COIN-OR::LEMON - Graph Library

source: lemon/lemon/network_simplex.h @ 660:b1811c363299

Last change on this file since 660:b1811c363299 was 660:b1811c363299, checked in by Peter Kovacs <kpeter@…>, 11 years ago

Bug fix in NetworkSimplex? (#234)

File size: 48.4 KB
RevLine 
[648]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
[652]25/// \brief Network Simplex algorithm for finding a minimum cost flow.
[648]26
27#include <vector>
28#include <limits>
29#include <algorithm>
30
[650]31#include <lemon/core.h>
[648]32#include <lemon/math.h>
[656]33#include <lemon/maps.h>
34#include <lemon/circulation.h>
35#include <lemon/adaptors.h>
[648]36
37namespace lemon {
38
39  /// \addtogroup min_cost_flow
40  /// @{
41
[652]42  /// \brief Implementation of the primal Network Simplex algorithm
[648]43  /// for finding a \ref min_cost_flow "minimum cost flow".
44  ///
[652]45  /// \ref NetworkSimplex implements the primal Network Simplex algorithm
[648]46  /// for finding a \ref min_cost_flow "minimum cost flow".
[653]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.
[656]53  /// Moreover it supports both direction of the supply/demand inequality
54  /// constraints. For more information see \ref ProblemType.
[648]55  ///
[652]56  /// \tparam GR The digraph type the algorithm runs on.
[654]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.
[648]61  ///
[655]62  /// \warning Both value types must be signed and all input data must
63  /// be integer.
[648]64  ///
[652]65  /// \note %NetworkSimplex provides five different pivot rule
[656]66  /// implementations, from which the most efficient one is used
67  /// by default. For more information see \ref PivotRule.
[654]68  template <typename GR, typename F = int, typename C = F>
[648]69  class NetworkSimplex
70  {
[652]71  public:
[648]72
[654]73    /// The flow type of the algorithm
74    typedef F Flow;
75    /// The cost type of the algorithm
76    typedef C Cost;
[656]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
[652]83    /// The type of the flow map
[654]84    typedef typename GR::template ArcMap<Flow> FlowMap;
[652]85    /// The type of the potential map
[654]86    typedef typename GR::template NodeMap<Cost> PotentialMap;
[656]87#endif
[652]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    };
[656]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    };
[652]185
186  private:
187
188    TEMPLATE_DIGRAPH_TYPEDEFS(GR);
189
[654]190    typedef typename GR::template ArcMap<Flow> FlowArcMap;
191    typedef typename GR::template ArcMap<Cost> CostArcMap;
192    typedef typename GR::template NodeMap<Flow> FlowNodeMap;
[648]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;
[654]198    typedef std::vector<Flow> FlowVector;
199    typedef std::vector<Cost> CostVector;
[648]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
[652]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
[654]216    FlowArcMap *_plower;
217    FlowArcMap *_pupper;
218    CostArcMap *_pcost;
219    FlowNodeMap *_psupply;
[652]220    bool _pstsup;
221    Node _psource, _ptarget;
[654]222    Flow _pstflow;
[656]223    ProblemType _ptype;
[648]224
225    // Result maps
[650]226    FlowMap *_flow_map;
227    PotentialMap *_potential_map;
[648]228    bool _local_flow;
229    bool _local_potential;
230
[652]231    // Data structures for storing the digraph
[650]232    IntNodeMap _node_id;
233    ArcVector _arc_ref;
234    IntVector _source;
235    IntVector _target;
236
[652]237    // Node and arc data
[654]238    FlowVector _cap;
239    CostVector _cost;
240    FlowVector _supply;
241    FlowVector _flow;
242    CostVector _pi;
[648]243
[650]244    // Data for storing the spanning tree structure
[648]245    IntVector _parent;
246    IntVector _pred;
247    IntVector _thread;
[651]248    IntVector _rev_thread;
249    IntVector _succ_num;
250    IntVector _last_succ;
251    IntVector _dirty_revs;
[648]252    BoolVector _forward;
253    IntVector _state;
254    int _root;
255
256    // Temporary data used in the current pivot iteration
[650]257    int in_arc, join, u_in, v_in, u_out, v_out;
258    int first, second, right, last;
[648]259    int stem, par_stem, new_stem;
[654]260    Flow delta;
[648]261
262  private:
263
[652]264    // Implementation of the First Eligible pivot rule
[648]265    class FirstEligiblePivotRule
266    {
267    private:
268
269      // References to the NetworkSimplex class
270      const IntVector  &_source;
271      const IntVector  &_target;
[654]272      const CostVector &_cost;
[648]273      const IntVector  &_state;
[654]274      const CostVector &_pi;
[648]275      int &_in_arc;
276      int _arc_num;
277
278      // Pivot rule data
279      int _next_arc;
280
281    public:
282
[652]283      // Constructor
[648]284      FirstEligiblePivotRule(NetworkSimplex &ns) :
[650]285        _source(ns._source), _target(ns._target),
[648]286        _cost(ns._cost), _state(ns._state), _pi(ns._pi),
[650]287        _in_arc(ns.in_arc), _arc_num(ns._arc_num), _next_arc(0)
[648]288      {}
289
[652]290      // Find next entering arc
[648]291      bool findEnteringArc() {
[654]292        Cost c;
[648]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
[652]315    // Implementation of the Best Eligible pivot rule
[648]316    class BestEligiblePivotRule
317    {
318    private:
319
320      // References to the NetworkSimplex class
321      const IntVector  &_source;
322      const IntVector  &_target;
[654]323      const CostVector &_cost;
[648]324      const IntVector  &_state;
[654]325      const CostVector &_pi;
[648]326      int &_in_arc;
327      int _arc_num;
328
329    public:
330
[652]331      // Constructor
[648]332      BestEligiblePivotRule(NetworkSimplex &ns) :
[650]333        _source(ns._source), _target(ns._target),
[648]334        _cost(ns._cost), _state(ns._state), _pi(ns._pi),
[650]335        _in_arc(ns.in_arc), _arc_num(ns._arc_num)
[648]336      {}
337
[652]338      // Find next entering arc
[648]339      bool findEnteringArc() {
[654]340        Cost c, min = 0;
[648]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
[652]354    // Implementation of the Block Search pivot rule
[648]355    class BlockSearchPivotRule
356    {
357    private:
358
359      // References to the NetworkSimplex class
360      const IntVector  &_source;
361      const IntVector  &_target;
[654]362      const CostVector &_cost;
[648]363      const IntVector  &_state;
[654]364      const CostVector &_pi;
[648]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
[652]374      // Constructor
[648]375      BlockSearchPivotRule(NetworkSimplex &ns) :
[650]376        _source(ns._source), _target(ns._target),
[648]377        _cost(ns._cost), _state(ns._state), _pi(ns._pi),
[650]378        _in_arc(ns.in_arc), _arc_num(ns._arc_num), _next_arc(0)
[648]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 * sqrt(_arc_num)),
385                                MIN_BLOCK_SIZE );
386      }
387
[652]388      // Find next entering arc
[648]389      bool findEnteringArc() {
[654]390        Cost c, min = 0;
[648]391        int cnt = _block_size;
392        int e, min_arc = _next_arc;
393        for (e = _next_arc; e < _arc_num; ++e) {
394          c = _state[e] * (_cost[e] + _pi[_source[e]] - _pi[_target[e]]);
395          if (c < min) {
396            min = c;
397            min_arc = e;
398          }
399          if (--cnt == 0) {
400            if (min < 0) break;
401            cnt = _block_size;
402          }
403        }
404        if (min == 0 || cnt > 0) {
405          for (e = 0; e < _next_arc; ++e) {
406            c = _state[e] * (_cost[e] + _pi[_source[e]] - _pi[_target[e]]);
407            if (c < min) {
408              min = c;
409              min_arc = e;
410            }
411            if (--cnt == 0) {
412              if (min < 0) break;
413              cnt = _block_size;
414            }
415          }
416        }
417        if (min >= 0) return false;
418        _in_arc = min_arc;
419        _next_arc = e;
420        return true;
421      }
422
423    }; //class BlockSearchPivotRule
424
425
[652]426    // Implementation of the Candidate List pivot rule
[648]427    class CandidateListPivotRule
428    {
429    private:
430
431      // References to the NetworkSimplex class
432      const IntVector  &_source;
433      const IntVector  &_target;
[654]434      const CostVector &_cost;
[648]435      const IntVector  &_state;
[654]436      const CostVector &_pi;
[648]437      int &_in_arc;
438      int _arc_num;
439
440      // Pivot rule data
441      IntVector _candidates;
442      int _list_length, _minor_limit;
443      int _curr_length, _minor_count;
444      int _next_arc;
445
446    public:
447
448      /// Constructor
449      CandidateListPivotRule(NetworkSimplex &ns) :
[650]450        _source(ns._source), _target(ns._target),
[648]451        _cost(ns._cost), _state(ns._state), _pi(ns._pi),
[650]452        _in_arc(ns.in_arc), _arc_num(ns._arc_num), _next_arc(0)
[648]453      {
454        // The main parameters of the pivot rule
455        const double LIST_LENGTH_FACTOR = 1.0;
456        const int MIN_LIST_LENGTH = 10;
457        const double MINOR_LIMIT_FACTOR = 0.1;
458        const int MIN_MINOR_LIMIT = 3;
459
460        _list_length = std::max( int(LIST_LENGTH_FACTOR * sqrt(_arc_num)),
461                                 MIN_LIST_LENGTH );
462        _minor_limit = std::max( int(MINOR_LIMIT_FACTOR * _list_length),
463                                 MIN_MINOR_LIMIT );
464        _curr_length = _minor_count = 0;
465        _candidates.resize(_list_length);
466      }
467
468      /// Find next entering arc
469      bool findEnteringArc() {
[654]470        Cost min, c;
[648]471        int e, min_arc = _next_arc;
472        if (_curr_length > 0 && _minor_count < _minor_limit) {
473          // Minor iteration: select the best eligible arc from the
474          // current candidate list
475          ++_minor_count;
476          min = 0;
477          for (int i = 0; i < _curr_length; ++i) {
478            e = _candidates[i];
479            c = _state[e] * (_cost[e] + _pi[_source[e]] - _pi[_target[e]]);
480            if (c < min) {
481              min = c;
482              min_arc = e;
483            }
484            if (c >= 0) {
485              _candidates[i--] = _candidates[--_curr_length];
486            }
487          }
488          if (min < 0) {
489            _in_arc = min_arc;
490            return true;
491          }
492        }
493
494        // Major iteration: build a new candidate list
495        min = 0;
496        _curr_length = 0;
497        for (e = _next_arc; e < _arc_num; ++e) {
498          c = _state[e] * (_cost[e] + _pi[_source[e]] - _pi[_target[e]]);
499          if (c < 0) {
500            _candidates[_curr_length++] = e;
501            if (c < min) {
502              min = c;
503              min_arc = e;
504            }
505            if (_curr_length == _list_length) break;
506          }
507        }
508        if (_curr_length < _list_length) {
509          for (e = 0; e < _next_arc; ++e) {
510            c = _state[e] * (_cost[e] + _pi[_source[e]] - _pi[_target[e]]);
511            if (c < 0) {
512              _candidates[_curr_length++] = e;
513              if (c < min) {
514                min = c;
515                min_arc = e;
516              }
517              if (_curr_length == _list_length) break;
518            }
519          }
520        }
521        if (_curr_length == 0) return false;
522        _minor_count = 1;
523        _in_arc = min_arc;
524        _next_arc = e;
525        return true;
526      }
527
528    }; //class CandidateListPivotRule
529
530
[652]531    // Implementation of the Altering Candidate List pivot rule
[648]532    class AlteringListPivotRule
533    {
534    private:
535
536      // References to the NetworkSimplex class
537      const IntVector  &_source;
538      const IntVector  &_target;
[654]539      const CostVector &_cost;
[648]540      const IntVector  &_state;
[654]541      const CostVector &_pi;
[648]542      int &_in_arc;
543      int _arc_num;
544
545      // Pivot rule data
546      int _block_size, _head_length, _curr_length;
547      int _next_arc;
548      IntVector _candidates;
[654]549      CostVector _cand_cost;
[648]550
551      // Functor class to compare arcs during sort of the candidate list
552      class SortFunc
553      {
554      private:
[654]555        const CostVector &_map;
[648]556      public:
[654]557        SortFunc(const CostVector &map) : _map(map) {}
[648]558        bool operator()(int left, int right) {
559          return _map[left] > _map[right];
560        }
561      };
562
563      SortFunc _sort_func;
564
565    public:
566
[652]567      // Constructor
[648]568      AlteringListPivotRule(NetworkSimplex &ns) :
[650]569        _source(ns._source), _target(ns._target),
[648]570        _cost(ns._cost), _state(ns._state), _pi(ns._pi),
[650]571        _in_arc(ns.in_arc), _arc_num(ns._arc_num),
[648]572        _next_arc(0), _cand_cost(ns._arc_num), _sort_func(_cand_cost)
573      {
574        // The main parameters of the pivot rule
575        const double BLOCK_SIZE_FACTOR = 1.5;
576        const int MIN_BLOCK_SIZE = 10;
577        const double HEAD_LENGTH_FACTOR = 0.1;
578        const int MIN_HEAD_LENGTH = 3;
579
580        _block_size = std::max( int(BLOCK_SIZE_FACTOR * sqrt(_arc_num)),
581                                MIN_BLOCK_SIZE );
582        _head_length = std::max( int(HEAD_LENGTH_FACTOR * _block_size),
583                                 MIN_HEAD_LENGTH );
584        _candidates.resize(_head_length + _block_size);
585        _curr_length = 0;
586      }
587
[652]588      // Find next entering arc
[648]589      bool findEnteringArc() {
590        // Check the current candidate list
591        int e;
592        for (int i = 0; i < _curr_length; ++i) {
593          e = _candidates[i];
594          _cand_cost[e] = _state[e] *
595            (_cost[e] + _pi[_source[e]] - _pi[_target[e]]);
596          if (_cand_cost[e] >= 0) {
597            _candidates[i--] = _candidates[--_curr_length];
598          }
599        }
600
601        // Extend the list
602        int cnt = _block_size;
[650]603        int last_arc = 0;
[648]604        int limit = _head_length;
605
606        for (int e = _next_arc; e < _arc_num; ++e) {
607          _cand_cost[e] = _state[e] *
608            (_cost[e] + _pi[_source[e]] - _pi[_target[e]]);
609          if (_cand_cost[e] < 0) {
610            _candidates[_curr_length++] = e;
[650]611            last_arc = e;
[648]612          }
613          if (--cnt == 0) {
614            if (_curr_length > limit) break;
615            limit = 0;
616            cnt = _block_size;
617          }
618        }
619        if (_curr_length <= limit) {
620          for (int e = 0; e < _next_arc; ++e) {
621            _cand_cost[e] = _state[e] *
622              (_cost[e] + _pi[_source[e]] - _pi[_target[e]]);
623            if (_cand_cost[e] < 0) {
624              _candidates[_curr_length++] = e;
[650]625              last_arc = e;
[648]626            }
627            if (--cnt == 0) {
628              if (_curr_length > limit) break;
629              limit = 0;
630              cnt = _block_size;
631            }
632          }
633        }
634        if (_curr_length == 0) return false;
[650]635        _next_arc = last_arc + 1;
[648]636
637        // Make heap of the candidate list (approximating a partial sort)
638        make_heap( _candidates.begin(), _candidates.begin() + _curr_length,
639                   _sort_func );
640
641        // Pop the first element of the heap
642        _in_arc = _candidates[0];
643        pop_heap( _candidates.begin(), _candidates.begin() + _curr_length,
644                  _sort_func );
645        _curr_length = std::min(_head_length, _curr_length - 1);
646        return true;
647      }
648
649    }; //class AlteringListPivotRule
650
651  public:
652
[652]653    /// \brief Constructor.
[648]654    ///
[656]655    /// The constructor of the class.
[648]656    ///
[650]657    /// \param graph The digraph the algorithm runs on.
[652]658    NetworkSimplex(const GR& graph) :
659      _graph(graph),
660      _plower(NULL), _pupper(NULL), _pcost(NULL),
[656]661      _psupply(NULL), _pstsup(false), _ptype(GEQ),
[650]662      _flow_map(NULL), _potential_map(NULL),
[648]663      _local_flow(false), _local_potential(false),
[650]664      _node_id(graph)
[652]665    {
[654]666      LEMON_ASSERT(std::numeric_limits<Flow>::is_integer &&
667                   std::numeric_limits<Flow>::is_signed,
668        "The flow type of NetworkSimplex must be signed integer");
669      LEMON_ASSERT(std::numeric_limits<Cost>::is_integer &&
670                   std::numeric_limits<Cost>::is_signed,
671        "The cost type of NetworkSimplex must be signed integer");
[652]672    }
[648]673
674    /// Destructor.
675    ~NetworkSimplex() {
[650]676      if (_local_flow) delete _flow_map;
677      if (_local_potential) delete _potential_map;
[648]678    }
679
[656]680    /// \name Parameters
681    /// The parameters of the algorithm can be specified using these
682    /// functions.
683
684    /// @{
685
[652]686    /// \brief Set the lower bounds on the arcs.
687    ///
688    /// This function sets the lower bounds on the arcs.
689    /// If neither this function nor \ref boundMaps() is used before
690    /// calling \ref run(), the lower bounds will be set to zero
691    /// on all arcs.
692    ///
693    /// \param map An arc map storing the lower bounds.
[654]694    /// Its \c Value type must be convertible to the \c Flow type
[652]695    /// of the algorithm.
696    ///
697    /// \return <tt>(*this)</tt>
698    template <typename LOWER>
699    NetworkSimplex& lowerMap(const LOWER& map) {
700      delete _plower;
[654]701      _plower = new FlowArcMap(_graph);
[652]702      for (ArcIt a(_graph); a != INVALID; ++a) {
703        (*_plower)[a] = map[a];
704      }
705      return *this;
706    }
707
708    /// \brief Set the upper bounds (capacities) on the arcs.
709    ///
710    /// This function sets the upper bounds (capacities) on the arcs.
711    /// If none of the functions \ref upperMap(), \ref capacityMap()
712    /// and \ref boundMaps() is used before calling \ref run(),
713    /// the upper bounds (capacities) will be set to
[654]714    /// \c std::numeric_limits<Flow>::max() on all arcs.
[652]715    ///
716    /// \param map An arc map storing the upper bounds.
[654]717    /// Its \c Value type must be convertible to the \c Flow type
[652]718    /// of the algorithm.
719    ///
720    /// \return <tt>(*this)</tt>
721    template<typename UPPER>
722    NetworkSimplex& upperMap(const UPPER& map) {
723      delete _pupper;
[654]724      _pupper = new FlowArcMap(_graph);
[652]725      for (ArcIt a(_graph); a != INVALID; ++a) {
726        (*_pupper)[a] = map[a];
727      }
728      return *this;
729    }
730
731    /// \brief Set the upper bounds (capacities) on the arcs.
732    ///
733    /// This function sets the upper bounds (capacities) on the arcs.
734    /// It is just an alias for \ref upperMap().
735    ///
736    /// \return <tt>(*this)</tt>
737    template<typename CAP>
738    NetworkSimplex& capacityMap(const CAP& map) {
739      return upperMap(map);
740    }
741
742    /// \brief Set the lower and upper bounds on the arcs.
743    ///
744    /// This function sets the lower and upper bounds on the arcs.
745    /// If neither this function nor \ref lowerMap() is used before
746    /// calling \ref run(), the lower bounds will be set to zero
747    /// on all arcs.
748    /// If none of the functions \ref upperMap(), \ref capacityMap()
749    /// and \ref boundMaps() is used before calling \ref run(),
750    /// the upper bounds (capacities) will be set to
[654]751    /// \c std::numeric_limits<Flow>::max() on all arcs.
[652]752    ///
753    /// \param lower An arc map storing the lower bounds.
754    /// \param upper An arc map storing the upper bounds.
755    ///
756    /// The \c Value type of the maps must be convertible to the
[654]757    /// \c Flow type of the algorithm.
[652]758    ///
759    /// \note This function is just a shortcut of calling \ref lowerMap()
760    /// and \ref upperMap() separately.
761    ///
762    /// \return <tt>(*this)</tt>
763    template <typename LOWER, typename UPPER>
764    NetworkSimplex& boundMaps(const LOWER& lower, const UPPER& upper) {
765      return lowerMap(lower).upperMap(upper);
766    }
767
768    /// \brief Set the costs of the arcs.
769    ///
770    /// This function sets the costs of the arcs.
771    /// If it is not used before calling \ref run(), the costs
772    /// will be set to \c 1 on all arcs.
773    ///
774    /// \param map An arc map storing the costs.
[654]775    /// Its \c Value type must be convertible to the \c Cost type
[652]776    /// of the algorithm.
777    ///
778    /// \return <tt>(*this)</tt>
779    template<typename COST>
780    NetworkSimplex& costMap(const COST& map) {
781      delete _pcost;
[654]782      _pcost = new CostArcMap(_graph);
[652]783      for (ArcIt a(_graph); a != INVALID; ++a) {
784        (*_pcost)[a] = map[a];
785      }
786      return *this;
787    }
788
789    /// \brief Set the supply values of the nodes.
790    ///
791    /// This function sets the supply values of the nodes.
792    /// If neither this function nor \ref stSupply() is used before
793    /// calling \ref run(), the supply of each node will be set to zero.
794    /// (It makes sense only if non-zero lower bounds are given.)
795    ///
796    /// \param map A node map storing the supply values.
[654]797    /// Its \c Value type must be convertible to the \c Flow type
[652]798    /// of the algorithm.
799    ///
800    /// \return <tt>(*this)</tt>
801    template<typename SUP>
802    NetworkSimplex& supplyMap(const SUP& map) {
803      delete _psupply;
804      _pstsup = false;
[654]805      _psupply = new FlowNodeMap(_graph);
[652]806      for (NodeIt n(_graph); n != INVALID; ++n) {
807        (*_psupply)[n] = map[n];
808      }
809      return *this;
810    }
811
812    /// \brief Set single source and target nodes and a supply value.
813    ///
814    /// This function sets a single source node and a single target node
815    /// and the required flow value.
816    /// If neither this function nor \ref supplyMap() is used before
817    /// calling \ref run(), the supply of each node will be set to zero.
818    /// (It makes sense only if non-zero lower bounds are given.)
819    ///
820    /// \param s The source node.
821    /// \param t The target node.
822    /// \param k The required amount of flow from node \c s to node \c t
823    /// (i.e. the supply of \c s and the demand of \c t).
824    ///
825    /// \return <tt>(*this)</tt>
[654]826    NetworkSimplex& stSupply(const Node& s, const Node& t, Flow k) {
[652]827      delete _psupply;
828      _psupply = NULL;
829      _pstsup = true;
830      _psource = s;
831      _ptarget = t;
832      _pstflow = k;
833      return *this;
834    }
[656]835   
836    /// \brief Set the problem type.
837    ///
838    /// This function sets the problem type for the algorithm.
839    /// If it is not used before calling \ref run(), the \ref GEQ problem
840    /// type will be used.
841    ///
842    /// For more information see \ref ProblemType.
843    ///
844    /// \return <tt>(*this)</tt>
845    NetworkSimplex& problemType(ProblemType problem_type) {
846      _ptype = problem_type;
847      return *this;
848    }
[652]849
[648]850    /// \brief Set the flow map.
851    ///
852    /// This function sets the flow map.
[652]853    /// If it is not used before calling \ref run(), an instance will
854    /// be allocated automatically. The destructor deallocates this
855    /// automatically allocated map, of course.
[648]856    ///
857    /// \return <tt>(*this)</tt>
[652]858    NetworkSimplex& flowMap(FlowMap& map) {
[648]859      if (_local_flow) {
[650]860        delete _flow_map;
[648]861        _local_flow = false;
862      }
[650]863      _flow_map = &map;
[648]864      return *this;
865    }
866
867    /// \brief Set the potential map.
868    ///
[652]869    /// This function sets the potential map, which is used for storing
870    /// the dual solution.
871    /// If it is not used before calling \ref run(), an instance will
872    /// be allocated automatically. The destructor deallocates this
873    /// automatically allocated map, of course.
[648]874    ///
875    /// \return <tt>(*this)</tt>
[652]876    NetworkSimplex& potentialMap(PotentialMap& map) {
[648]877      if (_local_potential) {
[650]878        delete _potential_map;
[648]879        _local_potential = false;
880      }
[650]881      _potential_map = &map;
[648]882      return *this;
883    }
[656]884   
885    /// @}
[648]886
[652]887    /// \name Execution Control
888    /// The algorithm can be executed using \ref run().
889
[648]890    /// @{
891
892    /// \brief Run the algorithm.
893    ///
894    /// This function runs the algorithm.
[656]895    /// The paramters can be specified using functions \ref lowerMap(),
[653]896    /// \ref upperMap(), \ref capacityMap(), \ref boundMaps(),
[656]897    /// \ref costMap(), \ref supplyMap(), \ref stSupply(),
898    /// \ref problemType(), \ref flowMap() and \ref potentialMap().
899    /// For example,
[652]900    /// \code
901    ///   NetworkSimplex<ListDigraph> ns(graph);
902    ///   ns.boundMaps(lower, upper).costMap(cost)
903    ///     .supplyMap(sup).run();
904    /// \endcode
[648]905    ///
[653]906    /// This function can be called more than once. All the parameters
907    /// that have been given are kept for the next call, unless
908    /// \ref reset() is called, thus only the modified parameters
909    /// have to be set again. See \ref reset() for examples.
910    ///
[652]911    /// \param pivot_rule The pivot rule that will be used during the
912    /// algorithm. For more information see \ref PivotRule.
[648]913    ///
914    /// \return \c true if a feasible flow can be found.
[652]915    bool run(PivotRule pivot_rule = BLOCK_SEARCH) {
[648]916      return init() && start(pivot_rule);
917    }
918
[653]919    /// \brief Reset all the parameters that have been given before.
920    ///
921    /// This function resets all the paramaters that have been given
[656]922    /// before using functions \ref lowerMap(), \ref upperMap(),
923    /// \ref capacityMap(), \ref boundMaps(), \ref costMap(),
924    /// \ref supplyMap(), \ref stSupply(), \ref problemType(),
925    /// \ref flowMap() and \ref potentialMap().
[653]926    ///
927    /// It is useful for multiple run() calls. If this function is not
928    /// used, all the parameters given before are kept for the next
929    /// \ref run() call.
930    ///
931    /// For example,
932    /// \code
933    ///   NetworkSimplex<ListDigraph> ns(graph);
934    ///
935    ///   // First run
936    ///   ns.lowerMap(lower).capacityMap(cap).costMap(cost)
937    ///     .supplyMap(sup).run();
938    ///
939    ///   // Run again with modified cost map (reset() is not called,
940    ///   // so only the cost map have to be set again)
941    ///   cost[e] += 100;
942    ///   ns.costMap(cost).run();
943    ///
944    ///   // Run again from scratch using reset()
945    ///   // (the lower bounds will be set to zero on all arcs)
946    ///   ns.reset();
947    ///   ns.capacityMap(cap).costMap(cost)
948    ///     .supplyMap(sup).run();
949    /// \endcode
950    ///
951    /// \return <tt>(*this)</tt>
952    NetworkSimplex& reset() {
953      delete _plower;
954      delete _pupper;
955      delete _pcost;
956      delete _psupply;
957      _plower = NULL;
958      _pupper = NULL;
959      _pcost = NULL;
960      _psupply = NULL;
961      _pstsup = false;
[656]962      _ptype = GEQ;
963      if (_local_flow) delete _flow_map;
964      if (_local_potential) delete _potential_map;
965      _flow_map = NULL;
966      _potential_map = NULL;
967      _local_flow = false;
968      _local_potential = false;
969
[653]970      return *this;
971    }
972
[648]973    /// @}
974
975    /// \name Query Functions
976    /// The results of the algorithm can be obtained using these
977    /// functions.\n
[652]978    /// The \ref run() function must be called before using them.
979
[648]980    /// @{
981
[652]982    /// \brief Return the total cost of the found flow.
983    ///
984    /// This function returns the total cost of the found flow.
[654]985    /// The complexity of the function is O(e).
[652]986    ///
987    /// \note The return type of the function can be specified as a
988    /// template parameter. For example,
989    /// \code
990    ///   ns.totalCost<double>();
991    /// \endcode
[654]992    /// It is useful if the total cost cannot be stored in the \c Cost
[652]993    /// type of the algorithm, which is the default return type of the
994    /// function.
995    ///
996    /// \pre \ref run() must be called before using this function.
997    template <typename Num>
998    Num totalCost() const {
999      Num c = 0;
1000      if (_pcost) {
1001        for (ArcIt e(_graph); e != INVALID; ++e)
1002          c += (*_flow_map)[e] * (*_pcost)[e];
1003      } else {
1004        for (ArcIt e(_graph); e != INVALID; ++e)
1005          c += (*_flow_map)[e];
1006      }
1007      return c;
1008    }
1009
1010#ifndef DOXYGEN
[654]1011    Cost totalCost() const {
1012      return totalCost<Cost>();
[652]1013    }
1014#endif
1015
1016    /// \brief Return the flow on the given arc.
1017    ///
1018    /// This function returns the flow on the given arc.
1019    ///
1020    /// \pre \ref run() must be called before using this function.
[654]1021    Flow flow(const Arc& a) const {
[652]1022      return (*_flow_map)[a];
1023    }
1024
[648]1025    /// \brief Return a const reference to the flow map.
1026    ///
1027    /// This function returns a const reference to an arc map storing
1028    /// the found flow.
1029    ///
1030    /// \pre \ref run() must be called before using this function.
1031    const FlowMap& flowMap() const {
[650]1032      return *_flow_map;
[648]1033    }
1034
[652]1035    /// \brief Return the potential (dual value) of the given node.
1036    ///
1037    /// This function returns the potential (dual value) of the
1038    /// given node.
1039    ///
1040    /// \pre \ref run() must be called before using this function.
[654]1041    Cost potential(const Node& n) const {
[652]1042      return (*_potential_map)[n];
1043    }
1044
[648]1045    /// \brief Return a const reference to the potential map
1046    /// (the dual solution).
1047    ///
1048    /// This function returns a const reference to a node map storing
[652]1049    /// the found potentials, which form the dual solution of the
1050    /// \ref min_cost_flow "minimum cost flow" problem.
[648]1051    ///
1052    /// \pre \ref run() must be called before using this function.
1053    const PotentialMap& potentialMap() const {
[650]1054      return *_potential_map;
[648]1055    }
1056
1057    /// @}
1058
1059  private:
1060
1061    // Initialize internal data structures
1062    bool init() {
1063      // Initialize result maps
[650]1064      if (!_flow_map) {
1065        _flow_map = new FlowMap(_graph);
[648]1066        _local_flow = true;
1067      }
[650]1068      if (!_potential_map) {
1069        _potential_map = new PotentialMap(_graph);
[648]1070        _local_potential = true;
1071      }
1072
1073      // Initialize vectors
[650]1074      _node_num = countNodes(_graph);
1075      _arc_num = countArcs(_graph);
[648]1076      int all_node_num = _node_num + 1;
[650]1077      int all_arc_num = _arc_num + _node_num;
[652]1078      if (_node_num == 0) return false;
[648]1079
[650]1080      _arc_ref.resize(_arc_num);
1081      _source.resize(all_arc_num);
1082      _target.resize(all_arc_num);
[648]1083
[650]1084      _cap.resize(all_arc_num);
1085      _cost.resize(all_arc_num);
[648]1086      _supply.resize(all_node_num);
[653]1087      _flow.resize(all_arc_num);
1088      _pi.resize(all_node_num);
[648]1089
1090      _parent.resize(all_node_num);
1091      _pred.resize(all_node_num);
[650]1092      _forward.resize(all_node_num);
[648]1093      _thread.resize(all_node_num);
[651]1094      _rev_thread.resize(all_node_num);
1095      _succ_num.resize(all_node_num);
1096      _last_succ.resize(all_node_num);
[653]1097      _state.resize(all_arc_num);
[648]1098
1099      // Initialize node related data
1100      bool valid_supply = true;
[656]1101      Flow sum_supply = 0;
[652]1102      if (!_pstsup && !_psupply) {
1103        _pstsup = true;
1104        _psource = _ptarget = NodeIt(_graph);
1105        _pstflow = 0;
1106      }
1107      if (_psupply) {
[648]1108        int i = 0;
[650]1109        for (NodeIt n(_graph); n != INVALID; ++n, ++i) {
[648]1110          _node_id[n] = i;
[652]1111          _supply[i] = (*_psupply)[n];
[656]1112          sum_supply += _supply[i];
[648]1113        }
[656]1114        valid_supply = (_ptype == GEQ && sum_supply <= 0) ||
1115                       (_ptype == LEQ && sum_supply >= 0);
[648]1116      } else {
1117        int i = 0;
[650]1118        for (NodeIt n(_graph); n != INVALID; ++n, ++i) {
[648]1119          _node_id[n] = i;
1120          _supply[i] = 0;
1121        }
[652]1122        _supply[_node_id[_psource]] =  _pstflow;
[656]1123        _supply[_node_id[_ptarget]] = -_pstflow;
[648]1124      }
1125      if (!valid_supply) return false;
1126
[656]1127      // Infinite capacity value
1128      Flow inf_cap =
1129        std::numeric_limits<Flow>::has_infinity ?
1130        std::numeric_limits<Flow>::infinity() :
1131        std::numeric_limits<Flow>::max();
1132
1133      // Initialize artifical cost
1134      Cost art_cost;
1135      if (std::numeric_limits<Cost>::is_exact) {
1136        art_cost = std::numeric_limits<Cost>::max() / 4 + 1;
1137      } else {
1138        art_cost = std::numeric_limits<Cost>::min();
1139        for (int i = 0; i != _arc_num; ++i) {
1140          if (_cost[i] > art_cost) art_cost = _cost[i];
1141        }
1142        art_cost = (art_cost + 1) * _node_num;
1143      }
1144
1145      // Run Circulation to check if a feasible solution exists
1146      typedef ConstMap<Arc, Flow> ConstArcMap;
[660]1147      ConstArcMap zero_arc_map(0), inf_arc_map(inf_cap);
[656]1148      FlowNodeMap *csup = NULL;
1149      bool local_csup = false;
1150      if (_psupply) {
1151        csup = _psupply;
1152      } else {
1153        csup = new FlowNodeMap(_graph, 0);
1154        (*csup)[_psource] =  _pstflow;
1155        (*csup)[_ptarget] = -_pstflow;
1156        local_csup = true;
1157      }
1158      bool circ_result = false;
1159      if (_ptype == GEQ || (_ptype == LEQ && sum_supply == 0)) {
1160        // GEQ problem type
1161        if (_plower) {
1162          if (_pupper) {
1163            Circulation<GR, FlowArcMap, FlowArcMap, FlowNodeMap>
1164              circ(_graph, *_plower, *_pupper, *csup);
1165            circ_result = circ.run();
1166          } else {
1167            Circulation<GR, FlowArcMap, ConstArcMap, FlowNodeMap>
[660]1168              circ(_graph, *_plower, inf_arc_map, *csup);
[656]1169            circ_result = circ.run();
1170          }
1171        } else {
1172          if (_pupper) {
1173            Circulation<GR, ConstArcMap, FlowArcMap, FlowNodeMap>
[660]1174              circ(_graph, zero_arc_map, *_pupper, *csup);
[656]1175            circ_result = circ.run();
1176          } else {
1177            Circulation<GR, ConstArcMap, ConstArcMap, FlowNodeMap>
[660]1178              circ(_graph, zero_arc_map, inf_arc_map, *csup);
[656]1179            circ_result = circ.run();
1180          }
1181        }
1182      } else {
1183        // LEQ problem type
1184        typedef ReverseDigraph<const GR> RevGraph;
1185        typedef NegMap<FlowNodeMap> NegNodeMap;
1186        RevGraph rgraph(_graph);
1187        NegNodeMap neg_csup(*csup);
1188        if (_plower) {
1189          if (_pupper) {
1190            Circulation<RevGraph, FlowArcMap, FlowArcMap, NegNodeMap>
1191              circ(rgraph, *_plower, *_pupper, neg_csup);
1192            circ_result = circ.run();
1193          } else {
1194            Circulation<RevGraph, FlowArcMap, ConstArcMap, NegNodeMap>
[660]1195              circ(rgraph, *_plower, inf_arc_map, neg_csup);
[656]1196            circ_result = circ.run();
1197          }
1198        } else {
1199          if (_pupper) {
1200            Circulation<RevGraph, ConstArcMap, FlowArcMap, NegNodeMap>
[660]1201              circ(rgraph, zero_arc_map, *_pupper, neg_csup);
[656]1202            circ_result = circ.run();
1203          } else {
1204            Circulation<RevGraph, ConstArcMap, ConstArcMap, NegNodeMap>
[660]1205              circ(rgraph, zero_arc_map, inf_arc_map, neg_csup);
[656]1206            circ_result = circ.run();
1207          }
1208        }
1209      }
1210      if (local_csup) delete csup;
1211      if (!circ_result) return false;
1212
[648]1213      // Set data for the artificial root node
1214      _root = _node_num;
1215      _parent[_root] = -1;
1216      _pred[_root] = -1;
1217      _thread[_root] = 0;
[651]1218      _rev_thread[0] = _root;
1219      _succ_num[_root] = all_node_num;
1220      _last_succ[_root] = _root - 1;
[656]1221      _supply[_root] = -sum_supply;
1222      if (sum_supply < 0) {
1223        _pi[_root] = -art_cost;
1224      } else {
1225        _pi[_root] = art_cost;
1226      }
[648]1227
1228      // Store the arcs in a mixed order
1229      int k = std::max(int(sqrt(_arc_num)), 10);
1230      int i = 0;
[650]1231      for (ArcIt e(_graph); e != INVALID; ++e) {
1232        _arc_ref[i] = e;
[648]1233        if ((i += k) >= _arc_num) i = (i % k) + 1;
1234      }
1235
1236      // Initialize arc maps
[652]1237      if (_pupper && _pcost) {
1238        for (int i = 0; i != _arc_num; ++i) {
1239          Arc e = _arc_ref[i];
1240          _source[i] = _node_id[_graph.source(e)];
1241          _target[i] = _node_id[_graph.target(e)];
1242          _cap[i] = (*_pupper)[e];
1243          _cost[i] = (*_pcost)[e];
[653]1244          _flow[i] = 0;
1245          _state[i] = STATE_LOWER;
[652]1246        }
1247      } else {
1248        for (int i = 0; i != _arc_num; ++i) {
1249          Arc e = _arc_ref[i];
1250          _source[i] = _node_id[_graph.source(e)];
1251          _target[i] = _node_id[_graph.target(e)];
[653]1252          _flow[i] = 0;
1253          _state[i] = STATE_LOWER;
[652]1254        }
1255        if (_pupper) {
1256          for (int i = 0; i != _arc_num; ++i)
1257            _cap[i] = (*_pupper)[_arc_ref[i]];
1258        } else {
1259          for (int i = 0; i != _arc_num; ++i)
[655]1260            _cap[i] = inf_cap;
[652]1261        }
1262        if (_pcost) {
1263          for (int i = 0; i != _arc_num; ++i)
1264            _cost[i] = (*_pcost)[_arc_ref[i]];
1265        } else {
1266          for (int i = 0; i != _arc_num; ++i)
1267            _cost[i] = 1;
1268        }
[648]1269      }
[655]1270     
[648]1271      // Remove non-zero lower bounds
[652]1272      if (_plower) {
[648]1273        for (int i = 0; i != _arc_num; ++i) {
[654]1274          Flow c = (*_plower)[_arc_ref[i]];
[648]1275          if (c != 0) {
1276            _cap[i] -= c;
1277            _supply[_source[i]] -= c;
1278            _supply[_target[i]] += c;
1279          }
1280        }
1281      }
1282
1283      // Add artificial arcs and initialize the spanning tree data structure
1284      for (int u = 0, e = _arc_num; u != _node_num; ++u, ++e) {
1285        _thread[u] = u + 1;
[651]1286        _rev_thread[u + 1] = u;
1287        _succ_num[u] = 1;
1288        _last_succ[u] = u;
[648]1289        _parent[u] = _root;
1290        _pred[u] = e;
[655]1291        _cost[e] = art_cost;
1292        _cap[e] = inf_cap;
[653]1293        _state[e] = STATE_TREE;
[656]1294        if (_supply[u] > 0 || (_supply[u] == 0 && sum_supply <= 0)) {
[648]1295          _flow[e] = _supply[u];
1296          _forward[u] = true;
[656]1297          _pi[u] = -art_cost + _pi[_root];
[648]1298        } else {
1299          _flow[e] = -_supply[u];
1300          _forward[u] = false;
[656]1301          _pi[u] = art_cost + _pi[_root];
[648]1302        }
1303      }
1304
1305      return true;
1306    }
1307
1308    // Find the join node
1309    void findJoinNode() {
[650]1310      int u = _source[in_arc];
1311      int v = _target[in_arc];
[648]1312      while (u != v) {
[651]1313        if (_succ_num[u] < _succ_num[v]) {
1314          u = _parent[u];
1315        } else {
1316          v = _parent[v];
1317        }
[648]1318      }
1319      join = u;
1320    }
1321
1322    // Find the leaving arc of the cycle and returns true if the
1323    // leaving arc is not the same as the entering arc
1324    bool findLeavingArc() {
1325      // Initialize first and second nodes according to the direction
1326      // of the cycle
[650]1327      if (_state[in_arc] == STATE_LOWER) {
1328        first  = _source[in_arc];
1329        second = _target[in_arc];
[648]1330      } else {
[650]1331        first  = _target[in_arc];
1332        second = _source[in_arc];
[648]1333      }
[650]1334      delta = _cap[in_arc];
[648]1335      int result = 0;
[654]1336      Flow d;
[648]1337      int e;
1338
1339      // Search the cycle along the path form the first node to the root
1340      for (int u = first; u != join; u = _parent[u]) {
1341        e = _pred[u];
1342        d = _forward[u] ? _flow[e] : _cap[e] - _flow[e];
1343        if (d < delta) {
1344          delta = d;
1345          u_out = u;
1346          result = 1;
1347        }
1348      }
1349      // Search the cycle along the path form the second node to the root
1350      for (int u = second; u != join; u = _parent[u]) {
1351        e = _pred[u];
1352        d = _forward[u] ? _cap[e] - _flow[e] : _flow[e];
1353        if (d <= delta) {
1354          delta = d;
1355          u_out = u;
1356          result = 2;
1357        }
1358      }
1359
1360      if (result == 1) {
1361        u_in = first;
1362        v_in = second;
1363      } else {
1364        u_in = second;
1365        v_in = first;
1366      }
1367      return result != 0;
1368    }
1369
1370    // Change _flow and _state vectors
1371    void changeFlow(bool change) {
1372      // Augment along the cycle
1373      if (delta > 0) {
[654]1374        Flow val = _state[in_arc] * delta;
[650]1375        _flow[in_arc] += val;
1376        for (int u = _source[in_arc]; u != join; u = _parent[u]) {
[648]1377          _flow[_pred[u]] += _forward[u] ? -val : val;
1378        }
[650]1379        for (int u = _target[in_arc]; u != join; u = _parent[u]) {
[648]1380          _flow[_pred[u]] += _forward[u] ? val : -val;
1381        }
1382      }
1383      // Update the state of the entering and leaving arcs
1384      if (change) {
[650]1385        _state[in_arc] = STATE_TREE;
[648]1386        _state[_pred[u_out]] =
1387          (_flow[_pred[u_out]] == 0) ? STATE_LOWER : STATE_UPPER;
1388      } else {
[650]1389        _state[in_arc] = -_state[in_arc];
[648]1390      }
1391    }
1392
[651]1393    // Update the tree structure
1394    void updateTreeStructure() {
1395      int u, w;
1396      int old_rev_thread = _rev_thread[u_out];
1397      int old_succ_num = _succ_num[u_out];
1398      int old_last_succ = _last_succ[u_out];
[648]1399      v_out = _parent[u_out];
1400
[651]1401      u = _last_succ[u_in];  // the last successor of u_in
1402      right = _thread[u];    // the node after it
1403
1404      // Handle the case when old_rev_thread equals to v_in
1405      // (it also means that join and v_out coincide)
1406      if (old_rev_thread == v_in) {
1407        last = _thread[_last_succ[u_out]];
1408      } else {
1409        last = _thread[v_in];
[648]1410      }
1411
[651]1412      // Update _thread and _parent along the stem nodes (i.e. the nodes
1413      // between u_in and u_out, whose parent have to be changed)
[648]1414      _thread[v_in] = stem = u_in;
[651]1415      _dirty_revs.clear();
1416      _dirty_revs.push_back(v_in);
[648]1417      par_stem = v_in;
1418      while (stem != u_out) {
[651]1419        // Insert the next stem node into the thread list
1420        new_stem = _parent[stem];
1421        _thread[u] = new_stem;
1422        _dirty_revs.push_back(u);
[648]1423
[651]1424        // Remove the subtree of stem from the thread list
1425        w = _rev_thread[stem];
1426        _thread[w] = right;
1427        _rev_thread[right] = w;
[648]1428
[651]1429        // Change the parent node and shift stem nodes
[648]1430        _parent[stem] = par_stem;
1431        par_stem = stem;
1432        stem = new_stem;
1433
[651]1434        // Update u and right
1435        u = _last_succ[stem] == _last_succ[par_stem] ?
1436          _rev_thread[par_stem] : _last_succ[stem];
[648]1437        right = _thread[u];
1438      }
1439      _parent[u_out] = par_stem;
1440      _thread[u] = last;
[651]1441      _rev_thread[last] = u;
1442      _last_succ[u_out] = u;
[648]1443
[651]1444      // Remove the subtree of u_out from the thread list except for
1445      // the case when old_rev_thread equals to v_in
1446      // (it also means that join and v_out coincide)
1447      if (old_rev_thread != v_in) {
1448        _thread[old_rev_thread] = right;
1449        _rev_thread[right] = old_rev_thread;
1450      }
1451
1452      // Update _rev_thread using the new _thread values
1453      for (int i = 0; i < int(_dirty_revs.size()); ++i) {
1454        u = _dirty_revs[i];
1455        _rev_thread[_thread[u]] = u;
1456      }
1457
1458      // Update _pred, _forward, _last_succ and _succ_num for the
1459      // stem nodes from u_out to u_in
1460      int tmp_sc = 0, tmp_ls = _last_succ[u_out];
1461      u = u_out;
1462      while (u != u_in) {
1463        w = _parent[u];
1464        _pred[u] = _pred[w];
1465        _forward[u] = !_forward[w];
1466        tmp_sc += _succ_num[u] - _succ_num[w];
1467        _succ_num[u] = tmp_sc;
1468        _last_succ[w] = tmp_ls;
1469        u = w;
1470      }
1471      _pred[u_in] = in_arc;
1472      _forward[u_in] = (u_in == _source[in_arc]);
1473      _succ_num[u_in] = old_succ_num;
1474
1475      // Set limits for updating _last_succ form v_in and v_out
1476      // towards the root
1477      int up_limit_in = -1;
1478      int up_limit_out = -1;
1479      if (_last_succ[join] == v_in) {
1480        up_limit_out = join;
[648]1481      } else {
[651]1482        up_limit_in = join;
1483      }
1484
1485      // Update _last_succ from v_in towards the root
1486      for (u = v_in; u != up_limit_in && _last_succ[u] == v_in;
1487           u = _parent[u]) {
1488        _last_succ[u] = _last_succ[u_out];
1489      }
1490      // Update _last_succ from v_out towards the root
1491      if (join != old_rev_thread && v_in != old_rev_thread) {
1492        for (u = v_out; u != up_limit_out && _last_succ[u] == old_last_succ;
1493             u = _parent[u]) {
1494          _last_succ[u] = old_rev_thread;
1495        }
1496      } else {
1497        for (u = v_out; u != up_limit_out && _last_succ[u] == old_last_succ;
1498             u = _parent[u]) {
1499          _last_succ[u] = _last_succ[u_out];
1500        }
1501      }
1502
1503      // Update _succ_num from v_in to join
1504      for (u = v_in; u != join; u = _parent[u]) {
1505        _succ_num[u] += old_succ_num;
1506      }
1507      // Update _succ_num from v_out to join
1508      for (u = v_out; u != join; u = _parent[u]) {
1509        _succ_num[u] -= old_succ_num;
[648]1510      }
1511    }
1512
[651]1513    // Update potentials
1514    void updatePotential() {
[654]1515      Cost sigma = _forward[u_in] ?
[648]1516        _pi[v_in] - _pi[u_in] - _cost[_pred[u_in]] :
1517        _pi[v_in] - _pi[u_in] + _cost[_pred[u_in]];
[655]1518      // Update potentials in the subtree, which has been moved
1519      int end = _thread[_last_succ[u_in]];
1520      for (int u = u_in; u != end; u = _thread[u]) {
1521        _pi[u] += sigma;
[648]1522      }
1523    }
1524
1525    // Execute the algorithm
[652]1526    bool start(PivotRule pivot_rule) {
[648]1527      // Select the pivot rule implementation
1528      switch (pivot_rule) {
[652]1529        case FIRST_ELIGIBLE:
[648]1530          return start<FirstEligiblePivotRule>();
[652]1531        case BEST_ELIGIBLE:
[648]1532          return start<BestEligiblePivotRule>();
[652]1533        case BLOCK_SEARCH:
[648]1534          return start<BlockSearchPivotRule>();
[652]1535        case CANDIDATE_LIST:
[648]1536          return start<CandidateListPivotRule>();
[652]1537        case ALTERING_LIST:
[648]1538          return start<AlteringListPivotRule>();
1539      }
1540      return false;
1541    }
1542
[652]1543    template <typename PivotRuleImpl>
[648]1544    bool start() {
[652]1545      PivotRuleImpl pivot(*this);
[648]1546
[652]1547      // Execute the Network Simplex algorithm
[648]1548      while (pivot.findEnteringArc()) {
1549        findJoinNode();
1550        bool change = findLeavingArc();
1551        changeFlow(change);
1552        if (change) {
[651]1553          updateTreeStructure();
1554          updatePotential();
[648]1555        }
1556      }
1557
[650]1558      // Copy flow values to _flow_map
[652]1559      if (_plower) {
[648]1560        for (int i = 0; i != _arc_num; ++i) {
[650]1561          Arc e = _arc_ref[i];
[652]1562          _flow_map->set(e, (*_plower)[e] + _flow[i]);
[648]1563        }
1564      } else {
1565        for (int i = 0; i != _arc_num; ++i) {
[650]1566          _flow_map->set(_arc_ref[i], _flow[i]);
[648]1567        }
1568      }
[650]1569      // Copy potential values to _potential_map
1570      for (NodeIt n(_graph); n != INVALID; ++n) {
1571        _potential_map->set(n, _pi[_node_id[n]]);
[648]1572      }
1573
1574      return true;
1575    }
1576
1577  }; //class NetworkSimplex
1578
1579  ///@}
1580
1581} //namespace lemon
1582
1583#endif //LEMON_NETWORK_SIMPLEX_H
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