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source: lemon/lemon/cycle_canceling.h @ 1270:dceba191c00d

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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) 2003-2013
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_CYCLE_CANCELING_H
20#define LEMON_CYCLE_CANCELING_H
21
22/// \ingroup min_cost_flow_algs
23/// \file
24/// \brief Cycle-canceling algorithms for finding a minimum cost flow.
25
26#include <vector>
27#include <limits>
28
29#include <lemon/core.h>
30#include <lemon/maps.h>
31#include <lemon/path.h>
32#include <lemon/math.h>
33#include <lemon/static_graph.h>
34#include <lemon/adaptors.h>
35#include <lemon/circulation.h>
36#include <lemon/bellman_ford.h>
37#include <lemon/howard_mmc.h>
38#include <lemon/hartmann_orlin_mmc.h>
39
40namespace lemon {
41
42  /// \addtogroup min_cost_flow_algs
43  /// @{
44
45  /// \brief Implementation of cycle-canceling algorithms for
46  /// finding a \ref min_cost_flow "minimum cost flow".
47  ///
48  /// \ref CycleCanceling implements three different cycle-canceling
49  /// algorithms for finding a \ref min_cost_flow "minimum cost flow"
50  /// \cite amo93networkflows, \cite klein67primal,
51  /// \cite goldberg89cyclecanceling.
52  /// The most efficent one is the \ref CANCEL_AND_TIGHTEN
53  /// "Cancel-and-Tighten" algorithm, thus it is the default method.
54  /// It runs in strongly polynomial time \f$O(n^2 m^2 \log n)\f$,
55  /// but in practice, it is typically orders of magnitude slower than
56  /// the scaling algorithms and \ref NetworkSimplex.
57  /// (For more information, see \ref min_cost_flow_algs "the module page".)
58  ///
59  /// Most of the parameters of the problem (except for the digraph)
60  /// can be given using separate functions, and the algorithm can be
61  /// executed using the \ref run() function. If some parameters are not
62  /// specified, then default values will be used.
63  ///
64  /// \tparam GR The digraph type the algorithm runs on.
65  /// \tparam V The number type used for flow amounts, capacity bounds
66  /// and supply values in the algorithm. By default, it is \c int.
67  /// \tparam C The number type used for costs and potentials in the
68  /// algorithm. By default, it is the same as \c V.
69  ///
70  /// \warning Both \c V and \c C must be signed number types.
71  /// \warning All input data (capacities, supply values, and costs) must
72  /// be integer.
73  /// \warning This algorithm does not support negative costs for
74  /// arcs having infinite upper bound.
75  ///
76  /// \note For more information about the three available methods,
77  /// see \ref Method.
78#ifdef DOXYGEN
79  template <typename GR, typename V, typename C>
80#else
81  template <typename GR, typename V = int, typename C = V>
82#endif
83  class CycleCanceling
84  {
85  public:
86
87    /// The type of the digraph
88    typedef GR Digraph;
89    /// The type of the flow amounts, capacity bounds and supply values
90    typedef V Value;
91    /// The type of the arc costs
92    typedef C Cost;
93
94  public:
95
96    /// \brief Problem type constants for the \c run() function.
97    ///
98    /// Enum type containing the problem type constants that can be
99    /// returned by the \ref run() function of the algorithm.
100    enum ProblemType {
101      /// The problem has no feasible solution (flow).
102      INFEASIBLE,
103      /// The problem has optimal solution (i.e. it is feasible and
104      /// bounded), and the algorithm has found optimal flow and node
105      /// potentials (primal and dual solutions).
106      OPTIMAL,
107      /// The digraph contains an arc of negative cost and infinite
108      /// upper bound. It means that the objective function is unbounded
109      /// on that arc, however, note that it could actually be bounded
110      /// over the feasible flows, but this algroithm cannot handle
111      /// these cases.
112      UNBOUNDED
113    };
114
115    /// \brief Constants for selecting the used method.
116    ///
117    /// Enum type containing constants for selecting the used method
118    /// for the \ref run() function.
119    ///
120    /// \ref CycleCanceling provides three different cycle-canceling
121    /// methods. By default, \ref CANCEL_AND_TIGHTEN "Cancel-and-Tighten"
122    /// is used, which is by far the most efficient and the most robust.
123    /// However, the other methods can be selected using the \ref run()
124    /// function with the proper parameter.
125    enum Method {
126      /// A simple cycle-canceling method, which uses the
127      /// \ref BellmanFord "Bellman-Ford" algorithm for detecting negative
128      /// cycles in the residual network.
129      /// The number of Bellman-Ford iterations is bounded by a successively
130      /// increased limit.
131      SIMPLE_CYCLE_CANCELING,
132      /// The "Minimum Mean Cycle-Canceling" algorithm, which is a
133      /// well-known strongly polynomial method
134      /// \cite goldberg89cyclecanceling. It improves along a
135      /// \ref min_mean_cycle "minimum mean cycle" in each iteration.
136      /// Its running time complexity is \f$O(n^2 m^3 \log n)\f$.
137      MINIMUM_MEAN_CYCLE_CANCELING,
138      /// The "Cancel-and-Tighten" algorithm, which can be viewed as an
139      /// improved version of the previous method
140      /// \cite goldberg89cyclecanceling.
141      /// It is faster both in theory and in practice, its running time
142      /// complexity is \f$O(n^2 m^2 \log n)\f$.
143      CANCEL_AND_TIGHTEN
144    };
145
146  private:
147
148    TEMPLATE_DIGRAPH_TYPEDEFS(GR);
149
150    typedef std::vector<int> IntVector;
151    typedef std::vector<double> DoubleVector;
152    typedef std::vector<Value> ValueVector;
153    typedef std::vector<Cost> CostVector;
154    typedef std::vector<char> BoolVector;
155    // Note: vector<char> is used instead of vector<bool> for efficiency reasons
156
157  private:
158
159    template <typename KT, typename VT>
160    class StaticVectorMap {
161    public:
162      typedef KT Key;
163      typedef VT Value;
164
165      StaticVectorMap(std::vector<Value>& v) : _v(v) {}
166
167      const Value& operator[](const Key& key) const {
168        return _v[StaticDigraph::id(key)];
169      }
170
171      Value& operator[](const Key& key) {
172        return _v[StaticDigraph::id(key)];
173      }
174
175      void set(const Key& key, const Value& val) {
176        _v[StaticDigraph::id(key)] = val;
177      }
178
179    private:
180      std::vector<Value>& _v;
181    };
182
183    typedef StaticVectorMap<StaticDigraph::Node, Cost> CostNodeMap;
184    typedef StaticVectorMap<StaticDigraph::Arc, Cost> CostArcMap;
185
186  private:
187
188
189    // Data related to the underlying digraph
190    const GR &_graph;
191    int _node_num;
192    int _arc_num;
193    int _res_node_num;
194    int _res_arc_num;
195    int _root;
196
197    // Parameters of the problem
198    bool _have_lower;
199    Value _sum_supply;
200
201    // Data structures for storing the digraph
202    IntNodeMap _node_id;
203    IntArcMap _arc_idf;
204    IntArcMap _arc_idb;
205    IntVector _first_out;
206    BoolVector _forward;
207    IntVector _source;
208    IntVector _target;
209    IntVector _reverse;
210
211    // Node and arc data
212    ValueVector _lower;
213    ValueVector _upper;
214    CostVector _cost;
215    ValueVector _supply;
216
217    ValueVector _res_cap;
218    CostVector _pi;
219
220    // Data for a StaticDigraph structure
221    typedef std::pair<int, int> IntPair;
222    StaticDigraph _sgr;
223    std::vector<IntPair> _arc_vec;
224    std::vector<Cost> _cost_vec;
225    IntVector _id_vec;
226    CostArcMap _cost_map;
227    CostNodeMap _pi_map;
228
229  public:
230
231    /// \brief Constant for infinite upper bounds (capacities).
232    ///
233    /// Constant for infinite upper bounds (capacities).
234    /// It is \c std::numeric_limits<Value>::infinity() if available,
235    /// \c std::numeric_limits<Value>::max() otherwise.
236    const Value INF;
237
238  public:
239
240    /// \brief Constructor.
241    ///
242    /// The constructor of the class.
243    ///
244    /// \param graph The digraph the algorithm runs on.
245    CycleCanceling(const GR& graph) :
246      _graph(graph), _node_id(graph), _arc_idf(graph), _arc_idb(graph),
247      _cost_map(_cost_vec), _pi_map(_pi),
248      INF(std::numeric_limits<Value>::has_infinity ?
249          std::numeric_limits<Value>::infinity() :
250          std::numeric_limits<Value>::max())
251    {
252      // Check the number types
253      LEMON_ASSERT(std::numeric_limits<Value>::is_signed,
254        "The flow type of CycleCanceling must be signed");
255      LEMON_ASSERT(std::numeric_limits<Cost>::is_signed,
256        "The cost type of CycleCanceling must be signed");
257
258      // Reset data structures
259      reset();
260    }
261
262    /// \name Parameters
263    /// The parameters of the algorithm can be specified using these
264    /// functions.
265
266    /// @{
267
268    /// \brief Set the lower bounds on the arcs.
269    ///
270    /// This function sets the lower bounds on the arcs.
271    /// If it is not used before calling \ref run(), the lower bounds
272    /// will be set to zero on all arcs.
273    ///
274    /// \param map An arc map storing the lower bounds.
275    /// Its \c Value type must be convertible to the \c Value type
276    /// of the algorithm.
277    ///
278    /// \return <tt>(*this)</tt>
279    template <typename LowerMap>
280    CycleCanceling& lowerMap(const LowerMap& map) {
281      _have_lower = true;
282      for (ArcIt a(_graph); a != INVALID; ++a) {
283        _lower[_arc_idf[a]] = map[a];
284        _lower[_arc_idb[a]] = map[a];
285      }
286      return *this;
287    }
288
289    /// \brief Set the upper bounds (capacities) on the arcs.
290    ///
291    /// This function sets the upper bounds (capacities) on the arcs.
292    /// If it is not used before calling \ref run(), the upper bounds
293    /// will be set to \ref INF on all arcs (i.e. the flow value will be
294    /// unbounded from above).
295    ///
296    /// \param map An arc map storing the upper bounds.
297    /// Its \c Value type must be convertible to the \c Value type
298    /// of the algorithm.
299    ///
300    /// \return <tt>(*this)</tt>
301    template<typename UpperMap>
302    CycleCanceling& upperMap(const UpperMap& map) {
303      for (ArcIt a(_graph); a != INVALID; ++a) {
304        _upper[_arc_idf[a]] = map[a];
305      }
306      return *this;
307    }
308
309    /// \brief Set the costs of the arcs.
310    ///
311    /// This function sets the costs of the arcs.
312    /// If it is not used before calling \ref run(), the costs
313    /// will be set to \c 1 on all arcs.
314    ///
315    /// \param map An arc map storing the costs.
316    /// Its \c Value type must be convertible to the \c Cost type
317    /// of the algorithm.
318    ///
319    /// \return <tt>(*this)</tt>
320    template<typename CostMap>
321    CycleCanceling& costMap(const CostMap& map) {
322      for (ArcIt a(_graph); a != INVALID; ++a) {
323        _cost[_arc_idf[a]] =  map[a];
324        _cost[_arc_idb[a]] = -map[a];
325      }
326      return *this;
327    }
328
329    /// \brief Set the supply values of the nodes.
330    ///
331    /// This function sets the supply values of the nodes.
332    /// If neither this function nor \ref stSupply() is used before
333    /// calling \ref run(), the supply of each node will be set to zero.
334    ///
335    /// \param map A node map storing the supply values.
336    /// Its \c Value type must be convertible to the \c Value type
337    /// of the algorithm.
338    ///
339    /// \return <tt>(*this)</tt>
340    template<typename SupplyMap>
341    CycleCanceling& supplyMap(const SupplyMap& map) {
342      for (NodeIt n(_graph); n != INVALID; ++n) {
343        _supply[_node_id[n]] = map[n];
344      }
345      return *this;
346    }
347
348    /// \brief Set single source and target nodes and a supply value.
349    ///
350    /// This function sets a single source node and a single target node
351    /// and the required flow value.
352    /// If neither this function nor \ref supplyMap() is used before
353    /// calling \ref run(), the supply of each node will be set to zero.
354    ///
355    /// Using this function has the same effect as using \ref supplyMap()
356    /// with a map in which \c k is assigned to \c s, \c -k is
357    /// assigned to \c t and all other nodes have zero supply value.
358    ///
359    /// \param s The source node.
360    /// \param t The target node.
361    /// \param k The required amount of flow from node \c s to node \c t
362    /// (i.e. the supply of \c s and the demand of \c t).
363    ///
364    /// \return <tt>(*this)</tt>
365    CycleCanceling& stSupply(const Node& s, const Node& t, Value k) {
366      for (int i = 0; i != _res_node_num; ++i) {
367        _supply[i] = 0;
368      }
369      _supply[_node_id[s]] =  k;
370      _supply[_node_id[t]] = -k;
371      return *this;
372    }
373
374    /// @}
375
376    /// \name Execution control
377    /// The algorithm can be executed using \ref run().
378
379    /// @{
380
381    /// \brief Run the algorithm.
382    ///
383    /// This function runs the algorithm.
384    /// The paramters can be specified using functions \ref lowerMap(),
385    /// \ref upperMap(), \ref costMap(), \ref supplyMap(), \ref stSupply().
386    /// For example,
387    /// \code
388    ///   CycleCanceling<ListDigraph> cc(graph);
389    ///   cc.lowerMap(lower).upperMap(upper).costMap(cost)
390    ///     .supplyMap(sup).run();
391    /// \endcode
392    ///
393    /// This function can be called more than once. All the given parameters
394    /// are kept for the next call, unless \ref resetParams() or \ref reset()
395    /// is used, thus only the modified parameters have to be set again.
396    /// If the underlying digraph was also modified after the construction
397    /// of the class (or the last \ref reset() call), then the \ref reset()
398    /// function must be called.
399    ///
400    /// \param method The cycle-canceling method that will be used.
401    /// For more information, see \ref Method.
402    ///
403    /// \return \c INFEASIBLE if no feasible flow exists,
404    /// \n \c OPTIMAL if the problem has optimal solution
405    /// (i.e. it is feasible and bounded), and the algorithm has found
406    /// optimal flow and node potentials (primal and dual solutions),
407    /// \n \c UNBOUNDED if the digraph contains an arc of negative cost
408    /// and infinite upper bound. It means that the objective function
409    /// is unbounded on that arc, however, note that it could actually be
410    /// bounded over the feasible flows, but this algroithm cannot handle
411    /// these cases.
412    ///
413    /// \see ProblemType, Method
414    /// \see resetParams(), reset()
415    ProblemType run(Method method = CANCEL_AND_TIGHTEN) {
416      ProblemType pt = init();
417      if (pt != OPTIMAL) return pt;
418      start(method);
419      return OPTIMAL;
420    }
421
422    /// \brief Reset all the parameters that have been given before.
423    ///
424    /// This function resets all the paramaters that have been given
425    /// before using functions \ref lowerMap(), \ref upperMap(),
426    /// \ref costMap(), \ref supplyMap(), \ref stSupply().
427    ///
428    /// It is useful for multiple \ref run() calls. Basically, all the given
429    /// parameters are kept for the next \ref run() call, unless
430    /// \ref resetParams() or \ref reset() is used.
431    /// If the underlying digraph was also modified after the construction
432    /// of the class or the last \ref reset() call, then the \ref reset()
433    /// function must be used, otherwise \ref resetParams() is sufficient.
434    ///
435    /// For example,
436    /// \code
437    ///   CycleCanceling<ListDigraph> cs(graph);
438    ///
439    ///   // First run
440    ///   cc.lowerMap(lower).upperMap(upper).costMap(cost)
441    ///     .supplyMap(sup).run();
442    ///
443    ///   // Run again with modified cost map (resetParams() is not called,
444    ///   // so only the cost map have to be set again)
445    ///   cost[e] += 100;
446    ///   cc.costMap(cost).run();
447    ///
448    ///   // Run again from scratch using resetParams()
449    ///   // (the lower bounds will be set to zero on all arcs)
450    ///   cc.resetParams();
451    ///   cc.upperMap(capacity).costMap(cost)
452    ///     .supplyMap(sup).run();
453    /// \endcode
454    ///
455    /// \return <tt>(*this)</tt>
456    ///
457    /// \see reset(), run()
458    CycleCanceling& resetParams() {
459      for (int i = 0; i != _res_node_num; ++i) {
460        _supply[i] = 0;
461      }
462      int limit = _first_out[_root];
463      for (int j = 0; j != limit; ++j) {
464        _lower[j] = 0;
465        _upper[j] = INF;
466        _cost[j] = _forward[j] ? 1 : -1;
467      }
468      for (int j = limit; j != _res_arc_num; ++j) {
469        _lower[j] = 0;
470        _upper[j] = INF;
471        _cost[j] = 0;
472        _cost[_reverse[j]] = 0;
473      }
474      _have_lower = false;
475      return *this;
476    }
477
478    /// \brief Reset the internal data structures and all the parameters
479    /// that have been given before.
480    ///
481    /// This function resets the internal data structures and all the
482    /// paramaters that have been given before using functions \ref lowerMap(),
483    /// \ref upperMap(), \ref costMap(), \ref supplyMap(), \ref stSupply().
484    ///
485    /// It is useful for multiple \ref run() calls. Basically, all the given
486    /// parameters are kept for the next \ref run() call, unless
487    /// \ref resetParams() or \ref reset() is used.
488    /// If the underlying digraph was also modified after the construction
489    /// of the class or the last \ref reset() call, then the \ref reset()
490    /// function must be used, otherwise \ref resetParams() is sufficient.
491    ///
492    /// See \ref resetParams() for examples.
493    ///
494    /// \return <tt>(*this)</tt>
495    ///
496    /// \see resetParams(), run()
497    CycleCanceling& reset() {
498      // Resize vectors
499      _node_num = countNodes(_graph);
500      _arc_num = countArcs(_graph);
501      _res_node_num = _node_num + 1;
502      _res_arc_num = 2 * (_arc_num + _node_num);
503      _root = _node_num;
504
505      _first_out.resize(_res_node_num + 1);
506      _forward.resize(_res_arc_num);
507      _source.resize(_res_arc_num);
508      _target.resize(_res_arc_num);
509      _reverse.resize(_res_arc_num);
510
511      _lower.resize(_res_arc_num);
512      _upper.resize(_res_arc_num);
513      _cost.resize(_res_arc_num);
514      _supply.resize(_res_node_num);
515
516      _res_cap.resize(_res_arc_num);
517      _pi.resize(_res_node_num);
518
519      _arc_vec.reserve(_res_arc_num);
520      _cost_vec.reserve(_res_arc_num);
521      _id_vec.reserve(_res_arc_num);
522
523      // Copy the graph
524      int i = 0, j = 0, k = 2 * _arc_num + _node_num;
525      for (NodeIt n(_graph); n != INVALID; ++n, ++i) {
526        _node_id[n] = i;
527      }
528      i = 0;
529      for (NodeIt n(_graph); n != INVALID; ++n, ++i) {
530        _first_out[i] = j;
531        for (OutArcIt a(_graph, n); a != INVALID; ++a, ++j) {
532          _arc_idf[a] = j;
533          _forward[j] = true;
534          _source[j] = i;
535          _target[j] = _node_id[_graph.runningNode(a)];
536        }
537        for (InArcIt a(_graph, n); a != INVALID; ++a, ++j) {
538          _arc_idb[a] = j;
539          _forward[j] = false;
540          _source[j] = i;
541          _target[j] = _node_id[_graph.runningNode(a)];
542        }
543        _forward[j] = false;
544        _source[j] = i;
545        _target[j] = _root;
546        _reverse[j] = k;
547        _forward[k] = true;
548        _source[k] = _root;
549        _target[k] = i;
550        _reverse[k] = j;
551        ++j; ++k;
552      }
553      _first_out[i] = j;
554      _first_out[_res_node_num] = k;
555      for (ArcIt a(_graph); a != INVALID; ++a) {
556        int fi = _arc_idf[a];
557        int bi = _arc_idb[a];
558        _reverse[fi] = bi;
559        _reverse[bi] = fi;
560      }
561
562      // Reset parameters
563      resetParams();
564      return *this;
565    }
566
567    /// @}
568
569    /// \name Query Functions
570    /// The results of the algorithm can be obtained using these
571    /// functions.\n
572    /// The \ref run() function must be called before using them.
573
574    /// @{
575
576    /// \brief Return the total cost of the found flow.
577    ///
578    /// This function returns the total cost of the found flow.
579    /// Its complexity is O(m).
580    ///
581    /// \note The return type of the function can be specified as a
582    /// template parameter. For example,
583    /// \code
584    ///   cc.totalCost<double>();
585    /// \endcode
586    /// It is useful if the total cost cannot be stored in the \c Cost
587    /// type of the algorithm, which is the default return type of the
588    /// function.
589    ///
590    /// \pre \ref run() must be called before using this function.
591    template <typename Number>
592    Number totalCost() const {
593      Number c = 0;
594      for (ArcIt a(_graph); a != INVALID; ++a) {
595        int i = _arc_idb[a];
596        c += static_cast<Number>(_res_cap[i]) *
597             (-static_cast<Number>(_cost[i]));
598      }
599      return c;
600    }
601
602#ifndef DOXYGEN
603    Cost totalCost() const {
604      return totalCost<Cost>();
605    }
606#endif
607
608    /// \brief Return the flow on the given arc.
609    ///
610    /// This function returns the flow on the given arc.
611    ///
612    /// \pre \ref run() must be called before using this function.
613    Value flow(const Arc& a) const {
614      return _res_cap[_arc_idb[a]];
615    }
616
617    /// \brief Copy the flow values (the primal solution) into the
618    /// given map.
619    ///
620    /// This function copies the flow value on each arc into the given
621    /// map. The \c Value type of the algorithm must be convertible to
622    /// the \c Value type of the map.
623    ///
624    /// \pre \ref run() must be called before using this function.
625    template <typename FlowMap>
626    void flowMap(FlowMap &map) const {
627      for (ArcIt a(_graph); a != INVALID; ++a) {
628        map.set(a, _res_cap[_arc_idb[a]]);
629      }
630    }
631
632    /// \brief Return the potential (dual value) of the given node.
633    ///
634    /// This function returns the potential (dual value) of the
635    /// given node.
636    ///
637    /// \pre \ref run() must be called before using this function.
638    Cost potential(const Node& n) const {
639      return static_cast<Cost>(_pi[_node_id[n]]);
640    }
641
642    /// \brief Copy the potential values (the dual solution) into the
643    /// given map.
644    ///
645    /// This function copies the potential (dual value) of each node
646    /// into the given map.
647    /// The \c Cost type of the algorithm must be convertible to the
648    /// \c Value type of the map.
649    ///
650    /// \pre \ref run() must be called before using this function.
651    template <typename PotentialMap>
652    void potentialMap(PotentialMap &map) const {
653      for (NodeIt n(_graph); n != INVALID; ++n) {
654        map.set(n, static_cast<Cost>(_pi[_node_id[n]]));
655      }
656    }
657
658    /// @}
659
660  private:
661
662    // Initialize the algorithm
663    ProblemType init() {
664      if (_res_node_num <= 1) return INFEASIBLE;
665
666      // Check the sum of supply values
667      _sum_supply = 0;
668      for (int i = 0; i != _root; ++i) {
669        _sum_supply += _supply[i];
670      }
671      if (_sum_supply > 0) return INFEASIBLE;
672
673      // Check lower and upper bounds
674      LEMON_DEBUG(checkBoundMaps(),
675          "Upper bounds must be greater or equal to the lower bounds");
676
677
678      // Initialize vectors
679      for (int i = 0; i != _res_node_num; ++i) {
680        _pi[i] = 0;
681      }
682      ValueVector excess(_supply);
683
684      // Remove infinite upper bounds and check negative arcs
685      const Value MAX = std::numeric_limits<Value>::max();
686      int last_out;
687      if (_have_lower) {
688        for (int i = 0; i != _root; ++i) {
689          last_out = _first_out[i+1];
690          for (int j = _first_out[i]; j != last_out; ++j) {
691            if (_forward[j]) {
692              Value c = _cost[j] < 0 ? _upper[j] : _lower[j];
693              if (c >= MAX) return UNBOUNDED;
694              excess[i] -= c;
695              excess[_target[j]] += c;
696            }
697          }
698        }
699      } else {
700        for (int i = 0; i != _root; ++i) {
701          last_out = _first_out[i+1];
702          for (int j = _first_out[i]; j != last_out; ++j) {
703            if (_forward[j] && _cost[j] < 0) {
704              Value c = _upper[j];
705              if (c >= MAX) return UNBOUNDED;
706              excess[i] -= c;
707              excess[_target[j]] += c;
708            }
709          }
710        }
711      }
712      Value ex, max_cap = 0;
713      for (int i = 0; i != _res_node_num; ++i) {
714        ex = excess[i];
715        if (ex < 0) max_cap -= ex;
716      }
717      for (int j = 0; j != _res_arc_num; ++j) {
718        if (_upper[j] >= MAX) _upper[j] = max_cap;
719      }
720
721      // Initialize maps for Circulation and remove non-zero lower bounds
722      ConstMap<Arc, Value> low(0);
723      typedef typename Digraph::template ArcMap<Value> ValueArcMap;
724      typedef typename Digraph::template NodeMap<Value> ValueNodeMap;
725      ValueArcMap cap(_graph), flow(_graph);
726      ValueNodeMap sup(_graph);
727      for (NodeIt n(_graph); n != INVALID; ++n) {
728        sup[n] = _supply[_node_id[n]];
729      }
730      if (_have_lower) {
731        for (ArcIt a(_graph); a != INVALID; ++a) {
732          int j = _arc_idf[a];
733          Value c = _lower[j];
734          cap[a] = _upper[j] - c;
735          sup[_graph.source(a)] -= c;
736          sup[_graph.target(a)] += c;
737        }
738      } else {
739        for (ArcIt a(_graph); a != INVALID; ++a) {
740          cap[a] = _upper[_arc_idf[a]];
741        }
742      }
743
744      // Find a feasible flow using Circulation
745      Circulation<Digraph, ConstMap<Arc, Value>, ValueArcMap, ValueNodeMap>
746        circ(_graph, low, cap, sup);
747      if (!circ.flowMap(flow).run()) return INFEASIBLE;
748
749      // Set residual capacities and handle GEQ supply type
750      if (_sum_supply < 0) {
751        for (ArcIt a(_graph); a != INVALID; ++a) {
752          Value fa = flow[a];
753          _res_cap[_arc_idf[a]] = cap[a] - fa;
754          _res_cap[_arc_idb[a]] = fa;
755          sup[_graph.source(a)] -= fa;
756          sup[_graph.target(a)] += fa;
757        }
758        for (NodeIt n(_graph); n != INVALID; ++n) {
759          excess[_node_id[n]] = sup[n];
760        }
761        for (int a = _first_out[_root]; a != _res_arc_num; ++a) {
762          int u = _target[a];
763          int ra = _reverse[a];
764          _res_cap[a] = -_sum_supply + 1;
765          _res_cap[ra] = -excess[u];
766          _cost[a] = 0;
767          _cost[ra] = 0;
768        }
769      } else {
770        for (ArcIt a(_graph); a != INVALID; ++a) {
771          Value fa = flow[a];
772          _res_cap[_arc_idf[a]] = cap[a] - fa;
773          _res_cap[_arc_idb[a]] = fa;
774        }
775        for (int a = _first_out[_root]; a != _res_arc_num; ++a) {
776          int ra = _reverse[a];
777          _res_cap[a] = 1;
778          _res_cap[ra] = 0;
779          _cost[a] = 0;
780          _cost[ra] = 0;
781        }
782      }
783
784      return OPTIMAL;
785    }
786
787    // Check if the upper bound is greater or equal to the lower bound
788    // on each arc.
789    bool checkBoundMaps() {
790      for (int j = 0; j != _res_arc_num; ++j) {
791        if (_upper[j] < _lower[j]) return false;
792      }
793      return true;
794    }
795
796    // Build a StaticDigraph structure containing the current
797    // residual network
798    void buildResidualNetwork() {
799      _arc_vec.clear();
800      _cost_vec.clear();
801      _id_vec.clear();
802      for (int j = 0; j != _res_arc_num; ++j) {
803        if (_res_cap[j] > 0) {
804          _arc_vec.push_back(IntPair(_source[j], _target[j]));
805          _cost_vec.push_back(_cost[j]);
806          _id_vec.push_back(j);
807        }
808      }
809      _sgr.build(_res_node_num, _arc_vec.begin(), _arc_vec.end());
810    }
811
812    // Execute the algorithm and transform the results
813    void start(Method method) {
814      // Execute the algorithm
815      switch (method) {
816        case SIMPLE_CYCLE_CANCELING:
817          startSimpleCycleCanceling();
818          break;
819        case MINIMUM_MEAN_CYCLE_CANCELING:
820          startMinMeanCycleCanceling();
821          break;
822        case CANCEL_AND_TIGHTEN:
823          startCancelAndTighten();
824          break;
825      }
826
827      // Compute node potentials
828      if (method != SIMPLE_CYCLE_CANCELING) {
829        buildResidualNetwork();
830        typename BellmanFord<StaticDigraph, CostArcMap>
831          ::template SetDistMap<CostNodeMap>::Create bf(_sgr, _cost_map);
832        bf.distMap(_pi_map);
833        bf.init(0);
834        bf.start();
835      }
836
837      // Handle non-zero lower bounds
838      if (_have_lower) {
839        int limit = _first_out[_root];
840        for (int j = 0; j != limit; ++j) {
841          if (!_forward[j]) _res_cap[j] += _lower[j];
842        }
843      }
844    }
845
846    // Execute the "Simple Cycle Canceling" method
847    void startSimpleCycleCanceling() {
848      // Constants for computing the iteration limits
849      const int BF_FIRST_LIMIT  = 2;
850      const double BF_LIMIT_FACTOR = 1.5;
851
852      typedef StaticVectorMap<StaticDigraph::Arc, Value> FilterMap;
853      typedef FilterArcs<StaticDigraph, FilterMap> ResDigraph;
854      typedef StaticVectorMap<StaticDigraph::Node, StaticDigraph::Arc> PredMap;
855      typedef typename BellmanFord<ResDigraph, CostArcMap>
856        ::template SetDistMap<CostNodeMap>
857        ::template SetPredMap<PredMap>::Create BF;
858
859      // Build the residual network
860      _arc_vec.clear();
861      _cost_vec.clear();
862      for (int j = 0; j != _res_arc_num; ++j) {
863        _arc_vec.push_back(IntPair(_source[j], _target[j]));
864        _cost_vec.push_back(_cost[j]);
865      }
866      _sgr.build(_res_node_num, _arc_vec.begin(), _arc_vec.end());
867
868      FilterMap filter_map(_res_cap);
869      ResDigraph rgr(_sgr, filter_map);
870      std::vector<int> cycle;
871      std::vector<StaticDigraph::Arc> pred(_res_arc_num);
872      PredMap pred_map(pred);
873      BF bf(rgr, _cost_map);
874      bf.distMap(_pi_map).predMap(pred_map);
875
876      int length_bound = BF_FIRST_LIMIT;
877      bool optimal = false;
878      while (!optimal) {
879        bf.init(0);
880        int iter_num = 0;
881        bool cycle_found = false;
882        while (!cycle_found) {
883          // Perform some iterations of the Bellman-Ford algorithm
884          int curr_iter_num = iter_num + length_bound <= _node_num ?
885            length_bound : _node_num - iter_num;
886          iter_num += curr_iter_num;
887          int real_iter_num = curr_iter_num;
888          for (int i = 0; i < curr_iter_num; ++i) {
889            if (bf.processNextWeakRound()) {
890              real_iter_num = i;
891              break;
892            }
893          }
894          if (real_iter_num < curr_iter_num) {
895            // Optimal flow is found
896            optimal = true;
897            break;
898          } else {
899            // Search for node disjoint negative cycles
900            std::vector<int> state(_res_node_num, 0);
901            int id = 0;
902            for (int u = 0; u != _res_node_num; ++u) {
903              if (state[u] != 0) continue;
904              ++id;
905              int v = u;
906              for (; v != -1 && state[v] == 0; v = pred[v] == INVALID ?
907                   -1 : rgr.id(rgr.source(pred[v]))) {
908                state[v] = id;
909              }
910              if (v != -1 && state[v] == id) {
911                // A negative cycle is found
912                cycle_found = true;
913                cycle.clear();
914                StaticDigraph::Arc a = pred[v];
915                Value d, delta = _res_cap[rgr.id(a)];
916                cycle.push_back(rgr.id(a));
917                while (rgr.id(rgr.source(a)) != v) {
918                  a = pred_map[rgr.source(a)];
919                  d = _res_cap[rgr.id(a)];
920                  if (d < delta) delta = d;
921                  cycle.push_back(rgr.id(a));
922                }
923
924                // Augment along the cycle
925                for (int i = 0; i < int(cycle.size()); ++i) {
926                  int j = cycle[i];
927                  _res_cap[j] -= delta;
928                  _res_cap[_reverse[j]] += delta;
929                }
930              }
931            }
932          }
933
934          // Increase iteration limit if no cycle is found
935          if (!cycle_found) {
936            length_bound = static_cast<int>(length_bound * BF_LIMIT_FACTOR);
937          }
938        }
939      }
940    }
941
942    // Execute the "Minimum Mean Cycle Canceling" method
943    void startMinMeanCycleCanceling() {
944      typedef Path<StaticDigraph> SPath;
945      typedef typename SPath::ArcIt SPathArcIt;
946      typedef typename HowardMmc<StaticDigraph, CostArcMap>
947        ::template SetPath<SPath>::Create HwMmc;
948      typedef typename HartmannOrlinMmc<StaticDigraph, CostArcMap>
949        ::template SetPath<SPath>::Create HoMmc;
950
951      const double HW_ITER_LIMIT_FACTOR = 1.0;
952      const int HW_ITER_LIMIT_MIN_VALUE = 5;
953
954      const int hw_iter_limit =
955          std::max(static_cast<int>(HW_ITER_LIMIT_FACTOR * _node_num),
956                   HW_ITER_LIMIT_MIN_VALUE);
957
958      SPath cycle;
959      HwMmc hw_mmc(_sgr, _cost_map);
960      hw_mmc.cycle(cycle);
961      buildResidualNetwork();
962      while (true) {
963
964        typename HwMmc::TerminationCause hw_tc =
965            hw_mmc.findCycleMean(hw_iter_limit);
966        if (hw_tc == HwMmc::ITERATION_LIMIT) {
967          // Howard's algorithm reached the iteration limit, start a
968          // strongly polynomial algorithm instead
969          HoMmc ho_mmc(_sgr, _cost_map);
970          ho_mmc.cycle(cycle);
971          // Find a minimum mean cycle (Hartmann-Orlin algorithm)
972          if (!(ho_mmc.findCycleMean() && ho_mmc.cycleCost() < 0)) break;
973          ho_mmc.findCycle();
974        } else {
975          // Find a minimum mean cycle (Howard algorithm)
976          if (!(hw_tc == HwMmc::OPTIMAL && hw_mmc.cycleCost() < 0)) break;
977          hw_mmc.findCycle();
978        }
979
980        // Compute delta value
981        Value delta = INF;
982        for (SPathArcIt a(cycle); a != INVALID; ++a) {
983          Value d = _res_cap[_id_vec[_sgr.id(a)]];
984          if (d < delta) delta = d;
985        }
986
987        // Augment along the cycle
988        for (SPathArcIt a(cycle); a != INVALID; ++a) {
989          int j = _id_vec[_sgr.id(a)];
990          _res_cap[j] -= delta;
991          _res_cap[_reverse[j]] += delta;
992        }
993
994        // Rebuild the residual network
995        buildResidualNetwork();
996      }
997    }
998
999    // Execute the "Cancel-and-Tighten" method
1000    void startCancelAndTighten() {
1001      // Constants for the min mean cycle computations
1002      const double LIMIT_FACTOR = 1.0;
1003      const int MIN_LIMIT = 5;
1004      const double HW_ITER_LIMIT_FACTOR = 1.0;
1005      const int HW_ITER_LIMIT_MIN_VALUE = 5;
1006
1007      const int hw_iter_limit =
1008          std::max(static_cast<int>(HW_ITER_LIMIT_FACTOR * _node_num),
1009                   HW_ITER_LIMIT_MIN_VALUE);
1010
1011      // Contruct auxiliary data vectors
1012      DoubleVector pi(_res_node_num, 0.0);
1013      IntVector level(_res_node_num);
1014      BoolVector reached(_res_node_num);
1015      BoolVector processed(_res_node_num);
1016      IntVector pred_node(_res_node_num);
1017      IntVector pred_arc(_res_node_num);
1018      std::vector<int> stack(_res_node_num);
1019      std::vector<int> proc_vector(_res_node_num);
1020
1021      // Initialize epsilon
1022      double epsilon = 0;
1023      for (int a = 0; a != _res_arc_num; ++a) {
1024        if (_res_cap[a] > 0 && -_cost[a] > epsilon)
1025          epsilon = -_cost[a];
1026      }
1027
1028      // Start phases
1029      Tolerance<double> tol;
1030      tol.epsilon(1e-6);
1031      int limit = int(LIMIT_FACTOR * std::sqrt(double(_res_node_num)));
1032      if (limit < MIN_LIMIT) limit = MIN_LIMIT;
1033      int iter = limit;
1034      while (epsilon * _res_node_num >= 1) {
1035        // Find and cancel cycles in the admissible network using DFS
1036        for (int u = 0; u != _res_node_num; ++u) {
1037          reached[u] = false;
1038          processed[u] = false;
1039        }
1040        int stack_head = -1;
1041        int proc_head = -1;
1042        for (int start = 0; start != _res_node_num; ++start) {
1043          if (reached[start]) continue;
1044
1045          // New start node
1046          reached[start] = true;
1047          pred_arc[start] = -1;
1048          pred_node[start] = -1;
1049
1050          // Find the first admissible outgoing arc
1051          double p = pi[start];
1052          int a = _first_out[start];
1053          int last_out = _first_out[start+1];
1054          for (; a != last_out && (_res_cap[a] == 0 ||
1055               !tol.negative(_cost[a] + p - pi[_target[a]])); ++a) ;
1056          if (a == last_out) {
1057            processed[start] = true;
1058            proc_vector[++proc_head] = start;
1059            continue;
1060          }
1061          stack[++stack_head] = a;
1062
1063          while (stack_head >= 0) {
1064            int sa = stack[stack_head];
1065            int u = _source[sa];
1066            int v = _target[sa];
1067
1068            if (!reached[v]) {
1069              // A new node is reached
1070              reached[v] = true;
1071              pred_node[v] = u;
1072              pred_arc[v] = sa;
1073              p = pi[v];
1074              a = _first_out[v];
1075              last_out = _first_out[v+1];
1076              for (; a != last_out && (_res_cap[a] == 0 ||
1077                   !tol.negative(_cost[a] + p - pi[_target[a]])); ++a) ;
1078              stack[++stack_head] = a == last_out ? -1 : a;
1079            } else {
1080              if (!processed[v]) {
1081                // A cycle is found
1082                int n, w = u;
1083                Value d, delta = _res_cap[sa];
1084                for (n = u; n != v; n = pred_node[n]) {
1085                  d = _res_cap[pred_arc[n]];
1086                  if (d <= delta) {
1087                    delta = d;
1088                    w = pred_node[n];
1089                  }
1090                }
1091
1092                // Augment along the cycle
1093                _res_cap[sa] -= delta;
1094                _res_cap[_reverse[sa]] += delta;
1095                for (n = u; n != v; n = pred_node[n]) {
1096                  int pa = pred_arc[n];
1097                  _res_cap[pa] -= delta;
1098                  _res_cap[_reverse[pa]] += delta;
1099                }
1100                for (n = u; stack_head > 0 && n != w; n = pred_node[n]) {
1101                  --stack_head;
1102                  reached[n] = false;
1103                }
1104                u = w;
1105              }
1106              v = u;
1107
1108              // Find the next admissible outgoing arc
1109              p = pi[v];
1110              a = stack[stack_head] + 1;
1111              last_out = _first_out[v+1];
1112              for (; a != last_out && (_res_cap[a] == 0 ||
1113                   !tol.negative(_cost[a] + p - pi[_target[a]])); ++a) ;
1114              stack[stack_head] = a == last_out ? -1 : a;
1115            }
1116
1117            while (stack_head >= 0 && stack[stack_head] == -1) {
1118              processed[v] = true;
1119              proc_vector[++proc_head] = v;
1120              if (--stack_head >= 0) {
1121                // Find the next admissible outgoing arc
1122                v = _source[stack[stack_head]];
1123                p = pi[v];
1124                a = stack[stack_head] + 1;
1125                last_out = _first_out[v+1];
1126                for (; a != last_out && (_res_cap[a] == 0 ||
1127                     !tol.negative(_cost[a] + p - pi[_target[a]])); ++a) ;
1128                stack[stack_head] = a == last_out ? -1 : a;
1129              }
1130            }
1131          }
1132        }
1133
1134        // Tighten potentials and epsilon
1135        if (--iter > 0) {
1136          for (int u = 0; u != _res_node_num; ++u) {
1137            level[u] = 0;
1138          }
1139          for (int i = proc_head; i > 0; --i) {
1140            int u = proc_vector[i];
1141            double p = pi[u];
1142            int l = level[u] + 1;
1143            int last_out = _first_out[u+1];
1144            for (int a = _first_out[u]; a != last_out; ++a) {
1145              int v = _target[a];
1146              if (_res_cap[a] > 0 && tol.negative(_cost[a] + p - pi[v]) &&
1147                  l > level[v]) level[v] = l;
1148            }
1149          }
1150
1151          // Modify potentials
1152          double q = std::numeric_limits<double>::max();
1153          for (int u = 0; u != _res_node_num; ++u) {
1154            int lu = level[u];
1155            double p, pu = pi[u];
1156            int last_out = _first_out[u+1];
1157            for (int a = _first_out[u]; a != last_out; ++a) {
1158              if (_res_cap[a] == 0) continue;
1159              int v = _target[a];
1160              int ld = lu - level[v];
1161              if (ld > 0) {
1162                p = (_cost[a] + pu - pi[v] + epsilon) / (ld + 1);
1163                if (p < q) q = p;
1164              }
1165            }
1166          }
1167          for (int u = 0; u != _res_node_num; ++u) {
1168            pi[u] -= q * level[u];
1169          }
1170
1171          // Modify epsilon
1172          epsilon = 0;
1173          for (int u = 0; u != _res_node_num; ++u) {
1174            double curr, pu = pi[u];
1175            int last_out = _first_out[u+1];
1176            for (int a = _first_out[u]; a != last_out; ++a) {
1177              if (_res_cap[a] == 0) continue;
1178              curr = _cost[a] + pu - pi[_target[a]];
1179              if (-curr > epsilon) epsilon = -curr;
1180            }
1181          }
1182        } else {
1183          typedef HowardMmc<StaticDigraph, CostArcMap> HwMmc;
1184          typedef HartmannOrlinMmc<StaticDigraph, CostArcMap> HoMmc;
1185          typedef typename BellmanFord<StaticDigraph, CostArcMap>
1186            ::template SetDistMap<CostNodeMap>::Create BF;
1187
1188          // Set epsilon to the minimum cycle mean
1189          Cost cycle_cost = 0;
1190          int cycle_size = 1;
1191          buildResidualNetwork();
1192          HwMmc hw_mmc(_sgr, _cost_map);
1193          if (hw_mmc.findCycleMean(hw_iter_limit) == HwMmc::ITERATION_LIMIT) {
1194            // Howard's algorithm reached the iteration limit, start a
1195            // strongly polynomial algorithm instead
1196            HoMmc ho_mmc(_sgr, _cost_map);
1197            ho_mmc.findCycleMean();
1198            epsilon = -ho_mmc.cycleMean();
1199            cycle_cost = ho_mmc.cycleCost();
1200            cycle_size = ho_mmc.cycleSize();
1201          } else {
1202            // Set epsilon
1203            epsilon = -hw_mmc.cycleMean();
1204            cycle_cost = hw_mmc.cycleCost();
1205            cycle_size = hw_mmc.cycleSize();
1206          }
1207
1208          // Compute feasible potentials for the current epsilon
1209          for (int i = 0; i != int(_cost_vec.size()); ++i) {
1210            _cost_vec[i] = cycle_size * _cost_vec[i] - cycle_cost;
1211          }
1212          BF bf(_sgr, _cost_map);
1213          bf.distMap(_pi_map);
1214          bf.init(0);
1215          bf.start();
1216          for (int u = 0; u != _res_node_num; ++u) {
1217            pi[u] = static_cast<double>(_pi[u]) / cycle_size;
1218          }
1219
1220          iter = limit;
1221        }
1222      }
1223    }
1224
1225  }; //class CycleCanceling
1226
1227  ///@}
1228
1229} //namespace lemon
1230
1231#endif //LEMON_CYCLE_CANCELING_H
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