COIN-OR::LEMON - Graph Library

source: lemon/lemon/network_simplex.h @ 911:2914b6f0fde0

Last change on this file since 911:2914b6f0fde0 was 911:2914b6f0fde0, checked in by Alpar Juttner <alpar@…>, 10 years ago

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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_algs
23///
24/// \file
25/// \brief Network Simplex algorithm for finding a minimum cost flow.
26
27#include <vector>
28#include <limits>
29#include <algorithm>
30
31#include <lemon/core.h>
32#include <lemon/math.h>
33
34namespace lemon {
35
36  /// \addtogroup min_cost_flow_algs
37  /// @{
38
39  /// \brief Implementation of the primal Network Simplex algorithm
40  /// for finding a \ref min_cost_flow "minimum cost flow".
41  ///
42  /// \ref NetworkSimplex implements the primal Network Simplex algorithm
43  /// for finding a \ref min_cost_flow "minimum cost flow"
44  /// \ref amo93networkflows, \ref dantzig63linearprog,
45  /// \ref kellyoneill91netsimplex.
46  /// This algorithm is a highly efficient specialized version of the
47  /// linear programming simplex method directly for the minimum cost
48  /// flow problem.
49  ///
50  /// In general, %NetworkSimplex is the fastest implementation available
51  /// in LEMON for this problem.
52  /// Moreover, it supports both directions of the supply/demand inequality
53  /// constraints. For more information, see \ref SupplyType.
54  ///
55  /// Most of the parameters of the problem (except for the digraph)
56  /// can be given using separate functions, and the algorithm can be
57  /// executed using the \ref run() function. If some parameters are not
58  /// specified, then default values will be used.
59  ///
60  /// \tparam GR The digraph type the algorithm runs on.
61  /// \tparam V The number type used for flow amounts, capacity bounds
62  /// and supply values in the algorithm. By default, it is \c int.
63  /// \tparam C The number type used for costs and potentials in the
64  /// algorithm. By default, it is the same as \c V.
65  ///
66  /// \warning Both number types must be signed and all input data must
67  /// be integer.
68  ///
69  /// \note %NetworkSimplex provides five different pivot rule
70  /// implementations, from which the most efficient one is used
71  /// by default. For more information, see \ref PivotRule.
72  template <typename GR, typename V = int, typename C = V>
73  class NetworkSimplex
74  {
75  public:
76
77    /// The type of the flow amounts, capacity bounds and supply values
78    typedef V Value;
79    /// The type of the arc costs
80    typedef C Cost;
81
82  public:
83
84    /// \brief Problem type constants for the \c run() function.
85    ///
86    /// Enum type containing the problem type constants that can be
87    /// returned by the \ref run() function of the algorithm.
88    enum ProblemType {
89      /// The problem has no feasible solution (flow).
90      INFEASIBLE,
91      /// The problem has optimal solution (i.e. it is feasible and
92      /// bounded), and the algorithm has found optimal flow and node
93      /// potentials (primal and dual solutions).
94      OPTIMAL,
95      /// The objective function of the problem is unbounded, i.e.
96      /// there is a directed cycle having negative total cost and
97      /// infinite upper bound.
98      UNBOUNDED
99    };
100   
101    /// \brief Constants for selecting the type of the supply constraints.
102    ///
103    /// Enum type containing constants for selecting the supply type,
104    /// i.e. the direction of the inequalities in the supply/demand
105    /// constraints of the \ref min_cost_flow "minimum cost flow problem".
106    ///
107    /// The default supply type is \c GEQ, the \c LEQ type can be
108    /// selected using \ref supplyType().
109    /// The equality form is a special case of both supply types.
110    enum SupplyType {
111      /// This option means that there are <em>"greater or equal"</em>
112      /// supply/demand constraints in the definition of the problem.
113      GEQ,
114      /// This option means that there are <em>"less or equal"</em>
115      /// supply/demand constraints in the definition of the problem.
116      LEQ
117    };
118   
119    /// \brief Constants for selecting the pivot rule.
120    ///
121    /// Enum type containing constants for selecting the pivot rule for
122    /// the \ref run() function.
123    ///
124    /// \ref NetworkSimplex provides five different pivot rule
125    /// implementations that significantly affect the running time
126    /// of the algorithm.
127    /// By default, \ref BLOCK_SEARCH "Block Search" is used, which
128    /// proved to be the most efficient and the most robust on various
129    /// test inputs.
130    /// However, another pivot rule can be selected using the \ref run()
131    /// function with the proper parameter.
132    enum PivotRule {
133
134      /// The \e First \e Eligible pivot rule.
135      /// The next eligible arc is selected in a wraparound fashion
136      /// in every iteration.
137      FIRST_ELIGIBLE,
138
139      /// The \e Best \e Eligible pivot rule.
140      /// The best eligible arc is selected in every iteration.
141      BEST_ELIGIBLE,
142
143      /// The \e Block \e Search pivot rule.
144      /// A specified number of arcs are examined in every iteration
145      /// in a wraparound fashion and the best eligible arc is selected
146      /// from this block.
147      BLOCK_SEARCH,
148
149      /// The \e Candidate \e List pivot rule.
150      /// In a major iteration a candidate list is built from eligible arcs
151      /// in a wraparound fashion and in the following minor iterations
152      /// the best eligible arc is selected from this list.
153      CANDIDATE_LIST,
154
155      /// The \e Altering \e Candidate \e List pivot rule.
156      /// It is a modified version of the Candidate List method.
157      /// It keeps only the several best eligible arcs from the former
158      /// candidate list and extends this list in every iteration.
159      ALTERING_LIST
160    };
161   
162  private:
163
164    TEMPLATE_DIGRAPH_TYPEDEFS(GR);
165
166    typedef std::vector<int> IntVector;
167    typedef std::vector<Value> ValueVector;
168    typedef std::vector<Cost> CostVector;
169    typedef std::vector<char> BoolVector;
170    // Note: vector<char> is used instead of vector<bool> for efficiency reasons
171
172    // State constants for arcs
173    enum ArcStateEnum {
174      STATE_UPPER = -1,
175      STATE_TREE  =  0,
176      STATE_LOWER =  1
177    };
178
179  private:
180
181    // Data related to the underlying digraph
182    const GR &_graph;
183    int _node_num;
184    int _arc_num;
185    int _all_arc_num;
186    int _search_arc_num;
187
188    // Parameters of the problem
189    bool _have_lower;
190    SupplyType _stype;
191    Value _sum_supply;
192
193    // Data structures for storing the digraph
194    IntNodeMap _node_id;
195    IntArcMap _arc_id;
196    IntVector _source;
197    IntVector _target;
198    bool _arc_mixing;
199
200    // Node and arc data
201    ValueVector _lower;
202    ValueVector _upper;
203    ValueVector _cap;
204    CostVector _cost;
205    ValueVector _supply;
206    ValueVector _flow;
207    CostVector _pi;
208
209    // Data for storing the spanning tree structure
210    IntVector _parent;
211    IntVector _pred;
212    IntVector _thread;
213    IntVector _rev_thread;
214    IntVector _succ_num;
215    IntVector _last_succ;
216    IntVector _dirty_revs;
217    BoolVector _forward;
218    BoolVector _state;
219    int _root;
220
221    // Temporary data used in the current pivot iteration
222    int in_arc, join, u_in, v_in, u_out, v_out;
223    int first, second, right, last;
224    int stem, par_stem, new_stem;
225    Value delta;
226   
227    const Value MAX;
228
229  public:
230 
231    /// \brief Constant for infinite upper bounds (capacities).
232    ///
233    /// Constant for infinite upper bounds (capacities).
234    /// It is \c std::numeric_limits<Value>::infinity() if available,
235    /// \c std::numeric_limits<Value>::max() otherwise.
236    const Value INF;
237
238  private:
239
240    // Implementation of the First Eligible pivot rule
241    class FirstEligiblePivotRule
242    {
243    private:
244
245      // References to the NetworkSimplex class
246      const IntVector  &_source;
247      const IntVector  &_target;
248      const CostVector &_cost;
249      const BoolVector &_state;
250      const CostVector &_pi;
251      int &_in_arc;
252      int _search_arc_num;
253
254      // Pivot rule data
255      int _next_arc;
256
257    public:
258
259      // Constructor
260      FirstEligiblePivotRule(NetworkSimplex &ns) :
261        _source(ns._source), _target(ns._target),
262        _cost(ns._cost), _state(ns._state), _pi(ns._pi),
263        _in_arc(ns.in_arc), _search_arc_num(ns._search_arc_num),
264        _next_arc(0)
265      {}
266
267      // Find next entering arc
268      bool findEnteringArc() {
269        Cost c;
270        for (int e = _next_arc; e != _search_arc_num; ++e) {
271          c = _state[e] * (_cost[e] + _pi[_source[e]] - _pi[_target[e]]);
272          if (c < 0) {
273            _in_arc = e;
274            _next_arc = e + 1;
275            return true;
276          }
277        }
278        for (int e = 0; e != _next_arc; ++e) {
279          c = _state[e] * (_cost[e] + _pi[_source[e]] - _pi[_target[e]]);
280          if (c < 0) {
281            _in_arc = e;
282            _next_arc = e + 1;
283            return true;
284          }
285        }
286        return false;
287      }
288
289    }; //class FirstEligiblePivotRule
290
291
292    // Implementation of the Best Eligible pivot rule
293    class BestEligiblePivotRule
294    {
295    private:
296
297      // References to the NetworkSimplex class
298      const IntVector  &_source;
299      const IntVector  &_target;
300      const CostVector &_cost;
301      const BoolVector &_state;
302      const CostVector &_pi;
303      int &_in_arc;
304      int _search_arc_num;
305
306    public:
307
308      // Constructor
309      BestEligiblePivotRule(NetworkSimplex &ns) :
310        _source(ns._source), _target(ns._target),
311        _cost(ns._cost), _state(ns._state), _pi(ns._pi),
312        _in_arc(ns.in_arc), _search_arc_num(ns._search_arc_num)
313      {}
314
315      // Find next entering arc
316      bool findEnteringArc() {
317        Cost c, min = 0;
318        for (int e = 0; e != _search_arc_num; ++e) {
319          c = _state[e] * (_cost[e] + _pi[_source[e]] - _pi[_target[e]]);
320          if (c < min) {
321            min = c;
322            _in_arc = e;
323          }
324        }
325        return min < 0;
326      }
327
328    }; //class BestEligiblePivotRule
329
330
331    // Implementation of the Block Search pivot rule
332    class BlockSearchPivotRule
333    {
334    private:
335
336      // References to the NetworkSimplex class
337      const IntVector  &_source;
338      const IntVector  &_target;
339      const CostVector &_cost;
340      const BoolVector &_state;
341      const CostVector &_pi;
342      int &_in_arc;
343      int _search_arc_num;
344
345      // Pivot rule data
346      int _block_size;
347      int _next_arc;
348
349    public:
350
351      // Constructor
352      BlockSearchPivotRule(NetworkSimplex &ns) :
353        _source(ns._source), _target(ns._target),
354        _cost(ns._cost), _state(ns._state), _pi(ns._pi),
355        _in_arc(ns.in_arc), _search_arc_num(ns._search_arc_num),
356        _next_arc(0)
357      {
358        // The main parameters of the pivot rule
359        const double BLOCK_SIZE_FACTOR = 1.0;
360        const int MIN_BLOCK_SIZE = 10;
361
362        _block_size = std::max( int(BLOCK_SIZE_FACTOR *
363                                    std::sqrt(double(_search_arc_num))),
364                                MIN_BLOCK_SIZE );
365      }
366
367      // Find next entering arc
368      bool findEnteringArc() {
369        Cost c, min = 0;
370        int cnt = _block_size;
371        int e;
372        for (e = _next_arc; e != _search_arc_num; ++e) {
373          c = _state[e] * (_cost[e] + _pi[_source[e]] - _pi[_target[e]]);
374          if (c < min) {
375            min = c;
376            _in_arc = e;
377          }
378          if (--cnt == 0) {
379            if (min < 0) goto search_end;
380            cnt = _block_size;
381          }
382        }
383        for (e = 0; e != _next_arc; ++e) {
384          c = _state[e] * (_cost[e] + _pi[_source[e]] - _pi[_target[e]]);
385          if (c < min) {
386            min = c;
387            _in_arc = e;
388          }
389          if (--cnt == 0) {
390            if (min < 0) goto search_end;
391            cnt = _block_size;
392          }
393        }
394        if (min >= 0) return false;
395
396      search_end:
397        _next_arc = e;
398        return true;
399      }
400
401    }; //class BlockSearchPivotRule
402
403
404    // Implementation of the Candidate List pivot rule
405    class CandidateListPivotRule
406    {
407    private:
408
409      // References to the NetworkSimplex class
410      const IntVector  &_source;
411      const IntVector  &_target;
412      const CostVector &_cost;
413      const BoolVector &_state;
414      const CostVector &_pi;
415      int &_in_arc;
416      int _search_arc_num;
417
418      // Pivot rule data
419      IntVector _candidates;
420      int _list_length, _minor_limit;
421      int _curr_length, _minor_count;
422      int _next_arc;
423
424    public:
425
426      /// Constructor
427      CandidateListPivotRule(NetworkSimplex &ns) :
428        _source(ns._source), _target(ns._target),
429        _cost(ns._cost), _state(ns._state), _pi(ns._pi),
430        _in_arc(ns.in_arc), _search_arc_num(ns._search_arc_num),
431        _next_arc(0)
432      {
433        // The main parameters of the pivot rule
434        const double LIST_LENGTH_FACTOR = 0.25;
435        const int MIN_LIST_LENGTH = 10;
436        const double MINOR_LIMIT_FACTOR = 0.1;
437        const int MIN_MINOR_LIMIT = 3;
438
439        _list_length = std::max( int(LIST_LENGTH_FACTOR *
440                                     std::sqrt(double(_search_arc_num))),
441                                 MIN_LIST_LENGTH );
442        _minor_limit = std::max( int(MINOR_LIMIT_FACTOR * _list_length),
443                                 MIN_MINOR_LIMIT );
444        _curr_length = _minor_count = 0;
445        _candidates.resize(_list_length);
446      }
447
448      /// Find next entering arc
449      bool findEnteringArc() {
450        Cost min, c;
451        int e;
452        if (_curr_length > 0 && _minor_count < _minor_limit) {
453          // Minor iteration: select the best eligible arc from the
454          // current candidate list
455          ++_minor_count;
456          min = 0;
457          for (int i = 0; i < _curr_length; ++i) {
458            e = _candidates[i];
459            c = _state[e] * (_cost[e] + _pi[_source[e]] - _pi[_target[e]]);
460            if (c < min) {
461              min = c;
462              _in_arc = e;
463            }
464            else if (c >= 0) {
465              _candidates[i--] = _candidates[--_curr_length];
466            }
467          }
468          if (min < 0) return true;
469        }
470
471        // Major iteration: build a new candidate list
472        min = 0;
473        _curr_length = 0;
474        for (e = _next_arc; e != _search_arc_num; ++e) {
475          c = _state[e] * (_cost[e] + _pi[_source[e]] - _pi[_target[e]]);
476          if (c < 0) {
477            _candidates[_curr_length++] = e;
478            if (c < min) {
479              min = c;
480              _in_arc = e;
481            }
482            if (_curr_length == _list_length) goto search_end;
483          }
484        }
485        for (e = 0; e != _next_arc; ++e) {
486          c = _state[e] * (_cost[e] + _pi[_source[e]] - _pi[_target[e]]);
487          if (c < 0) {
488            _candidates[_curr_length++] = e;
489            if (c < min) {
490              min = c;
491              _in_arc = e;
492            }
493            if (_curr_length == _list_length) goto search_end;
494          }
495        }
496        if (_curr_length == 0) return false;
497     
498      search_end:       
499        _minor_count = 1;
500        _next_arc = e;
501        return true;
502      }
503
504    }; //class CandidateListPivotRule
505
506
507    // Implementation of the Altering Candidate List pivot rule
508    class AlteringListPivotRule
509    {
510    private:
511
512      // References to the NetworkSimplex class
513      const IntVector  &_source;
514      const IntVector  &_target;
515      const CostVector &_cost;
516      const BoolVector &_state;
517      const CostVector &_pi;
518      int &_in_arc;
519      int _search_arc_num;
520
521      // Pivot rule data
522      int _block_size, _head_length, _curr_length;
523      int _next_arc;
524      IntVector _candidates;
525      CostVector _cand_cost;
526
527      // Functor class to compare arcs during sort of the candidate list
528      class SortFunc
529      {
530      private:
531        const CostVector &_map;
532      public:
533        SortFunc(const CostVector &map) : _map(map) {}
534        bool operator()(int left, int right) {
535          return _map[left] > _map[right];
536        }
537      };
538
539      SortFunc _sort_func;
540
541    public:
542
543      // Constructor
544      AlteringListPivotRule(NetworkSimplex &ns) :
545        _source(ns._source), _target(ns._target),
546        _cost(ns._cost), _state(ns._state), _pi(ns._pi),
547        _in_arc(ns.in_arc), _search_arc_num(ns._search_arc_num),
548        _next_arc(0), _cand_cost(ns._search_arc_num), _sort_func(_cand_cost)
549      {
550        // The main parameters of the pivot rule
551        const double BLOCK_SIZE_FACTOR = 1.0;
552        const int MIN_BLOCK_SIZE = 10;
553        const double HEAD_LENGTH_FACTOR = 0.1;
554        const int MIN_HEAD_LENGTH = 3;
555
556        _block_size = std::max( int(BLOCK_SIZE_FACTOR *
557                                    std::sqrt(double(_search_arc_num))),
558                                MIN_BLOCK_SIZE );
559        _head_length = std::max( int(HEAD_LENGTH_FACTOR * _block_size),
560                                 MIN_HEAD_LENGTH );
561        _candidates.resize(_head_length + _block_size);
562        _curr_length = 0;
563      }
564
565      // Find next entering arc
566      bool findEnteringArc() {
567        // Check the current candidate list
568        int e;
569        for (int i = 0; i != _curr_length; ++i) {
570          e = _candidates[i];
571          _cand_cost[e] = _state[e] *
572            (_cost[e] + _pi[_source[e]] - _pi[_target[e]]);
573          if (_cand_cost[e] >= 0) {
574            _candidates[i--] = _candidates[--_curr_length];
575          }
576        }
577
578        // Extend the list
579        int cnt = _block_size;
580        int limit = _head_length;
581
582        for (e = _next_arc; e != _search_arc_num; ++e) {
583          _cand_cost[e] = _state[e] *
584            (_cost[e] + _pi[_source[e]] - _pi[_target[e]]);
585          if (_cand_cost[e] < 0) {
586            _candidates[_curr_length++] = e;
587          }
588          if (--cnt == 0) {
589            if (_curr_length > limit) goto search_end;
590            limit = 0;
591            cnt = _block_size;
592          }
593        }
594        for (e = 0; e != _next_arc; ++e) {
595          _cand_cost[e] = _state[e] *
596            (_cost[e] + _pi[_source[e]] - _pi[_target[e]]);
597          if (_cand_cost[e] < 0) {
598            _candidates[_curr_length++] = e;
599          }
600          if (--cnt == 0) {
601            if (_curr_length > limit) goto search_end;
602            limit = 0;
603            cnt = _block_size;
604          }
605        }
606        if (_curr_length == 0) return false;
607       
608      search_end:
609
610        // Make heap of the candidate list (approximating a partial sort)
611        make_heap( _candidates.begin(), _candidates.begin() + _curr_length,
612                   _sort_func );
613
614        // Pop the first element of the heap
615        _in_arc = _candidates[0];
616        _next_arc = e;
617        pop_heap( _candidates.begin(), _candidates.begin() + _curr_length,
618                  _sort_func );
619        _curr_length = std::min(_head_length, _curr_length - 1);
620        return true;
621      }
622
623    }; //class AlteringListPivotRule
624
625  public:
626
627    /// \brief Constructor.
628    ///
629    /// The constructor of the class.
630    ///
631    /// \param graph The digraph the algorithm runs on.
632    /// \param arc_mixing Indicate if the arcs have to be stored in a
633    /// mixed order in the internal data structure.
634    /// In special cases, it could lead to better overall performance,
635    /// but it is usually slower. Therefore it is disabled by default.
636    NetworkSimplex(const GR& graph, bool arc_mixing = false) :
637      _graph(graph), _node_id(graph), _arc_id(graph),
638      _arc_mixing(arc_mixing),
639      MAX(std::numeric_limits<Value>::max()),
640      INF(std::numeric_limits<Value>::has_infinity ?
641          std::numeric_limits<Value>::infinity() : MAX)
642    {
643      // Check the number types
644      LEMON_ASSERT(std::numeric_limits<Value>::is_signed,
645        "The flow type of NetworkSimplex must be signed");
646      LEMON_ASSERT(std::numeric_limits<Cost>::is_signed,
647        "The cost type of NetworkSimplex must be signed");
648       
649      // Reset data structures
650      reset();
651    }
652
653    /// \name Parameters
654    /// The parameters of the algorithm can be specified using these
655    /// functions.
656
657    /// @{
658
659    /// \brief Set the lower bounds on the arcs.
660    ///
661    /// This function sets the lower bounds on the arcs.
662    /// If it is not used before calling \ref run(), the lower bounds
663    /// will be set to zero on all arcs.
664    ///
665    /// \param map An arc map storing the lower bounds.
666    /// Its \c Value type must be convertible to the \c Value type
667    /// of the algorithm.
668    ///
669    /// \return <tt>(*this)</tt>
670    template <typename LowerMap>
671    NetworkSimplex& lowerMap(const LowerMap& map) {
672      _have_lower = true;
673      for (ArcIt a(_graph); a != INVALID; ++a) {
674        _lower[_arc_id[a]] = map[a];
675      }
676      return *this;
677    }
678
679    /// \brief Set the upper bounds (capacities) on the arcs.
680    ///
681    /// This function sets the upper bounds (capacities) on the arcs.
682    /// If it is not used before calling \ref run(), the upper bounds
683    /// will be set to \ref INF on all arcs (i.e. the flow value will be
684    /// unbounded from above).
685    ///
686    /// \param map An arc map storing the upper bounds.
687    /// Its \c Value type must be convertible to the \c Value type
688    /// of the algorithm.
689    ///
690    /// \return <tt>(*this)</tt>
691    template<typename UpperMap>
692    NetworkSimplex& upperMap(const UpperMap& map) {
693      for (ArcIt a(_graph); a != INVALID; ++a) {
694        _upper[_arc_id[a]] = map[a];
695      }
696      return *this;
697    }
698
699    /// \brief Set the costs of the arcs.
700    ///
701    /// This function sets the costs of the arcs.
702    /// If it is not used before calling \ref run(), the costs
703    /// will be set to \c 1 on all arcs.
704    ///
705    /// \param map An arc map storing the costs.
706    /// Its \c Value type must be convertible to the \c Cost type
707    /// of the algorithm.
708    ///
709    /// \return <tt>(*this)</tt>
710    template<typename CostMap>
711    NetworkSimplex& costMap(const CostMap& map) {
712      for (ArcIt a(_graph); a != INVALID; ++a) {
713        _cost[_arc_id[a]] = map[a];
714      }
715      return *this;
716    }
717
718    /// \brief Set the supply values of the nodes.
719    ///
720    /// This function sets the supply values of the nodes.
721    /// If neither this function nor \ref stSupply() is used before
722    /// calling \ref run(), the supply of each node will be set to zero.
723    ///
724    /// \param map A node map storing the supply values.
725    /// Its \c Value type must be convertible to the \c Value type
726    /// of the algorithm.
727    ///
728    /// \return <tt>(*this)</tt>
729    template<typename SupplyMap>
730    NetworkSimplex& supplyMap(const SupplyMap& map) {
731      for (NodeIt n(_graph); n != INVALID; ++n) {
732        _supply[_node_id[n]] = map[n];
733      }
734      return *this;
735    }
736
737    /// \brief Set single source and target nodes and a supply value.
738    ///
739    /// This function sets a single source node and a single target node
740    /// and the required flow value.
741    /// If neither this function nor \ref supplyMap() is used before
742    /// calling \ref run(), the supply of each node will be set to zero.
743    ///
744    /// Using this function has the same effect as using \ref supplyMap()
745    /// with such a map in which \c k is assigned to \c s, \c -k is
746    /// assigned to \c t and all other nodes have zero supply value.
747    ///
748    /// \param s The source node.
749    /// \param t The target node.
750    /// \param k The required amount of flow from node \c s to node \c t
751    /// (i.e. the supply of \c s and the demand of \c t).
752    ///
753    /// \return <tt>(*this)</tt>
754    NetworkSimplex& stSupply(const Node& s, const Node& t, Value k) {
755      for (int i = 0; i != _node_num; ++i) {
756        _supply[i] = 0;
757      }
758      _supply[_node_id[s]] =  k;
759      _supply[_node_id[t]] = -k;
760      return *this;
761    }
762   
763    /// \brief Set the type of the supply constraints.
764    ///
765    /// This function sets the type of the supply/demand constraints.
766    /// If it is not used before calling \ref run(), the \ref GEQ supply
767    /// type will be used.
768    ///
769    /// For more information, see \ref SupplyType.
770    ///
771    /// \return <tt>(*this)</tt>
772    NetworkSimplex& supplyType(SupplyType supply_type) {
773      _stype = supply_type;
774      return *this;
775    }
776
777    /// @}
778
779    /// \name Execution Control
780    /// The algorithm can be executed using \ref run().
781
782    /// @{
783
784    /// \brief Run the algorithm.
785    ///
786    /// This function runs the algorithm.
787    /// The paramters can be specified using functions \ref lowerMap(),
788    /// \ref upperMap(), \ref costMap(), \ref supplyMap(), \ref stSupply(),
789    /// \ref supplyType().
790    /// For example,
791    /// \code
792    ///   NetworkSimplex<ListDigraph> ns(graph);
793    ///   ns.lowerMap(lower).upperMap(upper).costMap(cost)
794    ///     .supplyMap(sup).run();
795    /// \endcode
796    ///
797    /// This function can be called more than once. All the given parameters
798    /// are kept for the next call, unless \ref resetParams() or \ref reset()
799    /// is used, thus only the modified parameters have to be set again.
800    /// If the underlying digraph was also modified after the construction
801    /// of the class (or the last \ref reset() call), then the \ref reset()
802    /// function must be called.
803    ///
804    /// \param pivot_rule The pivot rule that will be used during the
805    /// algorithm. For more information, see \ref PivotRule.
806    ///
807    /// \return \c INFEASIBLE if no feasible flow exists,
808    /// \n \c OPTIMAL if the problem has optimal solution
809    /// (i.e. it is feasible and bounded), and the algorithm has found
810    /// optimal flow and node potentials (primal and dual solutions),
811    /// \n \c UNBOUNDED if the objective function of the problem is
812    /// unbounded, i.e. there is a directed cycle having negative total
813    /// cost and infinite upper bound.
814    ///
815    /// \see ProblemType, PivotRule
816    /// \see resetParams(), reset()
817    ProblemType run(PivotRule pivot_rule = BLOCK_SEARCH) {
818      if (!init()) return INFEASIBLE;
819      return start(pivot_rule);
820    }
821
822    /// \brief Reset all the parameters that have been given before.
823    ///
824    /// This function resets all the paramaters that have been given
825    /// before using functions \ref lowerMap(), \ref upperMap(),
826    /// \ref costMap(), \ref supplyMap(), \ref stSupply(), \ref supplyType().
827    ///
828    /// It is useful for multiple \ref run() calls. Basically, all the given
829    /// parameters are kept for the next \ref run() call, unless
830    /// \ref resetParams() or \ref reset() is used.
831    /// If the underlying digraph was also modified after the construction
832    /// of the class or the last \ref reset() call, then the \ref reset()
833    /// function must be used, otherwise \ref resetParams() is sufficient.
834    ///
835    /// For example,
836    /// \code
837    ///   NetworkSimplex<ListDigraph> ns(graph);
838    ///
839    ///   // First run
840    ///   ns.lowerMap(lower).upperMap(upper).costMap(cost)
841    ///     .supplyMap(sup).run();
842    ///
843    ///   // Run again with modified cost map (resetParams() is not called,
844    ///   // so only the cost map have to be set again)
845    ///   cost[e] += 100;
846    ///   ns.costMap(cost).run();
847    ///
848    ///   // Run again from scratch using resetParams()
849    ///   // (the lower bounds will be set to zero on all arcs)
850    ///   ns.resetParams();
851    ///   ns.upperMap(capacity).costMap(cost)
852    ///     .supplyMap(sup).run();
853    /// \endcode
854    ///
855    /// \return <tt>(*this)</tt>
856    ///
857    /// \see reset(), run()
858    NetworkSimplex& resetParams() {
859      for (int i = 0; i != _node_num; ++i) {
860        _supply[i] = 0;
861      }
862      for (int i = 0; i != _arc_num; ++i) {
863        _lower[i] = 0;
864        _upper[i] = INF;
865        _cost[i] = 1;
866      }
867      _have_lower = false;
868      _stype = GEQ;
869      return *this;
870    }
871
872    /// \brief Reset the internal data structures and all the parameters
873    /// that have been given before.
874    ///
875    /// This function resets the internal data structures and all the
876    /// paramaters that have been given before using functions \ref lowerMap(),
877    /// \ref upperMap(), \ref costMap(), \ref supplyMap(), \ref stSupply(),
878    /// \ref supplyType().
879    ///
880    /// It is useful for multiple \ref run() calls. Basically, all the given
881    /// parameters are kept for the next \ref run() call, unless
882    /// \ref resetParams() or \ref reset() is used.
883    /// If the underlying digraph was also modified after the construction
884    /// of the class or the last \ref reset() call, then the \ref reset()
885    /// function must be used, otherwise \ref resetParams() is sufficient.
886    ///
887    /// See \ref resetParams() for examples.
888    ///
889    /// \return <tt>(*this)</tt>
890    ///
891    /// \see resetParams(), run()
892    NetworkSimplex& reset() {
893      // Resize vectors
894      _node_num = countNodes(_graph);
895      _arc_num = countArcs(_graph);
896      int all_node_num = _node_num + 1;
897      int max_arc_num = _arc_num + 2 * _node_num;
898
899      _source.resize(max_arc_num);
900      _target.resize(max_arc_num);
901
902      _lower.resize(_arc_num);
903      _upper.resize(_arc_num);
904      _cap.resize(max_arc_num);
905      _cost.resize(max_arc_num);
906      _supply.resize(all_node_num);
907      _flow.resize(max_arc_num);
908      _pi.resize(all_node_num);
909
910      _parent.resize(all_node_num);
911      _pred.resize(all_node_num);
912      _forward.resize(all_node_num);
913      _thread.resize(all_node_num);
914      _rev_thread.resize(all_node_num);
915      _succ_num.resize(all_node_num);
916      _last_succ.resize(all_node_num);
917      _state.resize(max_arc_num);
918
919      // Copy the graph
920      int i = 0;
921      for (NodeIt n(_graph); n != INVALID; ++n, ++i) {
922        _node_id[n] = i;
923      }
924      if (_arc_mixing) {
925        // Store the arcs in a mixed order
926        int k = std::max(int(std::sqrt(double(_arc_num))), 10);
927        int i = 0, j = 0;
928        for (ArcIt a(_graph); a != INVALID; ++a) {
929          _arc_id[a] = i;
930          _source[i] = _node_id[_graph.source(a)];
931          _target[i] = _node_id[_graph.target(a)];
932          if ((i += k) >= _arc_num) i = ++j;
933        }
934      } else {
935        // Store the arcs in the original order
936        int i = 0;
937        for (ArcIt a(_graph); a != INVALID; ++a, ++i) {
938          _arc_id[a] = i;
939          _source[i] = _node_id[_graph.source(a)];
940          _target[i] = _node_id[_graph.target(a)];
941        }
942      }
943     
944      // Reset parameters
945      resetParams();
946      return *this;
947    }
948   
949    /// @}
950
951    /// \name Query Functions
952    /// The results of the algorithm can be obtained using these
953    /// functions.\n
954    /// The \ref run() function must be called before using them.
955
956    /// @{
957
958    /// \brief Return the total cost of the found flow.
959    ///
960    /// This function returns the total cost of the found flow.
961    /// Its complexity is O(e).
962    ///
963    /// \note The return type of the function can be specified as a
964    /// template parameter. For example,
965    /// \code
966    ///   ns.totalCost<double>();
967    /// \endcode
968    /// It is useful if the total cost cannot be stored in the \c Cost
969    /// type of the algorithm, which is the default return type of the
970    /// function.
971    ///
972    /// \pre \ref run() must be called before using this function.
973    template <typename Number>
974    Number totalCost() const {
975      Number c = 0;
976      for (ArcIt a(_graph); a != INVALID; ++a) {
977        int i = _arc_id[a];
978        c += Number(_flow[i]) * Number(_cost[i]);
979      }
980      return c;
981    }
982
983#ifndef DOXYGEN
984    Cost totalCost() const {
985      return totalCost<Cost>();
986    }
987#endif
988
989    /// \brief Return the flow on the given arc.
990    ///
991    /// This function returns the flow on the given arc.
992    ///
993    /// \pre \ref run() must be called before using this function.
994    Value flow(const Arc& a) const {
995      return _flow[_arc_id[a]];
996    }
997
998    /// \brief Return the flow map (the primal solution).
999    ///
1000    /// This function copies the flow value on each arc into the given
1001    /// map. The \c Value type of the algorithm must be convertible to
1002    /// the \c Value type of the map.
1003    ///
1004    /// \pre \ref run() must be called before using this function.
1005    template <typename FlowMap>
1006    void flowMap(FlowMap &map) const {
1007      for (ArcIt a(_graph); a != INVALID; ++a) {
1008        map.set(a, _flow[_arc_id[a]]);
1009      }
1010    }
1011
1012    /// \brief Return the potential (dual value) of the given node.
1013    ///
1014    /// This function returns the potential (dual value) of the
1015    /// given node.
1016    ///
1017    /// \pre \ref run() must be called before using this function.
1018    Cost potential(const Node& n) const {
1019      return _pi[_node_id[n]];
1020    }
1021
1022    /// \brief Return the potential map (the dual solution).
1023    ///
1024    /// This function copies the potential (dual value) of each node
1025    /// into the given map.
1026    /// The \c Cost type of the algorithm must be convertible to the
1027    /// \c Value type of the map.
1028    ///
1029    /// \pre \ref run() must be called before using this function.
1030    template <typename PotentialMap>
1031    void potentialMap(PotentialMap &map) const {
1032      for (NodeIt n(_graph); n != INVALID; ++n) {
1033        map.set(n, _pi[_node_id[n]]);
1034      }
1035    }
1036
1037    /// @}
1038
1039  private:
1040
1041    // Initialize internal data structures
1042    bool init() {
1043      if (_node_num == 0) return false;
1044
1045      // Check the sum of supply values
1046      _sum_supply = 0;
1047      for (int i = 0; i != _node_num; ++i) {
1048        _sum_supply += _supply[i];
1049      }
1050      if ( !((_stype == GEQ && _sum_supply <= 0) ||
1051             (_stype == LEQ && _sum_supply >= 0)) ) return false;
1052
1053      // Remove non-zero lower bounds
1054      if (_have_lower) {
1055        for (int i = 0; i != _arc_num; ++i) {
1056          Value c = _lower[i];
1057          if (c >= 0) {
1058            _cap[i] = _upper[i] < MAX ? _upper[i] - c : INF;
1059          } else {
1060            _cap[i] = _upper[i] < MAX + c ? _upper[i] - c : INF;
1061          }
1062          _supply[_source[i]] -= c;
1063          _supply[_target[i]] += c;
1064        }
1065      } else {
1066        for (int i = 0; i != _arc_num; ++i) {
1067          _cap[i] = _upper[i];
1068        }
1069      }
1070
1071      // Initialize artifical cost
1072      Cost ART_COST;
1073      if (std::numeric_limits<Cost>::is_exact) {
1074        ART_COST = std::numeric_limits<Cost>::max() / 2 + 1;
1075      } else {
1076        ART_COST = std::numeric_limits<Cost>::min();
1077        for (int i = 0; i != _arc_num; ++i) {
1078          if (_cost[i] > ART_COST) ART_COST = _cost[i];
1079        }
1080        ART_COST = (ART_COST + 1) * _node_num;
1081      }
1082
1083      // Initialize arc maps
1084      for (int i = 0; i != _arc_num; ++i) {
1085        _flow[i] = 0;
1086        _state[i] = STATE_LOWER;
1087      }
1088     
1089      // Set data for the artificial root node
1090      _root = _node_num;
1091      _parent[_root] = -1;
1092      _pred[_root] = -1;
1093      _thread[_root] = 0;
1094      _rev_thread[0] = _root;
1095      _succ_num[_root] = _node_num + 1;
1096      _last_succ[_root] = _root - 1;
1097      _supply[_root] = -_sum_supply;
1098      _pi[_root] = 0;
1099
1100      // Add artificial arcs and initialize the spanning tree data structure
1101      if (_sum_supply == 0) {
1102        // EQ supply constraints
1103        _search_arc_num = _arc_num;
1104        _all_arc_num = _arc_num + _node_num;
1105        for (int u = 0, e = _arc_num; u != _node_num; ++u, ++e) {
1106          _parent[u] = _root;
1107          _pred[u] = e;
1108          _thread[u] = u + 1;
1109          _rev_thread[u + 1] = u;
1110          _succ_num[u] = 1;
1111          _last_succ[u] = u;
1112          _cap[e] = INF;
1113          _state[e] = STATE_TREE;
1114          if (_supply[u] >= 0) {
1115            _forward[u] = true;
1116            _pi[u] = 0;
1117            _source[e] = u;
1118            _target[e] = _root;
1119            _flow[e] = _supply[u];
1120            _cost[e] = 0;
1121          } else {
1122            _forward[u] = false;
1123            _pi[u] = ART_COST;
1124            _source[e] = _root;
1125            _target[e] = u;
1126            _flow[e] = -_supply[u];
1127            _cost[e] = ART_COST;
1128          }
1129        }
1130      }
1131      else if (_sum_supply > 0) {
1132        // LEQ supply constraints
1133        _search_arc_num = _arc_num + _node_num;
1134        int f = _arc_num + _node_num;
1135        for (int u = 0, e = _arc_num; u != _node_num; ++u, ++e) {
1136          _parent[u] = _root;
1137          _thread[u] = u + 1;
1138          _rev_thread[u + 1] = u;
1139          _succ_num[u] = 1;
1140          _last_succ[u] = u;
1141          if (_supply[u] >= 0) {
1142            _forward[u] = true;
1143            _pi[u] = 0;
1144            _pred[u] = e;
1145            _source[e] = u;
1146            _target[e] = _root;
1147            _cap[e] = INF;
1148            _flow[e] = _supply[u];
1149            _cost[e] = 0;
1150            _state[e] = STATE_TREE;
1151          } else {
1152            _forward[u] = false;
1153            _pi[u] = ART_COST;
1154            _pred[u] = f;
1155            _source[f] = _root;
1156            _target[f] = u;
1157            _cap[f] = INF;
1158            _flow[f] = -_supply[u];
1159            _cost[f] = ART_COST;
1160            _state[f] = STATE_TREE;
1161            _source[e] = u;
1162            _target[e] = _root;
1163            _cap[e] = INF;
1164            _flow[e] = 0;
1165            _cost[e] = 0;
1166            _state[e] = STATE_LOWER;
1167            ++f;
1168          }
1169        }
1170        _all_arc_num = f;
1171      }
1172      else {
1173        // GEQ supply constraints
1174        _search_arc_num = _arc_num + _node_num;
1175        int f = _arc_num + _node_num;
1176        for (int u = 0, e = _arc_num; u != _node_num; ++u, ++e) {
1177          _parent[u] = _root;
1178          _thread[u] = u + 1;
1179          _rev_thread[u + 1] = u;
1180          _succ_num[u] = 1;
1181          _last_succ[u] = u;
1182          if (_supply[u] <= 0) {
1183            _forward[u] = false;
1184            _pi[u] = 0;
1185            _pred[u] = e;
1186            _source[e] = _root;
1187            _target[e] = u;
1188            _cap[e] = INF;
1189            _flow[e] = -_supply[u];
1190            _cost[e] = 0;
1191            _state[e] = STATE_TREE;
1192          } else {
1193            _forward[u] = true;
1194            _pi[u] = -ART_COST;
1195            _pred[u] = f;
1196            _source[f] = u;
1197            _target[f] = _root;
1198            _cap[f] = INF;
1199            _flow[f] = _supply[u];
1200            _state[f] = STATE_TREE;
1201            _cost[f] = ART_COST;
1202            _source[e] = _root;
1203            _target[e] = u;
1204            _cap[e] = INF;
1205            _flow[e] = 0;
1206            _cost[e] = 0;
1207            _state[e] = STATE_LOWER;
1208            ++f;
1209          }
1210        }
1211        _all_arc_num = f;
1212      }
1213
1214      return true;
1215    }
1216
1217    // Find the join node
1218    void findJoinNode() {
1219      int u = _source[in_arc];
1220      int v = _target[in_arc];
1221      while (u != v) {
1222        if (_succ_num[u] < _succ_num[v]) {
1223          u = _parent[u];
1224        } else {
1225          v = _parent[v];
1226        }
1227      }
1228      join = u;
1229    }
1230
1231    // Find the leaving arc of the cycle and returns true if the
1232    // leaving arc is not the same as the entering arc
1233    bool findLeavingArc() {
1234      // Initialize first and second nodes according to the direction
1235      // of the cycle
1236      if (_state[in_arc] == STATE_LOWER) {
1237        first  = _source[in_arc];
1238        second = _target[in_arc];
1239      } else {
1240        first  = _target[in_arc];
1241        second = _source[in_arc];
1242      }
1243      delta = _cap[in_arc];
1244      int result = 0;
1245      Value d;
1246      int e;
1247
1248      // Search the cycle along the path form the first node to the root
1249      for (int u = first; u != join; u = _parent[u]) {
1250        e = _pred[u];
1251        d = _forward[u] ?
1252          _flow[e] : (_cap[e] >= MAX ? INF : _cap[e] - _flow[e]);
1253        if (d < delta) {
1254          delta = d;
1255          u_out = u;
1256          result = 1;
1257        }
1258      }
1259      // Search the cycle along the path form the second node to the root
1260      for (int u = second; u != join; u = _parent[u]) {
1261        e = _pred[u];
1262        d = _forward[u] ?
1263          (_cap[e] >= MAX ? INF : _cap[e] - _flow[e]) : _flow[e];
1264        if (d <= delta) {
1265          delta = d;
1266          u_out = u;
1267          result = 2;
1268        }
1269      }
1270
1271      if (result == 1) {
1272        u_in = first;
1273        v_in = second;
1274      } else {
1275        u_in = second;
1276        v_in = first;
1277      }
1278      return result != 0;
1279    }
1280
1281    // Change _flow and _state vectors
1282    void changeFlow(bool change) {
1283      // Augment along the cycle
1284      if (delta > 0) {
1285        Value val = _state[in_arc] * delta;
1286        _flow[in_arc] += val;
1287        for (int u = _source[in_arc]; u != join; u = _parent[u]) {
1288          _flow[_pred[u]] += _forward[u] ? -val : val;
1289        }
1290        for (int u = _target[in_arc]; u != join; u = _parent[u]) {
1291          _flow[_pred[u]] += _forward[u] ? val : -val;
1292        }
1293      }
1294      // Update the state of the entering and leaving arcs
1295      if (change) {
1296        _state[in_arc] = STATE_TREE;
1297        _state[_pred[u_out]] =
1298          (_flow[_pred[u_out]] == 0) ? STATE_LOWER : STATE_UPPER;
1299      } else {
1300        _state[in_arc] = -_state[in_arc];
1301      }
1302    }
1303
1304    // Update the tree structure
1305    void updateTreeStructure() {
1306      int u, w;
1307      int old_rev_thread = _rev_thread[u_out];
1308      int old_succ_num = _succ_num[u_out];
1309      int old_last_succ = _last_succ[u_out];
1310      v_out = _parent[u_out];
1311
1312      u = _last_succ[u_in];  // the last successor of u_in
1313      right = _thread[u];    // the node after it
1314
1315      // Handle the case when old_rev_thread equals to v_in
1316      // (it also means that join and v_out coincide)
1317      if (old_rev_thread == v_in) {
1318        last = _thread[_last_succ[u_out]];
1319      } else {
1320        last = _thread[v_in];
1321      }
1322
1323      // Update _thread and _parent along the stem nodes (i.e. the nodes
1324      // between u_in and u_out, whose parent have to be changed)
1325      _thread[v_in] = stem = u_in;
1326      _dirty_revs.clear();
1327      _dirty_revs.push_back(v_in);
1328      par_stem = v_in;
1329      while (stem != u_out) {
1330        // Insert the next stem node into the thread list
1331        new_stem = _parent[stem];
1332        _thread[u] = new_stem;
1333        _dirty_revs.push_back(u);
1334
1335        // Remove the subtree of stem from the thread list
1336        w = _rev_thread[stem];
1337        _thread[w] = right;
1338        _rev_thread[right] = w;
1339
1340        // Change the parent node and shift stem nodes
1341        _parent[stem] = par_stem;
1342        par_stem = stem;
1343        stem = new_stem;
1344
1345        // Update u and right
1346        u = _last_succ[stem] == _last_succ[par_stem] ?
1347          _rev_thread[par_stem] : _last_succ[stem];
1348        right = _thread[u];
1349      }
1350      _parent[u_out] = par_stem;
1351      _thread[u] = last;
1352      _rev_thread[last] = u;
1353      _last_succ[u_out] = u;
1354
1355      // Remove the subtree of u_out from the thread list except for
1356      // the case when old_rev_thread equals to v_in
1357      // (it also means that join and v_out coincide)
1358      if (old_rev_thread != v_in) {
1359        _thread[old_rev_thread] = right;
1360        _rev_thread[right] = old_rev_thread;
1361      }
1362
1363      // Update _rev_thread using the new _thread values
1364      for (int i = 0; i != int(_dirty_revs.size()); ++i) {
1365        u = _dirty_revs[i];
1366        _rev_thread[_thread[u]] = u;
1367      }
1368
1369      // Update _pred, _forward, _last_succ and _succ_num for the
1370      // stem nodes from u_out to u_in
1371      int tmp_sc = 0, tmp_ls = _last_succ[u_out];
1372      u = u_out;
1373      while (u != u_in) {
1374        w = _parent[u];
1375        _pred[u] = _pred[w];
1376        _forward[u] = !_forward[w];
1377        tmp_sc += _succ_num[u] - _succ_num[w];
1378        _succ_num[u] = tmp_sc;
1379        _last_succ[w] = tmp_ls;
1380        u = w;
1381      }
1382      _pred[u_in] = in_arc;
1383      _forward[u_in] = (u_in == _source[in_arc]);
1384      _succ_num[u_in] = old_succ_num;
1385
1386      // Set limits for updating _last_succ form v_in and v_out
1387      // towards the root
1388      int up_limit_in = -1;
1389      int up_limit_out = -1;
1390      if (_last_succ[join] == v_in) {
1391        up_limit_out = join;
1392      } else {
1393        up_limit_in = join;
1394      }
1395
1396      // Update _last_succ from v_in towards the root
1397      for (u = v_in; u != up_limit_in && _last_succ[u] == v_in;
1398           u = _parent[u]) {
1399        _last_succ[u] = _last_succ[u_out];
1400      }
1401      // Update _last_succ from v_out towards the root
1402      if (join != old_rev_thread && v_in != old_rev_thread) {
1403        for (u = v_out; u != up_limit_out && _last_succ[u] == old_last_succ;
1404             u = _parent[u]) {
1405          _last_succ[u] = old_rev_thread;
1406        }
1407      } else {
1408        for (u = v_out; u != up_limit_out && _last_succ[u] == old_last_succ;
1409             u = _parent[u]) {
1410          _last_succ[u] = _last_succ[u_out];
1411        }
1412      }
1413
1414      // Update _succ_num from v_in to join
1415      for (u = v_in; u != join; u = _parent[u]) {
1416        _succ_num[u] += old_succ_num;
1417      }
1418      // Update _succ_num from v_out to join
1419      for (u = v_out; u != join; u = _parent[u]) {
1420        _succ_num[u] -= old_succ_num;
1421      }
1422    }
1423
1424    // Update potentials
1425    void updatePotential() {
1426      Cost sigma = _forward[u_in] ?
1427        _pi[v_in] - _pi[u_in] - _cost[_pred[u_in]] :
1428        _pi[v_in] - _pi[u_in] + _cost[_pred[u_in]];
1429      // Update potentials in the subtree, which has been moved
1430      int end = _thread[_last_succ[u_in]];
1431      for (int u = u_in; u != end; u = _thread[u]) {
1432        _pi[u] += sigma;
1433      }
1434    }
1435
1436    // Heuristic initial pivots
1437    bool initialPivots() {
1438      Value curr, total = 0;
1439      std::vector<Node> supply_nodes, demand_nodes;
1440      for (NodeIt u(_graph); u != INVALID; ++u) {
1441        curr = _supply[_node_id[u]];
1442        if (curr > 0) {
1443          total += curr;
1444          supply_nodes.push_back(u);
1445        }
1446        else if (curr < 0) {
1447          demand_nodes.push_back(u);
1448        }
1449      }
1450      if (_sum_supply > 0) total -= _sum_supply;
1451      if (total <= 0) return true;
1452
1453      IntVector arc_vector;
1454      if (_sum_supply >= 0) {
1455        if (supply_nodes.size() == 1 && demand_nodes.size() == 1) {
1456          // Perform a reverse graph search from the sink to the source
1457          typename GR::template NodeMap<bool> reached(_graph, false);
1458          Node s = supply_nodes[0], t = demand_nodes[0];
1459          std::vector<Node> stack;
1460          reached[t] = true;
1461          stack.push_back(t);
1462          while (!stack.empty()) {
1463            Node u, v = stack.back();
1464            stack.pop_back();
1465            if (v == s) break;
1466            for (InArcIt a(_graph, v); a != INVALID; ++a) {
1467              if (reached[u = _graph.source(a)]) continue;
1468              int j = _arc_id[a];
1469              if (_cap[j] >= total) {
1470                arc_vector.push_back(j);
1471                reached[u] = true;
1472                stack.push_back(u);
1473              }
1474            }
1475          }
1476        } else {
1477          // Find the min. cost incomming arc for each demand node
1478          for (int i = 0; i != int(demand_nodes.size()); ++i) {
1479            Node v = demand_nodes[i];
1480            Cost c, min_cost = std::numeric_limits<Cost>::max();
1481            Arc min_arc = INVALID;
1482            for (InArcIt a(_graph, v); a != INVALID; ++a) {
1483              c = _cost[_arc_id[a]];
1484              if (c < min_cost) {
1485                min_cost = c;
1486                min_arc = a;
1487              }
1488            }
1489            if (min_arc != INVALID) {
1490              arc_vector.push_back(_arc_id[min_arc]);
1491            }
1492          }
1493        }
1494      } else {
1495        // Find the min. cost outgoing arc for each supply node
1496        for (int i = 0; i != int(supply_nodes.size()); ++i) {
1497          Node u = supply_nodes[i];
1498          Cost c, min_cost = std::numeric_limits<Cost>::max();
1499          Arc min_arc = INVALID;
1500          for (OutArcIt a(_graph, u); a != INVALID; ++a) {
1501            c = _cost[_arc_id[a]];
1502            if (c < min_cost) {
1503              min_cost = c;
1504              min_arc = a;
1505            }
1506          }
1507          if (min_arc != INVALID) {
1508            arc_vector.push_back(_arc_id[min_arc]);
1509          }
1510        }
1511      }
1512
1513      // Perform heuristic initial pivots
1514      for (int i = 0; i != int(arc_vector.size()); ++i) {
1515        in_arc = arc_vector[i];
1516        if (_state[in_arc] * (_cost[in_arc] + _pi[_source[in_arc]] -
1517            _pi[_target[in_arc]]) >= 0) continue;
1518        findJoinNode();
1519        bool change = findLeavingArc();
1520        if (delta >= MAX) return false;
1521        changeFlow(change);
1522        if (change) {
1523          updateTreeStructure();
1524          updatePotential();
1525        }
1526      }
1527      return true;
1528    }
1529
1530    // Execute the algorithm
1531    ProblemType start(PivotRule pivot_rule) {
1532      // Select the pivot rule implementation
1533      switch (pivot_rule) {
1534        case FIRST_ELIGIBLE:
1535          return start<FirstEligiblePivotRule>();
1536        case BEST_ELIGIBLE:
1537          return start<BestEligiblePivotRule>();
1538        case BLOCK_SEARCH:
1539          return start<BlockSearchPivotRule>();
1540        case CANDIDATE_LIST:
1541          return start<CandidateListPivotRule>();
1542        case ALTERING_LIST:
1543          return start<AlteringListPivotRule>();
1544      }
1545      return INFEASIBLE; // avoid warning
1546    }
1547
1548    template <typename PivotRuleImpl>
1549    ProblemType start() {
1550      PivotRuleImpl pivot(*this);
1551
1552      // Perform heuristic initial pivots
1553      if (!initialPivots()) return UNBOUNDED;
1554
1555      // Execute the Network Simplex algorithm
1556      while (pivot.findEnteringArc()) {
1557        findJoinNode();
1558        bool change = findLeavingArc();
1559        if (delta >= MAX) return UNBOUNDED;
1560        changeFlow(change);
1561        if (change) {
1562          updateTreeStructure();
1563          updatePotential();
1564        }
1565      }
1566     
1567      // Check feasibility
1568      for (int e = _search_arc_num; e != _all_arc_num; ++e) {
1569        if (_flow[e] != 0) return INFEASIBLE;
1570      }
1571
1572      // Transform the solution and the supply map to the original form
1573      if (_have_lower) {
1574        for (int i = 0; i != _arc_num; ++i) {
1575          Value c = _lower[i];
1576          if (c != 0) {
1577            _flow[i] += c;
1578            _supply[_source[i]] += c;
1579            _supply[_target[i]] -= c;
1580          }
1581        }
1582      }
1583     
1584      // Shift potentials to meet the requirements of the GEQ/LEQ type
1585      // optimality conditions
1586      if (_sum_supply == 0) {
1587        if (_stype == GEQ) {
1588          Cost max_pot = std::numeric_limits<Cost>::min();
1589          for (int i = 0; i != _node_num; ++i) {
1590            if (_pi[i] > max_pot) max_pot = _pi[i];
1591          }
1592          if (max_pot > 0) {
1593            for (int i = 0; i != _node_num; ++i)
1594              _pi[i] -= max_pot;
1595          }
1596        } else {
1597          Cost min_pot = std::numeric_limits<Cost>::max();
1598          for (int i = 0; i != _node_num; ++i) {
1599            if (_pi[i] < min_pot) min_pot = _pi[i];
1600          }
1601          if (min_pot < 0) {
1602            for (int i = 0; i != _node_num; ++i)
1603              _pi[i] -= min_pot;
1604          }
1605        }
1606      }
1607
1608      return OPTIMAL;
1609    }
1610
1611  }; //class NetworkSimplex
1612
1613  ///@}
1614
1615} //namespace lemon
1616
1617#endif //LEMON_NETWORK_SIMPLEX_H
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