COIN-OR::LEMON - Graph Library

source: lemon-1.2/lemon/network_simplex.h @ 864:d3ea191c3412

Last change on this file since 864:d3ea191c3412 was 862:b6f76c95992e, checked in by Peter Kovacs <kpeter@…>, 10 years ago

Clarify type names in NetworkSimplex? (#353)

This patch clarifies the misleading effects of the renamings
in [f3bc4e9b5f3a].

File size: 50.0 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  /// \ref amo93networkflows, \ref dantzig63linearprog,
45  /// \ref kellyoneill91netsimplex.
46  /// This algorithm is a highly efficient specialized version of the
47  /// linear programming simplex method directly for the minimum cost
48  /// flow problem.
49  ///
50  /// In general, %NetworkSimplex is the fastest implementation available
51  /// in LEMON for this problem.
52  /// Moreover, it supports both directions of the supply/demand inequality
53  /// constraints. For more information, see \ref SupplyType.
54  ///
55  /// Most of the parameters of the problem (except for the digraph)
56  /// can be given using separate functions, and the algorithm can be
57  /// executed using the \ref run() function. If some parameters are not
58  /// specified, then default values will be used.
59  ///
60  /// \tparam GR The digraph type the algorithm runs on.
61  /// \tparam V The number type used for flow amounts, capacity bounds
62  /// and supply values in the algorithm. By default, it is \c int.
63  /// \tparam C The number type used for costs and potentials in the
64  /// algorithm. By default, it is the same as \c V.
65  ///
66  /// \warning Both number types must be signed and all input data must
67  /// be integer.
68  ///
69  /// \note %NetworkSimplex provides five different pivot rule
70  /// implementations, from which the most efficient one is used
71  /// by default. For more information, see \ref PivotRule.
72  template <typename GR, typename V = int, typename C = V>
73  class NetworkSimplex
74  {
75  public:
76
77    /// The type of the flow amounts, capacity bounds and supply values
78    typedef V Value;
79    /// The type of the arc costs
80    typedef C Cost;
81
82  public:
83
84    /// \brief Problem type constants for the \c run() function.
85    ///
86    /// Enum type containing the problem type constants that can be
87    /// returned by the \ref run() function of the algorithm.
88    enum ProblemType {
89      /// The problem has no feasible solution (flow).
90      INFEASIBLE,
91      /// The problem has optimal solution (i.e. it is feasible and
92      /// bounded), and the algorithm has found optimal flow and node
93      /// potentials (primal and dual solutions).
94      OPTIMAL,
95      /// The objective function of the problem is unbounded, i.e.
96      /// there is a directed cycle having negative total cost and
97      /// infinite upper bound.
98      UNBOUNDED
99    };
100   
101    /// \brief Constants for selecting the type of the supply constraints.
102    ///
103    /// Enum type containing constants for selecting the supply type,
104    /// i.e. the direction of the inequalities in the supply/demand
105    /// constraints of the \ref min_cost_flow "minimum cost flow problem".
106    ///
107    /// The default supply type is \c GEQ, the \c LEQ type can be
108    /// selected using \ref supplyType().
109    /// The equality form is a special case of both supply types.
110    enum SupplyType {
111      /// This option means that there are <em>"greater or equal"</em>
112      /// supply/demand constraints in the definition of the problem.
113      GEQ,
114      /// This option means that there are <em>"less or equal"</em>
115      /// supply/demand constraints in the definition of the problem.
116      LEQ
117    };
118   
119    /// \brief Constants for selecting the pivot rule.
120    ///
121    /// Enum type containing constants for selecting the pivot rule for
122    /// the \ref run() function.
123    ///
124    /// \ref NetworkSimplex provides five different pivot rule
125    /// implementations that significantly affect the running time
126    /// of the algorithm.
127    /// By default, \ref BLOCK_SEARCH "Block Search" is used, which
128    /// proved to be the most efficient and the most robust on various
129    /// test inputs.
130    /// However, another pivot rule can be selected using the \ref run()
131    /// function with the proper parameter.
132    enum PivotRule {
133
134      /// The \e First \e Eligible pivot rule.
135      /// The next eligible arc is selected in a wraparound fashion
136      /// in every iteration.
137      FIRST_ELIGIBLE,
138
139      /// The \e Best \e Eligible pivot rule.
140      /// The best eligible arc is selected in every iteration.
141      BEST_ELIGIBLE,
142
143      /// The \e Block \e Search pivot rule.
144      /// A specified number of arcs are examined in every iteration
145      /// in a wraparound fashion and the best eligible arc is selected
146      /// from this block.
147      BLOCK_SEARCH,
148
149      /// The \e Candidate \e List pivot rule.
150      /// In a major iteration a candidate list is built from eligible arcs
151      /// in a wraparound fashion and in the following minor iterations
152      /// the best eligible arc is selected from this list.
153      CANDIDATE_LIST,
154
155      /// The \e Altering \e Candidate \e List pivot rule.
156      /// It is a modified version of the Candidate List method.
157      /// It keeps only the several best eligible arcs from the former
158      /// candidate list and extends this list in every iteration.
159      ALTERING_LIST
160    };
161   
162  private:
163
164    TEMPLATE_DIGRAPH_TYPEDEFS(GR);
165
166    typedef std::vector<int> IntVector;
167    typedef std::vector<Value> ValueVector;
168    typedef std::vector<Cost> CostVector;
169    typedef std::vector<char> BoolVector;
170    // Note: vector<char> is used instead of vector<bool> for efficiency reasons
171
172    // State constants for arcs
173    enum ArcState {
174      STATE_UPPER = -1,
175      STATE_TREE  =  0,
176      STATE_LOWER =  1
177    };
178
179    typedef std::vector<signed char> StateVector;
180    // Note: vector<signed char> is used instead of vector<ArcState> for
181    // efficiency reasons
182
183  private:
184
185    // Data related to the underlying digraph
186    const GR &_graph;
187    int _node_num;
188    int _arc_num;
189    int _all_arc_num;
190    int _search_arc_num;
191
192    // Parameters of the problem
193    bool _have_lower;
194    SupplyType _stype;
195    Value _sum_supply;
196
197    // Data structures for storing the digraph
198    IntNodeMap _node_id;
199    IntArcMap _arc_id;
200    IntVector _source;
201    IntVector _target;
202    bool _arc_mixing;
203
204    // Node and arc data
205    ValueVector _lower;
206    ValueVector _upper;
207    ValueVector _cap;
208    CostVector _cost;
209    ValueVector _supply;
210    ValueVector _flow;
211    CostVector _pi;
212
213    // Data for storing the spanning tree structure
214    IntVector _parent;
215    IntVector _pred;
216    IntVector _thread;
217    IntVector _rev_thread;
218    IntVector _succ_num;
219    IntVector _last_succ;
220    IntVector _dirty_revs;
221    BoolVector _forward;
222    StateVector _state;
223    int _root;
224
225    // Temporary data used in the current pivot iteration
226    int in_arc, join, u_in, v_in, u_out, v_out;
227    int first, second, right, last;
228    int stem, par_stem, new_stem;
229    Value delta;
230   
231    const Value MAX;
232
233  public:
234 
235    /// \brief Constant for infinite upper bounds (capacities).
236    ///
237    /// Constant for infinite upper bounds (capacities).
238    /// It is \c std::numeric_limits<Value>::infinity() if available,
239    /// \c std::numeric_limits<Value>::max() otherwise.
240    const Value INF;
241
242  private:
243
244    // Implementation of the First Eligible pivot rule
245    class FirstEligiblePivotRule
246    {
247    private:
248
249      // References to the NetworkSimplex class
250      const IntVector  &_source;
251      const IntVector  &_target;
252      const CostVector &_cost;
253      const StateVector &_state;
254      const CostVector &_pi;
255      int &_in_arc;
256      int _search_arc_num;
257
258      // Pivot rule data
259      int _next_arc;
260
261    public:
262
263      // Constructor
264      FirstEligiblePivotRule(NetworkSimplex &ns) :
265        _source(ns._source), _target(ns._target),
266        _cost(ns._cost), _state(ns._state), _pi(ns._pi),
267        _in_arc(ns.in_arc), _search_arc_num(ns._search_arc_num),
268        _next_arc(0)
269      {}
270
271      // Find next entering arc
272      bool findEnteringArc() {
273        Cost c;
274        for (int e = _next_arc; e != _search_arc_num; ++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        for (int e = 0; e != _next_arc; ++e) {
283          c = _state[e] * (_cost[e] + _pi[_source[e]] - _pi[_target[e]]);
284          if (c < 0) {
285            _in_arc = e;
286            _next_arc = e + 1;
287            return true;
288          }
289        }
290        return false;
291      }
292
293    }; //class FirstEligiblePivotRule
294
295
296    // Implementation of the Best Eligible pivot rule
297    class BestEligiblePivotRule
298    {
299    private:
300
301      // References to the NetworkSimplex class
302      const IntVector  &_source;
303      const IntVector  &_target;
304      const CostVector &_cost;
305      const StateVector &_state;
306      const CostVector &_pi;
307      int &_in_arc;
308      int _search_arc_num;
309
310    public:
311
312      // Constructor
313      BestEligiblePivotRule(NetworkSimplex &ns) :
314        _source(ns._source), _target(ns._target),
315        _cost(ns._cost), _state(ns._state), _pi(ns._pi),
316        _in_arc(ns.in_arc), _search_arc_num(ns._search_arc_num)
317      {}
318
319      // Find next entering arc
320      bool findEnteringArc() {
321        Cost c, min = 0;
322        for (int e = 0; e != _search_arc_num; ++e) {
323          c = _state[e] * (_cost[e] + _pi[_source[e]] - _pi[_target[e]]);
324          if (c < min) {
325            min = c;
326            _in_arc = e;
327          }
328        }
329        return min < 0;
330      }
331
332    }; //class BestEligiblePivotRule
333
334
335    // Implementation of the Block Search pivot rule
336    class BlockSearchPivotRule
337    {
338    private:
339
340      // References to the NetworkSimplex class
341      const IntVector  &_source;
342      const IntVector  &_target;
343      const CostVector &_cost;
344      const StateVector &_state;
345      const CostVector &_pi;
346      int &_in_arc;
347      int _search_arc_num;
348
349      // Pivot rule data
350      int _block_size;
351      int _next_arc;
352
353    public:
354
355      // Constructor
356      BlockSearchPivotRule(NetworkSimplex &ns) :
357        _source(ns._source), _target(ns._target),
358        _cost(ns._cost), _state(ns._state), _pi(ns._pi),
359        _in_arc(ns.in_arc), _search_arc_num(ns._search_arc_num),
360        _next_arc(0)
361      {
362        // The main parameters of the pivot rule
363        const double BLOCK_SIZE_FACTOR = 1.0;
364        const int MIN_BLOCK_SIZE = 10;
365
366        _block_size = std::max( int(BLOCK_SIZE_FACTOR *
367                                    std::sqrt(double(_search_arc_num))),
368                                MIN_BLOCK_SIZE );
369      }
370
371      // Find next entering arc
372      bool findEnteringArc() {
373        Cost c, min = 0;
374        int cnt = _block_size;
375        int e;
376        for (e = _next_arc; e != _search_arc_num; ++e) {
377          c = _state[e] * (_cost[e] + _pi[_source[e]] - _pi[_target[e]]);
378          if (c < min) {
379            min = c;
380            _in_arc = e;
381          }
382          if (--cnt == 0) {
383            if (min < 0) goto search_end;
384            cnt = _block_size;
385          }
386        }
387        for (e = 0; e != _next_arc; ++e) {
388          c = _state[e] * (_cost[e] + _pi[_source[e]] - _pi[_target[e]]);
389          if (c < min) {
390            min = c;
391            _in_arc = e;
392          }
393          if (--cnt == 0) {
394            if (min < 0) goto search_end;
395            cnt = _block_size;
396          }
397        }
398        if (min >= 0) return false;
399
400      search_end:
401        _next_arc = e;
402        return true;
403      }
404
405    }; //class BlockSearchPivotRule
406
407
408    // Implementation of the Candidate List pivot rule
409    class CandidateListPivotRule
410    {
411    private:
412
413      // References to the NetworkSimplex class
414      const IntVector  &_source;
415      const IntVector  &_target;
416      const CostVector &_cost;
417      const StateVector &_state;
418      const CostVector &_pi;
419      int &_in_arc;
420      int _search_arc_num;
421
422      // Pivot rule data
423      IntVector _candidates;
424      int _list_length, _minor_limit;
425      int _curr_length, _minor_count;
426      int _next_arc;
427
428    public:
429
430      /// Constructor
431      CandidateListPivotRule(NetworkSimplex &ns) :
432        _source(ns._source), _target(ns._target),
433        _cost(ns._cost), _state(ns._state), _pi(ns._pi),
434        _in_arc(ns.in_arc), _search_arc_num(ns._search_arc_num),
435        _next_arc(0)
436      {
437        // The main parameters of the pivot rule
438        const double LIST_LENGTH_FACTOR = 0.25;
439        const int MIN_LIST_LENGTH = 10;
440        const double MINOR_LIMIT_FACTOR = 0.1;
441        const int MIN_MINOR_LIMIT = 3;
442
443        _list_length = std::max( int(LIST_LENGTH_FACTOR *
444                                     std::sqrt(double(_search_arc_num))),
445                                 MIN_LIST_LENGTH );
446        _minor_limit = std::max( int(MINOR_LIMIT_FACTOR * _list_length),
447                                 MIN_MINOR_LIMIT );
448        _curr_length = _minor_count = 0;
449        _candidates.resize(_list_length);
450      }
451
452      /// Find next entering arc
453      bool findEnteringArc() {
454        Cost min, c;
455        int e;
456        if (_curr_length > 0 && _minor_count < _minor_limit) {
457          // Minor iteration: select the best eligible arc from the
458          // current candidate list
459          ++_minor_count;
460          min = 0;
461          for (int i = 0; i < _curr_length; ++i) {
462            e = _candidates[i];
463            c = _state[e] * (_cost[e] + _pi[_source[e]] - _pi[_target[e]]);
464            if (c < min) {
465              min = c;
466              _in_arc = e;
467            }
468            else if (c >= 0) {
469              _candidates[i--] = _candidates[--_curr_length];
470            }
471          }
472          if (min < 0) return true;
473        }
474
475        // Major iteration: build a new candidate list
476        min = 0;
477        _curr_length = 0;
478        for (e = _next_arc; e != _search_arc_num; ++e) {
479          c = _state[e] * (_cost[e] + _pi[_source[e]] - _pi[_target[e]]);
480          if (c < 0) {
481            _candidates[_curr_length++] = e;
482            if (c < min) {
483              min = c;
484              _in_arc = e;
485            }
486            if (_curr_length == _list_length) goto search_end;
487          }
488        }
489        for (e = 0; e != _next_arc; ++e) {
490          c = _state[e] * (_cost[e] + _pi[_source[e]] - _pi[_target[e]]);
491          if (c < 0) {
492            _candidates[_curr_length++] = e;
493            if (c < min) {
494              min = c;
495              _in_arc = e;
496            }
497            if (_curr_length == _list_length) goto search_end;
498          }
499        }
500        if (_curr_length == 0) return false;
501     
502      search_end:       
503        _minor_count = 1;
504        _next_arc = e;
505        return true;
506      }
507
508    }; //class CandidateListPivotRule
509
510
511    // Implementation of the Altering Candidate List pivot rule
512    class AlteringListPivotRule
513    {
514    private:
515
516      // References to the NetworkSimplex class
517      const IntVector  &_source;
518      const IntVector  &_target;
519      const CostVector &_cost;
520      const StateVector &_state;
521      const CostVector &_pi;
522      int &_in_arc;
523      int _search_arc_num;
524
525      // Pivot rule data
526      int _block_size, _head_length, _curr_length;
527      int _next_arc;
528      IntVector _candidates;
529      CostVector _cand_cost;
530
531      // Functor class to compare arcs during sort of the candidate list
532      class SortFunc
533      {
534      private:
535        const CostVector &_map;
536      public:
537        SortFunc(const CostVector &map) : _map(map) {}
538        bool operator()(int left, int right) {
539          return _map[left] > _map[right];
540        }
541      };
542
543      SortFunc _sort_func;
544
545    public:
546
547      // Constructor
548      AlteringListPivotRule(NetworkSimplex &ns) :
549        _source(ns._source), _target(ns._target),
550        _cost(ns._cost), _state(ns._state), _pi(ns._pi),
551        _in_arc(ns.in_arc), _search_arc_num(ns._search_arc_num),
552        _next_arc(0), _cand_cost(ns._search_arc_num), _sort_func(_cand_cost)
553      {
554        // The main parameters of the pivot rule
555        const double BLOCK_SIZE_FACTOR = 1.0;
556        const int MIN_BLOCK_SIZE = 10;
557        const double HEAD_LENGTH_FACTOR = 0.1;
558        const int MIN_HEAD_LENGTH = 3;
559
560        _block_size = std::max( int(BLOCK_SIZE_FACTOR *
561                                    std::sqrt(double(_search_arc_num))),
562                                MIN_BLOCK_SIZE );
563        _head_length = std::max( int(HEAD_LENGTH_FACTOR * _block_size),
564                                 MIN_HEAD_LENGTH );
565        _candidates.resize(_head_length + _block_size);
566        _curr_length = 0;
567      }
568
569      // Find next entering arc
570      bool findEnteringArc() {
571        // Check the current candidate list
572        int e;
573        for (int i = 0; i != _curr_length; ++i) {
574          e = _candidates[i];
575          _cand_cost[e] = _state[e] *
576            (_cost[e] + _pi[_source[e]] - _pi[_target[e]]);
577          if (_cand_cost[e] >= 0) {
578            _candidates[i--] = _candidates[--_curr_length];
579          }
580        }
581
582        // Extend the list
583        int cnt = _block_size;
584        int limit = _head_length;
585
586        for (e = _next_arc; e != _search_arc_num; ++e) {
587          _cand_cost[e] = _state[e] *
588            (_cost[e] + _pi[_source[e]] - _pi[_target[e]]);
589          if (_cand_cost[e] < 0) {
590            _candidates[_curr_length++] = e;
591          }
592          if (--cnt == 0) {
593            if (_curr_length > limit) goto search_end;
594            limit = 0;
595            cnt = _block_size;
596          }
597        }
598        for (e = 0; e != _next_arc; ++e) {
599          _cand_cost[e] = _state[e] *
600            (_cost[e] + _pi[_source[e]] - _pi[_target[e]]);
601          if (_cand_cost[e] < 0) {
602            _candidates[_curr_length++] = e;
603          }
604          if (--cnt == 0) {
605            if (_curr_length > limit) goto search_end;
606            limit = 0;
607            cnt = _block_size;
608          }
609        }
610        if (_curr_length == 0) return false;
611       
612      search_end:
613
614        // Make heap of the candidate list (approximating a partial sort)
615        make_heap( _candidates.begin(), _candidates.begin() + _curr_length,
616                   _sort_func );
617
618        // Pop the first element of the heap
619        _in_arc = _candidates[0];
620        _next_arc = e;
621        pop_heap( _candidates.begin(), _candidates.begin() + _curr_length,
622                  _sort_func );
623        _curr_length = std::min(_head_length, _curr_length - 1);
624        return true;
625      }
626
627    }; //class AlteringListPivotRule
628
629  public:
630
631    /// \brief Constructor.
632    ///
633    /// The constructor of the class.
634    ///
635    /// \param graph The digraph the algorithm runs on.
636    /// \param arc_mixing Indicate if the arcs have to be stored in a
637    /// mixed order in the internal data structure.
638    /// In special cases, it could lead to better overall performance,
639    /// but it is usually slower. Therefore it is disabled by default.
640    NetworkSimplex(const GR& graph, bool arc_mixing = false) :
641      _graph(graph), _node_id(graph), _arc_id(graph),
642      _arc_mixing(arc_mixing),
643      MAX(std::numeric_limits<Value>::max()),
644      INF(std::numeric_limits<Value>::has_infinity ?
645          std::numeric_limits<Value>::infinity() : MAX)
646    {
647      // Check the number types
648      LEMON_ASSERT(std::numeric_limits<Value>::is_signed,
649        "The flow type of NetworkSimplex must be signed");
650      LEMON_ASSERT(std::numeric_limits<Cost>::is_signed,
651        "The cost type of NetworkSimplex must be signed");
652       
653      // Reset data structures
654      reset();
655    }
656
657    /// \name Parameters
658    /// The parameters of the algorithm can be specified using these
659    /// functions.
660
661    /// @{
662
663    /// \brief Set the lower bounds on the arcs.
664    ///
665    /// This function sets the lower bounds on the arcs.
666    /// If it is not used before calling \ref run(), the lower bounds
667    /// will be set to zero on all arcs.
668    ///
669    /// \param map An arc map storing the lower bounds.
670    /// Its \c Value type must be convertible to the \c Value type
671    /// of the algorithm.
672    ///
673    /// \return <tt>(*this)</tt>
674    template <typename LowerMap>
675    NetworkSimplex& lowerMap(const LowerMap& map) {
676      _have_lower = true;
677      for (ArcIt a(_graph); a != INVALID; ++a) {
678        _lower[_arc_id[a]] = map[a];
679      }
680      return *this;
681    }
682
683    /// \brief Set the upper bounds (capacities) on the arcs.
684    ///
685    /// This function sets the upper bounds (capacities) on the arcs.
686    /// If it is not used before calling \ref run(), the upper bounds
687    /// will be set to \ref INF on all arcs (i.e. the flow value will be
688    /// unbounded from above).
689    ///
690    /// \param map An arc map storing the upper bounds.
691    /// Its \c Value type must be convertible to the \c Value type
692    /// of the algorithm.
693    ///
694    /// \return <tt>(*this)</tt>
695    template<typename UpperMap>
696    NetworkSimplex& upperMap(const UpperMap& map) {
697      for (ArcIt a(_graph); a != INVALID; ++a) {
698        _upper[_arc_id[a]] = map[a];
699      }
700      return *this;
701    }
702
703    /// \brief Set the costs of the arcs.
704    ///
705    /// This function sets the costs of the arcs.
706    /// If it is not used before calling \ref run(), the costs
707    /// will be set to \c 1 on all arcs.
708    ///
709    /// \param map An arc map storing the costs.
710    /// Its \c Value type must be convertible to the \c Cost type
711    /// of the algorithm.
712    ///
713    /// \return <tt>(*this)</tt>
714    template<typename CostMap>
715    NetworkSimplex& costMap(const CostMap& map) {
716      for (ArcIt a(_graph); a != INVALID; ++a) {
717        _cost[_arc_id[a]] = map[a];
718      }
719      return *this;
720    }
721
722    /// \brief Set the supply values of the nodes.
723    ///
724    /// This function sets the supply values of the nodes.
725    /// If neither this function nor \ref stSupply() is used before
726    /// calling \ref run(), the supply of each node will be set to zero.
727    ///
728    /// \param map A node map storing the supply values.
729    /// Its \c Value type must be convertible to the \c Value type
730    /// of the algorithm.
731    ///
732    /// \return <tt>(*this)</tt>
733    template<typename SupplyMap>
734    NetworkSimplex& supplyMap(const SupplyMap& map) {
735      for (NodeIt n(_graph); n != INVALID; ++n) {
736        _supply[_node_id[n]] = map[n];
737      }
738      return *this;
739    }
740
741    /// \brief Set single source and target nodes and a supply value.
742    ///
743    /// This function sets a single source node and a single target node
744    /// and the required flow value.
745    /// If neither this function nor \ref supplyMap() is used before
746    /// calling \ref run(), the supply of each node will be set to zero.
747    ///
748    /// Using this function has the same effect as using \ref supplyMap()
749    /// with such a map in which \c k is assigned to \c s, \c -k is
750    /// assigned to \c t and all other nodes have zero supply value.
751    ///
752    /// \param s The source node.
753    /// \param t The target node.
754    /// \param k The required amount of flow from node \c s to node \c t
755    /// (i.e. the supply of \c s and the demand of \c t).
756    ///
757    /// \return <tt>(*this)</tt>
758    NetworkSimplex& stSupply(const Node& s, const Node& t, Value k) {
759      for (int i = 0; i != _node_num; ++i) {
760        _supply[i] = 0;
761      }
762      _supply[_node_id[s]] =  k;
763      _supply[_node_id[t]] = -k;
764      return *this;
765    }
766   
767    /// \brief Set the type of the supply constraints.
768    ///
769    /// This function sets the type of the supply/demand constraints.
770    /// If it is not used before calling \ref run(), the \ref GEQ supply
771    /// type will be used.
772    ///
773    /// For more information, see \ref SupplyType.
774    ///
775    /// \return <tt>(*this)</tt>
776    NetworkSimplex& supplyType(SupplyType supply_type) {
777      _stype = supply_type;
778      return *this;
779    }
780
781    /// @}
782
783    /// \name Execution Control
784    /// The algorithm can be executed using \ref run().
785
786    /// @{
787
788    /// \brief Run the algorithm.
789    ///
790    /// This function runs the algorithm.
791    /// The paramters can be specified using functions \ref lowerMap(),
792    /// \ref upperMap(), \ref costMap(), \ref supplyMap(), \ref stSupply(),
793    /// \ref supplyType().
794    /// For example,
795    /// \code
796    ///   NetworkSimplex<ListDigraph> ns(graph);
797    ///   ns.lowerMap(lower).upperMap(upper).costMap(cost)
798    ///     .supplyMap(sup).run();
799    /// \endcode
800    ///
801    /// This function can be called more than once. All the given parameters
802    /// are kept for the next call, unless \ref resetParams() or \ref reset()
803    /// is used, thus only the modified parameters have to be set again.
804    /// If the underlying digraph was also modified after the construction
805    /// of the class (or the last \ref reset() call), then the \ref reset()
806    /// function must be called.
807    ///
808    /// \param pivot_rule The pivot rule that will be used during the
809    /// algorithm. For more information, see \ref PivotRule.
810    ///
811    /// \return \c INFEASIBLE if no feasible flow exists,
812    /// \n \c OPTIMAL if the problem has optimal solution
813    /// (i.e. it is feasible and bounded), and the algorithm has found
814    /// optimal flow and node potentials (primal and dual solutions),
815    /// \n \c UNBOUNDED if the objective function of the problem is
816    /// unbounded, i.e. there is a directed cycle having negative total
817    /// cost and infinite upper bound.
818    ///
819    /// \see ProblemType, PivotRule
820    /// \see resetParams(), reset()
821    ProblemType run(PivotRule pivot_rule = BLOCK_SEARCH) {
822      if (!init()) return INFEASIBLE;
823      return start(pivot_rule);
824    }
825
826    /// \brief Reset all the parameters that have been given before.
827    ///
828    /// This function resets all the paramaters that have been given
829    /// before using functions \ref lowerMap(), \ref upperMap(),
830    /// \ref costMap(), \ref supplyMap(), \ref stSupply(), \ref supplyType().
831    ///
832    /// It is useful for multiple \ref run() calls. Basically, all the given
833    /// parameters are kept for the next \ref run() call, unless
834    /// \ref resetParams() or \ref reset() is used.
835    /// If the underlying digraph was also modified after the construction
836    /// of the class or the last \ref reset() call, then the \ref reset()
837    /// function must be used, otherwise \ref resetParams() is sufficient.
838    ///
839    /// For example,
840    /// \code
841    ///   NetworkSimplex<ListDigraph> ns(graph);
842    ///
843    ///   // First run
844    ///   ns.lowerMap(lower).upperMap(upper).costMap(cost)
845    ///     .supplyMap(sup).run();
846    ///
847    ///   // Run again with modified cost map (resetParams() is not called,
848    ///   // so only the cost map have to be set again)
849    ///   cost[e] += 100;
850    ///   ns.costMap(cost).run();
851    ///
852    ///   // Run again from scratch using resetParams()
853    ///   // (the lower bounds will be set to zero on all arcs)
854    ///   ns.resetParams();
855    ///   ns.upperMap(capacity).costMap(cost)
856    ///     .supplyMap(sup).run();
857    /// \endcode
858    ///
859    /// \return <tt>(*this)</tt>
860    ///
861    /// \see reset(), run()
862    NetworkSimplex& resetParams() {
863      for (int i = 0; i != _node_num; ++i) {
864        _supply[i] = 0;
865      }
866      for (int i = 0; i != _arc_num; ++i) {
867        _lower[i] = 0;
868        _upper[i] = INF;
869        _cost[i] = 1;
870      }
871      _have_lower = false;
872      _stype = GEQ;
873      return *this;
874    }
875
876    /// \brief Reset the internal data structures and all the parameters
877    /// that have been given before.
878    ///
879    /// This function resets the internal data structures and all the
880    /// paramaters that have been given before using functions \ref lowerMap(),
881    /// \ref upperMap(), \ref costMap(), \ref supplyMap(), \ref stSupply(),
882    /// \ref supplyType().
883    ///
884    /// It is useful for multiple \ref run() calls. Basically, all the given
885    /// parameters are kept for the next \ref run() call, unless
886    /// \ref resetParams() or \ref reset() is used.
887    /// If the underlying digraph was also modified after the construction
888    /// of the class or the last \ref reset() call, then the \ref reset()
889    /// function must be used, otherwise \ref resetParams() is sufficient.
890    ///
891    /// See \ref resetParams() for examples.
892    ///
893    /// \return <tt>(*this)</tt>
894    ///
895    /// \see resetParams(), run()
896    NetworkSimplex& reset() {
897      // Resize vectors
898      _node_num = countNodes(_graph);
899      _arc_num = countArcs(_graph);
900      int all_node_num = _node_num + 1;
901      int max_arc_num = _arc_num + 2 * _node_num;
902
903      _source.resize(max_arc_num);
904      _target.resize(max_arc_num);
905
906      _lower.resize(_arc_num);
907      _upper.resize(_arc_num);
908      _cap.resize(max_arc_num);
909      _cost.resize(max_arc_num);
910      _supply.resize(all_node_num);
911      _flow.resize(max_arc_num);
912      _pi.resize(all_node_num);
913
914      _parent.resize(all_node_num);
915      _pred.resize(all_node_num);
916      _forward.resize(all_node_num);
917      _thread.resize(all_node_num);
918      _rev_thread.resize(all_node_num);
919      _succ_num.resize(all_node_num);
920      _last_succ.resize(all_node_num);
921      _state.resize(max_arc_num);
922
923      // Copy the graph
924      int i = 0;
925      for (NodeIt n(_graph); n != INVALID; ++n, ++i) {
926        _node_id[n] = i;
927      }
928      if (_arc_mixing) {
929        // Store the arcs in a mixed order
930        int k = std::max(int(std::sqrt(double(_arc_num))), 10);
931        int i = 0, j = 0;
932        for (ArcIt a(_graph); a != INVALID; ++a) {
933          _arc_id[a] = i;
934          _source[i] = _node_id[_graph.source(a)];
935          _target[i] = _node_id[_graph.target(a)];
936          if ((i += k) >= _arc_num) i = ++j;
937        }
938      } else {
939        // Store the arcs in the original order
940        int i = 0;
941        for (ArcIt a(_graph); a != INVALID; ++a, ++i) {
942          _arc_id[a] = i;
943          _source[i] = _node_id[_graph.source(a)];
944          _target[i] = _node_id[_graph.target(a)];
945        }
946      }
947     
948      // Reset parameters
949      resetParams();
950      return *this;
951    }
952   
953    /// @}
954
955    /// \name Query Functions
956    /// The results of the algorithm can be obtained using these
957    /// functions.\n
958    /// The \ref run() function must be called before using them.
959
960    /// @{
961
962    /// \brief Return the total cost of the found flow.
963    ///
964    /// This function returns the total cost of the found flow.
965    /// Its complexity is O(e).
966    ///
967    /// \note The return type of the function can be specified as a
968    /// template parameter. For example,
969    /// \code
970    ///   ns.totalCost<double>();
971    /// \endcode
972    /// It is useful if the total cost cannot be stored in the \c Cost
973    /// type of the algorithm, which is the default return type of the
974    /// function.
975    ///
976    /// \pre \ref run() must be called before using this function.
977    template <typename Number>
978    Number totalCost() const {
979      Number c = 0;
980      for (ArcIt a(_graph); a != INVALID; ++a) {
981        int i = _arc_id[a];
982        c += Number(_flow[i]) * Number(_cost[i]);
983      }
984      return c;
985    }
986
987#ifndef DOXYGEN
988    Cost totalCost() const {
989      return totalCost<Cost>();
990    }
991#endif
992
993    /// \brief Return the flow on the given arc.
994    ///
995    /// This function returns the flow on the given arc.
996    ///
997    /// \pre \ref run() must be called before using this function.
998    Value flow(const Arc& a) const {
999      return _flow[_arc_id[a]];
1000    }
1001
1002    /// \brief Return the flow map (the primal solution).
1003    ///
1004    /// This function copies the flow value on each arc into the given
1005    /// map. The \c Value type of the algorithm must be convertible to
1006    /// the \c Value type of the map.
1007    ///
1008    /// \pre \ref run() must be called before using this function.
1009    template <typename FlowMap>
1010    void flowMap(FlowMap &map) const {
1011      for (ArcIt a(_graph); a != INVALID; ++a) {
1012        map.set(a, _flow[_arc_id[a]]);
1013      }
1014    }
1015
1016    /// \brief Return the potential (dual value) of the given node.
1017    ///
1018    /// This function returns the potential (dual value) of the
1019    /// given node.
1020    ///
1021    /// \pre \ref run() must be called before using this function.
1022    Cost potential(const Node& n) const {
1023      return _pi[_node_id[n]];
1024    }
1025
1026    /// \brief Return the potential map (the dual solution).
1027    ///
1028    /// This function copies the potential (dual value) of each node
1029    /// into the given map.
1030    /// The \c Cost type of the algorithm must be convertible to the
1031    /// \c Value type of the map.
1032    ///
1033    /// \pre \ref run() must be called before using this function.
1034    template <typename PotentialMap>
1035    void potentialMap(PotentialMap &map) const {
1036      for (NodeIt n(_graph); n != INVALID; ++n) {
1037        map.set(n, _pi[_node_id[n]]);
1038      }
1039    }
1040
1041    /// @}
1042
1043  private:
1044
1045    // Initialize internal data structures
1046    bool init() {
1047      if (_node_num == 0) return false;
1048
1049      // Check the sum of supply values
1050      _sum_supply = 0;
1051      for (int i = 0; i != _node_num; ++i) {
1052        _sum_supply += _supply[i];
1053      }
1054      if ( !((_stype == GEQ && _sum_supply <= 0) ||
1055             (_stype == LEQ && _sum_supply >= 0)) ) return false;
1056
1057      // Remove non-zero lower bounds
1058      if (_have_lower) {
1059        for (int i = 0; i != _arc_num; ++i) {
1060          Value c = _lower[i];
1061          if (c >= 0) {
1062            _cap[i] = _upper[i] < MAX ? _upper[i] - c : INF;
1063          } else {
1064            _cap[i] = _upper[i] < MAX + c ? _upper[i] - c : INF;
1065          }
1066          _supply[_source[i]] -= c;
1067          _supply[_target[i]] += c;
1068        }
1069      } else {
1070        for (int i = 0; i != _arc_num; ++i) {
1071          _cap[i] = _upper[i];
1072        }
1073      }
1074
1075      // Initialize artifical cost
1076      Cost ART_COST;
1077      if (std::numeric_limits<Cost>::is_exact) {
1078        ART_COST = std::numeric_limits<Cost>::max() / 2 + 1;
1079      } else {
1080        ART_COST = std::numeric_limits<Cost>::min();
1081        for (int i = 0; i != _arc_num; ++i) {
1082          if (_cost[i] > ART_COST) ART_COST = _cost[i];
1083        }
1084        ART_COST = (ART_COST + 1) * _node_num;
1085      }
1086
1087      // Initialize arc maps
1088      for (int i = 0; i != _arc_num; ++i) {
1089        _flow[i] = 0;
1090        _state[i] = STATE_LOWER;
1091      }
1092     
1093      // Set data for the artificial root node
1094      _root = _node_num;
1095      _parent[_root] = -1;
1096      _pred[_root] = -1;
1097      _thread[_root] = 0;
1098      _rev_thread[0] = _root;
1099      _succ_num[_root] = _node_num + 1;
1100      _last_succ[_root] = _root - 1;
1101      _supply[_root] = -_sum_supply;
1102      _pi[_root] = 0;
1103
1104      // Add artificial arcs and initialize the spanning tree data structure
1105      if (_sum_supply == 0) {
1106        // EQ supply constraints
1107        _search_arc_num = _arc_num;
1108        _all_arc_num = _arc_num + _node_num;
1109        for (int u = 0, e = _arc_num; u != _node_num; ++u, ++e) {
1110          _parent[u] = _root;
1111          _pred[u] = e;
1112          _thread[u] = u + 1;
1113          _rev_thread[u + 1] = u;
1114          _succ_num[u] = 1;
1115          _last_succ[u] = u;
1116          _cap[e] = INF;
1117          _state[e] = STATE_TREE;
1118          if (_supply[u] >= 0) {
1119            _forward[u] = true;
1120            _pi[u] = 0;
1121            _source[e] = u;
1122            _target[e] = _root;
1123            _flow[e] = _supply[u];
1124            _cost[e] = 0;
1125          } else {
1126            _forward[u] = false;
1127            _pi[u] = ART_COST;
1128            _source[e] = _root;
1129            _target[e] = u;
1130            _flow[e] = -_supply[u];
1131            _cost[e] = ART_COST;
1132          }
1133        }
1134      }
1135      else if (_sum_supply > 0) {
1136        // LEQ supply constraints
1137        _search_arc_num = _arc_num + _node_num;
1138        int f = _arc_num + _node_num;
1139        for (int u = 0, e = _arc_num; u != _node_num; ++u, ++e) {
1140          _parent[u] = _root;
1141          _thread[u] = u + 1;
1142          _rev_thread[u + 1] = u;
1143          _succ_num[u] = 1;
1144          _last_succ[u] = u;
1145          if (_supply[u] >= 0) {
1146            _forward[u] = true;
1147            _pi[u] = 0;
1148            _pred[u] = e;
1149            _source[e] = u;
1150            _target[e] = _root;
1151            _cap[e] = INF;
1152            _flow[e] = _supply[u];
1153            _cost[e] = 0;
1154            _state[e] = STATE_TREE;
1155          } else {
1156            _forward[u] = false;
1157            _pi[u] = ART_COST;
1158            _pred[u] = f;
1159            _source[f] = _root;
1160            _target[f] = u;
1161            _cap[f] = INF;
1162            _flow[f] = -_supply[u];
1163            _cost[f] = ART_COST;
1164            _state[f] = STATE_TREE;
1165            _source[e] = u;
1166            _target[e] = _root;
1167            _cap[e] = INF;
1168            _flow[e] = 0;
1169            _cost[e] = 0;
1170            _state[e] = STATE_LOWER;
1171            ++f;
1172          }
1173        }
1174        _all_arc_num = f;
1175      }
1176      else {
1177        // GEQ supply constraints
1178        _search_arc_num = _arc_num + _node_num;
1179        int f = _arc_num + _node_num;
1180        for (int u = 0, e = _arc_num; u != _node_num; ++u, ++e) {
1181          _parent[u] = _root;
1182          _thread[u] = u + 1;
1183          _rev_thread[u + 1] = u;
1184          _succ_num[u] = 1;
1185          _last_succ[u] = u;
1186          if (_supply[u] <= 0) {
1187            _forward[u] = false;
1188            _pi[u] = 0;
1189            _pred[u] = e;
1190            _source[e] = _root;
1191            _target[e] = u;
1192            _cap[e] = INF;
1193            _flow[e] = -_supply[u];
1194            _cost[e] = 0;
1195            _state[e] = STATE_TREE;
1196          } else {
1197            _forward[u] = true;
1198            _pi[u] = -ART_COST;
1199            _pred[u] = f;
1200            _source[f] = u;
1201            _target[f] = _root;
1202            _cap[f] = INF;
1203            _flow[f] = _supply[u];
1204            _state[f] = STATE_TREE;
1205            _cost[f] = ART_COST;
1206            _source[e] = _root;
1207            _target[e] = u;
1208            _cap[e] = INF;
1209            _flow[e] = 0;
1210            _cost[e] = 0;
1211            _state[e] = STATE_LOWER;
1212            ++f;
1213          }
1214        }
1215        _all_arc_num = f;
1216      }
1217
1218      return true;
1219    }
1220
1221    // Find the join node
1222    void findJoinNode() {
1223      int u = _source[in_arc];
1224      int v = _target[in_arc];
1225      while (u != v) {
1226        if (_succ_num[u] < _succ_num[v]) {
1227          u = _parent[u];
1228        } else {
1229          v = _parent[v];
1230        }
1231      }
1232      join = u;
1233    }
1234
1235    // Find the leaving arc of the cycle and returns true if the
1236    // leaving arc is not the same as the entering arc
1237    bool findLeavingArc() {
1238      // Initialize first and second nodes according to the direction
1239      // of the cycle
1240      if (_state[in_arc] == STATE_LOWER) {
1241        first  = _source[in_arc];
1242        second = _target[in_arc];
1243      } else {
1244        first  = _target[in_arc];
1245        second = _source[in_arc];
1246      }
1247      delta = _cap[in_arc];
1248      int result = 0;
1249      Value d;
1250      int e;
1251
1252      // Search the cycle along the path form the first node to the root
1253      for (int u = first; u != join; u = _parent[u]) {
1254        e = _pred[u];
1255        d = _forward[u] ?
1256          _flow[e] : (_cap[e] >= MAX ? INF : _cap[e] - _flow[e]);
1257        if (d < delta) {
1258          delta = d;
1259          u_out = u;
1260          result = 1;
1261        }
1262      }
1263      // Search the cycle along the path form the second node to the root
1264      for (int u = second; u != join; u = _parent[u]) {
1265        e = _pred[u];
1266        d = _forward[u] ?
1267          (_cap[e] >= MAX ? INF : _cap[e] - _flow[e]) : _flow[e];
1268        if (d <= delta) {
1269          delta = d;
1270          u_out = u;
1271          result = 2;
1272        }
1273      }
1274
1275      if (result == 1) {
1276        u_in = first;
1277        v_in = second;
1278      } else {
1279        u_in = second;
1280        v_in = first;
1281      }
1282      return result != 0;
1283    }
1284
1285    // Change _flow and _state vectors
1286    void changeFlow(bool change) {
1287      // Augment along the cycle
1288      if (delta > 0) {
1289        Value val = _state[in_arc] * delta;
1290        _flow[in_arc] += val;
1291        for (int u = _source[in_arc]; u != join; u = _parent[u]) {
1292          _flow[_pred[u]] += _forward[u] ? -val : val;
1293        }
1294        for (int u = _target[in_arc]; u != join; u = _parent[u]) {
1295          _flow[_pred[u]] += _forward[u] ? val : -val;
1296        }
1297      }
1298      // Update the state of the entering and leaving arcs
1299      if (change) {
1300        _state[in_arc] = STATE_TREE;
1301        _state[_pred[u_out]] =
1302          (_flow[_pred[u_out]] == 0) ? STATE_LOWER : STATE_UPPER;
1303      } else {
1304        _state[in_arc] = -_state[in_arc];
1305      }
1306    }
1307
1308    // Update the tree structure
1309    void updateTreeStructure() {
1310      int u, w;
1311      int old_rev_thread = _rev_thread[u_out];
1312      int old_succ_num = _succ_num[u_out];
1313      int old_last_succ = _last_succ[u_out];
1314      v_out = _parent[u_out];
1315
1316      u = _last_succ[u_in];  // the last successor of u_in
1317      right = _thread[u];    // the node after it
1318
1319      // Handle the case when old_rev_thread equals to v_in
1320      // (it also means that join and v_out coincide)
1321      if (old_rev_thread == v_in) {
1322        last = _thread[_last_succ[u_out]];
1323      } else {
1324        last = _thread[v_in];
1325      }
1326
1327      // Update _thread and _parent along the stem nodes (i.e. the nodes
1328      // between u_in and u_out, whose parent have to be changed)
1329      _thread[v_in] = stem = u_in;
1330      _dirty_revs.clear();
1331      _dirty_revs.push_back(v_in);
1332      par_stem = v_in;
1333      while (stem != u_out) {
1334        // Insert the next stem node into the thread list
1335        new_stem = _parent[stem];
1336        _thread[u] = new_stem;
1337        _dirty_revs.push_back(u);
1338
1339        // Remove the subtree of stem from the thread list
1340        w = _rev_thread[stem];
1341        _thread[w] = right;
1342        _rev_thread[right] = w;
1343
1344        // Change the parent node and shift stem nodes
1345        _parent[stem] = par_stem;
1346        par_stem = stem;
1347        stem = new_stem;
1348
1349        // Update u and right
1350        u = _last_succ[stem] == _last_succ[par_stem] ?
1351          _rev_thread[par_stem] : _last_succ[stem];
1352        right = _thread[u];
1353      }
1354      _parent[u_out] = par_stem;
1355      _thread[u] = last;
1356      _rev_thread[last] = u;
1357      _last_succ[u_out] = u;
1358
1359      // Remove the subtree of u_out from the thread list except for
1360      // the case when old_rev_thread equals to v_in
1361      // (it also means that join and v_out coincide)
1362      if (old_rev_thread != v_in) {
1363        _thread[old_rev_thread] = right;
1364        _rev_thread[right] = old_rev_thread;
1365      }
1366
1367      // Update _rev_thread using the new _thread values
1368      for (int i = 0; i != int(_dirty_revs.size()); ++i) {
1369        u = _dirty_revs[i];
1370        _rev_thread[_thread[u]] = u;
1371      }
1372
1373      // Update _pred, _forward, _last_succ and _succ_num for the
1374      // stem nodes from u_out to u_in
1375      int tmp_sc = 0, tmp_ls = _last_succ[u_out];
1376      u = u_out;
1377      while (u != u_in) {
1378        w = _parent[u];
1379        _pred[u] = _pred[w];
1380        _forward[u] = !_forward[w];
1381        tmp_sc += _succ_num[u] - _succ_num[w];
1382        _succ_num[u] = tmp_sc;
1383        _last_succ[w] = tmp_ls;
1384        u = w;
1385      }
1386      _pred[u_in] = in_arc;
1387      _forward[u_in] = (u_in == _source[in_arc]);
1388      _succ_num[u_in] = old_succ_num;
1389
1390      // Set limits for updating _last_succ form v_in and v_out
1391      // towards the root
1392      int up_limit_in = -1;
1393      int up_limit_out = -1;
1394      if (_last_succ[join] == v_in) {
1395        up_limit_out = join;
1396      } else {
1397        up_limit_in = join;
1398      }
1399
1400      // Update _last_succ from v_in towards the root
1401      for (u = v_in; u != up_limit_in && _last_succ[u] == v_in;
1402           u = _parent[u]) {
1403        _last_succ[u] = _last_succ[u_out];
1404      }
1405      // Update _last_succ from v_out towards the root
1406      if (join != old_rev_thread && v_in != old_rev_thread) {
1407        for (u = v_out; u != up_limit_out && _last_succ[u] == old_last_succ;
1408             u = _parent[u]) {
1409          _last_succ[u] = old_rev_thread;
1410        }
1411      } else {
1412        for (u = v_out; u != up_limit_out && _last_succ[u] == old_last_succ;
1413             u = _parent[u]) {
1414          _last_succ[u] = _last_succ[u_out];
1415        }
1416      }
1417
1418      // Update _succ_num from v_in to join
1419      for (u = v_in; u != join; u = _parent[u]) {
1420        _succ_num[u] += old_succ_num;
1421      }
1422      // Update _succ_num from v_out to join
1423      for (u = v_out; u != join; u = _parent[u]) {
1424        _succ_num[u] -= old_succ_num;
1425      }
1426    }
1427
1428    // Update potentials
1429    void updatePotential() {
1430      Cost sigma = _forward[u_in] ?
1431        _pi[v_in] - _pi[u_in] - _cost[_pred[u_in]] :
1432        _pi[v_in] - _pi[u_in] + _cost[_pred[u_in]];
1433      // Update potentials in the subtree, which has been moved
1434      int end = _thread[_last_succ[u_in]];
1435      for (int u = u_in; u != end; u = _thread[u]) {
1436        _pi[u] += sigma;
1437      }
1438    }
1439
1440    // Heuristic initial pivots
1441    bool initialPivots() {
1442      Value curr, total = 0;
1443      std::vector<Node> supply_nodes, demand_nodes;
1444      for (NodeIt u(_graph); u != INVALID; ++u) {
1445        curr = _supply[_node_id[u]];
1446        if (curr > 0) {
1447          total += curr;
1448          supply_nodes.push_back(u);
1449        }
1450        else if (curr < 0) {
1451          demand_nodes.push_back(u);
1452        }
1453      }
1454      if (_sum_supply > 0) total -= _sum_supply;
1455      if (total <= 0) return true;
1456
1457      IntVector arc_vector;
1458      if (_sum_supply >= 0) {
1459        if (supply_nodes.size() == 1 && demand_nodes.size() == 1) {
1460          // Perform a reverse graph search from the sink to the source
1461          typename GR::template NodeMap<bool> reached(_graph, false);
1462          Node s = supply_nodes[0], t = demand_nodes[0];
1463          std::vector<Node> stack;
1464          reached[t] = true;
1465          stack.push_back(t);
1466          while (!stack.empty()) {
1467            Node u, v = stack.back();
1468            stack.pop_back();
1469            if (v == s) break;
1470            for (InArcIt a(_graph, v); a != INVALID; ++a) {
1471              if (reached[u = _graph.source(a)]) continue;
1472              int j = _arc_id[a];
1473              if (_cap[j] >= total) {
1474                arc_vector.push_back(j);
1475                reached[u] = true;
1476                stack.push_back(u);
1477              }
1478            }
1479          }
1480        } else {
1481          // Find the min. cost incomming arc for each demand node
1482          for (int i = 0; i != int(demand_nodes.size()); ++i) {
1483            Node v = demand_nodes[i];
1484            Cost c, min_cost = std::numeric_limits<Cost>::max();
1485            Arc min_arc = INVALID;
1486            for (InArcIt a(_graph, v); a != INVALID; ++a) {
1487              c = _cost[_arc_id[a]];
1488              if (c < min_cost) {
1489                min_cost = c;
1490                min_arc = a;
1491              }
1492            }
1493            if (min_arc != INVALID) {
1494              arc_vector.push_back(_arc_id[min_arc]);
1495            }
1496          }
1497        }
1498      } else {
1499        // Find the min. cost outgoing arc for each supply node
1500        for (int i = 0; i != int(supply_nodes.size()); ++i) {
1501          Node u = supply_nodes[i];
1502          Cost c, min_cost = std::numeric_limits<Cost>::max();
1503          Arc min_arc = INVALID;
1504          for (OutArcIt a(_graph, u); a != INVALID; ++a) {
1505            c = _cost[_arc_id[a]];
1506            if (c < min_cost) {
1507              min_cost = c;
1508              min_arc = a;
1509            }
1510          }
1511          if (min_arc != INVALID) {
1512            arc_vector.push_back(_arc_id[min_arc]);
1513          }
1514        }
1515      }
1516
1517      // Perform heuristic initial pivots
1518      for (int i = 0; i != int(arc_vector.size()); ++i) {
1519        in_arc = arc_vector[i];
1520        if (_state[in_arc] * (_cost[in_arc] + _pi[_source[in_arc]] -
1521            _pi[_target[in_arc]]) >= 0) continue;
1522        findJoinNode();
1523        bool change = findLeavingArc();
1524        if (delta >= MAX) return false;
1525        changeFlow(change);
1526        if (change) {
1527          updateTreeStructure();
1528          updatePotential();
1529        }
1530      }
1531      return true;
1532    }
1533
1534    // Execute the algorithm
1535    ProblemType start(PivotRule pivot_rule) {
1536      // Select the pivot rule implementation
1537      switch (pivot_rule) {
1538        case FIRST_ELIGIBLE:
1539          return start<FirstEligiblePivotRule>();
1540        case BEST_ELIGIBLE:
1541          return start<BestEligiblePivotRule>();
1542        case BLOCK_SEARCH:
1543          return start<BlockSearchPivotRule>();
1544        case CANDIDATE_LIST:
1545          return start<CandidateListPivotRule>();
1546        case ALTERING_LIST:
1547          return start<AlteringListPivotRule>();
1548      }
1549      return INFEASIBLE; // avoid warning
1550    }
1551
1552    template <typename PivotRuleImpl>
1553    ProblemType start() {
1554      PivotRuleImpl pivot(*this);
1555
1556      // Perform heuristic initial pivots
1557      if (!initialPivots()) return UNBOUNDED;
1558
1559      // Execute the Network Simplex algorithm
1560      while (pivot.findEnteringArc()) {
1561        findJoinNode();
1562        bool change = findLeavingArc();
1563        if (delta >= MAX) return UNBOUNDED;
1564        changeFlow(change);
1565        if (change) {
1566          updateTreeStructure();
1567          updatePotential();
1568        }
1569      }
1570     
1571      // Check feasibility
1572      for (int e = _search_arc_num; e != _all_arc_num; ++e) {
1573        if (_flow[e] != 0) return INFEASIBLE;
1574      }
1575
1576      // Transform the solution and the supply map to the original form
1577      if (_have_lower) {
1578        for (int i = 0; i != _arc_num; ++i) {
1579          Value c = _lower[i];
1580          if (c != 0) {
1581            _flow[i] += c;
1582            _supply[_source[i]] += c;
1583            _supply[_target[i]] -= c;
1584          }
1585        }
1586      }
1587     
1588      // Shift potentials to meet the requirements of the GEQ/LEQ type
1589      // optimality conditions
1590      if (_sum_supply == 0) {
1591        if (_stype == GEQ) {
1592          Cost max_pot = std::numeric_limits<Cost>::min();
1593          for (int i = 0; i != _node_num; ++i) {
1594            if (_pi[i] > max_pot) max_pot = _pi[i];
1595          }
1596          if (max_pot > 0) {
1597            for (int i = 0; i != _node_num; ++i)
1598              _pi[i] -= max_pot;
1599          }
1600        } else {
1601          Cost min_pot = std::numeric_limits<Cost>::max();
1602          for (int i = 0; i != _node_num; ++i) {
1603            if (_pi[i] < min_pot) min_pot = _pi[i];
1604          }
1605          if (min_pot < 0) {
1606            for (int i = 0; i != _node_num; ++i)
1607              _pi[i] -= min_pot;
1608          }
1609        }
1610      }
1611
1612      return OPTIMAL;
1613    }
1614
1615  }; //class NetworkSimplex
1616
1617  ///@}
1618
1619} //namespace lemon
1620
1621#endif //LEMON_NETWORK_SIMPLEX_H
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