source:lemon/lemon/network_simplex.h@665:b95898314e09

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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
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#include <lemon/maps.h>
34#include <lemon/circulation.h>
36
37namespace lemon {
38
39  /// \addtogroup min_cost_flow
40  /// @{
41
42  /// \brief Implementation of the primal Network Simplex algorithm
43  /// for finding a \ref min_cost_flow "minimum cost flow".
44  ///
45  /// \ref NetworkSimplex implements the primal Network Simplex algorithm
46  /// for finding a \ref min_cost_flow "minimum cost flow".
47  /// This algorithm is a specialized version of the linear programming
48  /// simplex method directly for the minimum cost flow problem.
49  /// It is one of the most efficient solution methods.
50  ///
51  /// In general this class is the fastest implementation available
52  /// in LEMON for the minimum cost flow problem.
53  /// Moreover it supports both direction of the supply/demand inequality
54  /// constraints. For more information see \ref ProblemType.
55  ///
56  /// \tparam GR The digraph type the algorithm runs on.
57  /// \tparam F The value type used for flow amounts, capacity bounds
58  /// and supply values in the algorithm. By default it is \c int.
59  /// \tparam C The value type used for costs and potentials in the
60  /// algorithm. By default it is the same as \c F.
61  ///
62  /// \warning Both value types must be signed and all input data must
63  /// be integer.
64  ///
65  /// \note %NetworkSimplex provides five different pivot rule
66  /// implementations, from which the most efficient one is used
67  /// by default. For more information see \ref PivotRule.
68  template <typename GR, typename F = int, typename C = F>
69  class NetworkSimplex
70  {
71  public:
72
73    /// The flow type of the algorithm
74    typedef F Flow;
75    /// The cost type of the algorithm
76    typedef C Cost;
77#ifdef DOXYGEN
78    /// The type of the flow map
79    typedef GR::ArcMap<Flow> FlowMap;
80    /// The type of the potential map
81    typedef GR::NodeMap<Cost> PotentialMap;
82#else
83    /// The type of the flow map
84    typedef typename GR::template ArcMap<Flow> FlowMap;
85    /// The type of the potential map
86    typedef typename GR::template NodeMap<Cost> PotentialMap;
87#endif
88
89  public:
90
91    /// \brief Enum type for selecting the pivot rule.
92    ///
93    /// Enum type for selecting the pivot rule for the \ref run()
94    /// function.
95    ///
96    /// \ref NetworkSimplex provides five different pivot rule
97    /// implementations that significantly affect the running time
98    /// of the algorithm.
99    /// By default \ref BLOCK_SEARCH "Block Search" is used, which
100    /// proved to be the most efficient and the most robust on various
101    /// test inputs according to our benchmark tests.
102    /// However another pivot rule can be selected using the \ref run()
103    /// function with the proper parameter.
104    enum PivotRule {
105
106      /// The First Eligible pivot rule.
107      /// The next eligible arc is selected in a wraparound fashion
108      /// in every iteration.
109      FIRST_ELIGIBLE,
110
111      /// The Best Eligible pivot rule.
112      /// The best eligible arc is selected in every iteration.
113      BEST_ELIGIBLE,
114
115      /// The Block Search pivot rule.
116      /// A specified number of arcs are examined in every iteration
117      /// in a wraparound fashion and the best eligible arc is selected
118      /// from this block.
119      BLOCK_SEARCH,
120
121      /// The Candidate List pivot rule.
122      /// In a major iteration a candidate list is built from eligible arcs
123      /// in a wraparound fashion and in the following minor iterations
124      /// the best eligible arc is selected from this list.
125      CANDIDATE_LIST,
126
127      /// The Altering Candidate List pivot rule.
128      /// It is a modified version of the Candidate List method.
129      /// It keeps only the several best eligible arcs from the former
130      /// candidate list and extends this list in every iteration.
131      ALTERING_LIST
132    };
133
134    /// \brief Enum type for selecting the problem type.
135    ///
136    /// Enum type for selecting the problem type, i.e. the direction of
137    /// the inequalities in the supply/demand constraints of the
138    /// \ref min_cost_flow "minimum cost flow problem".
139    ///
140    /// The default problem type is \c GEQ, since this form is supported
141    /// by other minimum cost flow algorithms and the \ref Circulation
142    /// algorithm as well.
143    /// The \c LEQ problem type can be selected using the \ref problemType()
144    /// function.
145    ///
146    /// Note that the equality form is a special case of both problem type.
147    enum ProblemType {
148
149      /// This option means that there are "<em>greater or equal</em>"
150      /// constraints in the defintion, i.e. the exact formulation of the
151      /// problem is the following.
152      /**
153          \f[ \min\sum_{uv\in A} f(uv) \cdot cost(uv) \f]
154          \f[ \sum_{uv\in A} f(uv) - \sum_{vu\in A} f(vu) \geq
155              sup(u) \quad \forall u\in V \f]
156          \f[ lower(uv) \leq f(uv) \leq upper(uv) \quad \forall uv\in A \f]
157      */
158      /// It means that the total demand must be greater or equal to the
159      /// total supply (i.e. \f$\sum_{u\in V} sup(u)\f$ must be zero or
160      /// negative) and all the supplies have to be carried out from
161      /// the supply nodes, but there could be demands that are not
162      /// satisfied.
163      GEQ,
164      /// It is just an alias for the \c GEQ option.
165      CARRY_SUPPLIES = GEQ,
166
167      /// This option means that there are "<em>less or equal</em>"
168      /// constraints in the defintion, i.e. the exact formulation of the
169      /// problem is the following.
170      /**
171          \f[ \min\sum_{uv\in A} f(uv) \cdot cost(uv) \f]
172          \f[ \sum_{uv\in A} f(uv) - \sum_{vu\in A} f(vu) \leq
173              sup(u) \quad \forall u\in V \f]
174          \f[ lower(uv) \leq f(uv) \leq upper(uv) \quad \forall uv\in A \f]
175      */
176      /// It means that the total demand must be less or equal to the
177      /// total supply (i.e. \f$\sum_{u\in V} sup(u)\f$ must be zero or
178      /// positive) and all the demands have to be satisfied, but there
179      /// could be supplies that are not carried out from the supply
180      /// nodes.
181      LEQ,
182      /// It is just an alias for the \c LEQ option.
183      SATISFY_DEMANDS = LEQ
184    };
185
186  private:
187
188    TEMPLATE_DIGRAPH_TYPEDEFS(GR);
189
190    typedef typename GR::template ArcMap<Flow> FlowArcMap;
191    typedef typename GR::template ArcMap<Cost> CostArcMap;
192    typedef typename GR::template NodeMap<Flow> FlowNodeMap;
193
194    typedef std::vector<Arc> ArcVector;
195    typedef std::vector<Node> NodeVector;
196    typedef std::vector<int> IntVector;
197    typedef std::vector<bool> BoolVector;
198    typedef std::vector<Flow> FlowVector;
199    typedef std::vector<Cost> CostVector;
200
201    // State constants for arcs
202    enum ArcStateEnum {
203      STATE_UPPER = -1,
204      STATE_TREE  =  0,
205      STATE_LOWER =  1
206    };
207
208  private:
209
210    // Data related to the underlying digraph
211    const GR &_graph;
212    int _node_num;
213    int _arc_num;
214
215    // Parameters of the problem
216    FlowArcMap *_plower;
217    FlowArcMap *_pupper;
218    CostArcMap *_pcost;
219    FlowNodeMap *_psupply;
220    bool _pstsup;
221    Node _psource, _ptarget;
222    Flow _pstflow;
223    ProblemType _ptype;
224
225    // Result maps
226    FlowMap *_flow_map;
227    PotentialMap *_potential_map;
228    bool _local_flow;
229    bool _local_potential;
230
231    // Data structures for storing the digraph
232    IntNodeMap _node_id;
233    ArcVector _arc_ref;
234    IntVector _source;
235    IntVector _target;
236
237    // Node and arc data
238    FlowVector _cap;
239    CostVector _cost;
240    FlowVector _supply;
241    FlowVector _flow;
242    CostVector _pi;
243
244    // Data for storing the spanning tree structure
245    IntVector _parent;
246    IntVector _pred;
249    IntVector _succ_num;
250    IntVector _last_succ;
251    IntVector _dirty_revs;
252    BoolVector _forward;
253    IntVector _state;
254    int _root;
255
256    // Temporary data used in the current pivot iteration
257    int in_arc, join, u_in, v_in, u_out, v_out;
258    int first, second, right, last;
259    int stem, par_stem, new_stem;
260    Flow delta;
261
262  private:
263
264    // Implementation of the First Eligible pivot rule
265    class FirstEligiblePivotRule
266    {
267    private:
268
269      // References to the NetworkSimplex class
270      const IntVector  &_source;
271      const IntVector  &_target;
272      const CostVector &_cost;
273      const IntVector  &_state;
274      const CostVector &_pi;
275      int &_in_arc;
276      int _arc_num;
277
278      // Pivot rule data
279      int _next_arc;
280
281    public:
282
283      // Constructor
284      FirstEligiblePivotRule(NetworkSimplex &ns) :
285        _source(ns._source), _target(ns._target),
286        _cost(ns._cost), _state(ns._state), _pi(ns._pi),
287        _in_arc(ns.in_arc), _arc_num(ns._arc_num), _next_arc(0)
288      {}
289
290      // Find next entering arc
291      bool findEnteringArc() {
292        Cost c;
293        for (int e = _next_arc; e < _arc_num; ++e) {
294          c = _state[e] * (_cost[e] + _pi[_source[e]] - _pi[_target[e]]);
295          if (c < 0) {
296            _in_arc = e;
297            _next_arc = e + 1;
298            return true;
299          }
300        }
301        for (int e = 0; e < _next_arc; ++e) {
302          c = _state[e] * (_cost[e] + _pi[_source[e]] - _pi[_target[e]]);
303          if (c < 0) {
304            _in_arc = e;
305            _next_arc = e + 1;
306            return true;
307          }
308        }
309        return false;
310      }
311
312    }; //class FirstEligiblePivotRule
313
314
315    // Implementation of the Best Eligible pivot rule
316    class BestEligiblePivotRule
317    {
318    private:
319
320      // References to the NetworkSimplex class
321      const IntVector  &_source;
322      const IntVector  &_target;
323      const CostVector &_cost;
324      const IntVector  &_state;
325      const CostVector &_pi;
326      int &_in_arc;
327      int _arc_num;
328
329    public:
330
331      // Constructor
332      BestEligiblePivotRule(NetworkSimplex &ns) :
333        _source(ns._source), _target(ns._target),
334        _cost(ns._cost), _state(ns._state), _pi(ns._pi),
335        _in_arc(ns.in_arc), _arc_num(ns._arc_num)
336      {}
337
338      // Find next entering arc
339      bool findEnteringArc() {
340        Cost c, min = 0;
341        for (int e = 0; e < _arc_num; ++e) {
342          c = _state[e] * (_cost[e] + _pi[_source[e]] - _pi[_target[e]]);
343          if (c < min) {
344            min = c;
345            _in_arc = e;
346          }
347        }
348        return min < 0;
349      }
350
351    }; //class BestEligiblePivotRule
352
353
354    // Implementation of the Block Search pivot rule
355    class BlockSearchPivotRule
356    {
357    private:
358
359      // References to the NetworkSimplex class
360      const IntVector  &_source;
361      const IntVector  &_target;
362      const CostVector &_cost;
363      const IntVector  &_state;
364      const CostVector &_pi;
365      int &_in_arc;
366      int _arc_num;
367
368      // Pivot rule data
369      int _block_size;
370      int _next_arc;
371
372    public:
373
374      // Constructor
375      BlockSearchPivotRule(NetworkSimplex &ns) :
376        _source(ns._source), _target(ns._target),
377        _cost(ns._cost), _state(ns._state), _pi(ns._pi),
378        _in_arc(ns.in_arc), _arc_num(ns._arc_num), _next_arc(0)
379      {
380        // The main parameters of the pivot rule
381        const double BLOCK_SIZE_FACTOR = 2.0;
382        const int MIN_BLOCK_SIZE = 10;
383
384        _block_size = std::max( int(BLOCK_SIZE_FACTOR *
385                                    std::sqrt(double(_arc_num))),
386                                MIN_BLOCK_SIZE );
387      }
388
389      // Find next entering arc
390      bool findEnteringArc() {
391        Cost c, min = 0;
392        int cnt = _block_size;
393        int e, min_arc = _next_arc;
394        for (e = _next_arc; e < _arc_num; ++e) {
395          c = _state[e] * (_cost[e] + _pi[_source[e]] - _pi[_target[e]]);
396          if (c < min) {
397            min = c;
398            min_arc = e;
399          }
400          if (--cnt == 0) {
401            if (min < 0) break;
402            cnt = _block_size;
403          }
404        }
405        if (min == 0 || cnt > 0) {
406          for (e = 0; e < _next_arc; ++e) {
407            c = _state[e] * (_cost[e] + _pi[_source[e]] - _pi[_target[e]]);
408            if (c < min) {
409              min = c;
410              min_arc = e;
411            }
412            if (--cnt == 0) {
413              if (min < 0) break;
414              cnt = _block_size;
415            }
416          }
417        }
418        if (min >= 0) return false;
419        _in_arc = min_arc;
420        _next_arc = e;
421        return true;
422      }
423
424    }; //class BlockSearchPivotRule
425
426
427    // Implementation of the Candidate List pivot rule
428    class CandidateListPivotRule
429    {
430    private:
431
432      // References to the NetworkSimplex class
433      const IntVector  &_source;
434      const IntVector  &_target;
435      const CostVector &_cost;
436      const IntVector  &_state;
437      const CostVector &_pi;
438      int &_in_arc;
439      int _arc_num;
440
441      // Pivot rule data
442      IntVector _candidates;
443      int _list_length, _minor_limit;
444      int _curr_length, _minor_count;
445      int _next_arc;
446
447    public:
448
449      /// Constructor
450      CandidateListPivotRule(NetworkSimplex &ns) :
451        _source(ns._source), _target(ns._target),
452        _cost(ns._cost), _state(ns._state), _pi(ns._pi),
453        _in_arc(ns.in_arc), _arc_num(ns._arc_num), _next_arc(0)
454      {
455        // The main parameters of the pivot rule
456        const double LIST_LENGTH_FACTOR = 1.0;
457        const int MIN_LIST_LENGTH = 10;
458        const double MINOR_LIMIT_FACTOR = 0.1;
459        const int MIN_MINOR_LIMIT = 3;
460
461        _list_length = std::max( int(LIST_LENGTH_FACTOR *
462                                     std::sqrt(double(_arc_num))),
463                                 MIN_LIST_LENGTH );
464        _minor_limit = std::max( int(MINOR_LIMIT_FACTOR * _list_length),
465                                 MIN_MINOR_LIMIT );
466        _curr_length = _minor_count = 0;
467        _candidates.resize(_list_length);
468      }
469
470      /// Find next entering arc
471      bool findEnteringArc() {
472        Cost min, c;
473        int e, min_arc = _next_arc;
474        if (_curr_length > 0 && _minor_count < _minor_limit) {
475          // Minor iteration: select the best eligible arc from the
476          // current candidate list
477          ++_minor_count;
478          min = 0;
479          for (int i = 0; i < _curr_length; ++i) {
480            e = _candidates[i];
481            c = _state[e] * (_cost[e] + _pi[_source[e]] - _pi[_target[e]]);
482            if (c < min) {
483              min = c;
484              min_arc = e;
485            }
486            if (c >= 0) {
487              _candidates[i--] = _candidates[--_curr_length];
488            }
489          }
490          if (min < 0) {
491            _in_arc = min_arc;
492            return true;
493          }
494        }
495
496        // Major iteration: build a new candidate list
497        min = 0;
498        _curr_length = 0;
499        for (e = _next_arc; e < _arc_num; ++e) {
500          c = _state[e] * (_cost[e] + _pi[_source[e]] - _pi[_target[e]]);
501          if (c < 0) {
502            _candidates[_curr_length++] = e;
503            if (c < min) {
504              min = c;
505              min_arc = e;
506            }
507            if (_curr_length == _list_length) break;
508          }
509        }
510        if (_curr_length < _list_length) {
511          for (e = 0; e < _next_arc; ++e) {
512            c = _state[e] * (_cost[e] + _pi[_source[e]] - _pi[_target[e]]);
513            if (c < 0) {
514              _candidates[_curr_length++] = e;
515              if (c < min) {
516                min = c;
517                min_arc = e;
518              }
519              if (_curr_length == _list_length) break;
520            }
521          }
522        }
523        if (_curr_length == 0) return false;
524        _minor_count = 1;
525        _in_arc = min_arc;
526        _next_arc = e;
527        return true;
528      }
529
530    }; //class CandidateListPivotRule
531
532
533    // Implementation of the Altering Candidate List pivot rule
534    class AlteringListPivotRule
535    {
536    private:
537
538      // References to the NetworkSimplex class
539      const IntVector  &_source;
540      const IntVector  &_target;
541      const CostVector &_cost;
542      const IntVector  &_state;
543      const CostVector &_pi;
544      int &_in_arc;
545      int _arc_num;
546
547      // Pivot rule data
548      int _block_size, _head_length, _curr_length;
549      int _next_arc;
550      IntVector _candidates;
551      CostVector _cand_cost;
552
553      // Functor class to compare arcs during sort of the candidate list
554      class SortFunc
555      {
556      private:
557        const CostVector &_map;
558      public:
559        SortFunc(const CostVector &map) : _map(map) {}
560        bool operator()(int left, int right) {
561          return _map[left] > _map[right];
562        }
563      };
564
565      SortFunc _sort_func;
566
567    public:
568
569      // Constructor
570      AlteringListPivotRule(NetworkSimplex &ns) :
571        _source(ns._source), _target(ns._target),
572        _cost(ns._cost), _state(ns._state), _pi(ns._pi),
573        _in_arc(ns.in_arc), _arc_num(ns._arc_num),
574        _next_arc(0), _cand_cost(ns._arc_num), _sort_func(_cand_cost)
575      {
576        // The main parameters of the pivot rule
577        const double BLOCK_SIZE_FACTOR = 1.5;
578        const int MIN_BLOCK_SIZE = 10;
579        const double HEAD_LENGTH_FACTOR = 0.1;
580        const int MIN_HEAD_LENGTH = 3;
581
582        _block_size = std::max( int(BLOCK_SIZE_FACTOR *
583                                    std::sqrt(double(_arc_num))),
584                                MIN_BLOCK_SIZE );
585        _head_length = std::max( int(HEAD_LENGTH_FACTOR * _block_size),
587        _candidates.resize(_head_length + _block_size);
588        _curr_length = 0;
589      }
590
591      // Find next entering arc
592      bool findEnteringArc() {
593        // Check the current candidate list
594        int e;
595        for (int i = 0; i < _curr_length; ++i) {
596          e = _candidates[i];
597          _cand_cost[e] = _state[e] *
598            (_cost[e] + _pi[_source[e]] - _pi[_target[e]]);
599          if (_cand_cost[e] >= 0) {
600            _candidates[i--] = _candidates[--_curr_length];
601          }
602        }
603
604        // Extend the list
605        int cnt = _block_size;
606        int last_arc = 0;
607        int limit = _head_length;
608
609        for (int e = _next_arc; e < _arc_num; ++e) {
610          _cand_cost[e] = _state[e] *
611            (_cost[e] + _pi[_source[e]] - _pi[_target[e]]);
612          if (_cand_cost[e] < 0) {
613            _candidates[_curr_length++] = e;
614            last_arc = e;
615          }
616          if (--cnt == 0) {
617            if (_curr_length > limit) break;
618            limit = 0;
619            cnt = _block_size;
620          }
621        }
622        if (_curr_length <= limit) {
623          for (int e = 0; e < _next_arc; ++e) {
624            _cand_cost[e] = _state[e] *
625              (_cost[e] + _pi[_source[e]] - _pi[_target[e]]);
626            if (_cand_cost[e] < 0) {
627              _candidates[_curr_length++] = e;
628              last_arc = e;
629            }
630            if (--cnt == 0) {
631              if (_curr_length > limit) break;
632              limit = 0;
633              cnt = _block_size;
634            }
635          }
636        }
637        if (_curr_length == 0) return false;
638        _next_arc = last_arc + 1;
639
640        // Make heap of the candidate list (approximating a partial sort)
641        make_heap( _candidates.begin(), _candidates.begin() + _curr_length,
642                   _sort_func );
643
644        // Pop the first element of the heap
645        _in_arc = _candidates[0];
646        pop_heap( _candidates.begin(), _candidates.begin() + _curr_length,
647                  _sort_func );
648        _curr_length = std::min(_head_length, _curr_length - 1);
649        return true;
650      }
651
652    }; //class AlteringListPivotRule
653
654  public:
655
656    /// \brief Constructor.
657    ///
658    /// The constructor of the class.
659    ///
660    /// \param graph The digraph the algorithm runs on.
661    NetworkSimplex(const GR& graph) :
662      _graph(graph),
663      _plower(NULL), _pupper(NULL), _pcost(NULL),
664      _psupply(NULL), _pstsup(false), _ptype(GEQ),
665      _flow_map(NULL), _potential_map(NULL),
666      _local_flow(false), _local_potential(false),
667      _node_id(graph)
668    {
669      LEMON_ASSERT(std::numeric_limits<Flow>::is_integer &&
670                   std::numeric_limits<Flow>::is_signed,
671        "The flow type of NetworkSimplex must be signed integer");
672      LEMON_ASSERT(std::numeric_limits<Cost>::is_integer &&
673                   std::numeric_limits<Cost>::is_signed,
674        "The cost type of NetworkSimplex must be signed integer");
675    }
676
677    /// Destructor.
678    ~NetworkSimplex() {
679      if (_local_flow) delete _flow_map;
680      if (_local_potential) delete _potential_map;
681    }
682
683    /// \name Parameters
684    /// The parameters of the algorithm can be specified using these
685    /// functions.
686
687    /// @{
688
689    /// \brief Set the lower bounds on the arcs.
690    ///
691    /// This function sets the lower bounds on the arcs.
692    /// If neither this function nor \ref boundMaps() is used before
693    /// calling \ref run(), the lower bounds will be set to zero
694    /// on all arcs.
695    ///
696    /// \param map An arc map storing the lower bounds.
697    /// Its \c Value type must be convertible to the \c Flow type
698    /// of the algorithm.
699    ///
700    /// \return <tt>(*this)</tt>
701    template <typename LOWER>
702    NetworkSimplex& lowerMap(const LOWER& map) {
703      delete _plower;
704      _plower = new FlowArcMap(_graph);
705      for (ArcIt a(_graph); a != INVALID; ++a) {
706        (*_plower)[a] = map[a];
707      }
708      return *this;
709    }
710
711    /// \brief Set the upper bounds (capacities) on the arcs.
712    ///
713    /// This function sets the upper bounds (capacities) on the arcs.
714    /// If none of the functions \ref upperMap(), \ref capacityMap()
715    /// and \ref boundMaps() is used before calling \ref run(),
716    /// the upper bounds (capacities) will be set to
717    /// \c std::numeric_limits<Flow>::max() on all arcs.
718    ///
719    /// \param map An arc map storing the upper bounds.
720    /// Its \c Value type must be convertible to the \c Flow type
721    /// of the algorithm.
722    ///
723    /// \return <tt>(*this)</tt>
724    template<typename UPPER>
725    NetworkSimplex& upperMap(const UPPER& map) {
726      delete _pupper;
727      _pupper = new FlowArcMap(_graph);
728      for (ArcIt a(_graph); a != INVALID; ++a) {
729        (*_pupper)[a] = map[a];
730      }
731      return *this;
732    }
733
734    /// \brief Set the upper bounds (capacities) on the arcs.
735    ///
736    /// This function sets the upper bounds (capacities) on the arcs.
737    /// It is just an alias for \ref upperMap().
738    ///
739    /// \return <tt>(*this)</tt>
740    template<typename CAP>
741    NetworkSimplex& capacityMap(const CAP& map) {
742      return upperMap(map);
743    }
744
745    /// \brief Set the lower and upper bounds on the arcs.
746    ///
747    /// This function sets the lower and upper bounds on the arcs.
748    /// If neither this function nor \ref lowerMap() is used before
749    /// calling \ref run(), the lower bounds will be set to zero
750    /// on all arcs.
751    /// If none of the functions \ref upperMap(), \ref capacityMap()
752    /// and \ref boundMaps() is used before calling \ref run(),
753    /// the upper bounds (capacities) will be set to
754    /// \c std::numeric_limits<Flow>::max() on all arcs.
755    ///
756    /// \param lower An arc map storing the lower bounds.
757    /// \param upper An arc map storing the upper bounds.
758    ///
759    /// The \c Value type of the maps must be convertible to the
760    /// \c Flow type of the algorithm.
761    ///
762    /// \note This function is just a shortcut of calling \ref lowerMap()
763    /// and \ref upperMap() separately.
764    ///
765    /// \return <tt>(*this)</tt>
766    template <typename LOWER, typename UPPER>
767    NetworkSimplex& boundMaps(const LOWER& lower, const UPPER& upper) {
768      return lowerMap(lower).upperMap(upper);
769    }
770
771    /// \brief Set the costs of the arcs.
772    ///
773    /// This function sets the costs of the arcs.
774    /// If it is not used before calling \ref run(), the costs
775    /// will be set to \c 1 on all arcs.
776    ///
777    /// \param map An arc map storing the costs.
778    /// Its \c Value type must be convertible to the \c Cost type
779    /// of the algorithm.
780    ///
781    /// \return <tt>(*this)</tt>
782    template<typename COST>
783    NetworkSimplex& costMap(const COST& map) {
784      delete _pcost;
785      _pcost = new CostArcMap(_graph);
786      for (ArcIt a(_graph); a != INVALID; ++a) {
787        (*_pcost)[a] = map[a];
788      }
789      return *this;
790    }
791
792    /// \brief Set the supply values of the nodes.
793    ///
794    /// This function sets the supply values of the nodes.
795    /// If neither this function nor \ref stSupply() is used before
796    /// calling \ref run(), the supply of each node will be set to zero.
797    /// (It makes sense only if non-zero lower bounds are given.)
798    ///
799    /// \param map A node map storing the supply values.
800    /// Its \c Value type must be convertible to the \c Flow type
801    /// of the algorithm.
802    ///
803    /// \return <tt>(*this)</tt>
804    template<typename SUP>
805    NetworkSimplex& supplyMap(const SUP& map) {
806      delete _psupply;
807      _pstsup = false;
808      _psupply = new FlowNodeMap(_graph);
809      for (NodeIt n(_graph); n != INVALID; ++n) {
810        (*_psupply)[n] = map[n];
811      }
812      return *this;
813    }
814
815    /// \brief Set single source and target nodes and a supply value.
816    ///
817    /// This function sets a single source node and a single target node
818    /// and the required flow value.
819    /// If neither this function nor \ref supplyMap() is used before
820    /// calling \ref run(), the supply of each node will be set to zero.
821    /// (It makes sense only if non-zero lower bounds are given.)
822    ///
823    /// \param s The source node.
824    /// \param t The target node.
825    /// \param k The required amount of flow from node \c s to node \c t
826    /// (i.e. the supply of \c s and the demand of \c t).
827    ///
828    /// \return <tt>(*this)</tt>
829    NetworkSimplex& stSupply(const Node& s, const Node& t, Flow k) {
830      delete _psupply;
831      _psupply = NULL;
832      _pstsup = true;
833      _psource = s;
834      _ptarget = t;
835      _pstflow = k;
836      return *this;
837    }
838
839    /// \brief Set the problem type.
840    ///
841    /// This function sets the problem type for the algorithm.
842    /// If it is not used before calling \ref run(), the \ref GEQ problem
843    /// type will be used.
844    ///
845    /// For more information see \ref ProblemType.
846    ///
847    /// \return <tt>(*this)</tt>
848    NetworkSimplex& problemType(ProblemType problem_type) {
849      _ptype = problem_type;
850      return *this;
851    }
852
853    /// \brief Set the flow map.
854    ///
855    /// This function sets the flow map.
856    /// If it is not used before calling \ref run(), an instance will
857    /// be allocated automatically. The destructor deallocates this
858    /// automatically allocated map, of course.
859    ///
860    /// \return <tt>(*this)</tt>
861    NetworkSimplex& flowMap(FlowMap& map) {
862      if (_local_flow) {
863        delete _flow_map;
864        _local_flow = false;
865      }
866      _flow_map = &map;
867      return *this;
868    }
869
870    /// \brief Set the potential map.
871    ///
872    /// This function sets the potential map, which is used for storing
873    /// the dual solution.
874    /// If it is not used before calling \ref run(), an instance will
875    /// be allocated automatically. The destructor deallocates this
876    /// automatically allocated map, of course.
877    ///
878    /// \return <tt>(*this)</tt>
879    NetworkSimplex& potentialMap(PotentialMap& map) {
880      if (_local_potential) {
881        delete _potential_map;
882        _local_potential = false;
883      }
884      _potential_map = &map;
885      return *this;
886    }
887
888    /// @}
889
890    /// \name Execution Control
891    /// The algorithm can be executed using \ref run().
892
893    /// @{
894
895    /// \brief Run the algorithm.
896    ///
897    /// This function runs the algorithm.
898    /// The paramters can be specified using functions \ref lowerMap(),
899    /// \ref upperMap(), \ref capacityMap(), \ref boundMaps(),
900    /// \ref costMap(), \ref supplyMap(), \ref stSupply(),
901    /// \ref problemType(), \ref flowMap() and \ref potentialMap().
902    /// For example,
903    /// \code
904    ///   NetworkSimplex<ListDigraph> ns(graph);
905    ///   ns.boundMaps(lower, upper).costMap(cost)
906    ///     .supplyMap(sup).run();
907    /// \endcode
908    ///
909    /// This function can be called more than once. All the parameters
910    /// that have been given are kept for the next call, unless
911    /// \ref reset() is called, thus only the modified parameters
912    /// have to be set again. See \ref reset() for examples.
913    ///
914    /// \param pivot_rule The pivot rule that will be used during the
915    /// algorithm. For more information see \ref PivotRule.
916    ///
917    /// \return \c true if a feasible flow can be found.
918    bool run(PivotRule pivot_rule = BLOCK_SEARCH) {
919      return init() && start(pivot_rule);
920    }
921
922    /// \brief Reset all the parameters that have been given before.
923    ///
924    /// This function resets all the paramaters that have been given
925    /// before using functions \ref lowerMap(), \ref upperMap(),
926    /// \ref capacityMap(), \ref boundMaps(), \ref costMap(),
927    /// \ref supplyMap(), \ref stSupply(), \ref problemType(),
928    /// \ref flowMap() and \ref potentialMap().
929    ///
930    /// It is useful for multiple run() calls. If this function is not
931    /// used, all the parameters given before are kept for the next
932    /// \ref run() call.
933    ///
934    /// For example,
935    /// \code
936    ///   NetworkSimplex<ListDigraph> ns(graph);
937    ///
938    ///   // First run
939    ///   ns.lowerMap(lower).capacityMap(cap).costMap(cost)
940    ///     .supplyMap(sup).run();
941    ///
942    ///   // Run again with modified cost map (reset() is not called,
943    ///   // so only the cost map have to be set again)
944    ///   cost[e] += 100;
945    ///   ns.costMap(cost).run();
946    ///
947    ///   // Run again from scratch using reset()
948    ///   // (the lower bounds will be set to zero on all arcs)
949    ///   ns.reset();
950    ///   ns.capacityMap(cap).costMap(cost)
951    ///     .supplyMap(sup).run();
952    /// \endcode
953    ///
954    /// \return <tt>(*this)</tt>
955    NetworkSimplex& reset() {
956      delete _plower;
957      delete _pupper;
958      delete _pcost;
959      delete _psupply;
960      _plower = NULL;
961      _pupper = NULL;
962      _pcost = NULL;
963      _psupply = NULL;
964      _pstsup = false;
965      _ptype = GEQ;
966      if (_local_flow) delete _flow_map;
967      if (_local_potential) delete _potential_map;
968      _flow_map = NULL;
969      _potential_map = NULL;
970      _local_flow = false;
971      _local_potential = false;
972
973      return *this;
974    }
975
976    /// @}
977
978    /// \name Query Functions
979    /// The results of the algorithm can be obtained using these
980    /// functions.\n
981    /// The \ref run() function must be called before using them.
982
983    /// @{
984
985    /// \brief Return the total cost of the found flow.
986    ///
987    /// This function returns the total cost of the found flow.
988    /// The complexity of the function is O(e).
989    ///
990    /// \note The return type of the function can be specified as a
991    /// template parameter. For example,
992    /// \code
993    ///   ns.totalCost<double>();
994    /// \endcode
995    /// It is useful if the total cost cannot be stored in the \c Cost
996    /// type of the algorithm, which is the default return type of the
997    /// function.
998    ///
999    /// \pre \ref run() must be called before using this function.
1000    template <typename Num>
1001    Num totalCost() const {
1002      Num c = 0;
1003      if (_pcost) {
1004        for (ArcIt e(_graph); e != INVALID; ++e)
1005          c += (*_flow_map)[e] * (*_pcost)[e];
1006      } else {
1007        for (ArcIt e(_graph); e != INVALID; ++e)
1008          c += (*_flow_map)[e];
1009      }
1010      return c;
1011    }
1012
1013#ifndef DOXYGEN
1014    Cost totalCost() const {
1016    }
1017#endif
1018
1019    /// \brief Return the flow on the given arc.
1020    ///
1021    /// This function returns the flow on the given arc.
1022    ///
1023    /// \pre \ref run() must be called before using this function.
1024    Flow flow(const Arc& a) const {
1025      return (*_flow_map)[a];
1026    }
1027
1028    /// \brief Return a const reference to the flow map.
1029    ///
1030    /// This function returns a const reference to an arc map storing
1031    /// the found flow.
1032    ///
1033    /// \pre \ref run() must be called before using this function.
1034    const FlowMap& flowMap() const {
1035      return *_flow_map;
1036    }
1037
1038    /// \brief Return the potential (dual value) of the given node.
1039    ///
1040    /// This function returns the potential (dual value) of the
1041    /// given node.
1042    ///
1043    /// \pre \ref run() must be called before using this function.
1044    Cost potential(const Node& n) const {
1045      return (*_potential_map)[n];
1046    }
1047
1048    /// \brief Return a const reference to the potential map
1049    /// (the dual solution).
1050    ///
1051    /// This function returns a const reference to a node map storing
1052    /// the found potentials, which form the dual solution of the
1053    /// \ref min_cost_flow "minimum cost flow" problem.
1054    ///
1055    /// \pre \ref run() must be called before using this function.
1056    const PotentialMap& potentialMap() const {
1057      return *_potential_map;
1058    }
1059
1060    /// @}
1061
1062  private:
1063
1064    // Initialize internal data structures
1065    bool init() {
1066      // Initialize result maps
1067      if (!_flow_map) {
1068        _flow_map = new FlowMap(_graph);
1069        _local_flow = true;
1070      }
1071      if (!_potential_map) {
1072        _potential_map = new PotentialMap(_graph);
1073        _local_potential = true;
1074      }
1075
1076      // Initialize vectors
1077      _node_num = countNodes(_graph);
1078      _arc_num = countArcs(_graph);
1079      int all_node_num = _node_num + 1;
1080      int all_arc_num = _arc_num + _node_num;
1081      if (_node_num == 0) return false;
1082
1083      _arc_ref.resize(_arc_num);
1084      _source.resize(all_arc_num);
1085      _target.resize(all_arc_num);
1086
1087      _cap.resize(all_arc_num);
1088      _cost.resize(all_arc_num);
1089      _supply.resize(all_node_num);
1090      _flow.resize(all_arc_num);
1091      _pi.resize(all_node_num);
1092
1093      _parent.resize(all_node_num);
1094      _pred.resize(all_node_num);
1095      _forward.resize(all_node_num);
1098      _succ_num.resize(all_node_num);
1099      _last_succ.resize(all_node_num);
1100      _state.resize(all_arc_num);
1101
1102      // Initialize node related data
1103      bool valid_supply = true;
1104      Flow sum_supply = 0;
1105      if (!_pstsup && !_psupply) {
1106        _pstsup = true;
1107        _psource = _ptarget = NodeIt(_graph);
1108        _pstflow = 0;
1109      }
1110      if (_psupply) {
1111        int i = 0;
1112        for (NodeIt n(_graph); n != INVALID; ++n, ++i) {
1113          _node_id[n] = i;
1114          _supply[i] = (*_psupply)[n];
1115          sum_supply += _supply[i];
1116        }
1117        valid_supply = (_ptype == GEQ && sum_supply <= 0) ||
1118                       (_ptype == LEQ && sum_supply >= 0);
1119      } else {
1120        int i = 0;
1121        for (NodeIt n(_graph); n != INVALID; ++n, ++i) {
1122          _node_id[n] = i;
1123          _supply[i] = 0;
1124        }
1125        _supply[_node_id[_psource]] =  _pstflow;
1126        _supply[_node_id[_ptarget]] = -_pstflow;
1127      }
1128      if (!valid_supply) return false;
1129
1130      // Infinite capacity value
1131      Flow inf_cap =
1132        std::numeric_limits<Flow>::has_infinity ?
1133        std::numeric_limits<Flow>::infinity() :
1134        std::numeric_limits<Flow>::max();
1135
1136      // Initialize artifical cost
1137      Cost art_cost;
1138      if (std::numeric_limits<Cost>::is_exact) {
1139        art_cost = std::numeric_limits<Cost>::max() / 4 + 1;
1140      } else {
1141        art_cost = std::numeric_limits<Cost>::min();
1142        for (int i = 0; i != _arc_num; ++i) {
1143          if (_cost[i] > art_cost) art_cost = _cost[i];
1144        }
1145        art_cost = (art_cost + 1) * _node_num;
1146      }
1147
1148      // Run Circulation to check if a feasible solution exists
1149      typedef ConstMap<Arc, Flow> ConstArcMap;
1150      ConstArcMap zero_arc_map(0), inf_arc_map(inf_cap);
1151      FlowNodeMap *csup = NULL;
1152      bool local_csup = false;
1153      if (_psupply) {
1154        csup = _psupply;
1155      } else {
1156        csup = new FlowNodeMap(_graph, 0);
1157        (*csup)[_psource] =  _pstflow;
1158        (*csup)[_ptarget] = -_pstflow;
1159        local_csup = true;
1160      }
1161      bool circ_result = false;
1162      if (_ptype == GEQ || (_ptype == LEQ && sum_supply == 0)) {
1163        // GEQ problem type
1164        if (_plower) {
1165          if (_pupper) {
1166            Circulation<GR, FlowArcMap, FlowArcMap, FlowNodeMap>
1167              circ(_graph, *_plower, *_pupper, *csup);
1168            circ_result = circ.run();
1169          } else {
1170            Circulation<GR, FlowArcMap, ConstArcMap, FlowNodeMap>
1171              circ(_graph, *_plower, inf_arc_map, *csup);
1172            circ_result = circ.run();
1173          }
1174        } else {
1175          if (_pupper) {
1176            Circulation<GR, ConstArcMap, FlowArcMap, FlowNodeMap>
1177              circ(_graph, zero_arc_map, *_pupper, *csup);
1178            circ_result = circ.run();
1179          } else {
1180            Circulation<GR, ConstArcMap, ConstArcMap, FlowNodeMap>
1181              circ(_graph, zero_arc_map, inf_arc_map, *csup);
1182            circ_result = circ.run();
1183          }
1184        }
1185      } else {
1186        // LEQ problem type
1187        typedef ReverseDigraph<const GR> RevGraph;
1188        typedef NegMap<FlowNodeMap> NegNodeMap;
1189        RevGraph rgraph(_graph);
1190        NegNodeMap neg_csup(*csup);
1191        if (_plower) {
1192          if (_pupper) {
1193            Circulation<RevGraph, FlowArcMap, FlowArcMap, NegNodeMap>
1194              circ(rgraph, *_plower, *_pupper, neg_csup);
1195            circ_result = circ.run();
1196          } else {
1197            Circulation<RevGraph, FlowArcMap, ConstArcMap, NegNodeMap>
1198              circ(rgraph, *_plower, inf_arc_map, neg_csup);
1199            circ_result = circ.run();
1200          }
1201        } else {
1202          if (_pupper) {
1203            Circulation<RevGraph, ConstArcMap, FlowArcMap, NegNodeMap>
1204              circ(rgraph, zero_arc_map, *_pupper, neg_csup);
1205            circ_result = circ.run();
1206          } else {
1207            Circulation<RevGraph, ConstArcMap, ConstArcMap, NegNodeMap>
1208              circ(rgraph, zero_arc_map, inf_arc_map, neg_csup);
1209            circ_result = circ.run();
1210          }
1211        }
1212      }
1213      if (local_csup) delete csup;
1214      if (!circ_result) return false;
1215
1216      // Set data for the artificial root node
1217      _root = _node_num;
1218      _parent[_root] = -1;
1219      _pred[_root] = -1;
1220      _thread[_root] = 0;
1221      _rev_thread[0] = _root;
1222      _succ_num[_root] = all_node_num;
1223      _last_succ[_root] = _root - 1;
1224      _supply[_root] = -sum_supply;
1225      if (sum_supply < 0) {
1226        _pi[_root] = -art_cost;
1227      } else {
1228        _pi[_root] = art_cost;
1229      }
1230
1231      // Store the arcs in a mixed order
1232      int k = std::max(int(std::sqrt(double(_arc_num))), 10);
1233      int i = 0;
1234      for (ArcIt e(_graph); e != INVALID; ++e) {
1235        _arc_ref[i] = e;
1236        if ((i += k) >= _arc_num) i = (i % k) + 1;
1237      }
1238
1239      // Initialize arc maps
1240      if (_pupper && _pcost) {
1241        for (int i = 0; i != _arc_num; ++i) {
1242          Arc e = _arc_ref[i];
1243          _source[i] = _node_id[_graph.source(e)];
1244          _target[i] = _node_id[_graph.target(e)];
1245          _cap[i] = (*_pupper)[e];
1246          _cost[i] = (*_pcost)[e];
1247          _flow[i] = 0;
1248          _state[i] = STATE_LOWER;
1249        }
1250      } else {
1251        for (int i = 0; i != _arc_num; ++i) {
1252          Arc e = _arc_ref[i];
1253          _source[i] = _node_id[_graph.source(e)];
1254          _target[i] = _node_id[_graph.target(e)];
1255          _flow[i] = 0;
1256          _state[i] = STATE_LOWER;
1257        }
1258        if (_pupper) {
1259          for (int i = 0; i != _arc_num; ++i)
1260            _cap[i] = (*_pupper)[_arc_ref[i]];
1261        } else {
1262          for (int i = 0; i != _arc_num; ++i)
1263            _cap[i] = inf_cap;
1264        }
1265        if (_pcost) {
1266          for (int i = 0; i != _arc_num; ++i)
1267            _cost[i] = (*_pcost)[_arc_ref[i]];
1268        } else {
1269          for (int i = 0; i != _arc_num; ++i)
1270            _cost[i] = 1;
1271        }
1272      }
1273
1274      // Remove non-zero lower bounds
1275      if (_plower) {
1276        for (int i = 0; i != _arc_num; ++i) {
1277          Flow c = (*_plower)[_arc_ref[i]];
1278          if (c != 0) {
1279            _cap[i] -= c;
1280            _supply[_source[i]] -= c;
1281            _supply[_target[i]] += c;
1282          }
1283        }
1284      }
1285
1286      // Add artificial arcs and initialize the spanning tree data structure
1287      for (int u = 0, e = _arc_num; u != _node_num; ++u, ++e) {
1288        _thread[u] = u + 1;
1289        _rev_thread[u + 1] = u;
1290        _succ_num[u] = 1;
1291        _last_succ[u] = u;
1292        _parent[u] = _root;
1293        _pred[u] = e;
1294        _cost[e] = art_cost;
1295        _cap[e] = inf_cap;
1296        _state[e] = STATE_TREE;
1297        if (_supply[u] > 0 || (_supply[u] == 0 && sum_supply <= 0)) {
1298          _flow[e] = _supply[u];
1299          _forward[u] = true;
1300          _pi[u] = -art_cost + _pi[_root];
1301        } else {
1302          _flow[e] = -_supply[u];
1303          _forward[u] = false;
1304          _pi[u] = art_cost + _pi[_root];
1305        }
1306      }
1307
1308      return true;
1309    }
1310
1311    // Find the join node
1312    void findJoinNode() {
1313      int u = _source[in_arc];
1314      int v = _target[in_arc];
1315      while (u != v) {
1316        if (_succ_num[u] < _succ_num[v]) {
1317          u = _parent[u];
1318        } else {
1319          v = _parent[v];
1320        }
1321      }
1322      join = u;
1323    }
1324
1325    // Find the leaving arc of the cycle and returns true if the
1326    // leaving arc is not the same as the entering arc
1327    bool findLeavingArc() {
1328      // Initialize first and second nodes according to the direction
1329      // of the cycle
1330      if (_state[in_arc] == STATE_LOWER) {
1331        first  = _source[in_arc];
1332        second = _target[in_arc];
1333      } else {
1334        first  = _target[in_arc];
1335        second = _source[in_arc];
1336      }
1337      delta = _cap[in_arc];
1338      int result = 0;
1339      Flow d;
1340      int e;
1341
1342      // Search the cycle along the path form the first node to the root
1343      for (int u = first; u != join; u = _parent[u]) {
1344        e = _pred[u];
1345        d = _forward[u] ? _flow[e] : _cap[e] - _flow[e];
1346        if (d < delta) {
1347          delta = d;
1348          u_out = u;
1349          result = 1;
1350        }
1351      }
1352      // Search the cycle along the path form the second node to the root
1353      for (int u = second; u != join; u = _parent[u]) {
1354        e = _pred[u];
1355        d = _forward[u] ? _cap[e] - _flow[e] : _flow[e];
1356        if (d <= delta) {
1357          delta = d;
1358          u_out = u;
1359          result = 2;
1360        }
1361      }
1362
1363      if (result == 1) {
1364        u_in = first;
1365        v_in = second;
1366      } else {
1367        u_in = second;
1368        v_in = first;
1369      }
1370      return result != 0;
1371    }
1372
1373    // Change _flow and _state vectors
1374    void changeFlow(bool change) {
1375      // Augment along the cycle
1376      if (delta > 0) {
1377        Flow val = _state[in_arc] * delta;
1378        _flow[in_arc] += val;
1379        for (int u = _source[in_arc]; u != join; u = _parent[u]) {
1380          _flow[_pred[u]] += _forward[u] ? -val : val;
1381        }
1382        for (int u = _target[in_arc]; u != join; u = _parent[u]) {
1383          _flow[_pred[u]] += _forward[u] ? val : -val;
1384        }
1385      }
1386      // Update the state of the entering and leaving arcs
1387      if (change) {
1388        _state[in_arc] = STATE_TREE;
1389        _state[_pred[u_out]] =
1390          (_flow[_pred[u_out]] == 0) ? STATE_LOWER : STATE_UPPER;
1391      } else {
1392        _state[in_arc] = -_state[in_arc];
1393      }
1394    }
1395
1396    // Update the tree structure
1397    void updateTreeStructure() {
1398      int u, w;
1400      int old_succ_num = _succ_num[u_out];
1401      int old_last_succ = _last_succ[u_out];
1402      v_out = _parent[u_out];
1403
1404      u = _last_succ[u_in];  // the last successor of u_in
1405      right = _thread[u];    // the node after it
1406
1407      // Handle the case when old_rev_thread equals to v_in
1408      // (it also means that join and v_out coincide)
1409      if (old_rev_thread == v_in) {
1410        last = _thread[_last_succ[u_out]];
1411      } else {
1412        last = _thread[v_in];
1413      }
1414
1415      // Update _thread and _parent along the stem nodes (i.e. the nodes
1416      // between u_in and u_out, whose parent have to be changed)
1417      _thread[v_in] = stem = u_in;
1418      _dirty_revs.clear();
1419      _dirty_revs.push_back(v_in);
1420      par_stem = v_in;
1421      while (stem != u_out) {
1422        // Insert the next stem node into the thread list
1423        new_stem = _parent[stem];
1424        _thread[u] = new_stem;
1425        _dirty_revs.push_back(u);
1426
1427        // Remove the subtree of stem from the thread list
1428        w = _rev_thread[stem];
1429        _thread[w] = right;
1430        _rev_thread[right] = w;
1431
1432        // Change the parent node and shift stem nodes
1433        _parent[stem] = par_stem;
1434        par_stem = stem;
1435        stem = new_stem;
1436
1437        // Update u and right
1438        u = _last_succ[stem] == _last_succ[par_stem] ?
1439          _rev_thread[par_stem] : _last_succ[stem];
1440        right = _thread[u];
1441      }
1442      _parent[u_out] = par_stem;
1443      _thread[u] = last;
1444      _rev_thread[last] = u;
1445      _last_succ[u_out] = u;
1446
1447      // Remove the subtree of u_out from the thread list except for
1448      // the case when old_rev_thread equals to v_in
1449      // (it also means that join and v_out coincide)
1450      if (old_rev_thread != v_in) {
1453      }
1454
1455      // Update _rev_thread using the new _thread values
1456      for (int i = 0; i < int(_dirty_revs.size()); ++i) {
1457        u = _dirty_revs[i];
1459      }
1460
1461      // Update _pred, _forward, _last_succ and _succ_num for the
1462      // stem nodes from u_out to u_in
1463      int tmp_sc = 0, tmp_ls = _last_succ[u_out];
1464      u = u_out;
1465      while (u != u_in) {
1466        w = _parent[u];
1467        _pred[u] = _pred[w];
1468        _forward[u] = !_forward[w];
1469        tmp_sc += _succ_num[u] - _succ_num[w];
1470        _succ_num[u] = tmp_sc;
1471        _last_succ[w] = tmp_ls;
1472        u = w;
1473      }
1474      _pred[u_in] = in_arc;
1475      _forward[u_in] = (u_in == _source[in_arc]);
1476      _succ_num[u_in] = old_succ_num;
1477
1478      // Set limits for updating _last_succ form v_in and v_out
1479      // towards the root
1480      int up_limit_in = -1;
1481      int up_limit_out = -1;
1482      if (_last_succ[join] == v_in) {
1483        up_limit_out = join;
1484      } else {
1485        up_limit_in = join;
1486      }
1487
1488      // Update _last_succ from v_in towards the root
1489      for (u = v_in; u != up_limit_in && _last_succ[u] == v_in;
1490           u = _parent[u]) {
1491        _last_succ[u] = _last_succ[u_out];
1492      }
1493      // Update _last_succ from v_out towards the root
1494      if (join != old_rev_thread && v_in != old_rev_thread) {
1495        for (u = v_out; u != up_limit_out && _last_succ[u] == old_last_succ;
1496             u = _parent[u]) {
1497          _last_succ[u] = old_rev_thread;
1498        }
1499      } else {
1500        for (u = v_out; u != up_limit_out && _last_succ[u] == old_last_succ;
1501             u = _parent[u]) {
1502          _last_succ[u] = _last_succ[u_out];
1503        }
1504      }
1505
1506      // Update _succ_num from v_in to join
1507      for (u = v_in; u != join; u = _parent[u]) {
1508        _succ_num[u] += old_succ_num;
1509      }
1510      // Update _succ_num from v_out to join
1511      for (u = v_out; u != join; u = _parent[u]) {
1512        _succ_num[u] -= old_succ_num;
1513      }
1514    }
1515
1516    // Update potentials
1517    void updatePotential() {
1518      Cost sigma = _forward[u_in] ?
1519        _pi[v_in] - _pi[u_in] - _cost[_pred[u_in]] :
1520        _pi[v_in] - _pi[u_in] + _cost[_pred[u_in]];
1521      // Update potentials in the subtree, which has been moved
1522      int end = _thread[_last_succ[u_in]];
1523      for (int u = u_in; u != end; u = _thread[u]) {
1524        _pi[u] += sigma;
1525      }
1526    }
1527
1528    // Execute the algorithm
1529    bool start(PivotRule pivot_rule) {
1530      // Select the pivot rule implementation
1531      switch (pivot_rule) {
1532        case FIRST_ELIGIBLE:
1533          return start<FirstEligiblePivotRule>();
1534        case BEST_ELIGIBLE:
1535          return start<BestEligiblePivotRule>();
1536        case BLOCK_SEARCH:
1537          return start<BlockSearchPivotRule>();
1538        case CANDIDATE_LIST:
1539          return start<CandidateListPivotRule>();
1540        case ALTERING_LIST:
1541          return start<AlteringListPivotRule>();
1542      }
1543      return false;
1544    }
1545
1546    template <typename PivotRuleImpl>
1547    bool start() {
1548      PivotRuleImpl pivot(*this);
1549
1550      // Execute the Network Simplex algorithm
1551      while (pivot.findEnteringArc()) {
1552        findJoinNode();
1553        bool change = findLeavingArc();
1554        changeFlow(change);
1555        if (change) {
1556          updateTreeStructure();
1557          updatePotential();
1558        }
1559      }
1560
1561      // Copy flow values to _flow_map
1562      if (_plower) {
1563        for (int i = 0; i != _arc_num; ++i) {
1564          Arc e = _arc_ref[i];
1565          _flow_map->set(e, (*_plower)[e] + _flow[i]);
1566        }
1567      } else {
1568        for (int i = 0; i != _arc_num; ++i) {
1569          _flow_map->set(_arc_ref[i], _flow[i]);
1570        }
1571      }
1572      // Copy potential values to _potential_map
1573      for (NodeIt n(_graph); n != INVALID; ++n) {
1574        _potential_map->set(n, _pi[_node_id[n]]);
1575      }
1576
1577      return true;
1578    }
1579
1580  }; //class NetworkSimplex
1581
1582  ///@}
1583
1584} //namespace lemon
1585
1586#endif //LEMON_NETWORK_SIMPLEX_H
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