COIN-OR::LEMON - Graph Library

source: lemon-1.2/lemon/network_simplex.h @ 729:be48a648d28f

Last change on this file since 729:be48a648d28f was 729:be48a648d28f, checked in by Peter Kovacs <kpeter@…>, 10 years ago

Small improvements for NetworkSimplex? (#298)

File size: 44.9 KB
Line 
1/* -*- mode: C++; indent-tabs-mode: nil; -*-
2 *
3 * This file is a part of LEMON, a generic C++ optimization library.
4 *
5 * Copyright (C) 2003-2009
6 * Egervary Jeno Kombinatorikus Optimalizalasi Kutatocsoport
7 * (Egervary Research Group on Combinatorial Optimization, EGRES).
8 *
9 * Permission to use, modify and distribute this software is granted
10 * provided that this copyright notice appears in all copies. For
11 * precise terms see the accompanying LICENSE file.
12 *
13 * This software is provided "AS IS" with no warranty of any kind,
14 * express or implied, and with no claim as to its suitability for any
15 * purpose.
16 *
17 */
18
19#ifndef LEMON_NETWORK_SIMPLEX_H
20#define LEMON_NETWORK_SIMPLEX_H
21
22/// \ingroup min_cost_flow_algs
23///
24/// \file
25/// \brief Network Simplex algorithm for finding a minimum cost flow.
26
27#include <vector>
28#include <limits>
29#include <algorithm>
30
31#include <lemon/core.h>
32#include <lemon/math.h>
33
34namespace lemon {
35
36  /// \addtogroup min_cost_flow_algs
37  /// @{
38
39  /// \brief Implementation of the primal Network Simplex algorithm
40  /// for finding a \ref min_cost_flow "minimum cost flow".
41  ///
42  /// \ref NetworkSimplex implements the primal Network Simplex algorithm
43  /// for finding a \ref min_cost_flow "minimum cost flow".
44  /// This algorithm is a specialized version of the linear programming
45  /// simplex method directly for the minimum cost flow problem.
46  /// It is one of the most efficient solution methods.
47  ///
48  /// In general this class is the fastest implementation available
49  /// in LEMON for the minimum cost flow problem.
50  /// Moreover it supports both directions of the supply/demand inequality
51  /// constraints. For more information see \ref SupplyType.
52  ///
53  /// Most of the parameters of the problem (except for the digraph)
54  /// can be given using separate functions, and the algorithm can be
55  /// executed using the \ref run() function. If some parameters are not
56  /// specified, then default values will be used.
57  ///
58  /// \tparam GR The digraph type the algorithm runs on.
59  /// \tparam V The value type used for flow amounts, capacity bounds
60  /// and supply values in the algorithm. By default it is \c int.
61  /// \tparam C The value type used for costs and potentials in the
62  /// algorithm. By default it is the same as \c V.
63  ///
64  /// \warning Both value types must be signed and all input data must
65  /// be integer.
66  ///
67  /// \note %NetworkSimplex provides five different pivot rule
68  /// implementations, from which the most efficient one is used
69  /// by default. For more information see \ref PivotRule.
70  template <typename GR, typename V = int, typename C = V>
71  class NetworkSimplex
72  {
73  public:
74
75    /// The type of the flow amounts, capacity bounds and supply values
76    typedef V Value;
77    /// The type of the arc costs
78    typedef C Cost;
79
80  public:
81
82    /// \brief Problem type constants for the \c run() function.
83    ///
84    /// Enum type containing the problem type constants that can be
85    /// returned by the \ref run() function of the algorithm.
86    enum ProblemType {
87      /// The problem has no feasible solution (flow).
88      INFEASIBLE,
89      /// The problem has optimal solution (i.e. it is feasible and
90      /// bounded), and the algorithm has found optimal flow and node
91      /// potentials (primal and dual solutions).
92      OPTIMAL,
93      /// The objective function of the problem is unbounded, i.e.
94      /// there is a directed cycle having negative total cost and
95      /// infinite upper bound.
96      UNBOUNDED
97    };
98   
99    /// \brief Constants for selecting the type of the supply constraints.
100    ///
101    /// Enum type containing constants for selecting the supply type,
102    /// i.e. the direction of the inequalities in the supply/demand
103    /// constraints of the \ref min_cost_flow "minimum cost flow problem".
104    ///
105    /// The default supply type is \c GEQ, the \c LEQ type can be
106    /// selected using \ref supplyType().
107    /// The equality form is a special case of both supply types.
108    enum SupplyType {
109      /// This option means that there are <em>"greater or equal"</em>
110      /// supply/demand constraints in the definition of the problem.
111      GEQ,
112      /// This option means that there are <em>"less or equal"</em>
113      /// supply/demand constraints in the definition of the problem.
114      LEQ
115    };
116   
117    /// \brief Constants for selecting the pivot rule.
118    ///
119    /// Enum type containing constants for selecting the pivot rule for
120    /// the \ref run() function.
121    ///
122    /// \ref NetworkSimplex provides five different pivot rule
123    /// implementations that significantly affect the running time
124    /// of the algorithm.
125    /// By default \ref BLOCK_SEARCH "Block Search" is used, which
126    /// proved to be the most efficient and the most robust on various
127    /// test inputs according to our benchmark tests.
128    /// However another pivot rule can be selected using the \ref run()
129    /// function with the proper parameter.
130    enum PivotRule {
131
132      /// The First Eligible pivot rule.
133      /// The next eligible arc is selected in a wraparound fashion
134      /// in every iteration.
135      FIRST_ELIGIBLE,
136
137      /// The Best Eligible pivot rule.
138      /// The best eligible arc is selected in every iteration.
139      BEST_ELIGIBLE,
140
141      /// The Block Search pivot rule.
142      /// A specified number of arcs are examined in every iteration
143      /// in a wraparound fashion and the best eligible arc is selected
144      /// from this block.
145      BLOCK_SEARCH,
146
147      /// The Candidate List pivot rule.
148      /// In a major iteration a candidate list is built from eligible arcs
149      /// in a wraparound fashion and in the following minor iterations
150      /// the best eligible arc is selected from this list.
151      CANDIDATE_LIST,
152
153      /// The Altering Candidate List pivot rule.
154      /// It is a modified version of the Candidate List method.
155      /// It keeps only the several best eligible arcs from the former
156      /// candidate list and extends this list in every iteration.
157      ALTERING_LIST
158    };
159   
160  private:
161
162    TEMPLATE_DIGRAPH_TYPEDEFS(GR);
163
164    typedef std::vector<int> IntVector;
165    typedef std::vector<bool> BoolVector;
166    typedef std::vector<Value> ValueVector;
167    typedef std::vector<Cost> CostVector;
168
169    // State constants for arcs
170    enum ArcStateEnum {
171      STATE_UPPER = -1,
172      STATE_TREE  =  0,
173      STATE_LOWER =  1
174    };
175
176  private:
177
178    // Data related to the underlying digraph
179    const GR &_graph;
180    int _node_num;
181    int _arc_num;
182    int _all_arc_num;
183    int _search_arc_num;
184
185    // Parameters of the problem
186    bool _have_lower;
187    SupplyType _stype;
188    Value _sum_supply;
189
190    // Data structures for storing the digraph
191    IntNodeMap _node_id;
192    IntArcMap _arc_id;
193    IntVector _source;
194    IntVector _target;
195
196    // Node and arc data
197    ValueVector _lower;
198    ValueVector _upper;
199    ValueVector _cap;
200    CostVector _cost;
201    ValueVector _supply;
202    ValueVector _flow;
203    CostVector _pi;
204
205    // Data for storing the spanning tree structure
206    IntVector _parent;
207    IntVector _pred;
208    IntVector _thread;
209    IntVector _rev_thread;
210    IntVector _succ_num;
211    IntVector _last_succ;
212    IntVector _dirty_revs;
213    BoolVector _forward;
214    IntVector _state;
215    int _root;
216
217    // Temporary data used in the current pivot iteration
218    int in_arc, join, u_in, v_in, u_out, v_out;
219    int first, second, right, last;
220    int stem, par_stem, new_stem;
221    Value delta;
222
223  public:
224 
225    /// \brief Constant for infinite upper bounds (capacities).
226    ///
227    /// Constant for infinite upper bounds (capacities).
228    /// It is \c std::numeric_limits<Value>::infinity() if available,
229    /// \c std::numeric_limits<Value>::max() otherwise.
230    const Value INF;
231
232  private:
233
234    // Implementation of the First Eligible pivot rule
235    class FirstEligiblePivotRule
236    {
237    private:
238
239      // References to the NetworkSimplex class
240      const IntVector  &_source;
241      const IntVector  &_target;
242      const CostVector &_cost;
243      const IntVector  &_state;
244      const CostVector &_pi;
245      int &_in_arc;
246      int _search_arc_num;
247
248      // Pivot rule data
249      int _next_arc;
250
251    public:
252
253      // Constructor
254      FirstEligiblePivotRule(NetworkSimplex &ns) :
255        _source(ns._source), _target(ns._target),
256        _cost(ns._cost), _state(ns._state), _pi(ns._pi),
257        _in_arc(ns.in_arc), _search_arc_num(ns._search_arc_num),
258        _next_arc(0)
259      {}
260
261      // Find next entering arc
262      bool findEnteringArc() {
263        Cost c;
264        for (int e = _next_arc; e < _search_arc_num; ++e) {
265          c = _state[e] * (_cost[e] + _pi[_source[e]] - _pi[_target[e]]);
266          if (c < 0) {
267            _in_arc = e;
268            _next_arc = e + 1;
269            return true;
270          }
271        }
272        for (int e = 0; e < _next_arc; ++e) {
273          c = _state[e] * (_cost[e] + _pi[_source[e]] - _pi[_target[e]]);
274          if (c < 0) {
275            _in_arc = e;
276            _next_arc = e + 1;
277            return true;
278          }
279        }
280        return false;
281      }
282
283    }; //class FirstEligiblePivotRule
284
285
286    // Implementation of the Best Eligible pivot rule
287    class BestEligiblePivotRule
288    {
289    private:
290
291      // References to the NetworkSimplex class
292      const IntVector  &_source;
293      const IntVector  &_target;
294      const CostVector &_cost;
295      const IntVector  &_state;
296      const CostVector &_pi;
297      int &_in_arc;
298      int _search_arc_num;
299
300    public:
301
302      // Constructor
303      BestEligiblePivotRule(NetworkSimplex &ns) :
304        _source(ns._source), _target(ns._target),
305        _cost(ns._cost), _state(ns._state), _pi(ns._pi),
306        _in_arc(ns.in_arc), _search_arc_num(ns._search_arc_num)
307      {}
308
309      // Find next entering arc
310      bool findEnteringArc() {
311        Cost c, min = 0;
312        for (int e = 0; e < _search_arc_num; ++e) {
313          c = _state[e] * (_cost[e] + _pi[_source[e]] - _pi[_target[e]]);
314          if (c < min) {
315            min = c;
316            _in_arc = e;
317          }
318        }
319        return min < 0;
320      }
321
322    }; //class BestEligiblePivotRule
323
324
325    // Implementation of the Block Search pivot rule
326    class BlockSearchPivotRule
327    {
328    private:
329
330      // References to the NetworkSimplex class
331      const IntVector  &_source;
332      const IntVector  &_target;
333      const CostVector &_cost;
334      const IntVector  &_state;
335      const CostVector &_pi;
336      int &_in_arc;
337      int _search_arc_num;
338
339      // Pivot rule data
340      int _block_size;
341      int _next_arc;
342
343    public:
344
345      // Constructor
346      BlockSearchPivotRule(NetworkSimplex &ns) :
347        _source(ns._source), _target(ns._target),
348        _cost(ns._cost), _state(ns._state), _pi(ns._pi),
349        _in_arc(ns.in_arc), _search_arc_num(ns._search_arc_num),
350        _next_arc(0)
351      {
352        // The main parameters of the pivot rule
353        const double BLOCK_SIZE_FACTOR = 0.5;
354        const int MIN_BLOCK_SIZE = 10;
355
356        _block_size = std::max( int(BLOCK_SIZE_FACTOR *
357                                    std::sqrt(double(_search_arc_num))),
358                                MIN_BLOCK_SIZE );
359      }
360
361      // Find next entering arc
362      bool findEnteringArc() {
363        Cost c, min = 0;
364        int cnt = _block_size;
365        int e, min_arc = _next_arc;
366        for (e = _next_arc; e < _search_arc_num; ++e) {
367          c = _state[e] * (_cost[e] + _pi[_source[e]] - _pi[_target[e]]);
368          if (c < min) {
369            min = c;
370            min_arc = e;
371          }
372          if (--cnt == 0) {
373            if (min < 0) break;
374            cnt = _block_size;
375          }
376        }
377        if (min == 0 || cnt > 0) {
378          for (e = 0; e < _next_arc; ++e) {
379            c = _state[e] * (_cost[e] + _pi[_source[e]] - _pi[_target[e]]);
380            if (c < min) {
381              min = c;
382              min_arc = e;
383            }
384            if (--cnt == 0) {
385              if (min < 0) break;
386              cnt = _block_size;
387            }
388          }
389        }
390        if (min >= 0) return false;
391        _in_arc = min_arc;
392        _next_arc = e;
393        return true;
394      }
395
396    }; //class BlockSearchPivotRule
397
398
399    // Implementation of the Candidate List pivot rule
400    class CandidateListPivotRule
401    {
402    private:
403
404      // References to the NetworkSimplex class
405      const IntVector  &_source;
406      const IntVector  &_target;
407      const CostVector &_cost;
408      const IntVector  &_state;
409      const CostVector &_pi;
410      int &_in_arc;
411      int _search_arc_num;
412
413      // Pivot rule data
414      IntVector _candidates;
415      int _list_length, _minor_limit;
416      int _curr_length, _minor_count;
417      int _next_arc;
418
419    public:
420
421      /// Constructor
422      CandidateListPivotRule(NetworkSimplex &ns) :
423        _source(ns._source), _target(ns._target),
424        _cost(ns._cost), _state(ns._state), _pi(ns._pi),
425        _in_arc(ns.in_arc), _search_arc_num(ns._search_arc_num),
426        _next_arc(0)
427      {
428        // The main parameters of the pivot rule
429        const double LIST_LENGTH_FACTOR = 1.0;
430        const int MIN_LIST_LENGTH = 10;
431        const double MINOR_LIMIT_FACTOR = 0.1;
432        const int MIN_MINOR_LIMIT = 3;
433
434        _list_length = std::max( int(LIST_LENGTH_FACTOR *
435                                     std::sqrt(double(_search_arc_num))),
436                                 MIN_LIST_LENGTH );
437        _minor_limit = std::max( int(MINOR_LIMIT_FACTOR * _list_length),
438                                 MIN_MINOR_LIMIT );
439        _curr_length = _minor_count = 0;
440        _candidates.resize(_list_length);
441      }
442
443      /// Find next entering arc
444      bool findEnteringArc() {
445        Cost min, c;
446        int e, min_arc = _next_arc;
447        if (_curr_length > 0 && _minor_count < _minor_limit) {
448          // Minor iteration: select the best eligible arc from the
449          // current candidate list
450          ++_minor_count;
451          min = 0;
452          for (int i = 0; i < _curr_length; ++i) {
453            e = _candidates[i];
454            c = _state[e] * (_cost[e] + _pi[_source[e]] - _pi[_target[e]]);
455            if (c < min) {
456              min = c;
457              min_arc = e;
458            }
459            if (c >= 0) {
460              _candidates[i--] = _candidates[--_curr_length];
461            }
462          }
463          if (min < 0) {
464            _in_arc = min_arc;
465            return true;
466          }
467        }
468
469        // Major iteration: build a new candidate list
470        min = 0;
471        _curr_length = 0;
472        for (e = _next_arc; e < _search_arc_num; ++e) {
473          c = _state[e] * (_cost[e] + _pi[_source[e]] - _pi[_target[e]]);
474          if (c < 0) {
475            _candidates[_curr_length++] = e;
476            if (c < min) {
477              min = c;
478              min_arc = e;
479            }
480            if (_curr_length == _list_length) break;
481          }
482        }
483        if (_curr_length < _list_length) {
484          for (e = 0; e < _next_arc; ++e) {
485            c = _state[e] * (_cost[e] + _pi[_source[e]] - _pi[_target[e]]);
486            if (c < 0) {
487              _candidates[_curr_length++] = e;
488              if (c < min) {
489                min = c;
490                min_arc = e;
491              }
492              if (_curr_length == _list_length) break;
493            }
494          }
495        }
496        if (_curr_length == 0) return false;
497        _minor_count = 1;
498        _in_arc = min_arc;
499        _next_arc = e;
500        return true;
501      }
502
503    }; //class CandidateListPivotRule
504
505
506    // Implementation of the Altering Candidate List pivot rule
507    class AlteringListPivotRule
508    {
509    private:
510
511      // References to the NetworkSimplex class
512      const IntVector  &_source;
513      const IntVector  &_target;
514      const CostVector &_cost;
515      const IntVector  &_state;
516      const CostVector &_pi;
517      int &_in_arc;
518      int _search_arc_num;
519
520      // Pivot rule data
521      int _block_size, _head_length, _curr_length;
522      int _next_arc;
523      IntVector _candidates;
524      CostVector _cand_cost;
525
526      // Functor class to compare arcs during sort of the candidate list
527      class SortFunc
528      {
529      private:
530        const CostVector &_map;
531      public:
532        SortFunc(const CostVector &map) : _map(map) {}
533        bool operator()(int left, int right) {
534          return _map[left] > _map[right];
535        }
536      };
537
538      SortFunc _sort_func;
539
540    public:
541
542      // Constructor
543      AlteringListPivotRule(NetworkSimplex &ns) :
544        _source(ns._source), _target(ns._target),
545        _cost(ns._cost), _state(ns._state), _pi(ns._pi),
546        _in_arc(ns.in_arc), _search_arc_num(ns._search_arc_num),
547        _next_arc(0), _cand_cost(ns._search_arc_num), _sort_func(_cand_cost)
548      {
549        // The main parameters of the pivot rule
550        const double BLOCK_SIZE_FACTOR = 1.5;
551        const int MIN_BLOCK_SIZE = 10;
552        const double HEAD_LENGTH_FACTOR = 0.1;
553        const int MIN_HEAD_LENGTH = 3;
554
555        _block_size = std::max( int(BLOCK_SIZE_FACTOR *
556                                    std::sqrt(double(_search_arc_num))),
557                                MIN_BLOCK_SIZE );
558        _head_length = std::max( int(HEAD_LENGTH_FACTOR * _block_size),
559                                 MIN_HEAD_LENGTH );
560        _candidates.resize(_head_length + _block_size);
561        _curr_length = 0;
562      }
563
564      // Find next entering arc
565      bool findEnteringArc() {
566        // Check the current candidate list
567        int e;
568        for (int i = 0; i < _curr_length; ++i) {
569          e = _candidates[i];
570          _cand_cost[e] = _state[e] *
571            (_cost[e] + _pi[_source[e]] - _pi[_target[e]]);
572          if (_cand_cost[e] >= 0) {
573            _candidates[i--] = _candidates[--_curr_length];
574          }
575        }
576
577        // Extend the list
578        int cnt = _block_size;
579        int last_arc = 0;
580        int limit = _head_length;
581
582        for (int 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            last_arc = e;
588          }
589          if (--cnt == 0) {
590            if (_curr_length > limit) break;
591            limit = 0;
592            cnt = _block_size;
593          }
594        }
595        if (_curr_length <= limit) {
596          for (int e = 0; e < _next_arc; ++e) {
597            _cand_cost[e] = _state[e] *
598              (_cost[e] + _pi[_source[e]] - _pi[_target[e]]);
599            if (_cand_cost[e] < 0) {
600              _candidates[_curr_length++] = e;
601              last_arc = e;
602            }
603            if (--cnt == 0) {
604              if (_curr_length > limit) break;
605              limit = 0;
606              cnt = _block_size;
607            }
608          }
609        }
610        if (_curr_length == 0) return false;
611        _next_arc = last_arc + 1;
612
613        // Make heap of the candidate list (approximating a partial sort)
614        make_heap( _candidates.begin(), _candidates.begin() + _curr_length,
615                   _sort_func );
616
617        // Pop the first element of the heap
618        _in_arc = _candidates[0];
619        pop_heap( _candidates.begin(), _candidates.begin() + _curr_length,
620                  _sort_func );
621        _curr_length = std::min(_head_length, _curr_length - 1);
622        return true;
623      }
624
625    }; //class AlteringListPivotRule
626
627  public:
628
629    /// \brief Constructor.
630    ///
631    /// The constructor of the class.
632    ///
633    /// \param graph The digraph the algorithm runs on.
634    NetworkSimplex(const GR& graph) :
635      _graph(graph), _node_id(graph), _arc_id(graph),
636      INF(std::numeric_limits<Value>::has_infinity ?
637          std::numeric_limits<Value>::infinity() :
638          std::numeric_limits<Value>::max())
639    {
640      // Check the value types
641      LEMON_ASSERT(std::numeric_limits<Value>::is_signed,
642        "The flow type of NetworkSimplex must be signed");
643      LEMON_ASSERT(std::numeric_limits<Cost>::is_signed,
644        "The cost type of NetworkSimplex must be signed");
645       
646      // Resize vectors
647      _node_num = countNodes(_graph);
648      _arc_num = countArcs(_graph);
649      int all_node_num = _node_num + 1;
650      int max_arc_num = _arc_num + 2 * _node_num;
651
652      _source.resize(max_arc_num);
653      _target.resize(max_arc_num);
654
655      _lower.resize(_arc_num);
656      _upper.resize(_arc_num);
657      _cap.resize(max_arc_num);
658      _cost.resize(max_arc_num);
659      _supply.resize(all_node_num);
660      _flow.resize(max_arc_num);
661      _pi.resize(all_node_num);
662
663      _parent.resize(all_node_num);
664      _pred.resize(all_node_num);
665      _forward.resize(all_node_num);
666      _thread.resize(all_node_num);
667      _rev_thread.resize(all_node_num);
668      _succ_num.resize(all_node_num);
669      _last_succ.resize(all_node_num);
670      _state.resize(max_arc_num);
671
672      // Copy the graph (store the arcs in a mixed order)
673      int i = 0;
674      for (NodeIt n(_graph); n != INVALID; ++n, ++i) {
675        _node_id[n] = i;
676      }
677      int k = std::max(int(std::sqrt(double(_arc_num))), 10);
678      i = 0;
679      for (ArcIt a(_graph); a != INVALID; ++a) {
680        _arc_id[a] = i;
681        _source[i] = _node_id[_graph.source(a)];
682        _target[i] = _node_id[_graph.target(a)];
683        if ((i += k) >= _arc_num) i = (i % k) + 1;
684      }
685     
686      // Reset parameters
687      reset();
688    }
689
690    /// \name Parameters
691    /// The parameters of the algorithm can be specified using these
692    /// functions.
693
694    /// @{
695
696    /// \brief Set the lower bounds on the arcs.
697    ///
698    /// This function sets the lower bounds on the arcs.
699    /// If it is not used before calling \ref run(), the lower bounds
700    /// will be set to zero on all arcs.
701    ///
702    /// \param map An arc map storing the lower bounds.
703    /// Its \c Value type must be convertible to the \c Value type
704    /// of the algorithm.
705    ///
706    /// \return <tt>(*this)</tt>
707    template <typename LowerMap>
708    NetworkSimplex& lowerMap(const LowerMap& map) {
709      _have_lower = true;
710      for (ArcIt a(_graph); a != INVALID; ++a) {
711        _lower[_arc_id[a]] = map[a];
712      }
713      return *this;
714    }
715
716    /// \brief Set the upper bounds (capacities) on the arcs.
717    ///
718    /// This function sets the upper bounds (capacities) on the arcs.
719    /// If it is not used before calling \ref run(), the upper bounds
720    /// will be set to \ref INF on all arcs (i.e. the flow value will be
721    /// unbounded from above on each arc).
722    ///
723    /// \param map An arc map storing the upper bounds.
724    /// Its \c Value type must be convertible to the \c Value type
725    /// of the algorithm.
726    ///
727    /// \return <tt>(*this)</tt>
728    template<typename UpperMap>
729    NetworkSimplex& upperMap(const UpperMap& map) {
730      for (ArcIt a(_graph); a != INVALID; ++a) {
731        _upper[_arc_id[a]] = map[a];
732      }
733      return *this;
734    }
735
736    /// \brief Set the costs of the arcs.
737    ///
738    /// This function sets the costs of the arcs.
739    /// If it is not used before calling \ref run(), the costs
740    /// will be set to \c 1 on all arcs.
741    ///
742    /// \param map An arc map storing the costs.
743    /// Its \c Value type must be convertible to the \c Cost type
744    /// of the algorithm.
745    ///
746    /// \return <tt>(*this)</tt>
747    template<typename CostMap>
748    NetworkSimplex& costMap(const CostMap& map) {
749      for (ArcIt a(_graph); a != INVALID; ++a) {
750        _cost[_arc_id[a]] = map[a];
751      }
752      return *this;
753    }
754
755    /// \brief Set the supply values of the nodes.
756    ///
757    /// This function sets the supply values of the nodes.
758    /// If neither this function nor \ref stSupply() is used before
759    /// calling \ref run(), the supply of each node will be set to zero.
760    ///
761    /// \param map A node map storing the supply values.
762    /// Its \c Value type must be convertible to the \c Value type
763    /// of the algorithm.
764    ///
765    /// \return <tt>(*this)</tt>
766    template<typename SupplyMap>
767    NetworkSimplex& supplyMap(const SupplyMap& map) {
768      for (NodeIt n(_graph); n != INVALID; ++n) {
769        _supply[_node_id[n]] = map[n];
770      }
771      return *this;
772    }
773
774    /// \brief Set single source and target nodes and a supply value.
775    ///
776    /// This function sets a single source node and a single target node
777    /// and the required flow value.
778    /// If neither this function nor \ref supplyMap() is used before
779    /// calling \ref run(), the supply of each node will be set to zero.
780    ///
781    /// Using this function has the same effect as using \ref supplyMap()
782    /// with such a map in which \c k is assigned to \c s, \c -k is
783    /// assigned to \c t and all other nodes have zero supply value.
784    ///
785    /// \param s The source node.
786    /// \param t The target node.
787    /// \param k The required amount of flow from node \c s to node \c t
788    /// (i.e. the supply of \c s and the demand of \c t).
789    ///
790    /// \return <tt>(*this)</tt>
791    NetworkSimplex& stSupply(const Node& s, const Node& t, Value k) {
792      for (int i = 0; i != _node_num; ++i) {
793        _supply[i] = 0;
794      }
795      _supply[_node_id[s]] =  k;
796      _supply[_node_id[t]] = -k;
797      return *this;
798    }
799   
800    /// \brief Set the type of the supply constraints.
801    ///
802    /// This function sets the type of the supply/demand constraints.
803    /// If it is not used before calling \ref run(), the \ref GEQ supply
804    /// type will be used.
805    ///
806    /// For more information see \ref SupplyType.
807    ///
808    /// \return <tt>(*this)</tt>
809    NetworkSimplex& supplyType(SupplyType supply_type) {
810      _stype = supply_type;
811      return *this;
812    }
813
814    /// @}
815
816    /// \name Execution Control
817    /// The algorithm can be executed using \ref run().
818
819    /// @{
820
821    /// \brief Run the algorithm.
822    ///
823    /// This function runs the algorithm.
824    /// The paramters can be specified using functions \ref lowerMap(),
825    /// \ref upperMap(), \ref costMap(), \ref supplyMap(), \ref stSupply(),
826    /// \ref supplyType().
827    /// For example,
828    /// \code
829    ///   NetworkSimplex<ListDigraph> ns(graph);
830    ///   ns.lowerMap(lower).upperMap(upper).costMap(cost)
831    ///     .supplyMap(sup).run();
832    /// \endcode
833    ///
834    /// This function can be called more than once. All the parameters
835    /// that have been given are kept for the next call, unless
836    /// \ref reset() is called, thus only the modified parameters
837    /// have to be set again. See \ref reset() for examples.
838    /// However the underlying digraph must not be modified after this
839    /// class have been constructed, since it copies and extends the graph.
840    ///
841    /// \param pivot_rule The pivot rule that will be used during the
842    /// algorithm. For more information see \ref PivotRule.
843    ///
844    /// \return \c INFEASIBLE if no feasible flow exists,
845    /// \n \c OPTIMAL if the problem has optimal solution
846    /// (i.e. it is feasible and bounded), and the algorithm has found
847    /// optimal flow and node potentials (primal and dual solutions),
848    /// \n \c UNBOUNDED if the objective function of the problem is
849    /// unbounded, i.e. there is a directed cycle having negative total
850    /// cost and infinite upper bound.
851    ///
852    /// \see ProblemType, PivotRule
853    ProblemType run(PivotRule pivot_rule = BLOCK_SEARCH) {
854      if (!init()) return INFEASIBLE;
855      return start(pivot_rule);
856    }
857
858    /// \brief Reset all the parameters that have been given before.
859    ///
860    /// This function resets all the paramaters that have been given
861    /// before using functions \ref lowerMap(), \ref upperMap(),
862    /// \ref costMap(), \ref supplyMap(), \ref stSupply(), \ref supplyType().
863    ///
864    /// It is useful for multiple run() calls. If this function is not
865    /// used, all the parameters given before are kept for the next
866    /// \ref run() call.
867    /// However the underlying digraph must not be modified after this
868    /// class have been constructed, since it copies and extends the graph.
869    ///
870    /// For example,
871    /// \code
872    ///   NetworkSimplex<ListDigraph> ns(graph);
873    ///
874    ///   // First run
875    ///   ns.lowerMap(lower).upperMap(upper).costMap(cost)
876    ///     .supplyMap(sup).run();
877    ///
878    ///   // Run again with modified cost map (reset() is not called,
879    ///   // so only the cost map have to be set again)
880    ///   cost[e] += 100;
881    ///   ns.costMap(cost).run();
882    ///
883    ///   // Run again from scratch using reset()
884    ///   // (the lower bounds will be set to zero on all arcs)
885    ///   ns.reset();
886    ///   ns.upperMap(capacity).costMap(cost)
887    ///     .supplyMap(sup).run();
888    /// \endcode
889    ///
890    /// \return <tt>(*this)</tt>
891    NetworkSimplex& reset() {
892      for (int i = 0; i != _node_num; ++i) {
893        _supply[i] = 0;
894      }
895      for (int i = 0; i != _arc_num; ++i) {
896        _lower[i] = 0;
897        _upper[i] = INF;
898        _cost[i] = 1;
899      }
900      _have_lower = false;
901      _stype = GEQ;
902      return *this;
903    }
904
905    /// @}
906
907    /// \name Query Functions
908    /// The results of the algorithm can be obtained using these
909    /// functions.\n
910    /// The \ref run() function must be called before using them.
911
912    /// @{
913
914    /// \brief Return the total cost of the found flow.
915    ///
916    /// This function returns the total cost of the found flow.
917    /// Its complexity is O(e).
918    ///
919    /// \note The return type of the function can be specified as a
920    /// template parameter. For example,
921    /// \code
922    ///   ns.totalCost<double>();
923    /// \endcode
924    /// It is useful if the total cost cannot be stored in the \c Cost
925    /// type of the algorithm, which is the default return type of the
926    /// function.
927    ///
928    /// \pre \ref run() must be called before using this function.
929    template <typename Number>
930    Number totalCost() const {
931      Number c = 0;
932      for (ArcIt a(_graph); a != INVALID; ++a) {
933        int i = _arc_id[a];
934        c += Number(_flow[i]) * Number(_cost[i]);
935      }
936      return c;
937    }
938
939#ifndef DOXYGEN
940    Cost totalCost() const {
941      return totalCost<Cost>();
942    }
943#endif
944
945    /// \brief Return the flow on the given arc.
946    ///
947    /// This function returns the flow on the given arc.
948    ///
949    /// \pre \ref run() must be called before using this function.
950    Value flow(const Arc& a) const {
951      return _flow[_arc_id[a]];
952    }
953
954    /// \brief Return the flow map (the primal solution).
955    ///
956    /// This function copies the flow value on each arc into the given
957    /// map. The \c Value type of the algorithm must be convertible to
958    /// the \c Value type of the map.
959    ///
960    /// \pre \ref run() must be called before using this function.
961    template <typename FlowMap>
962    void flowMap(FlowMap &map) const {
963      for (ArcIt a(_graph); a != INVALID; ++a) {
964        map.set(a, _flow[_arc_id[a]]);
965      }
966    }
967
968    /// \brief Return the potential (dual value) of the given node.
969    ///
970    /// This function returns the potential (dual value) of the
971    /// given node.
972    ///
973    /// \pre \ref run() must be called before using this function.
974    Cost potential(const Node& n) const {
975      return _pi[_node_id[n]];
976    }
977
978    /// \brief Return the potential map (the dual solution).
979    ///
980    /// This function copies the potential (dual value) of each node
981    /// into the given map.
982    /// The \c Cost type of the algorithm must be convertible to the
983    /// \c Value type of the map.
984    ///
985    /// \pre \ref run() must be called before using this function.
986    template <typename PotentialMap>
987    void potentialMap(PotentialMap &map) const {
988      for (NodeIt n(_graph); n != INVALID; ++n) {
989        map.set(n, _pi[_node_id[n]]);
990      }
991    }
992
993    /// @}
994
995  private:
996
997    // Initialize internal data structures
998    bool init() {
999      if (_node_num == 0) return false;
1000
1001      // Check the sum of supply values
1002      _sum_supply = 0;
1003      for (int i = 0; i != _node_num; ++i) {
1004        _sum_supply += _supply[i];
1005      }
1006      if ( !((_stype == GEQ && _sum_supply <= 0) ||
1007             (_stype == LEQ && _sum_supply >= 0)) ) return false;
1008
1009      // Remove non-zero lower bounds
1010      if (_have_lower) {
1011        for (int i = 0; i != _arc_num; ++i) {
1012          Value c = _lower[i];
1013          if (c >= 0) {
1014            _cap[i] = _upper[i] < INF ? _upper[i] - c : INF;
1015          } else {
1016            _cap[i] = _upper[i] < INF + c ? _upper[i] - c : INF;
1017          }
1018          _supply[_source[i]] -= c;
1019          _supply[_target[i]] += c;
1020        }
1021      } else {
1022        for (int i = 0; i != _arc_num; ++i) {
1023          _cap[i] = _upper[i];
1024        }
1025      }
1026
1027      // Initialize artifical cost
1028      Cost ART_COST;
1029      if (std::numeric_limits<Cost>::is_exact) {
1030        ART_COST = std::numeric_limits<Cost>::max() / 2 + 1;
1031      } else {
1032        ART_COST = std::numeric_limits<Cost>::min();
1033        for (int i = 0; i != _arc_num; ++i) {
1034          if (_cost[i] > ART_COST) ART_COST = _cost[i];
1035        }
1036        ART_COST = (ART_COST + 1) * _node_num;
1037      }
1038
1039      // Initialize arc maps
1040      for (int i = 0; i != _arc_num; ++i) {
1041        _flow[i] = 0;
1042        _state[i] = STATE_LOWER;
1043      }
1044     
1045      // Set data for the artificial root node
1046      _root = _node_num;
1047      _parent[_root] = -1;
1048      _pred[_root] = -1;
1049      _thread[_root] = 0;
1050      _rev_thread[0] = _root;
1051      _succ_num[_root] = _node_num + 1;
1052      _last_succ[_root] = _root - 1;
1053      _supply[_root] = -_sum_supply;
1054      _pi[_root] = 0;
1055
1056      // Add artificial arcs and initialize the spanning tree data structure
1057      if (_sum_supply == 0) {
1058        // EQ supply constraints
1059        _search_arc_num = _arc_num;
1060        _all_arc_num = _arc_num + _node_num;
1061        for (int u = 0, e = _arc_num; u != _node_num; ++u, ++e) {
1062          _parent[u] = _root;
1063          _pred[u] = e;
1064          _thread[u] = u + 1;
1065          _rev_thread[u + 1] = u;
1066          _succ_num[u] = 1;
1067          _last_succ[u] = u;
1068          _cap[e] = INF;
1069          _state[e] = STATE_TREE;
1070          if (_supply[u] >= 0) {
1071            _forward[u] = true;
1072            _pi[u] = 0;
1073            _source[e] = u;
1074            _target[e] = _root;
1075            _flow[e] = _supply[u];
1076            _cost[e] = 0;
1077          } else {
1078            _forward[u] = false;
1079            _pi[u] = ART_COST;
1080            _source[e] = _root;
1081            _target[e] = u;
1082            _flow[e] = -_supply[u];
1083            _cost[e] = ART_COST;
1084          }
1085        }
1086      }
1087      else if (_sum_supply > 0) {
1088        // LEQ supply constraints
1089        _search_arc_num = _arc_num + _node_num;
1090        int f = _arc_num + _node_num;
1091        for (int u = 0, e = _arc_num; u != _node_num; ++u, ++e) {
1092          _parent[u] = _root;
1093          _thread[u] = u + 1;
1094          _rev_thread[u + 1] = u;
1095          _succ_num[u] = 1;
1096          _last_succ[u] = u;
1097          if (_supply[u] >= 0) {
1098            _forward[u] = true;
1099            _pi[u] = 0;
1100            _pred[u] = e;
1101            _source[e] = u;
1102            _target[e] = _root;
1103            _cap[e] = INF;
1104            _flow[e] = _supply[u];
1105            _cost[e] = 0;
1106            _state[e] = STATE_TREE;
1107          } else {
1108            _forward[u] = false;
1109            _pi[u] = ART_COST;
1110            _pred[u] = f;
1111            _source[f] = _root;
1112            _target[f] = u;
1113            _cap[f] = INF;
1114            _flow[f] = -_supply[u];
1115            _cost[f] = ART_COST;
1116            _state[f] = STATE_TREE;
1117            _source[e] = u;
1118            _target[e] = _root;
1119            _cap[e] = INF;
1120            _flow[e] = 0;
1121            _cost[e] = 0;
1122            _state[e] = STATE_LOWER;
1123            ++f;
1124          }
1125        }
1126        _all_arc_num = f;
1127      }
1128      else {
1129        // GEQ supply constraints
1130        _search_arc_num = _arc_num + _node_num;
1131        int f = _arc_num + _node_num;
1132        for (int u = 0, e = _arc_num; u != _node_num; ++u, ++e) {
1133          _parent[u] = _root;
1134          _thread[u] = u + 1;
1135          _rev_thread[u + 1] = u;
1136          _succ_num[u] = 1;
1137          _last_succ[u] = u;
1138          if (_supply[u] <= 0) {
1139            _forward[u] = false;
1140            _pi[u] = 0;
1141            _pred[u] = e;
1142            _source[e] = _root;
1143            _target[e] = u;
1144            _cap[e] = INF;
1145            _flow[e] = -_supply[u];
1146            _cost[e] = 0;
1147            _state[e] = STATE_TREE;
1148          } else {
1149            _forward[u] = true;
1150            _pi[u] = -ART_COST;
1151            _pred[u] = f;
1152            _source[f] = u;
1153            _target[f] = _root;
1154            _cap[f] = INF;
1155            _flow[f] = _supply[u];
1156            _state[f] = STATE_TREE;
1157            _cost[f] = ART_COST;
1158            _source[e] = _root;
1159            _target[e] = u;
1160            _cap[e] = INF;
1161            _flow[e] = 0;
1162            _cost[e] = 0;
1163            _state[e] = STATE_LOWER;
1164            ++f;
1165          }
1166        }
1167        _all_arc_num = f;
1168      }
1169
1170      return true;
1171    }
1172
1173    // Find the join node
1174    void findJoinNode() {
1175      int u = _source[in_arc];
1176      int v = _target[in_arc];
1177      while (u != v) {
1178        if (_succ_num[u] < _succ_num[v]) {
1179          u = _parent[u];
1180        } else {
1181          v = _parent[v];
1182        }
1183      }
1184      join = u;
1185    }
1186
1187    // Find the leaving arc of the cycle and returns true if the
1188    // leaving arc is not the same as the entering arc
1189    bool findLeavingArc() {
1190      // Initialize first and second nodes according to the direction
1191      // of the cycle
1192      if (_state[in_arc] == STATE_LOWER) {
1193        first  = _source[in_arc];
1194        second = _target[in_arc];
1195      } else {
1196        first  = _target[in_arc];
1197        second = _source[in_arc];
1198      }
1199      delta = _cap[in_arc];
1200      int result = 0;
1201      Value d;
1202      int e;
1203
1204      // Search the cycle along the path form the first node to the root
1205      for (int u = first; u != join; u = _parent[u]) {
1206        e = _pred[u];
1207        d = _forward[u] ?
1208          _flow[e] : (_cap[e] == INF ? INF : _cap[e] - _flow[e]);
1209        if (d < delta) {
1210          delta = d;
1211          u_out = u;
1212          result = 1;
1213        }
1214      }
1215      // Search the cycle along the path form the second node to the root
1216      for (int u = second; u != join; u = _parent[u]) {
1217        e = _pred[u];
1218        d = _forward[u] ?
1219          (_cap[e] == INF ? INF : _cap[e] - _flow[e]) : _flow[e];
1220        if (d <= delta) {
1221          delta = d;
1222          u_out = u;
1223          result = 2;
1224        }
1225      }
1226
1227      if (result == 1) {
1228        u_in = first;
1229        v_in = second;
1230      } else {
1231        u_in = second;
1232        v_in = first;
1233      }
1234      return result != 0;
1235    }
1236
1237    // Change _flow and _state vectors
1238    void changeFlow(bool change) {
1239      // Augment along the cycle
1240      if (delta > 0) {
1241        Value val = _state[in_arc] * delta;
1242        _flow[in_arc] += val;
1243        for (int u = _source[in_arc]; u != join; u = _parent[u]) {
1244          _flow[_pred[u]] += _forward[u] ? -val : val;
1245        }
1246        for (int u = _target[in_arc]; u != join; u = _parent[u]) {
1247          _flow[_pred[u]] += _forward[u] ? val : -val;
1248        }
1249      }
1250      // Update the state of the entering and leaving arcs
1251      if (change) {
1252        _state[in_arc] = STATE_TREE;
1253        _state[_pred[u_out]] =
1254          (_flow[_pred[u_out]] == 0) ? STATE_LOWER : STATE_UPPER;
1255      } else {
1256        _state[in_arc] = -_state[in_arc];
1257      }
1258    }
1259
1260    // Update the tree structure
1261    void updateTreeStructure() {
1262      int u, w;
1263      int old_rev_thread = _rev_thread[u_out];
1264      int old_succ_num = _succ_num[u_out];
1265      int old_last_succ = _last_succ[u_out];
1266      v_out = _parent[u_out];
1267
1268      u = _last_succ[u_in];  // the last successor of u_in
1269      right = _thread[u];    // the node after it
1270
1271      // Handle the case when old_rev_thread equals to v_in
1272      // (it also means that join and v_out coincide)
1273      if (old_rev_thread == v_in) {
1274        last = _thread[_last_succ[u_out]];
1275      } else {
1276        last = _thread[v_in];
1277      }
1278
1279      // Update _thread and _parent along the stem nodes (i.e. the nodes
1280      // between u_in and u_out, whose parent have to be changed)
1281      _thread[v_in] = stem = u_in;
1282      _dirty_revs.clear();
1283      _dirty_revs.push_back(v_in);
1284      par_stem = v_in;
1285      while (stem != u_out) {
1286        // Insert the next stem node into the thread list
1287        new_stem = _parent[stem];
1288        _thread[u] = new_stem;
1289        _dirty_revs.push_back(u);
1290
1291        // Remove the subtree of stem from the thread list
1292        w = _rev_thread[stem];
1293        _thread[w] = right;
1294        _rev_thread[right] = w;
1295
1296        // Change the parent node and shift stem nodes
1297        _parent[stem] = par_stem;
1298        par_stem = stem;
1299        stem = new_stem;
1300
1301        // Update u and right
1302        u = _last_succ[stem] == _last_succ[par_stem] ?
1303          _rev_thread[par_stem] : _last_succ[stem];
1304        right = _thread[u];
1305      }
1306      _parent[u_out] = par_stem;
1307      _thread[u] = last;
1308      _rev_thread[last] = u;
1309      _last_succ[u_out] = u;
1310
1311      // Remove the subtree of u_out from the thread list except for
1312      // the case when old_rev_thread equals to v_in
1313      // (it also means that join and v_out coincide)
1314      if (old_rev_thread != v_in) {
1315        _thread[old_rev_thread] = right;
1316        _rev_thread[right] = old_rev_thread;
1317      }
1318
1319      // Update _rev_thread using the new _thread values
1320      for (int i = 0; i < int(_dirty_revs.size()); ++i) {
1321        u = _dirty_revs[i];
1322        _rev_thread[_thread[u]] = u;
1323      }
1324
1325      // Update _pred, _forward, _last_succ and _succ_num for the
1326      // stem nodes from u_out to u_in
1327      int tmp_sc = 0, tmp_ls = _last_succ[u_out];
1328      u = u_out;
1329      while (u != u_in) {
1330        w = _parent[u];
1331        _pred[u] = _pred[w];
1332        _forward[u] = !_forward[w];
1333        tmp_sc += _succ_num[u] - _succ_num[w];
1334        _succ_num[u] = tmp_sc;
1335        _last_succ[w] = tmp_ls;
1336        u = w;
1337      }
1338      _pred[u_in] = in_arc;
1339      _forward[u_in] = (u_in == _source[in_arc]);
1340      _succ_num[u_in] = old_succ_num;
1341
1342      // Set limits for updating _last_succ form v_in and v_out
1343      // towards the root
1344      int up_limit_in = -1;
1345      int up_limit_out = -1;
1346      if (_last_succ[join] == v_in) {
1347        up_limit_out = join;
1348      } else {
1349        up_limit_in = join;
1350      }
1351
1352      // Update _last_succ from v_in towards the root
1353      for (u = v_in; u != up_limit_in && _last_succ[u] == v_in;
1354           u = _parent[u]) {
1355        _last_succ[u] = _last_succ[u_out];
1356      }
1357      // Update _last_succ from v_out towards the root
1358      if (join != old_rev_thread && v_in != old_rev_thread) {
1359        for (u = v_out; u != up_limit_out && _last_succ[u] == old_last_succ;
1360             u = _parent[u]) {
1361          _last_succ[u] = old_rev_thread;
1362        }
1363      } else {
1364        for (u = v_out; u != up_limit_out && _last_succ[u] == old_last_succ;
1365             u = _parent[u]) {
1366          _last_succ[u] = _last_succ[u_out];
1367        }
1368      }
1369
1370      // Update _succ_num from v_in to join
1371      for (u = v_in; u != join; u = _parent[u]) {
1372        _succ_num[u] += old_succ_num;
1373      }
1374      // Update _succ_num from v_out to join
1375      for (u = v_out; u != join; u = _parent[u]) {
1376        _succ_num[u] -= old_succ_num;
1377      }
1378    }
1379
1380    // Update potentials
1381    void updatePotential() {
1382      Cost sigma = _forward[u_in] ?
1383        _pi[v_in] - _pi[u_in] - _cost[_pred[u_in]] :
1384        _pi[v_in] - _pi[u_in] + _cost[_pred[u_in]];
1385      // Update potentials in the subtree, which has been moved
1386      int end = _thread[_last_succ[u_in]];
1387      for (int u = u_in; u != end; u = _thread[u]) {
1388        _pi[u] += sigma;
1389      }
1390    }
1391
1392    // Execute the algorithm
1393    ProblemType start(PivotRule pivot_rule) {
1394      // Select the pivot rule implementation
1395      switch (pivot_rule) {
1396        case FIRST_ELIGIBLE:
1397          return start<FirstEligiblePivotRule>();
1398        case BEST_ELIGIBLE:
1399          return start<BestEligiblePivotRule>();
1400        case BLOCK_SEARCH:
1401          return start<BlockSearchPivotRule>();
1402        case CANDIDATE_LIST:
1403          return start<CandidateListPivotRule>();
1404        case ALTERING_LIST:
1405          return start<AlteringListPivotRule>();
1406      }
1407      return INFEASIBLE; // avoid warning
1408    }
1409
1410    template <typename PivotRuleImpl>
1411    ProblemType start() {
1412      PivotRuleImpl pivot(*this);
1413
1414      // Execute the Network Simplex algorithm
1415      while (pivot.findEnteringArc()) {
1416        findJoinNode();
1417        bool change = findLeavingArc();
1418        if (delta >= INF) return UNBOUNDED;
1419        changeFlow(change);
1420        if (change) {
1421          updateTreeStructure();
1422          updatePotential();
1423        }
1424      }
1425     
1426      // Check feasibility
1427      for (int e = _search_arc_num; e != _all_arc_num; ++e) {
1428        if (_flow[e] != 0) return INFEASIBLE;
1429      }
1430
1431      // Transform the solution and the supply map to the original form
1432      if (_have_lower) {
1433        for (int i = 0; i != _arc_num; ++i) {
1434          Value c = _lower[i];
1435          if (c != 0) {
1436            _flow[i] += c;
1437            _supply[_source[i]] += c;
1438            _supply[_target[i]] -= c;
1439          }
1440        }
1441      }
1442     
1443      // Shift potentials to meet the requirements of the GEQ/LEQ type
1444      // optimality conditions
1445      if (_sum_supply == 0) {
1446        if (_stype == GEQ) {
1447          Cost max_pot = std::numeric_limits<Cost>::min();
1448          for (int i = 0; i != _node_num; ++i) {
1449            if (_pi[i] > max_pot) max_pot = _pi[i];
1450          }
1451          if (max_pot > 0) {
1452            for (int i = 0; i != _node_num; ++i)
1453              _pi[i] -= max_pot;
1454          }
1455        } else {
1456          Cost min_pot = std::numeric_limits<Cost>::max();
1457          for (int i = 0; i != _node_num; ++i) {
1458            if (_pi[i] < min_pot) min_pot = _pi[i];
1459          }
1460          if (min_pot < 0) {
1461            for (int i = 0; i != _node_num; ++i)
1462              _pi[i] -= min_pot;
1463          }
1464        }
1465      }
1466
1467      return OPTIMAL;
1468    }
1469
1470  }; //class NetworkSimplex
1471
1472  ///@}
1473
1474} //namespace lemon
1475
1476#endif //LEMON_NETWORK_SIMPLEX_H
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