supply;
1097      _pi[_root] = 0;
1098
1099      // Add artificial arcs and initialize the spanning tree data structure
1100      if (_sum_supply == 0) {
1101        // EQ supply constraints
1102        _search_arc_num = _arc_num;
1103        _all_arc_num = _arc_num + _node_num;
1104        for (int u = 0, e = _arc_num; u != _node_num; ++u, ++e) {
1105          _parent[u] = _root;
1106          _pred[u] = e;
1107          _thread[u] = u + 1;
1108          _rev_thread[u + 1] = u;
1109          _succ_num[u] = 1;
1110          _last_succ[u] = u;
1111          _cap[e] = INF;
1112          _state[e] = STATE_TREE;
1113          if (_supply[u] >= 0) {
1114            _forward[u] = true;
1115            _pi[u] = 0;
1116            _source[e] = u;
1117            _target[e] = _root;
1118            _flow[e] = _supply[u];
1119            _cost[e] = 0;
1120          } else {
1121            _forward[u] = false;
1122            _pi[u] = ART_COST;
1123            _source[e] = _root;
1124            _target[e] = u;
1125            _flow[e] = -_supply[u];
1126            _cost[e] = ART_COST;
1127          }
1128        }
1129      }
1130      else if (_sum_supply > 0) {
1131        // LEQ supply constraints
1132        _search_arc_num = _arc_num + _node_num;
1133        int f = _arc_num + _node_num;
1134        for (int u = 0, e = _arc_num; u != _node_num; ++u, ++e) {
1135          _parent[u] = _root;
1136          _thread[u] = u + 1;
1137          _rev_thread[u + 1] = u;
1138          _succ_num[u] = 1;
1139          _last_succ[u] = u;
1140          if (_supply[u] >= 0) {
1141            _forward[u] = true;
1142            _pi[u] = 0;
1143            _pred[u] = e;
1144            _source[e] = u;
1145            _target[e] = _root;
1146            _cap[e] = INF;
1147            _flow[e] = _supply[u];
1148            _cost[e] = 0;
1149            _state[e] = STATE_TREE;
1150          } else {
1151            _forward[u] = false;
1152            _pi[u] = ART_COST;
1153            _pred[u] = f;
1154            _source[f] = _root;
1155            _target[f] = u;
1156            _cap[f] = INF;
1157            _flow[f] = -_supply[u];
1158            _cost[f] = ART_COST;
1159            _state[f] = STATE_TREE;
1160            _source[e] = u;
1161            _target[e] = _root;
1162            _cap[e] = INF;
1163            _flow[e] = 0;
1164            _cost[e] = 0;
1165            _state[e] = STATE_LOWER;
1166            ++f;
1167          }
1168        }
1169        _all_arc_num = f;
1170      }
1171      else {
1172        // GEQ supply constraints
1173        _search_arc_num = _arc_num + _node_num;
1174        int f = _arc_num + _node_num;
1175        for (int u = 0, e = _arc_num; u != _node_num; ++u, ++e) {
1176          _parent[u] = _root;
1177          _thread[u] = u + 1;
1178          _rev_thread[u + 1] = u;
1179          _succ_num[u] = 1;
1180          _last_succ[u] = u;
1181          if (_supply[u] <= 0) {
1182            _forward[u] = false;
1183            _pi[u] = 0;
1184            _pred[u] = e;
1185            _source[e] = _root;
1186            _target[e] = u;
1187            _cap[e] = INF;
1188            _flow[e] = -_supply[u];
1189            _cost[e] = 0;
1190            _state[e] = STATE_TREE;
1191          } else {
1192            _forward[u] = true;
1193            _pi[u] = -ART_COST;
1194            _pred[u] = f;
1195            _source[f] = u;
1196            _target[f] = _root;
1197            _cap[f] = INF;
1198            _flow[f] = _supply[u];
1199            _state[f] = STATE_TREE;
1200            _cost[f] = ART_COST;
1201            _source[e] = _root;
1202            _target[e] = u;
1203            _cap[e] = INF;
1204            _flow[e] = 0;
1205            _cost[e] = 0;
1206            _state[e] = STATE_LOWER;
1207            ++f;
1208          }
1209        }
1210        _all_arc_num = f;
1211      }
1212
1213      return true;
1214    }
1215
1216    // Find the join node
1217    void findJoinNode() {
1218      int u = _source[in_arc];
1219      int v = _target[in_arc];
1220      while (u != v) {
1221        if (_succ_num[u] < _succ_num[v]) {
1222          u = _parent[u];
1223        } else {
1224          v = _parent[v];
1225        }
1226      }
1227      join = u;
1228    }
1229
1230    // Find the leaving arc of the cycle and returns true if the
1231    // leaving arc is not the same as the entering arc
1232    bool findLeavingArc() {
1233      // Initialize first and second nodes according to the direction
1234      // of the cycle
1235      if (_state[in_arc] == STATE_LOWER) {
1236        first  = _source[in_arc];
1237        second = _target[in_arc];
1238      } else {
1239        first  = _target[in_arc];
1240        second = _source[in_arc];
1241      }
1242      delta = _cap[in_arc];
1243      int result = 0;
1244      Value d;
1245      int e;
1246
1247      // Search the cycle along the path form the first node to the root
1248      for (int u = first; u != join; u = _parent[u]) {
1249        e = _pred[u];
1250        d = _forward[u] ?
1251          _flow[e] : (_cap[e] >= MAX ? INF : _cap[e] - _flow[e]);
1252        if (d < delta) {
1253          delta = d;
1254          u_out = u;
1255          result = 1;
1256        }
1257      }
1258      // Search the cycle along the path form the second node to the root
1259      for (int u = second; u != join; u = _parent[u]) {
1260        e = _pred[u];
1261        d = _forward[u] ?
1262          (_cap[e] >= MAX ? INF : _cap[e] - _flow[e]) : _flow[e];
1263        if (d <= delta) {
1264          delta = d;
1265          u_out = u;
1266          result = 2;
1267        }
1268      }
1269
1270      if (result == 1) {
1271        u_in = first;
1272        v_in = second;
1273      } else {
1274        u_in = second;
1275        v_in = first;
1276      }
1277      return result != 0;
1278    }
1279
1280    // Change _flow and _state vectors
1281    void changeFlow(bool change) {
1282      // Augment along the cycle
1283      if (delta > 0) {
1284        Value val = _state[in_arc] * delta;
1285        _flow[in_arc] += val;
1286        for (int u = _source[in_arc]; u != join; u = _parent[u]) {
1287          _flow[_pred[u]] += _forward[u] ? -val : val;
1288        }
1289        for (int u = _target[in_arc]; u != join; u = _parent[u]) {
1290          _flow[_pred[u]] += _forward[u] ? val : -val;
1291        }
1292      }
1293      // Update the state of the entering and leaving arcs
1294      if (change) {
1295        _state[in_arc] = STATE_TREE;
1296        _state[_pred[u_out]] =
1297          (_flow[_pred[u_out]] == 0) ? STATE_LOWER : STATE_UPPER;
1298      } else {
1299        _state[in_arc] = -_state[in_arc];
1300      }
1301    }
1302
1303    // Update the tree structure
1304    void updateTreeStructure() {
1305      int u, w;
1306      int old_rev_thread = _rev_thread[u_out];
1307      int old_succ_num = _succ_num[u_out];
1308      int old_last_succ = _last_succ[u_out];
1309      v_out = _parent[u_out];
1310
1311      u = _last_succ[u_in];  // the last successor of u_in
1312      right = _thread[u];    // the node after it
1313
1314      // Handle the case when old_rev_thread equals to v_in
1315      // (it also means that join and v_out coincide)
1316      if (old_rev_thread == v_in) {
1317        last = _thread[_last_succ[u_out]];
1318      } else {
1319        last = _thread[v_in];
1320      }
1321
1322      // Update _thread and _parent along the stem nodes (i.e. the nodes
1323      // between u_in and u_out, whose parent have to be changed)
1324      _thread[v_in] = stem = u_in;
1325      _dirty_revs.clear();
1326      _dirty_revs.push_back(v_in);
1327      par_stem = v_in;
1328      while (stem != u_out) {
1329        // Insert the next stem node into the thread list
1330        new_stem = _parent[stem];
1331        _thread[u] = new_stem;
1332        _dirty_revs.push_back(u);
1333
1334        // Remove the subtree of stem from the thread list
1335        w = _rev_thread[stem];
1336        _thread[w] = right;
1337        _rev_thread[right] = w;
1338
1339        // Change the parent node and shift stem nodes
1340        _parent[stem] = par_stem;
1341        par_stem = stem;
1342        stem = new_stem;
1343
1344        // Update u and right
1345        u = _last_succ[stem] == _last_succ[par_stem] ?
1346          _rev_thread[par_stem] : _last_succ[stem];
1347        right = _thread[u];
1348      }
1349      _parent[u_out] = par_stem;
1350      _thread[u] = last;
1351      _rev_thread[last] = u;
1352      _last_succ[u_out] = u;
1353
1354      // Remove the subtree of u_out from the thread list except for
1355      // the case when old_rev_thread equals to v_in
1356      // (it also means that join and v_out coincide)
1357      if (old_rev_thread != v_in) {
1358        _thread[old_rev_thread] = right;
1359        _rev_thread[right] = old_rev_thread;
1360      }
1361
1362      // Update _rev_thread using the new _thread values
1363      for (int i = 0; i < int(_dirty_revs.size()); ++i) {
1364        u = _dirty_revs[i];
1365        _rev_thread[_thread[u]] = u;
1366      }
1367
1368      // Update _pred, _forward, _last_succ and _succ_num for the
1369      // stem nodes from u_out to u_in
1370      int tmp_sc = 0, tmp_ls = _last_succ[u_out];
1371      u = u_out;
1372      while (u != u_in) {
1373        w = _parent[u];
1374        _pred[u] = _pred[w];
1375        _forward[u] = !_forward[w];
1376        tmp_sc += _succ_num[u] - _succ_num[w];
1377        _succ_num[u] = tmp_sc;
1378        _last_succ[w] = tmp_ls;
1379        u = w;
1380      }
1381      _pred[u_in] = in_arc;
1382      _forward[u_in] = (u_in == _source[in_arc]);
1383      _succ_num[u_in] = old_succ_num;
1384
1385      // Set limits for updating _last_succ form v_in and v_out
1386      // towards the root
1387      int up_limit_in = -1;
1388      int up_limit_out = -1;
1389      if (_last_succ[join] == v_in) {
1390        up_limit_out = join;
1391      } else {
1392        up_limit_in = join;
1393      }
1394
1395      // Update _last_succ from v_in towards the root
1396      for (u = v_in; u != up_limit_in && _last_succ[u] == v_in;
1397           u = _parent[u]) {
1398        _last_succ[u] = _last_succ[u_out];
1399      }
1400      // Update _last_succ from v_out towards the root
1401      if (join != old_rev_thread && v_in != old_rev_thread) {
1402        for (u = v_out; u != up_limit_out && _last_succ[u] == old_last_succ;
1403             u = _parent[u]) {
1404          _last_succ[u] = old_rev_thread;
1405        }
1406      } else {
1407        for (u = v_out; u != up_limit_out && _last_succ[u] == old_last_succ;
1408             u = _parent[u]) {
1409          _last_succ[u] = _last_succ[u_out];
1410        }
1411      }
1412
1413      // Update _succ_num from v_in to join
1414      for (u = v_in; u != join; u = _parent[u]) {
1415        _succ_num[u] += old_succ_num;
1416      }
1417      // Update _succ_num from v_out to join
1418      for (u = v_out; u != join; u = _parent[u]) {
1419        _succ_num[u] -= old_succ_num;
1420      }
1421    }
1422
1423    // Update potentials
1424    void updatePotential() {
1425      Cost sigma = _forward[u_in] ?
1426        _pi[v_in] - _pi[u_in] - _cost[_pred[u_in]] :
1427        _pi[v_in] - _pi[u_in] + _cost[_pred[u_in]];
1428      // Update potentials in the subtree, which has been moved
1429      int end = _thread[_last_succ[u_in]];
1430      for (int u = u_in; u != end; u = _thread[u]) {
1431        _pi[u] += sigma;
1432      }
1433    }
1434
1435    // Execute the algorithm
1436    ProblemType start(PivotRule pivot_rule) {
1437      // Select the pivot rule implementation
1438      switch (pivot_rule) {
1439        case FIRST_ELIGIBLE:
1440          return start<FirstEligiblePivotRule>();
1441        case BEST_ELIGIBLE:
1442          return start<BestEligiblePivotRule>();
1443        case BLOCK_SEARCH:
1444          return start<BlockSearchPivotRule>();
1445        case CANDIDATE_LIST:
1446          return start<CandidateListPivotRule>();
1447        case ALTERING_LIST:
1448          return start<AlteringListPivotRule>();
1449      }
1450      return INFEASIBLE; // avoid warning
1451    }
1452
1453    template <typename PivotRuleImpl>
1454    ProblemType start() {
1455      PivotRuleImpl pivot(*this);
1456
1457      // Execute the Network Simplex algorithm
1458      while (pivot.findEnteringArc()) {
1459        findJoinNode();
1460        bool change = findLeavingArc();
1461        if (delta >= MAX) return UNBOUNDED;
1462        changeFlow(change);
1463        if (change) {
1464          updateTreeStructure();
1465          updatePotential();
1466        }
1467      }
1468     
1469      // Check feasibility
1470      for (int e = _search_arc_num; e != _all_arc_num; ++e) {
1471        if (_flow[e] != 0) return INFEASIBLE;
1472      }
1473
1474      // Transform the solution and the supply map to the original form
1475      if (_have_lower) {
1476        for (int i = 0; i != _arc_num; ++i) {
1477          Value c = _lower[i];
1478          if (c != 0) {
1479            _flow[i] += c;
1480            _supply[_source[i]] += c;
1481            _supply[_target[i]] -= c;
1482          }
1483        }
1484      }
1485     
1486      // Shift potentials to meet the requirements of the GEQ/LEQ type
1487      // optimality conditions
1488      if (_sum_supply == 0) {
1489        if (_stype == GEQ) {
1490          Cost max_pot = std::numeric_limits<Cost>::min();
1491          for (int i = 0; i != _node_num; ++i) {
1492            if (_pi[i] > max_pot) max_pot = _pi[i];
1493          }
1494          if (max_pot > 0) {
1495            for (int i = 0; i != _node_num; ++i)
1496              _pi[i] -= max_pot;
1497          }
1498        } else {
1499          Cost min_pot = std::numeric_limits<Cost>::max();
1500          for (int i = 0; i != _node_num; ++i) {
1501            if (_pi[i] < min_pot) min_pot = _pi[i];
1502          }
1503          if (min_pot < 0) {
1504            for (int i = 0; i != _node_num; ++i)
1505              _pi[i] -= min_pot;
1506          }
1507        }
1508      }
1509
1510      return OPTIMAL;
1511    }
1512
1513  }; //class NetworkSimplex
1514
1515  ///@}
1516
1517} //namespace lemon
1518
1519#endif //LEMON_NETWORK_SIMPLEX_H
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