COIN-OR::LEMON - Graph Library

source: lemon-1.2/lemon/network_simplex.h @ 643:f3792d5bb294

Last change on this file since 643:f3792d5bb294 was 643:f3792d5bb294, checked in by Alpar Juttner <alpar@…>, 11 years ago

Resolve gcc-4.3 warning in lemon/network_simplex.h

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1/* -*- mode: C++; indent-tabs-mode: nil; -*-
2 *
3 * This file is a part of LEMON, a generic C++ optimization library.
4 *
5 * Copyright (C) 2003-2009
6 * Egervary Jeno Kombinatorikus Optimalizalasi Kutatocsoport
7 * (Egervary Research Group on Combinatorial Optimization, EGRES).
8 *
9 * Permission to use, modify and distribute this software is granted
10 * provided that this copyright notice appears in all copies. For
11 * precise terms see the accompanying LICENSE file.
12 *
13 * This software is provided "AS IS" with no warranty of any kind,
14 * express or implied, and with no claim as to its suitability for any
15 * purpose.
16 *
17 */
18
19#ifndef LEMON_NETWORK_SIMPLEX_H
20#define LEMON_NETWORK_SIMPLEX_H
21
22/// \ingroup min_cost_flow
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
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, since this form is supported
106    /// by other minimum cost flow algorithms and the \ref Circulation
107    /// algorithm, as well.
108    /// The \c LEQ problem type can be selected using the \ref supplyType()
109    /// function.
110    ///
111    /// Note that the equality form is a special case of both supply types.
112    enum SupplyType {
113
114      /// This option means that there are <em>"greater or equal"</em>
115      /// supply/demand constraints in the definition, i.e. the exact
116      /// formulation of the problem is the following.
117      /**
118          \f[ \min\sum_{uv\in A} f(uv) \cdot cost(uv) \f]
119          \f[ \sum_{uv\in A} f(uv) - \sum_{vu\in A} f(vu) \geq
120              sup(u) \quad \forall u\in V \f]
121          \f[ lower(uv) \leq f(uv) \leq upper(uv) \quad \forall uv\in A \f]
122      */
123      /// It means that the total demand must be greater or equal to the
124      /// total supply (i.e. \f$\sum_{u\in V} sup(u)\f$ must be zero or
125      /// negative) and all the supplies have to be carried out from
126      /// the supply nodes, but there could be demands that are not
127      /// satisfied.
128      GEQ,
129      /// It is just an alias for the \c GEQ option.
130      CARRY_SUPPLIES = GEQ,
131
132      /// This option means that there are <em>"less or equal"</em>
133      /// supply/demand constraints in the definition, i.e. the exact
134      /// formulation of the problem is the following.
135      /**
136          \f[ \min\sum_{uv\in A} f(uv) \cdot cost(uv) \f]
137          \f[ \sum_{uv\in A} f(uv) - \sum_{vu\in A} f(vu) \leq
138              sup(u) \quad \forall u\in V \f]
139          \f[ lower(uv) \leq f(uv) \leq upper(uv) \quad \forall uv\in A \f]
140      */
141      /// It means that the total demand must be less or equal to the
142      /// total supply (i.e. \f$\sum_{u\in V} sup(u)\f$ must be zero or
143      /// positive) and all the demands have to be satisfied, but there
144      /// could be supplies that are not carried out from the supply
145      /// nodes.
146      LEQ,
147      /// It is just an alias for the \c LEQ option.
148      SATISFY_DEMANDS = LEQ
149    };
150   
151    /// \brief Constants for selecting the pivot rule.
152    ///
153    /// Enum type containing constants for selecting the pivot rule for
154    /// the \ref run() function.
155    ///
156    /// \ref NetworkSimplex provides five different pivot rule
157    /// implementations that significantly affect the running time
158    /// of the algorithm.
159    /// By default \ref BLOCK_SEARCH "Block Search" is used, which
160    /// proved to be the most efficient and the most robust on various
161    /// test inputs according to our benchmark tests.
162    /// However another pivot rule can be selected using the \ref run()
163    /// function with the proper parameter.
164    enum PivotRule {
165
166      /// The First Eligible pivot rule.
167      /// The next eligible arc is selected in a wraparound fashion
168      /// in every iteration.
169      FIRST_ELIGIBLE,
170
171      /// The Best Eligible pivot rule.
172      /// The best eligible arc is selected in every iteration.
173      BEST_ELIGIBLE,
174
175      /// The Block Search pivot rule.
176      /// A specified number of arcs are examined in every iteration
177      /// in a wraparound fashion and the best eligible arc is selected
178      /// from this block.
179      BLOCK_SEARCH,
180
181      /// The Candidate List pivot rule.
182      /// In a major iteration a candidate list is built from eligible arcs
183      /// in a wraparound fashion and in the following minor iterations
184      /// the best eligible arc is selected from this list.
185      CANDIDATE_LIST,
186
187      /// The Altering Candidate List pivot rule.
188      /// It is a modified version of the Candidate List method.
189      /// It keeps only the several best eligible arcs from the former
190      /// candidate list and extends this list in every iteration.
191      ALTERING_LIST
192    };
193   
194  private:
195
196    TEMPLATE_DIGRAPH_TYPEDEFS(GR);
197
198    typedef std::vector<Arc> ArcVector;
199    typedef std::vector<Node> NodeVector;
200    typedef std::vector<int> IntVector;
201    typedef std::vector<bool> BoolVector;
202    typedef std::vector<Value> ValueVector;
203    typedef std::vector<Cost> CostVector;
204
205    // State constants for arcs
206    enum ArcStateEnum {
207      STATE_UPPER = -1,
208      STATE_TREE  =  0,
209      STATE_LOWER =  1
210    };
211
212  private:
213
214    // Data related to the underlying digraph
215    const GR &_graph;
216    int _node_num;
217    int _arc_num;
218
219    // Parameters of the problem
220    bool _have_lower;
221    SupplyType _stype;
222    Value _sum_supply;
223
224    // Data structures for storing the digraph
225    IntNodeMap _node_id;
226    IntArcMap _arc_id;
227    IntVector _source;
228    IntVector _target;
229
230    // Node and arc data
231    ValueVector _lower;
232    ValueVector _upper;
233    ValueVector _cap;
234    CostVector _cost;
235    ValueVector _supply;
236    ValueVector _flow;
237    CostVector _pi;
238
239    // Data for storing the spanning tree structure
240    IntVector _parent;
241    IntVector _pred;
242    IntVector _thread;
243    IntVector _rev_thread;
244    IntVector _succ_num;
245    IntVector _last_succ;
246    IntVector _dirty_revs;
247    BoolVector _forward;
248    IntVector _state;
249    int _root;
250
251    // Temporary data used in the current pivot iteration
252    int in_arc, join, u_in, v_in, u_out, v_out;
253    int first, second, right, last;
254    int stem, par_stem, new_stem;
255    Value delta;
256
257  public:
258 
259    /// \brief Constant for infinite upper bounds (capacities).
260    ///
261    /// Constant for infinite upper bounds (capacities).
262    /// It is \c std::numeric_limits<Value>::infinity() if available,
263    /// \c std::numeric_limits<Value>::max() otherwise.
264    const Value INF;
265
266  private:
267
268    // Implementation of the First Eligible pivot rule
269    class FirstEligiblePivotRule
270    {
271    private:
272
273      // References to the NetworkSimplex class
274      const IntVector  &_source;
275      const IntVector  &_target;
276      const CostVector &_cost;
277      const IntVector  &_state;
278      const CostVector &_pi;
279      int &_in_arc;
280      int _arc_num;
281
282      // Pivot rule data
283      int _next_arc;
284
285    public:
286
287      // Constructor
288      FirstEligiblePivotRule(NetworkSimplex &ns) :
289        _source(ns._source), _target(ns._target),
290        _cost(ns._cost), _state(ns._state), _pi(ns._pi),
291        _in_arc(ns.in_arc), _arc_num(ns._arc_num), _next_arc(0)
292      {}
293
294      // Find next entering arc
295      bool findEnteringArc() {
296        Cost c;
297        for (int e = _next_arc; e < _arc_num; ++e) {
298          c = _state[e] * (_cost[e] + _pi[_source[e]] - _pi[_target[e]]);
299          if (c < 0) {
300            _in_arc = e;
301            _next_arc = e + 1;
302            return true;
303          }
304        }
305        for (int e = 0; e < _next_arc; ++e) {
306          c = _state[e] * (_cost[e] + _pi[_source[e]] - _pi[_target[e]]);
307          if (c < 0) {
308            _in_arc = e;
309            _next_arc = e + 1;
310            return true;
311          }
312        }
313        return false;
314      }
315
316    }; //class FirstEligiblePivotRule
317
318
319    // Implementation of the Best Eligible pivot rule
320    class BestEligiblePivotRule
321    {
322    private:
323
324      // References to the NetworkSimplex class
325      const IntVector  &_source;
326      const IntVector  &_target;
327      const CostVector &_cost;
328      const IntVector  &_state;
329      const CostVector &_pi;
330      int &_in_arc;
331      int _arc_num;
332
333    public:
334
335      // Constructor
336      BestEligiblePivotRule(NetworkSimplex &ns) :
337        _source(ns._source), _target(ns._target),
338        _cost(ns._cost), _state(ns._state), _pi(ns._pi),
339        _in_arc(ns.in_arc), _arc_num(ns._arc_num)
340      {}
341
342      // Find next entering arc
343      bool findEnteringArc() {
344        Cost c, min = 0;
345        for (int e = 0; e < _arc_num; ++e) {
346          c = _state[e] * (_cost[e] + _pi[_source[e]] - _pi[_target[e]]);
347          if (c < min) {
348            min = c;
349            _in_arc = e;
350          }
351        }
352        return min < 0;
353      }
354
355    }; //class BestEligiblePivotRule
356
357
358    // Implementation of the Block Search pivot rule
359    class BlockSearchPivotRule
360    {
361    private:
362
363      // References to the NetworkSimplex class
364      const IntVector  &_source;
365      const IntVector  &_target;
366      const CostVector &_cost;
367      const IntVector  &_state;
368      const CostVector &_pi;
369      int &_in_arc;
370      int _arc_num;
371
372      // Pivot rule data
373      int _block_size;
374      int _next_arc;
375
376    public:
377
378      // Constructor
379      BlockSearchPivotRule(NetworkSimplex &ns) :
380        _source(ns._source), _target(ns._target),
381        _cost(ns._cost), _state(ns._state), _pi(ns._pi),
382        _in_arc(ns.in_arc), _arc_num(ns._arc_num), _next_arc(0)
383      {
384        // The main parameters of the pivot rule
385        const double BLOCK_SIZE_FACTOR = 2.0;
386        const int MIN_BLOCK_SIZE = 10;
387
388        _block_size = std::max( int(BLOCK_SIZE_FACTOR *
389                                    std::sqrt(double(_arc_num))),
390                                MIN_BLOCK_SIZE );
391      }
392
393      // Find next entering arc
394      bool findEnteringArc() {
395        Cost c, min = 0;
396        int cnt = _block_size;
397        int e, min_arc = _next_arc;
398        for (e = _next_arc; e < _arc_num; ++e) {
399          c = _state[e] * (_cost[e] + _pi[_source[e]] - _pi[_target[e]]);
400          if (c < min) {
401            min = c;
402            min_arc = e;
403          }
404          if (--cnt == 0) {
405            if (min < 0) break;
406            cnt = _block_size;
407          }
408        }
409        if (min == 0 || cnt > 0) {
410          for (e = 0; e < _next_arc; ++e) {
411            c = _state[e] * (_cost[e] + _pi[_source[e]] - _pi[_target[e]]);
412            if (c < min) {
413              min = c;
414              min_arc = e;
415            }
416            if (--cnt == 0) {
417              if (min < 0) break;
418              cnt = _block_size;
419            }
420          }
421        }
422        if (min >= 0) return false;
423        _in_arc = min_arc;
424        _next_arc = e;
425        return true;
426      }
427
428    }; //class BlockSearchPivotRule
429
430
431    // Implementation of the Candidate List pivot rule
432    class CandidateListPivotRule
433    {
434    private:
435
436      // References to the NetworkSimplex class
437      const IntVector  &_source;
438      const IntVector  &_target;
439      const CostVector &_cost;
440      const IntVector  &_state;
441      const CostVector &_pi;
442      int &_in_arc;
443      int _arc_num;
444
445      // Pivot rule data
446      IntVector _candidates;
447      int _list_length, _minor_limit;
448      int _curr_length, _minor_count;
449      int _next_arc;
450
451    public:
452
453      /// Constructor
454      CandidateListPivotRule(NetworkSimplex &ns) :
455        _source(ns._source), _target(ns._target),
456        _cost(ns._cost), _state(ns._state), _pi(ns._pi),
457        _in_arc(ns.in_arc), _arc_num(ns._arc_num), _next_arc(0)
458      {
459        // The main parameters of the pivot rule
460        const double LIST_LENGTH_FACTOR = 1.0;
461        const int MIN_LIST_LENGTH = 10;
462        const double MINOR_LIMIT_FACTOR = 0.1;
463        const int MIN_MINOR_LIMIT = 3;
464
465        _list_length = std::max( int(LIST_LENGTH_FACTOR *
466                                     std::sqrt(double(_arc_num))),
467                                 MIN_LIST_LENGTH );
468        _minor_limit = std::max( int(MINOR_LIMIT_FACTOR * _list_length),
469                                 MIN_MINOR_LIMIT );
470        _curr_length = _minor_count = 0;
471        _candidates.resize(_list_length);
472      }
473
474      /// Find next entering arc
475      bool findEnteringArc() {
476        Cost min, c;
477        int e, min_arc = _next_arc;
478        if (_curr_length > 0 && _minor_count < _minor_limit) {
479          // Minor iteration: select the best eligible arc from the
480          // current candidate list
481          ++_minor_count;
482          min = 0;
483          for (int i = 0; i < _curr_length; ++i) {
484            e = _candidates[i];
485            c = _state[e] * (_cost[e] + _pi[_source[e]] - _pi[_target[e]]);
486            if (c < min) {
487              min = c;
488              min_arc = e;
489            }
490            if (c >= 0) {
491              _candidates[i--] = _candidates[--_curr_length];
492            }
493          }
494          if (min < 0) {
495            _in_arc = min_arc;
496            return true;
497          }
498        }
499
500        // Major iteration: build a new candidate list
501        min = 0;
502        _curr_length = 0;
503        for (e = _next_arc; e < _arc_num; ++e) {
504          c = _state[e] * (_cost[e] + _pi[_source[e]] - _pi[_target[e]]);
505          if (c < 0) {
506            _candidates[_curr_length++] = e;
507            if (c < min) {
508              min = c;
509              min_arc = e;
510            }
511            if (_curr_length == _list_length) break;
512          }
513        }
514        if (_curr_length < _list_length) {
515          for (e = 0; e < _next_arc; ++e) {
516            c = _state[e] * (_cost[e] + _pi[_source[e]] - _pi[_target[e]]);
517            if (c < 0) {
518              _candidates[_curr_length++] = e;
519              if (c < min) {
520                min = c;
521                min_arc = e;
522              }
523              if (_curr_length == _list_length) break;
524            }
525          }
526        }
527        if (_curr_length == 0) return false;
528        _minor_count = 1;
529        _in_arc = min_arc;
530        _next_arc = e;
531        return true;
532      }
533
534    }; //class CandidateListPivotRule
535
536
537    // Implementation of the Altering Candidate List pivot rule
538    class AlteringListPivotRule
539    {
540    private:
541
542      // References to the NetworkSimplex class
543      const IntVector  &_source;
544      const IntVector  &_target;
545      const CostVector &_cost;
546      const IntVector  &_state;
547      const CostVector &_pi;
548      int &_in_arc;
549      int _arc_num;
550
551      // Pivot rule data
552      int _block_size, _head_length, _curr_length;
553      int _next_arc;
554      IntVector _candidates;
555      CostVector _cand_cost;
556
557      // Functor class to compare arcs during sort of the candidate list
558      class SortFunc
559      {
560      private:
561        const CostVector &_map;
562      public:
563        SortFunc(const CostVector &map) : _map(map) {}
564        bool operator()(int left, int right) {
565          return _map[left] > _map[right];
566        }
567      };
568
569      SortFunc _sort_func;
570
571    public:
572
573      // Constructor
574      AlteringListPivotRule(NetworkSimplex &ns) :
575        _source(ns._source), _target(ns._target),
576        _cost(ns._cost), _state(ns._state), _pi(ns._pi),
577        _in_arc(ns.in_arc), _arc_num(ns._arc_num),
578        _next_arc(0), _cand_cost(ns._arc_num), _sort_func(_cand_cost)
579      {
580        // The main parameters of the pivot rule
581        const double BLOCK_SIZE_FACTOR = 1.5;
582        const int MIN_BLOCK_SIZE = 10;
583        const double HEAD_LENGTH_FACTOR = 0.1;
584        const int MIN_HEAD_LENGTH = 3;
585
586        _block_size = std::max( int(BLOCK_SIZE_FACTOR *
587                                    std::sqrt(double(_arc_num))),
588                                MIN_BLOCK_SIZE );
589        _head_length = std::max( int(HEAD_LENGTH_FACTOR * _block_size),
590                                 MIN_HEAD_LENGTH );
591        _candidates.resize(_head_length + _block_size);
592        _curr_length = 0;
593      }
594
595      // Find next entering arc
596      bool findEnteringArc() {
597        // Check the current candidate list
598        int e;
599        for (int i = 0; i < _curr_length; ++i) {
600          e = _candidates[i];
601          _cand_cost[e] = _state[e] *
602            (_cost[e] + _pi[_source[e]] - _pi[_target[e]]);
603          if (_cand_cost[e] >= 0) {
604            _candidates[i--] = _candidates[--_curr_length];
605          }
606        }
607
608        // Extend the list
609        int cnt = _block_size;
610        int last_arc = 0;
611        int limit = _head_length;
612
613        for (int e = _next_arc; e < _arc_num; ++e) {
614          _cand_cost[e] = _state[e] *
615            (_cost[e] + _pi[_source[e]] - _pi[_target[e]]);
616          if (_cand_cost[e] < 0) {
617            _candidates[_curr_length++] = e;
618            last_arc = e;
619          }
620          if (--cnt == 0) {
621            if (_curr_length > limit) break;
622            limit = 0;
623            cnt = _block_size;
624          }
625        }
626        if (_curr_length <= limit) {
627          for (int e = 0; e < _next_arc; ++e) {
628            _cand_cost[e] = _state[e] *
629              (_cost[e] + _pi[_source[e]] - _pi[_target[e]]);
630            if (_cand_cost[e] < 0) {
631              _candidates[_curr_length++] = e;
632              last_arc = e;
633            }
634            if (--cnt == 0) {
635              if (_curr_length > limit) break;
636              limit = 0;
637              cnt = _block_size;
638            }
639          }
640        }
641        if (_curr_length == 0) return false;
642        _next_arc = last_arc + 1;
643
644        // Make heap of the candidate list (approximating a partial sort)
645        make_heap( _candidates.begin(), _candidates.begin() + _curr_length,
646                   _sort_func );
647
648        // Pop the first element of the heap
649        _in_arc = _candidates[0];
650        pop_heap( _candidates.begin(), _candidates.begin() + _curr_length,
651                  _sort_func );
652        _curr_length = std::min(_head_length, _curr_length - 1);
653        return true;
654      }
655
656    }; //class AlteringListPivotRule
657
658  public:
659
660    /// \brief Constructor.
661    ///
662    /// The constructor of the class.
663    ///
664    /// \param graph The digraph the algorithm runs on.
665    NetworkSimplex(const GR& graph) :
666      _graph(graph), _node_id(graph), _arc_id(graph),
667      INF(std::numeric_limits<Value>::has_infinity ?
668          std::numeric_limits<Value>::infinity() :
669          std::numeric_limits<Value>::max())
670    {
671      // Check the value types
672      LEMON_ASSERT(std::numeric_limits<Value>::is_signed,
673        "The flow type of NetworkSimplex must be signed");
674      LEMON_ASSERT(std::numeric_limits<Cost>::is_signed,
675        "The cost type of NetworkSimplex must be signed");
676       
677      // Resize vectors
678      _node_num = countNodes(_graph);
679      _arc_num = countArcs(_graph);
680      int all_node_num = _node_num + 1;
681      int all_arc_num = _arc_num + _node_num;
682
683      _source.resize(all_arc_num);
684      _target.resize(all_arc_num);
685
686      _lower.resize(all_arc_num);
687      _upper.resize(all_arc_num);
688      _cap.resize(all_arc_num);
689      _cost.resize(all_arc_num);
690      _supply.resize(all_node_num);
691      _flow.resize(all_arc_num);
692      _pi.resize(all_node_num);
693
694      _parent.resize(all_node_num);
695      _pred.resize(all_node_num);
696      _forward.resize(all_node_num);
697      _thread.resize(all_node_num);
698      _rev_thread.resize(all_node_num);
699      _succ_num.resize(all_node_num);
700      _last_succ.resize(all_node_num);
701      _state.resize(all_arc_num);
702
703      // Copy the graph (store the arcs in a mixed order)
704      int i = 0;
705      for (NodeIt n(_graph); n != INVALID; ++n, ++i) {
706        _node_id[n] = i;
707      }
708      int k = std::max(int(std::sqrt(double(_arc_num))), 10);
709      i = 0;
710      for (ArcIt a(_graph); a != INVALID; ++a) {
711        _arc_id[a] = i;
712        _source[i] = _node_id[_graph.source(a)];
713        _target[i] = _node_id[_graph.target(a)];
714        if ((i += k) >= _arc_num) i = (i % k) + 1;
715      }
716     
717      // Initialize maps
718      for (int i = 0; i != _node_num; ++i) {
719        _supply[i] = 0;
720      }
721      for (int i = 0; i != _arc_num; ++i) {
722        _lower[i] = 0;
723        _upper[i] = INF;
724        _cost[i] = 1;
725      }
726      _have_lower = false;
727      _stype = GEQ;
728    }
729
730    /// \name Parameters
731    /// The parameters of the algorithm can be specified using these
732    /// functions.
733
734    /// @{
735
736    /// \brief Set the lower bounds on the arcs.
737    ///
738    /// This function sets the lower bounds on the arcs.
739    /// If it is not used before calling \ref run(), the lower bounds
740    /// will be set to zero on all arcs.
741    ///
742    /// \param map An arc map storing the lower bounds.
743    /// Its \c Value type must be convertible to the \c Value type
744    /// of the algorithm.
745    ///
746    /// \return <tt>(*this)</tt>
747    template <typename LowerMap>
748    NetworkSimplex& lowerMap(const LowerMap& map) {
749      _have_lower = true;
750      for (ArcIt a(_graph); a != INVALID; ++a) {
751        _lower[_arc_id[a]] = map[a];
752      }
753      return *this;
754    }
755
756    /// \brief Set the upper bounds (capacities) on the arcs.
757    ///
758    /// This function sets the upper bounds (capacities) on the arcs.
759    /// If it is not used before calling \ref run(), the upper bounds
760    /// will be set to \ref INF on all arcs (i.e. the flow value will be
761    /// unbounded from above on each arc).
762    ///
763    /// \param map An arc map storing the upper bounds.
764    /// Its \c Value type must be convertible to the \c Value type
765    /// of the algorithm.
766    ///
767    /// \return <tt>(*this)</tt>
768    template<typename UpperMap>
769    NetworkSimplex& upperMap(const UpperMap& map) {
770      for (ArcIt a(_graph); a != INVALID; ++a) {
771        _upper[_arc_id[a]] = map[a];
772      }
773      return *this;
774    }
775
776    /// \brief Set the costs of the arcs.
777    ///
778    /// This function sets the costs of the arcs.
779    /// If it is not used before calling \ref run(), the costs
780    /// will be set to \c 1 on all arcs.
781    ///
782    /// \param map An arc map storing the costs.
783    /// Its \c Value type must be convertible to the \c Cost type
784    /// of the algorithm.
785    ///
786    /// \return <tt>(*this)</tt>
787    template<typename CostMap>
788    NetworkSimplex& costMap(const CostMap& map) {
789      for (ArcIt a(_graph); a != INVALID; ++a) {
790        _cost[_arc_id[a]] = map[a];
791      }
792      return *this;
793    }
794
795    /// \brief Set the supply values of the nodes.
796    ///
797    /// This function sets the supply values of the nodes.
798    /// If neither this function nor \ref stSupply() is used before
799    /// calling \ref run(), the supply of each node will be set to zero.
800    /// (It makes sense only if non-zero lower bounds are given.)
801    ///
802    /// \param map A node map storing the supply values.
803    /// Its \c Value type must be convertible to the \c Value type
804    /// of the algorithm.
805    ///
806    /// \return <tt>(*this)</tt>
807    template<typename SupplyMap>
808    NetworkSimplex& supplyMap(const SupplyMap& map) {
809      for (NodeIt n(_graph); n != INVALID; ++n) {
810        _supply[_node_id[n]] = map[n];
811      }
812      return *this;
813    }
814
815    /// \brief Set single source and target nodes and a supply value.
816    ///
817    /// This function sets a single source node and a single target node
818    /// and the required flow value.
819    /// If neither this function nor \ref supplyMap() is used before
820    /// calling \ref run(), the supply of each node will be set to zero.
821    /// (It makes sense only if non-zero lower bounds are given.)
822    ///
823    /// Using this function has the same effect as using \ref supplyMap()
824    /// with such a map in which \c k is assigned to \c s, \c -k is
825    /// assigned to \c t and all other nodes have zero supply value.
826    ///
827    /// \param s The source node.
828    /// \param t The target node.
829    /// \param k The required amount of flow from node \c s to node \c t
830    /// (i.e. the supply of \c s and the demand of \c t).
831    ///
832    /// \return <tt>(*this)</tt>
833    NetworkSimplex& stSupply(const Node& s, const Node& t, Value k) {
834      for (int i = 0; i != _node_num; ++i) {
835        _supply[i] = 0;
836      }
837      _supply[_node_id[s]] =  k;
838      _supply[_node_id[t]] = -k;
839      return *this;
840    }
841   
842    /// \brief Set the type of the supply constraints.
843    ///
844    /// This function sets the type of the supply/demand constraints.
845    /// If it is not used before calling \ref run(), the \ref GEQ supply
846    /// type will be used.
847    ///
848    /// For more information see \ref SupplyType.
849    ///
850    /// \return <tt>(*this)</tt>
851    NetworkSimplex& supplyType(SupplyType supply_type) {
852      _stype = supply_type;
853      return *this;
854    }
855
856    /// @}
857
858    /// \name Execution Control
859    /// The algorithm can be executed using \ref run().
860
861    /// @{
862
863    /// \brief Run the algorithm.
864    ///
865    /// This function runs the algorithm.
866    /// The paramters can be specified using functions \ref lowerMap(),
867    /// \ref upperMap(), \ref costMap(), \ref supplyMap(), \ref stSupply(),
868    /// \ref supplyType().
869    /// For example,
870    /// \code
871    ///   NetworkSimplex<ListDigraph> ns(graph);
872    ///   ns.lowerMap(lower).upperMap(upper).costMap(cost)
873    ///     .supplyMap(sup).run();
874    /// \endcode
875    ///
876    /// This function can be called more than once. All the parameters
877    /// that have been given are kept for the next call, unless
878    /// \ref reset() is called, thus only the modified parameters
879    /// have to be set again. See \ref reset() for examples.
880    /// However the underlying digraph must not be modified after this
881    /// class have been constructed, since it copies and extends the graph.
882    ///
883    /// \param pivot_rule The pivot rule that will be used during the
884    /// algorithm. For more information see \ref PivotRule.
885    ///
886    /// \return \c INFEASIBLE if no feasible flow exists,
887    /// \n \c OPTIMAL if the problem has optimal solution
888    /// (i.e. it is feasible and bounded), and the algorithm has found
889    /// optimal flow and node potentials (primal and dual solutions),
890    /// \n \c UNBOUNDED if the objective function of the problem is
891    /// unbounded, i.e. there is a directed cycle having negative total
892    /// cost and infinite upper bound.
893    ///
894    /// \see ProblemType, PivotRule
895    ProblemType run(PivotRule pivot_rule = BLOCK_SEARCH) {
896      if (!init()) return INFEASIBLE;
897      return start(pivot_rule);
898    }
899
900    /// \brief Reset all the parameters that have been given before.
901    ///
902    /// This function resets all the paramaters that have been given
903    /// before using functions \ref lowerMap(), \ref upperMap(),
904    /// \ref costMap(), \ref supplyMap(), \ref stSupply(), \ref supplyType().
905    ///
906    /// It is useful for multiple run() calls. If this function is not
907    /// used, all the parameters given before are kept for the next
908    /// \ref run() call.
909    /// However the underlying digraph must not be modified after this
910    /// class have been constructed, since it copies and extends the graph.
911    ///
912    /// For example,
913    /// \code
914    ///   NetworkSimplex<ListDigraph> ns(graph);
915    ///
916    ///   // First run
917    ///   ns.lowerMap(lower).upperMap(upper).costMap(cost)
918    ///     .supplyMap(sup).run();
919    ///
920    ///   // Run again with modified cost map (reset() is not called,
921    ///   // so only the cost map have to be set again)
922    ///   cost[e] += 100;
923    ///   ns.costMap(cost).run();
924    ///
925    ///   // Run again from scratch using reset()
926    ///   // (the lower bounds will be set to zero on all arcs)
927    ///   ns.reset();
928    ///   ns.upperMap(capacity).costMap(cost)
929    ///     .supplyMap(sup).run();
930    /// \endcode
931    ///
932    /// \return <tt>(*this)</tt>
933    NetworkSimplex& reset() {
934      for (int i = 0; i != _node_num; ++i) {
935        _supply[i] = 0;
936      }
937      for (int i = 0; i != _arc_num; ++i) {
938        _lower[i] = 0;
939        _upper[i] = INF;
940        _cost[i] = 1;
941      }
942      _have_lower = false;
943      _stype = GEQ;
944      return *this;
945    }
946
947    /// @}
948
949    /// \name Query Functions
950    /// The results of the algorithm can be obtained using these
951    /// functions.\n
952    /// The \ref run() function must be called before using them.
953
954    /// @{
955
956    /// \brief Return the total cost of the found flow.
957    ///
958    /// This function returns the total cost of the found flow.
959    /// Its complexity is O(e).
960    ///
961    /// \note The return type of the function can be specified as a
962    /// template parameter. For example,
963    /// \code
964    ///   ns.totalCost<double>();
965    /// \endcode
966    /// It is useful if the total cost cannot be stored in the \c Cost
967    /// type of the algorithm, which is the default return type of the
968    /// function.
969    ///
970    /// \pre \ref run() must be called before using this function.
971    template <typename Number>
972    Number totalCost() const {
973      Number c = 0;
974      for (ArcIt a(_graph); a != INVALID; ++a) {
975        int i = _arc_id[a];
976        c += Number(_flow[i]) * Number(_cost[i]);
977      }
978      return c;
979    }
980
981#ifndef DOXYGEN
982    Cost totalCost() const {
983      return totalCost<Cost>();
984    }
985#endif
986
987    /// \brief Return the flow on the given arc.
988    ///
989    /// This function returns the flow on the given arc.
990    ///
991    /// \pre \ref run() must be called before using this function.
992    Value flow(const Arc& a) const {
993      return _flow[_arc_id[a]];
994    }
995
996    /// \brief Return the flow map (the primal solution).
997    ///
998    /// This function copies the flow value on each arc into the given
999    /// map. The \c Value type of the algorithm must be convertible to
1000    /// the \c Value type of the map.
1001    ///
1002    /// \pre \ref run() must be called before using this function.
1003    template <typename FlowMap>
1004    void flowMap(FlowMap &map) const {
1005      for (ArcIt a(_graph); a != INVALID; ++a) {
1006        map.set(a, _flow[_arc_id[a]]);
1007      }
1008    }
1009
1010    /// \brief Return the potential (dual value) of the given node.
1011    ///
1012    /// This function returns the potential (dual value) of the
1013    /// given node.
1014    ///
1015    /// \pre \ref run() must be called before using this function.
1016    Cost potential(const Node& n) const {
1017      return _pi[_node_id[n]];
1018    }
1019
1020    /// \brief Return the potential map (the dual solution).
1021    ///
1022    /// This function copies the potential (dual value) of each node
1023    /// into the given map.
1024    /// The \c Cost type of the algorithm must be convertible to the
1025    /// \c Value type of the map.
1026    ///
1027    /// \pre \ref run() must be called before using this function.
1028    template <typename PotentialMap>
1029    void potentialMap(PotentialMap &map) const {
1030      for (NodeIt n(_graph); n != INVALID; ++n) {
1031        map.set(n, _pi[_node_id[n]]);
1032      }
1033    }
1034
1035    /// @}
1036
1037  private:
1038
1039    // Initialize internal data structures
1040    bool init() {
1041      if (_node_num == 0) return false;
1042
1043      // Check the sum of supply values
1044      _sum_supply = 0;
1045      for (int i = 0; i != _node_num; ++i) {
1046        _sum_supply += _supply[i];
1047      }
1048      if ( !((_stype == GEQ && _sum_supply <= 0) ||
1049             (_stype == LEQ && _sum_supply >= 0)) ) return false;
1050
1051      // Remove non-zero lower bounds
1052      if (_have_lower) {
1053        for (int i = 0; i != _arc_num; ++i) {
1054          Value c = _lower[i];
1055          if (c >= 0) {
1056            _cap[i] = _upper[i] < INF ? _upper[i] - c : INF;
1057          } else {
1058            _cap[i] = _upper[i] < INF + c ? _upper[i] - c : INF;
1059          }
1060          _supply[_source[i]] -= c;
1061          _supply[_target[i]] += c;
1062        }
1063      } else {
1064        for (int i = 0; i != _arc_num; ++i) {
1065          _cap[i] = _upper[i];
1066        }
1067      }
1068
1069      // Initialize artifical cost
1070      Cost ART_COST;
1071      if (std::numeric_limits<Cost>::is_exact) {
1072        ART_COST = std::numeric_limits<Cost>::max() / 4 + 1;
1073      } else {
1074        ART_COST = std::numeric_limits<Cost>::min();
1075        for (int i = 0; i != _arc_num; ++i) {
1076          if (_cost[i] > ART_COST) ART_COST = _cost[i];
1077        }
1078        ART_COST = (ART_COST + 1) * _node_num;
1079      }
1080
1081      // Initialize arc maps
1082      for (int i = 0; i != _arc_num; ++i) {
1083        _flow[i] = 0;
1084        _state[i] = STATE_LOWER;
1085      }
1086     
1087      // Set data for the artificial root node
1088      _root = _node_num;
1089      _parent[_root] = -1;
1090      _pred[_root] = -1;
1091      _thread[_root] = 0;
1092      _rev_thread[0] = _root;
1093      _succ_num[_root] = _node_num + 1;
1094      _last_succ[_root] = _root - 1;
1095      _supply[_root] = -_sum_supply;
1096      _pi[_root] = _sum_supply < 0 ? -ART_COST : ART_COST;
1097
1098      // Add artificial arcs and initialize the spanning tree data structure
1099      for (int u = 0, e = _arc_num; u != _node_num; ++u, ++e) {
1100        _parent[u] = _root;
1101        _pred[u] = e;
1102        _thread[u] = u + 1;
1103        _rev_thread[u + 1] = u;
1104        _succ_num[u] = 1;
1105        _last_succ[u] = u;
1106        _cost[e] = ART_COST;
1107        _cap[e] = INF;
1108        _state[e] = STATE_TREE;
1109        if (_supply[u] > 0 || (_supply[u] == 0 && _sum_supply <= 0)) {
1110          _flow[e] = _supply[u];
1111          _forward[u] = true;
1112          _pi[u] = -ART_COST + _pi[_root];
1113        } else {
1114          _flow[e] = -_supply[u];
1115          _forward[u] = false;
1116          _pi[u] = ART_COST + _pi[_root];
1117        }
1118      }
1119
1120      return true;
1121    }
1122
1123    // Find the join node
1124    void findJoinNode() {
1125      int u = _source[in_arc];
1126      int v = _target[in_arc];
1127      while (u != v) {
1128        if (_succ_num[u] < _succ_num[v]) {
1129          u = _parent[u];
1130        } else {
1131          v = _parent[v];
1132        }
1133      }
1134      join = u;
1135    }
1136
1137    // Find the leaving arc of the cycle and returns true if the
1138    // leaving arc is not the same as the entering arc
1139    bool findLeavingArc() {
1140      // Initialize first and second nodes according to the direction
1141      // of the cycle
1142      if (_state[in_arc] == STATE_LOWER) {
1143        first  = _source[in_arc];
1144        second = _target[in_arc];
1145      } else {
1146        first  = _target[in_arc];
1147        second = _source[in_arc];
1148      }
1149      delta = _cap[in_arc];
1150      int result = 0;
1151      Value d;
1152      int e;
1153
1154      // Search the cycle along the path form the first node to the root
1155      for (int u = first; u != join; u = _parent[u]) {
1156        e = _pred[u];
1157        d = _forward[u] ?
1158          _flow[e] : (_cap[e] == INF ? INF : _cap[e] - _flow[e]);
1159        if (d < delta) {
1160          delta = d;
1161          u_out = u;
1162          result = 1;
1163        }
1164      }
1165      // Search the cycle along the path form the second node to the root
1166      for (int u = second; u != join; u = _parent[u]) {
1167        e = _pred[u];
1168        d = _forward[u] ?
1169          (_cap[e] == INF ? INF : _cap[e] - _flow[e]) : _flow[e];
1170        if (d <= delta) {
1171          delta = d;
1172          u_out = u;
1173          result = 2;
1174        }
1175      }
1176
1177      if (result == 1) {
1178        u_in = first;
1179        v_in = second;
1180      } else {
1181        u_in = second;
1182        v_in = first;
1183      }
1184      return result != 0;
1185    }
1186
1187    // Change _flow and _state vectors
1188    void changeFlow(bool change) {
1189      // Augment along the cycle
1190      if (delta > 0) {
1191        Value val = _state[in_arc] * delta;
1192        _flow[in_arc] += val;
1193        for (int u = _source[in_arc]; u != join; u = _parent[u]) {
1194          _flow[_pred[u]] += _forward[u] ? -val : val;
1195        }
1196        for (int u = _target[in_arc]; u != join; u = _parent[u]) {
1197          _flow[_pred[u]] += _forward[u] ? val : -val;
1198        }
1199      }
1200      // Update the state of the entering and leaving arcs
1201      if (change) {
1202        _state[in_arc] = STATE_TREE;
1203        _state[_pred[u_out]] =
1204          (_flow[_pred[u_out]] == 0) ? STATE_LOWER : STATE_UPPER;
1205      } else {
1206        _state[in_arc] = -_state[in_arc];
1207      }
1208    }
1209
1210    // Update the tree structure
1211    void updateTreeStructure() {
1212      int u, w;
1213      int old_rev_thread = _rev_thread[u_out];
1214      int old_succ_num = _succ_num[u_out];
1215      int old_last_succ = _last_succ[u_out];
1216      v_out = _parent[u_out];
1217
1218      u = _last_succ[u_in];  // the last successor of u_in
1219      right = _thread[u];    // the node after it
1220
1221      // Handle the case when old_rev_thread equals to v_in
1222      // (it also means that join and v_out coincide)
1223      if (old_rev_thread == v_in) {
1224        last = _thread[_last_succ[u_out]];
1225      } else {
1226        last = _thread[v_in];
1227      }
1228
1229      // Update _thread and _parent along the stem nodes (i.e. the nodes
1230      // between u_in and u_out, whose parent have to be changed)
1231      _thread[v_in] = stem = u_in;
1232      _dirty_revs.clear();
1233      _dirty_revs.push_back(v_in);
1234      par_stem = v_in;
1235      while (stem != u_out) {
1236        // Insert the next stem node into the thread list
1237        new_stem = _parent[stem];
1238        _thread[u] = new_stem;
1239        _dirty_revs.push_back(u);
1240
1241        // Remove the subtree of stem from the thread list
1242        w = _rev_thread[stem];
1243        _thread[w] = right;
1244        _rev_thread[right] = w;
1245
1246        // Change the parent node and shift stem nodes
1247        _parent[stem] = par_stem;
1248        par_stem = stem;
1249        stem = new_stem;
1250
1251        // Update u and right
1252        u = _last_succ[stem] == _last_succ[par_stem] ?
1253          _rev_thread[par_stem] : _last_succ[stem];
1254        right = _thread[u];
1255      }
1256      _parent[u_out] = par_stem;
1257      _thread[u] = last;
1258      _rev_thread[last] = u;
1259      _last_succ[u_out] = u;
1260
1261      // Remove the subtree of u_out from the thread list except for
1262      // the case when old_rev_thread equals to v_in
1263      // (it also means that join and v_out coincide)
1264      if (old_rev_thread != v_in) {
1265        _thread[old_rev_thread] = right;
1266        _rev_thread[right] = old_rev_thread;
1267      }
1268
1269      // Update _rev_thread using the new _thread values
1270      for (int i = 0; i < int(_dirty_revs.size()); ++i) {
1271        u = _dirty_revs[i];
1272        _rev_thread[_thread[u]] = u;
1273      }
1274
1275      // Update _pred, _forward, _last_succ and _succ_num for the
1276      // stem nodes from u_out to u_in
1277      int tmp_sc = 0, tmp_ls = _last_succ[u_out];
1278      u = u_out;
1279      while (u != u_in) {
1280        w = _parent[u];
1281        _pred[u] = _pred[w];
1282        _forward[u] = !_forward[w];
1283        tmp_sc += _succ_num[u] - _succ_num[w];
1284        _succ_num[u] = tmp_sc;
1285        _last_succ[w] = tmp_ls;
1286        u = w;
1287      }
1288      _pred[u_in] = in_arc;
1289      _forward[u_in] = (u_in == _source[in_arc]);
1290      _succ_num[u_in] = old_succ_num;
1291
1292      // Set limits for updating _last_succ form v_in and v_out
1293      // towards the root
1294      int up_limit_in = -1;
1295      int up_limit_out = -1;
1296      if (_last_succ[join] == v_in) {
1297        up_limit_out = join;
1298      } else {
1299        up_limit_in = join;
1300      }
1301
1302      // Update _last_succ from v_in towards the root
1303      for (u = v_in; u != up_limit_in && _last_succ[u] == v_in;
1304           u = _parent[u]) {
1305        _last_succ[u] = _last_succ[u_out];
1306      }
1307      // Update _last_succ from v_out towards the root
1308      if (join != old_rev_thread && v_in != old_rev_thread) {
1309        for (u = v_out; u != up_limit_out && _last_succ[u] == old_last_succ;
1310             u = _parent[u]) {
1311          _last_succ[u] = old_rev_thread;
1312        }
1313      } else {
1314        for (u = v_out; u != up_limit_out && _last_succ[u] == old_last_succ;
1315             u = _parent[u]) {
1316          _last_succ[u] = _last_succ[u_out];
1317        }
1318      }
1319
1320      // Update _succ_num from v_in to join
1321      for (u = v_in; u != join; u = _parent[u]) {
1322        _succ_num[u] += old_succ_num;
1323      }
1324      // Update _succ_num from v_out to join
1325      for (u = v_out; u != join; u = _parent[u]) {
1326        _succ_num[u] -= old_succ_num;
1327      }
1328    }
1329
1330    // Update potentials
1331    void updatePotential() {
1332      Cost sigma = _forward[u_in] ?
1333        _pi[v_in] - _pi[u_in] - _cost[_pred[u_in]] :
1334        _pi[v_in] - _pi[u_in] + _cost[_pred[u_in]];
1335      // Update potentials in the subtree, which has been moved
1336      int end = _thread[_last_succ[u_in]];
1337      for (int u = u_in; u != end; u = _thread[u]) {
1338        _pi[u] += sigma;
1339      }
1340    }
1341
1342    // Execute the algorithm
1343    ProblemType start(PivotRule pivot_rule) {
1344      // Select the pivot rule implementation
1345      switch (pivot_rule) {
1346        case FIRST_ELIGIBLE:
1347          return start<FirstEligiblePivotRule>();
1348        case BEST_ELIGIBLE:
1349          return start<BestEligiblePivotRule>();
1350        case BLOCK_SEARCH:
1351          return start<BlockSearchPivotRule>();
1352        case CANDIDATE_LIST:
1353          return start<CandidateListPivotRule>();
1354        case ALTERING_LIST:
1355          return start<AlteringListPivotRule>();
1356      }
1357      return INFEASIBLE; // avoid warning
1358    }
1359
1360    template <typename PivotRuleImpl>
1361    ProblemType start() {
1362      PivotRuleImpl pivot(*this);
1363
1364      // Execute the Network Simplex algorithm
1365      while (pivot.findEnteringArc()) {
1366        findJoinNode();
1367        bool change = findLeavingArc();
1368        if (delta >= INF) return UNBOUNDED;
1369        changeFlow(change);
1370        if (change) {
1371          updateTreeStructure();
1372          updatePotential();
1373        }
1374      }
1375     
1376      // Check feasibility
1377      if (_sum_supply < 0) {
1378        for (int u = 0, e = _arc_num; u != _node_num; ++u, ++e) {
1379          if (_supply[u] >= 0 && _flow[e] != 0) return INFEASIBLE;
1380        }
1381      }
1382      else if (_sum_supply > 0) {
1383        for (int u = 0, e = _arc_num; u != _node_num; ++u, ++e) {
1384          if (_supply[u] <= 0 && _flow[e] != 0) return INFEASIBLE;
1385        }
1386      }
1387      else {
1388        for (int u = 0, e = _arc_num; u != _node_num; ++u, ++e) {
1389          if (_flow[e] != 0) return INFEASIBLE;
1390        }
1391      }
1392
1393      // Transform the solution and the supply map to the original form
1394      if (_have_lower) {
1395        for (int i = 0; i != _arc_num; ++i) {
1396          Value c = _lower[i];
1397          if (c != 0) {
1398            _flow[i] += c;
1399            _supply[_source[i]] += c;
1400            _supply[_target[i]] -= c;
1401          }
1402        }
1403      }
1404
1405      return OPTIMAL;
1406    }
1407
1408  }; //class NetworkSimplex
1409
1410  ///@}
1411
1412} //namespace lemon
1413
1414#endif //LEMON_NETWORK_SIMPLEX_H
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