COIN-OR::LEMON - Graph Library

source: lemon-1.2/lemon/network_simplex.h @ 728:e2bdd1a988f3

Last change on this file since 728:e2bdd1a988f3 was 728:e2bdd1a988f3, checked in by Peter Kovacs <kpeter@…>, 10 years ago

Add a parameter to control arc mixing in NS (#298)

File size: 45.7 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<Arc> ArcVector;
165    typedef std::vector<Node> NodeVector;
166    typedef std::vector<int> IntVector;
167    typedef std::vector<bool> BoolVector;
168    typedef std::vector<Value> ValueVector;
169    typedef std::vector<Cost> CostVector;
170
171    // State constants for arcs
172    enum ArcStateEnum {
173      STATE_UPPER = -1,
174      STATE_TREE  =  0,
175      STATE_LOWER =  1
176    };
177
178  private:
179
180    // Data related to the underlying digraph
181    const GR &_graph;
182    int _node_num;
183    int _arc_num;
184    int _all_arc_num;
185    int _search_arc_num;
186
187    // Parameters of the problem
188    bool _have_lower;
189    SupplyType _stype;
190    Value _sum_supply;
191
192    // Data structures for storing the digraph
193    IntNodeMap _node_id;
194    IntArcMap _arc_id;
195    IntVector _source;
196    IntVector _target;
197
198    // Node and arc data
199    ValueVector _lower;
200    ValueVector _upper;
201    ValueVector _cap;
202    CostVector _cost;
203    ValueVector _supply;
204    ValueVector _flow;
205    CostVector _pi;
206
207    // Data for storing the spanning tree structure
208    IntVector _parent;
209    IntVector _pred;
210    IntVector _thread;
211    IntVector _rev_thread;
212    IntVector _succ_num;
213    IntVector _last_succ;
214    IntVector _dirty_revs;
215    BoolVector _forward;
216    IntVector _state;
217    int _root;
218
219    // Temporary data used in the current pivot iteration
220    int in_arc, join, u_in, v_in, u_out, v_out;
221    int first, second, right, last;
222    int stem, par_stem, new_stem;
223    Value delta;
224
225  public:
226 
227    /// \brief Constant for infinite upper bounds (capacities).
228    ///
229    /// Constant for infinite upper bounds (capacities).
230    /// It is \c std::numeric_limits<Value>::infinity() if available,
231    /// \c std::numeric_limits<Value>::max() otherwise.
232    const Value INF;
233
234  private:
235
236    // Implementation of the First Eligible pivot rule
237    class FirstEligiblePivotRule
238    {
239    private:
240
241      // References to the NetworkSimplex class
242      const IntVector  &_source;
243      const IntVector  &_target;
244      const CostVector &_cost;
245      const IntVector  &_state;
246      const CostVector &_pi;
247      int &_in_arc;
248      int _search_arc_num;
249
250      // Pivot rule data
251      int _next_arc;
252
253    public:
254
255      // Constructor
256      FirstEligiblePivotRule(NetworkSimplex &ns) :
257        _source(ns._source), _target(ns._target),
258        _cost(ns._cost), _state(ns._state), _pi(ns._pi),
259        _in_arc(ns.in_arc), _search_arc_num(ns._search_arc_num),
260        _next_arc(0)
261      {}
262
263      // Find next entering arc
264      bool findEnteringArc() {
265        Cost c;
266        for (int e = _next_arc; e < _search_arc_num; ++e) {
267          c = _state[e] * (_cost[e] + _pi[_source[e]] - _pi[_target[e]]);
268          if (c < 0) {
269            _in_arc = e;
270            _next_arc = e + 1;
271            return true;
272          }
273        }
274        for (int e = 0; e < _next_arc; ++e) {
275          c = _state[e] * (_cost[e] + _pi[_source[e]] - _pi[_target[e]]);
276          if (c < 0) {
277            _in_arc = e;
278            _next_arc = e + 1;
279            return true;
280          }
281        }
282        return false;
283      }
284
285    }; //class FirstEligiblePivotRule
286
287
288    // Implementation of the Best Eligible pivot rule
289    class BestEligiblePivotRule
290    {
291    private:
292
293      // References to the NetworkSimplex class
294      const IntVector  &_source;
295      const IntVector  &_target;
296      const CostVector &_cost;
297      const IntVector  &_state;
298      const CostVector &_pi;
299      int &_in_arc;
300      int _search_arc_num;
301
302    public:
303
304      // Constructor
305      BestEligiblePivotRule(NetworkSimplex &ns) :
306        _source(ns._source), _target(ns._target),
307        _cost(ns._cost), _state(ns._state), _pi(ns._pi),
308        _in_arc(ns.in_arc), _search_arc_num(ns._search_arc_num)
309      {}
310
311      // Find next entering arc
312      bool findEnteringArc() {
313        Cost c, min = 0;
314        for (int e = 0; e < _search_arc_num; ++e) {
315          c = _state[e] * (_cost[e] + _pi[_source[e]] - _pi[_target[e]]);
316          if (c < min) {
317            min = c;
318            _in_arc = e;
319          }
320        }
321        return min < 0;
322      }
323
324    }; //class BestEligiblePivotRule
325
326
327    // Implementation of the Block Search pivot rule
328    class BlockSearchPivotRule
329    {
330    private:
331
332      // References to the NetworkSimplex class
333      const IntVector  &_source;
334      const IntVector  &_target;
335      const CostVector &_cost;
336      const IntVector  &_state;
337      const CostVector &_pi;
338      int &_in_arc;
339      int _search_arc_num;
340
341      // Pivot rule data
342      int _block_size;
343      int _next_arc;
344
345    public:
346
347      // Constructor
348      BlockSearchPivotRule(NetworkSimplex &ns) :
349        _source(ns._source), _target(ns._target),
350        _cost(ns._cost), _state(ns._state), _pi(ns._pi),
351        _in_arc(ns.in_arc), _search_arc_num(ns._search_arc_num),
352        _next_arc(0)
353      {
354        // The main parameters of the pivot rule
355        const double BLOCK_SIZE_FACTOR = 0.5;
356        const int MIN_BLOCK_SIZE = 10;
357
358        _block_size = std::max( int(BLOCK_SIZE_FACTOR *
359                                    std::sqrt(double(_search_arc_num))),
360                                MIN_BLOCK_SIZE );
361      }
362
363      // Find next entering arc
364      bool findEnteringArc() {
365        Cost c, min = 0;
366        int cnt = _block_size;
367        int e;
368        for (e = _next_arc; e < _search_arc_num; ++e) {
369          c = _state[e] * (_cost[e] + _pi[_source[e]] - _pi[_target[e]]);
370          if (c < min) {
371            min = c;
372            _in_arc = e;
373          }
374          if (--cnt == 0) {
375            if (min < 0) goto search_end;
376            cnt = _block_size;
377          }
378        }
379        for (e = 0; e < _next_arc; ++e) {
380          c = _state[e] * (_cost[e] + _pi[_source[e]] - _pi[_target[e]]);
381          if (c < min) {
382            min = c;
383            _in_arc = e;
384          }
385          if (--cnt == 0) {
386            if (min < 0) goto search_end;
387            cnt = _block_size;
388          }
389        }
390        if (min >= 0) return false;
391
392      search_end:
393        _next_arc = e;
394        return true;
395      }
396
397    }; //class BlockSearchPivotRule
398
399
400    // Implementation of the Candidate List pivot rule
401    class CandidateListPivotRule
402    {
403    private:
404
405      // References to the NetworkSimplex class
406      const IntVector  &_source;
407      const IntVector  &_target;
408      const CostVector &_cost;
409      const IntVector  &_state;
410      const CostVector &_pi;
411      int &_in_arc;
412      int _search_arc_num;
413
414      // Pivot rule data
415      IntVector _candidates;
416      int _list_length, _minor_limit;
417      int _curr_length, _minor_count;
418      int _next_arc;
419
420    public:
421
422      /// Constructor
423      CandidateListPivotRule(NetworkSimplex &ns) :
424        _source(ns._source), _target(ns._target),
425        _cost(ns._cost), _state(ns._state), _pi(ns._pi),
426        _in_arc(ns.in_arc), _search_arc_num(ns._search_arc_num),
427        _next_arc(0)
428      {
429        // The main parameters of the pivot rule
430        const double LIST_LENGTH_FACTOR = 0.25;
431        const int MIN_LIST_LENGTH = 10;
432        const double MINOR_LIMIT_FACTOR = 0.1;
433        const int MIN_MINOR_LIMIT = 3;
434
435        _list_length = std::max( int(LIST_LENGTH_FACTOR *
436                                     std::sqrt(double(_search_arc_num))),
437                                 MIN_LIST_LENGTH );
438        _minor_limit = std::max( int(MINOR_LIMIT_FACTOR * _list_length),
439                                 MIN_MINOR_LIMIT );
440        _curr_length = _minor_count = 0;
441        _candidates.resize(_list_length);
442      }
443
444      /// Find next entering arc
445      bool findEnteringArc() {
446        Cost min, c;
447        int e;
448        if (_curr_length > 0 && _minor_count < _minor_limit) {
449          // Minor iteration: select the best eligible arc from the
450          // current candidate list
451          ++_minor_count;
452          min = 0;
453          for (int i = 0; i < _curr_length; ++i) {
454            e = _candidates[i];
455            c = _state[e] * (_cost[e] + _pi[_source[e]] - _pi[_target[e]]);
456            if (c < min) {
457              min = c;
458              _in_arc = e;
459            }
460            else if (c >= 0) {
461              _candidates[i--] = _candidates[--_curr_length];
462            }
463          }
464          if (min < 0) return true;
465        }
466
467        // Major iteration: build a new candidate list
468        min = 0;
469        _curr_length = 0;
470        for (e = _next_arc; e < _search_arc_num; ++e) {
471          c = _state[e] * (_cost[e] + _pi[_source[e]] - _pi[_target[e]]);
472          if (c < 0) {
473            _candidates[_curr_length++] = e;
474            if (c < min) {
475              min = c;
476              _in_arc = e;
477            }
478            if (_curr_length == _list_length) goto search_end;
479          }
480        }
481        for (e = 0; e < _next_arc; ++e) {
482          c = _state[e] * (_cost[e] + _pi[_source[e]] - _pi[_target[e]]);
483          if (c < 0) {
484            _candidates[_curr_length++] = e;
485            if (c < min) {
486              min = c;
487              _in_arc = e;
488            }
489            if (_curr_length == _list_length) goto search_end;
490          }
491        }
492        if (_curr_length == 0) return false;
493     
494      search_end:       
495        _minor_count = 1;
496        _next_arc = e;
497        return true;
498      }
499
500    }; //class CandidateListPivotRule
501
502
503    // Implementation of the Altering Candidate List pivot rule
504    class AlteringListPivotRule
505    {
506    private:
507
508      // References to the NetworkSimplex class
509      const IntVector  &_source;
510      const IntVector  &_target;
511      const CostVector &_cost;
512      const IntVector  &_state;
513      const CostVector &_pi;
514      int &_in_arc;
515      int _search_arc_num;
516
517      // Pivot rule data
518      int _block_size, _head_length, _curr_length;
519      int _next_arc;
520      IntVector _candidates;
521      CostVector _cand_cost;
522
523      // Functor class to compare arcs during sort of the candidate list
524      class SortFunc
525      {
526      private:
527        const CostVector &_map;
528      public:
529        SortFunc(const CostVector &map) : _map(map) {}
530        bool operator()(int left, int right) {
531          return _map[left] > _map[right];
532        }
533      };
534
535      SortFunc _sort_func;
536
537    public:
538
539      // Constructor
540      AlteringListPivotRule(NetworkSimplex &ns) :
541        _source(ns._source), _target(ns._target),
542        _cost(ns._cost), _state(ns._state), _pi(ns._pi),
543        _in_arc(ns.in_arc), _search_arc_num(ns._search_arc_num),
544        _next_arc(0), _cand_cost(ns._search_arc_num), _sort_func(_cand_cost)
545      {
546        // The main parameters of the pivot rule
547        const double BLOCK_SIZE_FACTOR = 1.0;
548        const int MIN_BLOCK_SIZE = 10;
549        const double HEAD_LENGTH_FACTOR = 0.1;
550        const int MIN_HEAD_LENGTH = 3;
551
552        _block_size = std::max( int(BLOCK_SIZE_FACTOR *
553                                    std::sqrt(double(_search_arc_num))),
554                                MIN_BLOCK_SIZE );
555        _head_length = std::max( int(HEAD_LENGTH_FACTOR * _block_size),
556                                 MIN_HEAD_LENGTH );
557        _candidates.resize(_head_length + _block_size);
558        _curr_length = 0;
559      }
560
561      // Find next entering arc
562      bool findEnteringArc() {
563        // Check the current candidate list
564        int e;
565        for (int i = 0; i < _curr_length; ++i) {
566          e = _candidates[i];
567          _cand_cost[e] = _state[e] *
568            (_cost[e] + _pi[_source[e]] - _pi[_target[e]]);
569          if (_cand_cost[e] >= 0) {
570            _candidates[i--] = _candidates[--_curr_length];
571          }
572        }
573
574        // Extend the list
575        int cnt = _block_size;
576        int limit = _head_length;
577
578        for (e = _next_arc; e < _search_arc_num; ++e) {
579          _cand_cost[e] = _state[e] *
580            (_cost[e] + _pi[_source[e]] - _pi[_target[e]]);
581          if (_cand_cost[e] < 0) {
582            _candidates[_curr_length++] = e;
583          }
584          if (--cnt == 0) {
585            if (_curr_length > limit) goto search_end;
586            limit = 0;
587            cnt = _block_size;
588          }
589        }
590        for (e = 0; e < _next_arc; ++e) {
591          _cand_cost[e] = _state[e] *
592            (_cost[e] + _pi[_source[e]] - _pi[_target[e]]);
593          if (_cand_cost[e] < 0) {
594            _candidates[_curr_length++] = e;
595          }
596          if (--cnt == 0) {
597            if (_curr_length > limit) goto search_end;
598            limit = 0;
599            cnt = _block_size;
600          }
601        }
602        if (_curr_length == 0) return false;
603       
604      search_end:
605
606        // Make heap of the candidate list (approximating a partial sort)
607        make_heap( _candidates.begin(), _candidates.begin() + _curr_length,
608                   _sort_func );
609
610        // Pop the first element of the heap
611        _in_arc = _candidates[0];
612        _next_arc = e;
613        pop_heap( _candidates.begin(), _candidates.begin() + _curr_length,
614                  _sort_func );
615        _curr_length = std::min(_head_length, _curr_length - 1);
616        return true;
617      }
618
619    }; //class AlteringListPivotRule
620
621  public:
622
623    /// \brief Constructor.
624    ///
625    /// The constructor of the class.
626    ///
627    /// \param graph The digraph the algorithm runs on.
628    /// \param arc_mixing Indicate if the arcs have to be stored in a
629    /// mixed order in the internal data structure.
630    /// In special cases, it could lead to better overall performance,
631    /// but it is usually slower. Therefore it is disabled by default.
632    NetworkSimplex(const GR& graph, bool arc_mixing = false) :
633      _graph(graph), _node_id(graph), _arc_id(graph),
634      INF(std::numeric_limits<Value>::has_infinity ?
635          std::numeric_limits<Value>::infinity() :
636          std::numeric_limits<Value>::max())
637    {
638      // Check the value types
639      LEMON_ASSERT(std::numeric_limits<Value>::is_signed,
640        "The flow type of NetworkSimplex must be signed");
641      LEMON_ASSERT(std::numeric_limits<Cost>::is_signed,
642        "The cost type of NetworkSimplex must be signed");
643       
644      // Resize vectors
645      _node_num = countNodes(_graph);
646      _arc_num = countArcs(_graph);
647      int all_node_num = _node_num + 1;
648      int max_arc_num = _arc_num + 2 * _node_num;
649
650      _source.resize(max_arc_num);
651      _target.resize(max_arc_num);
652
653      _lower.resize(_arc_num);
654      _upper.resize(_arc_num);
655      _cap.resize(max_arc_num);
656      _cost.resize(max_arc_num);
657      _supply.resize(all_node_num);
658      _flow.resize(max_arc_num);
659      _pi.resize(all_node_num);
660
661      _parent.resize(all_node_num);
662      _pred.resize(all_node_num);
663      _forward.resize(all_node_num);
664      _thread.resize(all_node_num);
665      _rev_thread.resize(all_node_num);
666      _succ_num.resize(all_node_num);
667      _last_succ.resize(all_node_num);
668      _state.resize(max_arc_num);
669
670      // Copy the graph
671      int i = 0;
672      for (NodeIt n(_graph); n != INVALID; ++n, ++i) {
673        _node_id[n] = i;
674      }
675      if (arc_mixing) {
676        // Store the arcs in a mixed order
677        int k = std::max(int(std::sqrt(double(_arc_num))), 10);
678        int i = 0, j = 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 = ++j;
684        }
685      } else {
686        // Store the arcs in the original order
687        int i = 0;
688        for (ArcIt a(_graph); a != INVALID; ++a, ++i) {
689          _arc_id[a] = i;
690          _source[i] = _node_id[_graph.source(a)];
691          _target[i] = _node_id[_graph.target(a)];
692        }
693      }
694     
695      // Initialize maps
696      for (int i = 0; i != _node_num; ++i) {
697        _supply[i] = 0;
698      }
699      for (int i = 0; i != _arc_num; ++i) {
700        _lower[i] = 0;
701        _upper[i] = INF;
702        _cost[i] = 1;
703      }
704      _have_lower = false;
705      _stype = GEQ;
706    }
707
708    /// \name Parameters
709    /// The parameters of the algorithm can be specified using these
710    /// functions.
711
712    /// @{
713
714    /// \brief Set the lower bounds on the arcs.
715    ///
716    /// This function sets the lower bounds on the arcs.
717    /// If it is not used before calling \ref run(), the lower bounds
718    /// will be set to zero on all arcs.
719    ///
720    /// \param map An arc map storing the lower bounds.
721    /// Its \c Value type must be convertible to the \c Value type
722    /// of the algorithm.
723    ///
724    /// \return <tt>(*this)</tt>
725    template <typename LowerMap>
726    NetworkSimplex& lowerMap(const LowerMap& map) {
727      _have_lower = true;
728      for (ArcIt a(_graph); a != INVALID; ++a) {
729        _lower[_arc_id[a]] = map[a];
730      }
731      return *this;
732    }
733
734    /// \brief Set the upper bounds (capacities) on the arcs.
735    ///
736    /// This function sets the upper bounds (capacities) on the arcs.
737    /// If it is not used before calling \ref run(), the upper bounds
738    /// will be set to \ref INF on all arcs (i.e. the flow value will be
739    /// unbounded from above on each arc).
740    ///
741    /// \param map An arc map storing the upper bounds.
742    /// Its \c Value type must be convertible to the \c Value type
743    /// of the algorithm.
744    ///
745    /// \return <tt>(*this)</tt>
746    template<typename UpperMap>
747    NetworkSimplex& upperMap(const UpperMap& map) {
748      for (ArcIt a(_graph); a != INVALID; ++a) {
749        _upper[_arc_id[a]] = map[a];
750      }
751      return *this;
752    }
753
754    /// \brief Set the costs of the arcs.
755    ///
756    /// This function sets the costs of the arcs.
757    /// If it is not used before calling \ref run(), the costs
758    /// will be set to \c 1 on all arcs.
759    ///
760    /// \param map An arc map storing the costs.
761    /// Its \c Value type must be convertible to the \c Cost type
762    /// of the algorithm.
763    ///
764    /// \return <tt>(*this)</tt>
765    template<typename CostMap>
766    NetworkSimplex& costMap(const CostMap& map) {
767      for (ArcIt a(_graph); a != INVALID; ++a) {
768        _cost[_arc_id[a]] = map[a];
769      }
770      return *this;
771    }
772
773    /// \brief Set the supply values of the nodes.
774    ///
775    /// This function sets the supply values of the nodes.
776    /// If neither this function nor \ref stSupply() is used before
777    /// calling \ref run(), the supply of each node will be set to zero.
778    /// (It makes sense only if non-zero lower bounds are given.)
779    ///
780    /// \param map A node map storing the supply values.
781    /// Its \c Value type must be convertible to the \c Value type
782    /// of the algorithm.
783    ///
784    /// \return <tt>(*this)</tt>
785    template<typename SupplyMap>
786    NetworkSimplex& supplyMap(const SupplyMap& map) {
787      for (NodeIt n(_graph); n != INVALID; ++n) {
788        _supply[_node_id[n]] = map[n];
789      }
790      return *this;
791    }
792
793    /// \brief Set single source and target nodes and a supply value.
794    ///
795    /// This function sets a single source node and a single target node
796    /// and the required flow value.
797    /// If neither this function nor \ref supplyMap() is used before
798    /// calling \ref run(), the supply of each node will be set to zero.
799    /// (It makes sense only if non-zero lower bounds are given.)
800    ///
801    /// Using this function has the same effect as using \ref supplyMap()
802    /// with such a map in which \c k is assigned to \c s, \c -k is
803    /// assigned to \c t and all other nodes have zero supply value.
804    ///
805    /// \param s The source node.
806    /// \param t The target node.
807    /// \param k The required amount of flow from node \c s to node \c t
808    /// (i.e. the supply of \c s and the demand of \c t).
809    ///
810    /// \return <tt>(*this)</tt>
811    NetworkSimplex& stSupply(const Node& s, const Node& t, Value k) {
812      for (int i = 0; i != _node_num; ++i) {
813        _supply[i] = 0;
814      }
815      _supply[_node_id[s]] =  k;
816      _supply[_node_id[t]] = -k;
817      return *this;
818    }
819   
820    /// \brief Set the type of the supply constraints.
821    ///
822    /// This function sets the type of the supply/demand constraints.
823    /// If it is not used before calling \ref run(), the \ref GEQ supply
824    /// type will be used.
825    ///
826    /// For more information see \ref SupplyType.
827    ///
828    /// \return <tt>(*this)</tt>
829    NetworkSimplex& supplyType(SupplyType supply_type) {
830      _stype = supply_type;
831      return *this;
832    }
833
834    /// @}
835
836    /// \name Execution Control
837    /// The algorithm can be executed using \ref run().
838
839    /// @{
840
841    /// \brief Run the algorithm.
842    ///
843    /// This function runs the algorithm.
844    /// The paramters can be specified using functions \ref lowerMap(),
845    /// \ref upperMap(), \ref costMap(), \ref supplyMap(), \ref stSupply(),
846    /// \ref supplyType().
847    /// For example,
848    /// \code
849    ///   NetworkSimplex<ListDigraph> ns(graph);
850    ///   ns.lowerMap(lower).upperMap(upper).costMap(cost)
851    ///     .supplyMap(sup).run();
852    /// \endcode
853    ///
854    /// This function can be called more than once. All the parameters
855    /// that have been given are kept for the next call, unless
856    /// \ref reset() is called, thus only the modified parameters
857    /// have to be set again. See \ref reset() for examples.
858    /// However the underlying digraph must not be modified after this
859    /// class have been constructed, since it copies and extends the graph.
860    ///
861    /// \param pivot_rule The pivot rule that will be used during the
862    /// algorithm. For more information see \ref PivotRule.
863    ///
864    /// \return \c INFEASIBLE if no feasible flow exists,
865    /// \n \c OPTIMAL if the problem has optimal solution
866    /// (i.e. it is feasible and bounded), and the algorithm has found
867    /// optimal flow and node potentials (primal and dual solutions),
868    /// \n \c UNBOUNDED if the objective function of the problem is
869    /// unbounded, i.e. there is a directed cycle having negative total
870    /// cost and infinite upper bound.
871    ///
872    /// \see ProblemType, PivotRule
873    ProblemType run(PivotRule pivot_rule = BLOCK_SEARCH) {
874      if (!init()) return INFEASIBLE;
875      return start(pivot_rule);
876    }
877
878    /// \brief Reset all the parameters that have been given before.
879    ///
880    /// This function resets all the paramaters that have been given
881    /// before using functions \ref lowerMap(), \ref upperMap(),
882    /// \ref costMap(), \ref supplyMap(), \ref stSupply(), \ref supplyType().
883    ///
884    /// It is useful for multiple run() calls. If this function is not
885    /// used, all the parameters given before are kept for the next
886    /// \ref run() call.
887    /// However the underlying digraph must not be modified after this
888    /// class have been constructed, since it copies and extends the graph.
889    ///
890    /// For example,
891    /// \code
892    ///   NetworkSimplex<ListDigraph> ns(graph);
893    ///
894    ///   // First run
895    ///   ns.lowerMap(lower).upperMap(upper).costMap(cost)
896    ///     .supplyMap(sup).run();
897    ///
898    ///   // Run again with modified cost map (reset() is not called,
899    ///   // so only the cost map have to be set again)
900    ///   cost[e] += 100;
901    ///   ns.costMap(cost).run();
902    ///
903    ///   // Run again from scratch using reset()
904    ///   // (the lower bounds will be set to zero on all arcs)
905    ///   ns.reset();
906    ///   ns.upperMap(capacity).costMap(cost)
907    ///     .supplyMap(sup).run();
908    /// \endcode
909    ///
910    /// \return <tt>(*this)</tt>
911    NetworkSimplex& reset() {
912      for (int i = 0; i != _node_num; ++i) {
913        _supply[i] = 0;
914      }
915      for (int i = 0; i != _arc_num; ++i) {
916        _lower[i] = 0;
917        _upper[i] = INF;
918        _cost[i] = 1;
919      }
920      _have_lower = false;
921      _stype = GEQ;
922      return *this;
923    }
924
925    /// @}
926
927    /// \name Query Functions
928    /// The results of the algorithm can be obtained using these
929    /// functions.\n
930    /// The \ref run() function must be called before using them.
931
932    /// @{
933
934    /// \brief Return the total cost of the found flow.
935    ///
936    /// This function returns the total cost of the found flow.
937    /// Its complexity is O(e).
938    ///
939    /// \note The return type of the function can be specified as a
940    /// template parameter. For example,
941    /// \code
942    ///   ns.totalCost<double>();
943    /// \endcode
944    /// It is useful if the total cost cannot be stored in the \c Cost
945    /// type of the algorithm, which is the default return type of the
946    /// function.
947    ///
948    /// \pre \ref run() must be called before using this function.
949    template <typename Number>
950    Number totalCost() const {
951      Number c = 0;
952      for (ArcIt a(_graph); a != INVALID; ++a) {
953        int i = _arc_id[a];
954        c += Number(_flow[i]) * Number(_cost[i]);
955      }
956      return c;
957    }
958
959#ifndef DOXYGEN
960    Cost totalCost() const {
961      return totalCost<Cost>();
962    }
963#endif
964
965    /// \brief Return the flow on the given arc.
966    ///
967    /// This function returns the flow on the given arc.
968    ///
969    /// \pre \ref run() must be called before using this function.
970    Value flow(const Arc& a) const {
971      return _flow[_arc_id[a]];
972    }
973
974    /// \brief Return the flow map (the primal solution).
975    ///
976    /// This function copies the flow value on each arc into the given
977    /// map. The \c Value type of the algorithm must be convertible to
978    /// the \c Value type of the map.
979    ///
980    /// \pre \ref run() must be called before using this function.
981    template <typename FlowMap>
982    void flowMap(FlowMap &map) const {
983      for (ArcIt a(_graph); a != INVALID; ++a) {
984        map.set(a, _flow[_arc_id[a]]);
985      }
986    }
987
988    /// \brief Return the potential (dual value) of the given node.
989    ///
990    /// This function returns the potential (dual value) of the
991    /// given node.
992    ///
993    /// \pre \ref run() must be called before using this function.
994    Cost potential(const Node& n) const {
995      return _pi[_node_id[n]];
996    }
997
998    /// \brief Return the potential map (the dual solution).
999    ///
1000    /// This function copies the potential (dual value) of each node
1001    /// into the given map.
1002    /// The \c Cost type of the algorithm must be convertible to the
1003    /// \c Value type of the map.
1004    ///
1005    /// \pre \ref run() must be called before using this function.
1006    template <typename PotentialMap>
1007    void potentialMap(PotentialMap &map) const {
1008      for (NodeIt n(_graph); n != INVALID; ++n) {
1009        map.set(n, _pi[_node_id[n]]);
1010      }
1011    }
1012
1013    /// @}
1014
1015  private:
1016
1017    // Initialize internal data structures
1018    bool init() {
1019      if (_node_num == 0) return false;
1020
1021      // Check the sum of supply values
1022      _sum_supply = 0;
1023      for (int i = 0; i != _node_num; ++i) {
1024        _sum_supply += _supply[i];
1025      }
1026      if ( !((_stype == GEQ && _sum_supply <= 0) ||
1027             (_stype == LEQ && _sum_supply >= 0)) ) return false;
1028
1029      // Remove non-zero lower bounds
1030      if (_have_lower) {
1031        for (int i = 0; i != _arc_num; ++i) {
1032          Value c = _lower[i];
1033          if (c >= 0) {
1034            _cap[i] = _upper[i] < INF ? _upper[i] - c : INF;
1035          } else {
1036            _cap[i] = _upper[i] < INF + c ? _upper[i] - c : INF;
1037          }
1038          _supply[_source[i]] -= c;
1039          _supply[_target[i]] += c;
1040        }
1041      } else {
1042        for (int i = 0; i != _arc_num; ++i) {
1043          _cap[i] = _upper[i];
1044        }
1045      }
1046
1047      // Initialize artifical cost
1048      Cost ART_COST;
1049      if (std::numeric_limits<Cost>::is_exact) {
1050        ART_COST = std::numeric_limits<Cost>::max() / 2 + 1;
1051      } else {
1052        ART_COST = std::numeric_limits<Cost>::min();
1053        for (int i = 0; i != _arc_num; ++i) {
1054          if (_cost[i] > ART_COST) ART_COST = _cost[i];
1055        }
1056        ART_COST = (ART_COST + 1) * _node_num;
1057      }
1058
1059      // Initialize arc maps
1060      for (int i = 0; i != _arc_num; ++i) {
1061        _flow[i] = 0;
1062        _state[i] = STATE_LOWER;
1063      }
1064     
1065      // Set data for the artificial root node
1066      _root = _node_num;
1067      _parent[_root] = -1;
1068      _pred[_root] = -1;
1069      _thread[_root] = 0;
1070      _rev_thread[0] = _root;
1071      _succ_num[_root] = _node_num + 1;
1072      _last_succ[_root] = _root - 1;
1073      _supply[_root] = -_sum_supply;
1074      _pi[_root] = 0;
1075
1076      // Add artificial arcs and initialize the spanning tree data structure
1077      if (_sum_supply == 0) {
1078        // EQ supply constraints
1079        _search_arc_num = _arc_num;
1080        _all_arc_num = _arc_num + _node_num;
1081        for (int u = 0, e = _arc_num; u != _node_num; ++u, ++e) {
1082          _parent[u] = _root;
1083          _pred[u] = e;
1084          _thread[u] = u + 1;
1085          _rev_thread[u + 1] = u;
1086          _succ_num[u] = 1;
1087          _last_succ[u] = u;
1088          _cap[e] = INF;
1089          _state[e] = STATE_TREE;
1090          if (_supply[u] >= 0) {
1091            _forward[u] = true;
1092            _pi[u] = 0;
1093            _source[e] = u;
1094            _target[e] = _root;
1095            _flow[e] = _supply[u];
1096            _cost[e] = 0;
1097          } else {
1098            _forward[u] = false;
1099            _pi[u] = ART_COST;
1100            _source[e] = _root;
1101            _target[e] = u;
1102            _flow[e] = -_supply[u];
1103            _cost[e] = ART_COST;
1104          }
1105        }
1106      }
1107      else if (_sum_supply > 0) {
1108        // LEQ supply constraints
1109        _search_arc_num = _arc_num + _node_num;
1110        int f = _arc_num + _node_num;
1111        for (int u = 0, e = _arc_num; u != _node_num; ++u, ++e) {
1112          _parent[u] = _root;
1113          _thread[u] = u + 1;
1114          _rev_thread[u + 1] = u;
1115          _succ_num[u] = 1;
1116          _last_succ[u] = u;
1117          if (_supply[u] >= 0) {
1118            _forward[u] = true;
1119            _pi[u] = 0;
1120            _pred[u] = e;
1121            _source[e] = u;
1122            _target[e] = _root;
1123            _cap[e] = INF;
1124            _flow[e] = _supply[u];
1125            _cost[e] = 0;
1126            _state[e] = STATE_TREE;
1127          } else {
1128            _forward[u] = false;
1129            _pi[u] = ART_COST;
1130            _pred[u] = f;
1131            _source[f] = _root;
1132            _target[f] = u;
1133            _cap[f] = INF;
1134            _flow[f] = -_supply[u];
1135            _cost[f] = ART_COST;
1136            _state[f] = STATE_TREE;
1137            _source[e] = u;
1138            _target[e] = _root;
1139            _cap[e] = INF;
1140            _flow[e] = 0;
1141            _cost[e] = 0;
1142            _state[e] = STATE_LOWER;
1143            ++f;
1144          }
1145        }
1146        _all_arc_num = f;
1147      }
1148      else {
1149        // GEQ supply constraints
1150        _search_arc_num = _arc_num + _node_num;
1151        int f = _arc_num + _node_num;
1152        for (int u = 0, e = _arc_num; u != _node_num; ++u, ++e) {
1153          _parent[u] = _root;
1154          _thread[u] = u + 1;
1155          _rev_thread[u + 1] = u;
1156          _succ_num[u] = 1;
1157          _last_succ[u] = u;
1158          if (_supply[u] <= 0) {
1159            _forward[u] = false;
1160            _pi[u] = 0;
1161            _pred[u] = e;
1162            _source[e] = _root;
1163            _target[e] = u;
1164            _cap[e] = INF;
1165            _flow[e] = -_supply[u];
1166            _cost[e] = 0;
1167            _state[e] = STATE_TREE;
1168          } else {
1169            _forward[u] = true;
1170            _pi[u] = -ART_COST;
1171            _pred[u] = f;
1172            _source[f] = u;
1173            _target[f] = _root;
1174            _cap[f] = INF;
1175            _flow[f] = _supply[u];
1176            _state[f] = STATE_TREE;
1177            _cost[f] = ART_COST;
1178            _source[e] = _root;
1179            _target[e] = u;
1180            _cap[e] = INF;
1181            _flow[e] = 0;
1182            _cost[e] = 0;
1183            _state[e] = STATE_LOWER;
1184            ++f;
1185          }
1186        }
1187        _all_arc_num = f;
1188      }
1189
1190      return true;
1191    }
1192
1193    // Find the join node
1194    void findJoinNode() {
1195      int u = _source[in_arc];
1196      int v = _target[in_arc];
1197      while (u != v) {
1198        if (_succ_num[u] < _succ_num[v]) {
1199          u = _parent[u];
1200        } else {
1201          v = _parent[v];
1202        }
1203      }
1204      join = u;
1205    }
1206
1207    // Find the leaving arc of the cycle and returns true if the
1208    // leaving arc is not the same as the entering arc
1209    bool findLeavingArc() {
1210      // Initialize first and second nodes according to the direction
1211      // of the cycle
1212      if (_state[in_arc] == STATE_LOWER) {
1213        first  = _source[in_arc];
1214        second = _target[in_arc];
1215      } else {
1216        first  = _target[in_arc];
1217        second = _source[in_arc];
1218      }
1219      delta = _cap[in_arc];
1220      int result = 0;
1221      Value d;
1222      int e;
1223
1224      // Search the cycle along the path form the first node to the root
1225      for (int u = first; u != join; u = _parent[u]) {
1226        e = _pred[u];
1227        d = _forward[u] ?
1228          _flow[e] : (_cap[e] == INF ? INF : _cap[e] - _flow[e]);
1229        if (d < delta) {
1230          delta = d;
1231          u_out = u;
1232          result = 1;
1233        }
1234      }
1235      // Search the cycle along the path form the second node to the root
1236      for (int u = second; u != join; u = _parent[u]) {
1237        e = _pred[u];
1238        d = _forward[u] ?
1239          (_cap[e] == INF ? INF : _cap[e] - _flow[e]) : _flow[e];
1240        if (d <= delta) {
1241          delta = d;
1242          u_out = u;
1243          result = 2;
1244        }
1245      }
1246
1247      if (result == 1) {
1248        u_in = first;
1249        v_in = second;
1250      } else {
1251        u_in = second;
1252        v_in = first;
1253      }
1254      return result != 0;
1255    }
1256
1257    // Change _flow and _state vectors
1258    void changeFlow(bool change) {
1259      // Augment along the cycle
1260      if (delta > 0) {
1261        Value val = _state[in_arc] * delta;
1262        _flow[in_arc] += val;
1263        for (int u = _source[in_arc]; u != join; u = _parent[u]) {
1264          _flow[_pred[u]] += _forward[u] ? -val : val;
1265        }
1266        for (int u = _target[in_arc]; u != join; u = _parent[u]) {
1267          _flow[_pred[u]] += _forward[u] ? val : -val;
1268        }
1269      }
1270      // Update the state of the entering and leaving arcs
1271      if (change) {
1272        _state[in_arc] = STATE_TREE;
1273        _state[_pred[u_out]] =
1274          (_flow[_pred[u_out]] == 0) ? STATE_LOWER : STATE_UPPER;
1275      } else {
1276        _state[in_arc] = -_state[in_arc];
1277      }
1278    }
1279
1280    // Update the tree structure
1281    void updateTreeStructure() {
1282      int u, w;
1283      int old_rev_thread = _rev_thread[u_out];
1284      int old_succ_num = _succ_num[u_out];
1285      int old_last_succ = _last_succ[u_out];
1286      v_out = _parent[u_out];
1287
1288      u = _last_succ[u_in];  // the last successor of u_in
1289      right = _thread[u];    // the node after it
1290
1291      // Handle the case when old_rev_thread equals to v_in
1292      // (it also means that join and v_out coincide)
1293      if (old_rev_thread == v_in) {
1294        last = _thread[_last_succ[u_out]];
1295      } else {
1296        last = _thread[v_in];
1297      }
1298
1299      // Update _thread and _parent along the stem nodes (i.e. the nodes
1300      // between u_in and u_out, whose parent have to be changed)
1301      _thread[v_in] = stem = u_in;
1302      _dirty_revs.clear();
1303      _dirty_revs.push_back(v_in);
1304      par_stem = v_in;
1305      while (stem != u_out) {
1306        // Insert the next stem node into the thread list
1307        new_stem = _parent[stem];
1308        _thread[u] = new_stem;
1309        _dirty_revs.push_back(u);
1310
1311        // Remove the subtree of stem from the thread list
1312        w = _rev_thread[stem];
1313        _thread[w] = right;
1314        _rev_thread[right] = w;
1315
1316        // Change the parent node and shift stem nodes
1317        _parent[stem] = par_stem;
1318        par_stem = stem;
1319        stem = new_stem;
1320
1321        // Update u and right
1322        u = _last_succ[stem] == _last_succ[par_stem] ?
1323          _rev_thread[par_stem] : _last_succ[stem];
1324        right = _thread[u];
1325      }
1326      _parent[u_out] = par_stem;
1327      _thread[u] = last;
1328      _rev_thread[last] = u;
1329      _last_succ[u_out] = u;
1330
1331      // Remove the subtree of u_out from the thread list except for
1332      // the case when old_rev_thread equals to v_in
1333      // (it also means that join and v_out coincide)
1334      if (old_rev_thread != v_in) {
1335        _thread[old_rev_thread] = right;
1336        _rev_thread[right] = old_rev_thread;
1337      }
1338
1339      // Update _rev_thread using the new _thread values
1340      for (int i = 0; i < int(_dirty_revs.size()); ++i) {
1341        u = _dirty_revs[i];
1342        _rev_thread[_thread[u]] = u;
1343      }
1344
1345      // Update _pred, _forward, _last_succ and _succ_num for the
1346      // stem nodes from u_out to u_in
1347      int tmp_sc = 0, tmp_ls = _last_succ[u_out];
1348      u = u_out;
1349      while (u != u_in) {
1350        w = _parent[u];
1351        _pred[u] = _pred[w];
1352        _forward[u] = !_forward[w];
1353        tmp_sc += _succ_num[u] - _succ_num[w];
1354        _succ_num[u] = tmp_sc;
1355        _last_succ[w] = tmp_ls;
1356        u = w;
1357      }
1358      _pred[u_in] = in_arc;
1359      _forward[u_in] = (u_in == _source[in_arc]);
1360      _succ_num[u_in] = old_succ_num;
1361
1362      // Set limits for updating _last_succ form v_in and v_out
1363      // towards the root
1364      int up_limit_in = -1;
1365      int up_limit_out = -1;
1366      if (_last_succ[join] == v_in) {
1367        up_limit_out = join;
1368      } else {
1369        up_limit_in = join;
1370      }
1371
1372      // Update _last_succ from v_in towards the root
1373      for (u = v_in; u != up_limit_in && _last_succ[u] == v_in;
1374           u = _parent[u]) {
1375        _last_succ[u] = _last_succ[u_out];
1376      }
1377      // Update _last_succ from v_out towards the root
1378      if (join != old_rev_thread && v_in != old_rev_thread) {
1379        for (u = v_out; u != up_limit_out && _last_succ[u] == old_last_succ;
1380             u = _parent[u]) {
1381          _last_succ[u] = old_rev_thread;
1382        }
1383      } else {
1384        for (u = v_out; u != up_limit_out && _last_succ[u] == old_last_succ;
1385             u = _parent[u]) {
1386          _last_succ[u] = _last_succ[u_out];
1387        }
1388      }
1389
1390      // Update _succ_num from v_in to join
1391      for (u = v_in; u != join; u = _parent[u]) {
1392        _succ_num[u] += old_succ_num;
1393      }
1394      // Update _succ_num from v_out to join
1395      for (u = v_out; u != join; u = _parent[u]) {
1396        _succ_num[u] -= old_succ_num;
1397      }
1398    }
1399
1400    // Update potentials
1401    void updatePotential() {
1402      Cost sigma = _forward[u_in] ?
1403        _pi[v_in] - _pi[u_in] - _cost[_pred[u_in]] :
1404        _pi[v_in] - _pi[u_in] + _cost[_pred[u_in]];
1405      // Update potentials in the subtree, which has been moved
1406      int end = _thread[_last_succ[u_in]];
1407      for (int u = u_in; u != end; u = _thread[u]) {
1408        _pi[u] += sigma;
1409      }
1410    }
1411
1412    // Execute the algorithm
1413    ProblemType start(PivotRule pivot_rule) {
1414      // Select the pivot rule implementation
1415      switch (pivot_rule) {
1416        case FIRST_ELIGIBLE:
1417          return start<FirstEligiblePivotRule>();
1418        case BEST_ELIGIBLE:
1419          return start<BestEligiblePivotRule>();
1420        case BLOCK_SEARCH:
1421          return start<BlockSearchPivotRule>();
1422        case CANDIDATE_LIST:
1423          return start<CandidateListPivotRule>();
1424        case ALTERING_LIST:
1425          return start<AlteringListPivotRule>();
1426      }
1427      return INFEASIBLE; // avoid warning
1428    }
1429
1430    template <typename PivotRuleImpl>
1431    ProblemType start() {
1432      PivotRuleImpl pivot(*this);
1433
1434      // Execute the Network Simplex algorithm
1435      while (pivot.findEnteringArc()) {
1436        findJoinNode();
1437        bool change = findLeavingArc();
1438        if (delta >= INF) return UNBOUNDED;
1439        changeFlow(change);
1440        if (change) {
1441          updateTreeStructure();
1442          updatePotential();
1443        }
1444      }
1445     
1446      // Check feasibility
1447      for (int e = _search_arc_num; e != _all_arc_num; ++e) {
1448        if (_flow[e] != 0) return INFEASIBLE;
1449      }
1450
1451      // Transform the solution and the supply map to the original form
1452      if (_have_lower) {
1453        for (int i = 0; i != _arc_num; ++i) {
1454          Value c = _lower[i];
1455          if (c != 0) {
1456            _flow[i] += c;
1457            _supply[_source[i]] += c;
1458            _supply[_target[i]] -= c;
1459          }
1460        }
1461      }
1462     
1463      // Shift potentials to meet the requirements of the GEQ/LEQ type
1464      // optimality conditions
1465      if (_sum_supply == 0) {
1466        if (_stype == GEQ) {
1467          Cost max_pot = std::numeric_limits<Cost>::min();
1468          for (int i = 0; i != _node_num; ++i) {
1469            if (_pi[i] > max_pot) max_pot = _pi[i];
1470          }
1471          if (max_pot > 0) {
1472            for (int i = 0; i != _node_num; ++i)
1473              _pi[i] -= max_pot;
1474          }
1475        } else {
1476          Cost min_pot = std::numeric_limits<Cost>::max();
1477          for (int i = 0; i != _node_num; ++i) {
1478            if (_pi[i] < min_pot) min_pot = _pi[i];
1479          }
1480          if (min_pot < 0) {
1481            for (int i = 0; i != _node_num; ++i)
1482              _pi[i] -= min_pot;
1483          }
1484        }
1485      }
1486
1487      return OPTIMAL;
1488    }
1489
1490  }; //class NetworkSimplex
1491
1492  ///@}
1493
1494} //namespace lemon
1495
1496#endif //LEMON_NETWORK_SIMPLEX_H
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