COIN-OR::LEMON - Graph Library

source: lemon-1.2/lemon/cycle_canceling.h @ 820:7ef7a5fbb85d

Last change on this file since 820:7ef7a5fbb85d was 820:7ef7a5fbb85d, checked in by Peter Kovacs <kpeter@…>, 10 years ago

Rename a private type in MCF classes (#180)

The new MCF algorithms define a private map type VectorMap?,
which could be misleading, since there is an other VectorMap?
defined in lemon/bits/vector_map.h. Thus the new type is
is renamed to StaticVectorMap?.

File size: 37.0 KB
Line 
1/* -*- C++ -*-
2 *
3 * This file is a part of LEMON, a generic C++ optimization library
4 *
5 * Copyright (C) 2003-2008
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.h>
38
39namespace lemon {
40
41  /// \addtogroup min_cost_flow_algs
42  /// @{
43
44  /// \brief Implementation of cycle-canceling algorithms for
45  /// finding a \ref min_cost_flow "minimum cost flow".
46  ///
47  /// \ref CycleCanceling implements three different cycle-canceling
48  /// algorithms for finding a \ref min_cost_flow "minimum cost flow"
49  /// \ref amo93networkflows, \ref klein67primal,
50  /// \ref goldberg89cyclecanceling.
51  /// The most efficent one (both theoretically and practically)
52  /// is the \ref CANCEL_AND_TIGHTEN "Cancel and Tighten" algorithm,
53  /// thus it is the default method.
54  /// It is strongly polynomial, but in practice, it is typically much
55  /// slower than the scaling algorithms and NetworkSimplex.
56  ///
57  /// Most of the parameters of the problem (except for the digraph)
58  /// can be given using separate functions, and the algorithm can be
59  /// executed using the \ref run() function. If some parameters are not
60  /// specified, then default values will be used.
61  ///
62  /// \tparam GR The digraph type the algorithm runs on.
63  /// \tparam V The number type used for flow amounts, capacity bounds
64  /// and supply values in the algorithm. By default, it is \c int.
65  /// \tparam C The number type used for costs and potentials in the
66  /// algorithm. By default, it is the same as \c V.
67  ///
68  /// \warning Both number types must be signed and all input data must
69  /// be integer.
70  /// \warning This algorithm does not support negative costs for such
71  /// arcs that have infinite upper bound.
72  ///
73  /// \note For more information about the three available methods,
74  /// see \ref Method.
75#ifdef DOXYGEN
76  template <typename GR, typename V, typename C>
77#else
78  template <typename GR, typename V = int, typename C = V>
79#endif
80  class CycleCanceling
81  {
82  public:
83
84    /// The type of the digraph
85    typedef GR Digraph;
86    /// The type of the flow amounts, capacity bounds and supply values
87    typedef V Value;
88    /// The type of the arc costs
89    typedef C Cost;
90
91  public:
92
93    /// \brief Problem type constants for the \c run() function.
94    ///
95    /// Enum type containing the problem type constants that can be
96    /// returned by the \ref run() function of the algorithm.
97    enum ProblemType {
98      /// The problem has no feasible solution (flow).
99      INFEASIBLE,
100      /// The problem has optimal solution (i.e. it is feasible and
101      /// bounded), and the algorithm has found optimal flow and node
102      /// potentials (primal and dual solutions).
103      OPTIMAL,
104      /// The digraph contains an arc of negative cost and infinite
105      /// upper bound. It means that the objective function is unbounded
106      /// on that arc, however, note that it could actually be bounded
107      /// over the feasible flows, but this algroithm cannot handle
108      /// these cases.
109      UNBOUNDED
110    };
111
112    /// \brief Constants for selecting the used method.
113    ///
114    /// Enum type containing constants for selecting the used method
115    /// for the \ref run() function.
116    ///
117    /// \ref CycleCanceling provides three different cycle-canceling
118    /// methods. By default, \ref CANCEL_AND_TIGHTEN "Cancel and Tighten"
119    /// is used, which proved to be the most efficient and the most robust
120    /// on various test inputs.
121    /// However, the other methods can be selected using the \ref run()
122    /// function with the proper parameter.
123    enum Method {
124      /// A simple cycle-canceling method, which uses the
125      /// \ref BellmanFord "Bellman-Ford" algorithm with limited iteration
126      /// number for detecting negative cycles in the residual network.
127      SIMPLE_CYCLE_CANCELING,
128      /// The "Minimum Mean Cycle-Canceling" algorithm, which is a
129      /// well-known strongly polynomial method
130      /// \ref goldberg89cyclecanceling. It improves along a
131      /// \ref min_mean_cycle "minimum mean cycle" in each iteration.
132      /// Its running time complexity is O(n<sup>2</sup>m<sup>3</sup>log(n)).
133      MINIMUM_MEAN_CYCLE_CANCELING,
134      /// The "Cancel And Tighten" algorithm, which can be viewed as an
135      /// improved version of the previous method
136      /// \ref goldberg89cyclecanceling.
137      /// It is faster both in theory and in practice, its running time
138      /// complexity is O(n<sup>2</sup>m<sup>2</sup>log(n)).
139      CANCEL_AND_TIGHTEN
140    };
141
142  private:
143
144    TEMPLATE_DIGRAPH_TYPEDEFS(GR);
145   
146    typedef std::vector<int> IntVector;
147    typedef std::vector<char> CharVector;
148    typedef std::vector<double> DoubleVector;
149    typedef std::vector<Value> ValueVector;
150    typedef std::vector<Cost> CostVector;
151
152  private:
153 
154    template <typename KT, typename VT>
155    class StaticVectorMap {
156    public:
157      typedef KT Key;
158      typedef VT Value;
159     
160      StaticVectorMap(std::vector<Value>& v) : _v(v) {}
161     
162      const Value& operator[](const Key& key) const {
163        return _v[StaticDigraph::id(key)];
164      }
165
166      Value& operator[](const Key& key) {
167        return _v[StaticDigraph::id(key)];
168      }
169     
170      void set(const Key& key, const Value& val) {
171        _v[StaticDigraph::id(key)] = val;
172      }
173
174    private:
175      std::vector<Value>& _v;
176    };
177
178    typedef StaticVectorMap<StaticDigraph::Node, Cost> CostNodeMap;
179    typedef StaticVectorMap<StaticDigraph::Arc, Cost> CostArcMap;
180
181  private:
182
183
184    // Data related to the underlying digraph
185    const GR &_graph;
186    int _node_num;
187    int _arc_num;
188    int _res_node_num;
189    int _res_arc_num;
190    int _root;
191
192    // Parameters of the problem
193    bool _have_lower;
194    Value _sum_supply;
195
196    // Data structures for storing the digraph
197    IntNodeMap _node_id;
198    IntArcMap _arc_idf;
199    IntArcMap _arc_idb;
200    IntVector _first_out;
201    CharVector _forward;
202    IntVector _source;
203    IntVector _target;
204    IntVector _reverse;
205
206    // Node and arc data
207    ValueVector _lower;
208    ValueVector _upper;
209    CostVector _cost;
210    ValueVector _supply;
211
212    ValueVector _res_cap;
213    CostVector _pi;
214
215    // Data for a StaticDigraph structure
216    typedef std::pair<int, int> IntPair;
217    StaticDigraph _sgr;
218    std::vector<IntPair> _arc_vec;
219    std::vector<Cost> _cost_vec;
220    IntVector _id_vec;
221    CostArcMap _cost_map;
222    CostNodeMap _pi_map;
223 
224  public:
225 
226    /// \brief Constant for infinite upper bounds (capacities).
227    ///
228    /// Constant for infinite upper bounds (capacities).
229    /// It is \c std::numeric_limits<Value>::infinity() if available,
230    /// \c std::numeric_limits<Value>::max() otherwise.
231    const Value INF;
232
233  public:
234
235    /// \brief Constructor.
236    ///
237    /// The constructor of the class.
238    ///
239    /// \param graph The digraph the algorithm runs on.
240    CycleCanceling(const GR& graph) :
241      _graph(graph), _node_id(graph), _arc_idf(graph), _arc_idb(graph),
242      _cost_map(_cost_vec), _pi_map(_pi),
243      INF(std::numeric_limits<Value>::has_infinity ?
244          std::numeric_limits<Value>::infinity() :
245          std::numeric_limits<Value>::max())
246    {
247      // Check the number types
248      LEMON_ASSERT(std::numeric_limits<Value>::is_signed,
249        "The flow type of CycleCanceling must be signed");
250      LEMON_ASSERT(std::numeric_limits<Cost>::is_signed,
251        "The cost type of CycleCanceling must be signed");
252
253      // Resize vectors
254      _node_num = countNodes(_graph);
255      _arc_num = countArcs(_graph);
256      _res_node_num = _node_num + 1;
257      _res_arc_num = 2 * (_arc_num + _node_num);
258      _root = _node_num;
259
260      _first_out.resize(_res_node_num + 1);
261      _forward.resize(_res_arc_num);
262      _source.resize(_res_arc_num);
263      _target.resize(_res_arc_num);
264      _reverse.resize(_res_arc_num);
265
266      _lower.resize(_res_arc_num);
267      _upper.resize(_res_arc_num);
268      _cost.resize(_res_arc_num);
269      _supply.resize(_res_node_num);
270     
271      _res_cap.resize(_res_arc_num);
272      _pi.resize(_res_node_num);
273
274      _arc_vec.reserve(_res_arc_num);
275      _cost_vec.reserve(_res_arc_num);
276      _id_vec.reserve(_res_arc_num);
277
278      // Copy the graph
279      int i = 0, j = 0, k = 2 * _arc_num + _node_num;
280      for (NodeIt n(_graph); n != INVALID; ++n, ++i) {
281        _node_id[n] = i;
282      }
283      i = 0;
284      for (NodeIt n(_graph); n != INVALID; ++n, ++i) {
285        _first_out[i] = j;
286        for (OutArcIt a(_graph, n); a != INVALID; ++a, ++j) {
287          _arc_idf[a] = j;
288          _forward[j] = true;
289          _source[j] = i;
290          _target[j] = _node_id[_graph.runningNode(a)];
291        }
292        for (InArcIt a(_graph, n); a != INVALID; ++a, ++j) {
293          _arc_idb[a] = j;
294          _forward[j] = false;
295          _source[j] = i;
296          _target[j] = _node_id[_graph.runningNode(a)];
297        }
298        _forward[j] = false;
299        _source[j] = i;
300        _target[j] = _root;
301        _reverse[j] = k;
302        _forward[k] = true;
303        _source[k] = _root;
304        _target[k] = i;
305        _reverse[k] = j;
306        ++j; ++k;
307      }
308      _first_out[i] = j;
309      _first_out[_res_node_num] = k;
310      for (ArcIt a(_graph); a != INVALID; ++a) {
311        int fi = _arc_idf[a];
312        int bi = _arc_idb[a];
313        _reverse[fi] = bi;
314        _reverse[bi] = fi;
315      }
316     
317      // Reset parameters
318      reset();
319    }
320
321    /// \name Parameters
322    /// The parameters of the algorithm can be specified using these
323    /// functions.
324
325    /// @{
326
327    /// \brief Set the lower bounds on the arcs.
328    ///
329    /// This function sets the lower bounds on the arcs.
330    /// If it is not used before calling \ref run(), the lower bounds
331    /// will be set to zero on all arcs.
332    ///
333    /// \param map An arc map storing the lower bounds.
334    /// Its \c Value type must be convertible to the \c Value type
335    /// of the algorithm.
336    ///
337    /// \return <tt>(*this)</tt>
338    template <typename LowerMap>
339    CycleCanceling& lowerMap(const LowerMap& map) {
340      _have_lower = true;
341      for (ArcIt a(_graph); a != INVALID; ++a) {
342        _lower[_arc_idf[a]] = map[a];
343        _lower[_arc_idb[a]] = map[a];
344      }
345      return *this;
346    }
347
348    /// \brief Set the upper bounds (capacities) on the arcs.
349    ///
350    /// This function sets the upper bounds (capacities) on the arcs.
351    /// If it is not used before calling \ref run(), the upper bounds
352    /// will be set to \ref INF on all arcs (i.e. the flow value will be
353    /// unbounded from above).
354    ///
355    /// \param map An arc map storing the upper bounds.
356    /// Its \c Value type must be convertible to the \c Value type
357    /// of the algorithm.
358    ///
359    /// \return <tt>(*this)</tt>
360    template<typename UpperMap>
361    CycleCanceling& upperMap(const UpperMap& map) {
362      for (ArcIt a(_graph); a != INVALID; ++a) {
363        _upper[_arc_idf[a]] = map[a];
364      }
365      return *this;
366    }
367
368    /// \brief Set the costs of the arcs.
369    ///
370    /// This function sets the costs of the arcs.
371    /// If it is not used before calling \ref run(), the costs
372    /// will be set to \c 1 on all arcs.
373    ///
374    /// \param map An arc map storing the costs.
375    /// Its \c Value type must be convertible to the \c Cost type
376    /// of the algorithm.
377    ///
378    /// \return <tt>(*this)</tt>
379    template<typename CostMap>
380    CycleCanceling& costMap(const CostMap& map) {
381      for (ArcIt a(_graph); a != INVALID; ++a) {
382        _cost[_arc_idf[a]] =  map[a];
383        _cost[_arc_idb[a]] = -map[a];
384      }
385      return *this;
386    }
387
388    /// \brief Set the supply values of the nodes.
389    ///
390    /// This function sets the supply values of the nodes.
391    /// If neither this function nor \ref stSupply() is used before
392    /// calling \ref run(), the supply of each node will be set to zero.
393    ///
394    /// \param map A node map storing the supply values.
395    /// Its \c Value type must be convertible to the \c Value type
396    /// of the algorithm.
397    ///
398    /// \return <tt>(*this)</tt>
399    template<typename SupplyMap>
400    CycleCanceling& supplyMap(const SupplyMap& map) {
401      for (NodeIt n(_graph); n != INVALID; ++n) {
402        _supply[_node_id[n]] = map[n];
403      }
404      return *this;
405    }
406
407    /// \brief Set single source and target nodes and a supply value.
408    ///
409    /// This function sets a single source node and a single target node
410    /// and the required flow value.
411    /// If neither this function nor \ref supplyMap() is used before
412    /// calling \ref run(), the supply of each node will be set to zero.
413    ///
414    /// Using this function has the same effect as using \ref supplyMap()
415    /// with such a map in which \c k is assigned to \c s, \c -k is
416    /// assigned to \c t and all other nodes have zero supply value.
417    ///
418    /// \param s The source node.
419    /// \param t The target node.
420    /// \param k The required amount of flow from node \c s to node \c t
421    /// (i.e. the supply of \c s and the demand of \c t).
422    ///
423    /// \return <tt>(*this)</tt>
424    CycleCanceling& stSupply(const Node& s, const Node& t, Value k) {
425      for (int i = 0; i != _res_node_num; ++i) {
426        _supply[i] = 0;
427      }
428      _supply[_node_id[s]] =  k;
429      _supply[_node_id[t]] = -k;
430      return *this;
431    }
432   
433    /// @}
434
435    /// \name Execution control
436    /// The algorithm can be executed using \ref run().
437
438    /// @{
439
440    /// \brief Run the algorithm.
441    ///
442    /// This function runs the algorithm.
443    /// The paramters can be specified using functions \ref lowerMap(),
444    /// \ref upperMap(), \ref costMap(), \ref supplyMap(), \ref stSupply().
445    /// For example,
446    /// \code
447    ///   CycleCanceling<ListDigraph> cc(graph);
448    ///   cc.lowerMap(lower).upperMap(upper).costMap(cost)
449    ///     .supplyMap(sup).run();
450    /// \endcode
451    ///
452    /// This function can be called more than once. All the parameters
453    /// that have been given are kept for the next call, unless
454    /// \ref reset() is called, thus only the modified parameters
455    /// have to be set again. See \ref reset() for examples.
456    /// However, the underlying digraph must not be modified after this
457    /// class have been constructed, since it copies and extends the graph.
458    ///
459    /// \param method The cycle-canceling method that will be used.
460    /// For more information, see \ref Method.
461    ///
462    /// \return \c INFEASIBLE if no feasible flow exists,
463    /// \n \c OPTIMAL if the problem has optimal solution
464    /// (i.e. it is feasible and bounded), and the algorithm has found
465    /// optimal flow and node potentials (primal and dual solutions),
466    /// \n \c UNBOUNDED if the digraph contains an arc of negative cost
467    /// and infinite upper bound. It means that the objective function
468    /// is unbounded on that arc, however, note that it could actually be
469    /// bounded over the feasible flows, but this algroithm cannot handle
470    /// these cases.
471    ///
472    /// \see ProblemType, Method
473    ProblemType run(Method method = CANCEL_AND_TIGHTEN) {
474      ProblemType pt = init();
475      if (pt != OPTIMAL) return pt;
476      start(method);
477      return OPTIMAL;
478    }
479
480    /// \brief Reset all the parameters that have been given before.
481    ///
482    /// This function resets all the paramaters that have been given
483    /// before using functions \ref lowerMap(), \ref upperMap(),
484    /// \ref costMap(), \ref supplyMap(), \ref stSupply().
485    ///
486    /// It is useful for multiple run() calls. If this function is not
487    /// used, all the parameters given before are kept for the next
488    /// \ref run() call.
489    /// However, the underlying digraph must not be modified after this
490    /// class have been constructed, since it copies and extends the graph.
491    ///
492    /// For example,
493    /// \code
494    ///   CycleCanceling<ListDigraph> cs(graph);
495    ///
496    ///   // First run
497    ///   cc.lowerMap(lower).upperMap(upper).costMap(cost)
498    ///     .supplyMap(sup).run();
499    ///
500    ///   // Run again with modified cost map (reset() is not called,
501    ///   // so only the cost map have to be set again)
502    ///   cost[e] += 100;
503    ///   cc.costMap(cost).run();
504    ///
505    ///   // Run again from scratch using reset()
506    ///   // (the lower bounds will be set to zero on all arcs)
507    ///   cc.reset();
508    ///   cc.upperMap(capacity).costMap(cost)
509    ///     .supplyMap(sup).run();
510    /// \endcode
511    ///
512    /// \return <tt>(*this)</tt>
513    CycleCanceling& reset() {
514      for (int i = 0; i != _res_node_num; ++i) {
515        _supply[i] = 0;
516      }
517      int limit = _first_out[_root];
518      for (int j = 0; j != limit; ++j) {
519        _lower[j] = 0;
520        _upper[j] = INF;
521        _cost[j] = _forward[j] ? 1 : -1;
522      }
523      for (int j = limit; j != _res_arc_num; ++j) {
524        _lower[j] = 0;
525        _upper[j] = INF;
526        _cost[j] = 0;
527        _cost[_reverse[j]] = 0;
528      }     
529      _have_lower = false;
530      return *this;
531    }
532
533    /// @}
534
535    /// \name Query Functions
536    /// The results of the algorithm can be obtained using these
537    /// functions.\n
538    /// The \ref run() function must be called before using them.
539
540    /// @{
541
542    /// \brief Return the total cost of the found flow.
543    ///
544    /// This function returns the total cost of the found flow.
545    /// Its complexity is O(e).
546    ///
547    /// \note The return type of the function can be specified as a
548    /// template parameter. For example,
549    /// \code
550    ///   cc.totalCost<double>();
551    /// \endcode
552    /// It is useful if the total cost cannot be stored in the \c Cost
553    /// type of the algorithm, which is the default return type of the
554    /// function.
555    ///
556    /// \pre \ref run() must be called before using this function.
557    template <typename Number>
558    Number totalCost() const {
559      Number c = 0;
560      for (ArcIt a(_graph); a != INVALID; ++a) {
561        int i = _arc_idb[a];
562        c += static_cast<Number>(_res_cap[i]) *
563             (-static_cast<Number>(_cost[i]));
564      }
565      return c;
566    }
567
568#ifndef DOXYGEN
569    Cost totalCost() const {
570      return totalCost<Cost>();
571    }
572#endif
573
574    /// \brief Return the flow on the given arc.
575    ///
576    /// This function returns the flow on the given arc.
577    ///
578    /// \pre \ref run() must be called before using this function.
579    Value flow(const Arc& a) const {
580      return _res_cap[_arc_idb[a]];
581    }
582
583    /// \brief Return the flow map (the primal solution).
584    ///
585    /// This function copies the flow value on each arc into the given
586    /// map. The \c Value type of the algorithm must be convertible to
587    /// the \c Value type of the map.
588    ///
589    /// \pre \ref run() must be called before using this function.
590    template <typename FlowMap>
591    void flowMap(FlowMap &map) const {
592      for (ArcIt a(_graph); a != INVALID; ++a) {
593        map.set(a, _res_cap[_arc_idb[a]]);
594      }
595    }
596
597    /// \brief Return the potential (dual value) of the given node.
598    ///
599    /// This function returns the potential (dual value) of the
600    /// given node.
601    ///
602    /// \pre \ref run() must be called before using this function.
603    Cost potential(const Node& n) const {
604      return static_cast<Cost>(_pi[_node_id[n]]);
605    }
606
607    /// \brief Return the potential map (the dual solution).
608    ///
609    /// This function copies the potential (dual value) of each node
610    /// into the given map.
611    /// The \c Cost type of the algorithm must be convertible to the
612    /// \c Value type of the map.
613    ///
614    /// \pre \ref run() must be called before using this function.
615    template <typename PotentialMap>
616    void potentialMap(PotentialMap &map) const {
617      for (NodeIt n(_graph); n != INVALID; ++n) {
618        map.set(n, static_cast<Cost>(_pi[_node_id[n]]));
619      }
620    }
621
622    /// @}
623
624  private:
625
626    // Initialize the algorithm
627    ProblemType init() {
628      if (_res_node_num <= 1) return INFEASIBLE;
629
630      // Check the sum of supply values
631      _sum_supply = 0;
632      for (int i = 0; i != _root; ++i) {
633        _sum_supply += _supply[i];
634      }
635      if (_sum_supply > 0) return INFEASIBLE;
636     
637
638      // Initialize vectors
639      for (int i = 0; i != _res_node_num; ++i) {
640        _pi[i] = 0;
641      }
642      ValueVector excess(_supply);
643     
644      // Remove infinite upper bounds and check negative arcs
645      const Value MAX = std::numeric_limits<Value>::max();
646      int last_out;
647      if (_have_lower) {
648        for (int i = 0; i != _root; ++i) {
649          last_out = _first_out[i+1];
650          for (int j = _first_out[i]; j != last_out; ++j) {
651            if (_forward[j]) {
652              Value c = _cost[j] < 0 ? _upper[j] : _lower[j];
653              if (c >= MAX) return UNBOUNDED;
654              excess[i] -= c;
655              excess[_target[j]] += c;
656            }
657          }
658        }
659      } else {
660        for (int i = 0; i != _root; ++i) {
661          last_out = _first_out[i+1];
662          for (int j = _first_out[i]; j != last_out; ++j) {
663            if (_forward[j] && _cost[j] < 0) {
664              Value c = _upper[j];
665              if (c >= MAX) return UNBOUNDED;
666              excess[i] -= c;
667              excess[_target[j]] += c;
668            }
669          }
670        }
671      }
672      Value ex, max_cap = 0;
673      for (int i = 0; i != _res_node_num; ++i) {
674        ex = excess[i];
675        if (ex < 0) max_cap -= ex;
676      }
677      for (int j = 0; j != _res_arc_num; ++j) {
678        if (_upper[j] >= MAX) _upper[j] = max_cap;
679      }
680
681      // Initialize maps for Circulation and remove non-zero lower bounds
682      ConstMap<Arc, Value> low(0);
683      typedef typename Digraph::template ArcMap<Value> ValueArcMap;
684      typedef typename Digraph::template NodeMap<Value> ValueNodeMap;
685      ValueArcMap cap(_graph), flow(_graph);
686      ValueNodeMap sup(_graph);
687      for (NodeIt n(_graph); n != INVALID; ++n) {
688        sup[n] = _supply[_node_id[n]];
689      }
690      if (_have_lower) {
691        for (ArcIt a(_graph); a != INVALID; ++a) {
692          int j = _arc_idf[a];
693          Value c = _lower[j];
694          cap[a] = _upper[j] - c;
695          sup[_graph.source(a)] -= c;
696          sup[_graph.target(a)] += c;
697        }
698      } else {
699        for (ArcIt a(_graph); a != INVALID; ++a) {
700          cap[a] = _upper[_arc_idf[a]];
701        }
702      }
703
704      // Find a feasible flow using Circulation
705      Circulation<Digraph, ConstMap<Arc, Value>, ValueArcMap, ValueNodeMap>
706        circ(_graph, low, cap, sup);
707      if (!circ.flowMap(flow).run()) return INFEASIBLE;
708
709      // Set residual capacities and handle GEQ supply type
710      if (_sum_supply < 0) {
711        for (ArcIt a(_graph); a != INVALID; ++a) {
712          Value fa = flow[a];
713          _res_cap[_arc_idf[a]] = cap[a] - fa;
714          _res_cap[_arc_idb[a]] = fa;
715          sup[_graph.source(a)] -= fa;
716          sup[_graph.target(a)] += fa;
717        }
718        for (NodeIt n(_graph); n != INVALID; ++n) {
719          excess[_node_id[n]] = sup[n];
720        }
721        for (int a = _first_out[_root]; a != _res_arc_num; ++a) {
722          int u = _target[a];
723          int ra = _reverse[a];
724          _res_cap[a] = -_sum_supply + 1;
725          _res_cap[ra] = -excess[u];
726          _cost[a] = 0;
727          _cost[ra] = 0;
728        }
729      } else {
730        for (ArcIt a(_graph); a != INVALID; ++a) {
731          Value fa = flow[a];
732          _res_cap[_arc_idf[a]] = cap[a] - fa;
733          _res_cap[_arc_idb[a]] = fa;
734        }
735        for (int a = _first_out[_root]; a != _res_arc_num; ++a) {
736          int ra = _reverse[a];
737          _res_cap[a] = 1;
738          _res_cap[ra] = 0;
739          _cost[a] = 0;
740          _cost[ra] = 0;
741        }
742      }
743     
744      return OPTIMAL;
745    }
746   
747    // Build a StaticDigraph structure containing the current
748    // residual network
749    void buildResidualNetwork() {
750      _arc_vec.clear();
751      _cost_vec.clear();
752      _id_vec.clear();
753      for (int j = 0; j != _res_arc_num; ++j) {
754        if (_res_cap[j] > 0) {
755          _arc_vec.push_back(IntPair(_source[j], _target[j]));
756          _cost_vec.push_back(_cost[j]);
757          _id_vec.push_back(j);
758        }
759      }
760      _sgr.build(_res_node_num, _arc_vec.begin(), _arc_vec.end());
761    }
762
763    // Execute the algorithm and transform the results
764    void start(Method method) {
765      // Execute the algorithm
766      switch (method) {
767        case SIMPLE_CYCLE_CANCELING:
768          startSimpleCycleCanceling();
769          break;
770        case MINIMUM_MEAN_CYCLE_CANCELING:
771          startMinMeanCycleCanceling();
772          break;
773        case CANCEL_AND_TIGHTEN:
774          startCancelAndTighten();
775          break;
776      }
777
778      // Compute node potentials
779      if (method != SIMPLE_CYCLE_CANCELING) {
780        buildResidualNetwork();
781        typename BellmanFord<StaticDigraph, CostArcMap>
782          ::template SetDistMap<CostNodeMap>::Create bf(_sgr, _cost_map);
783        bf.distMap(_pi_map);
784        bf.init(0);
785        bf.start();
786      }
787
788      // Handle non-zero lower bounds
789      if (_have_lower) {
790        int limit = _first_out[_root];
791        for (int j = 0; j != limit; ++j) {
792          if (!_forward[j]) _res_cap[j] += _lower[j];
793        }
794      }
795    }
796
797    // Execute the "Simple Cycle Canceling" method
798    void startSimpleCycleCanceling() {
799      // Constants for computing the iteration limits
800      const int BF_FIRST_LIMIT  = 2;
801      const double BF_LIMIT_FACTOR = 1.5;
802     
803      typedef StaticVectorMap<StaticDigraph::Arc, Value> FilterMap;
804      typedef FilterArcs<StaticDigraph, FilterMap> ResDigraph;
805      typedef StaticVectorMap<StaticDigraph::Node, StaticDigraph::Arc> PredMap;
806      typedef typename BellmanFord<ResDigraph, CostArcMap>
807        ::template SetDistMap<CostNodeMap>
808        ::template SetPredMap<PredMap>::Create BF;
809     
810      // Build the residual network
811      _arc_vec.clear();
812      _cost_vec.clear();
813      for (int j = 0; j != _res_arc_num; ++j) {
814        _arc_vec.push_back(IntPair(_source[j], _target[j]));
815        _cost_vec.push_back(_cost[j]);
816      }
817      _sgr.build(_res_node_num, _arc_vec.begin(), _arc_vec.end());
818
819      FilterMap filter_map(_res_cap);
820      ResDigraph rgr(_sgr, filter_map);
821      std::vector<int> cycle;
822      std::vector<StaticDigraph::Arc> pred(_res_arc_num);
823      PredMap pred_map(pred);
824      BF bf(rgr, _cost_map);
825      bf.distMap(_pi_map).predMap(pred_map);
826
827      int length_bound = BF_FIRST_LIMIT;
828      bool optimal = false;
829      while (!optimal) {
830        bf.init(0);
831        int iter_num = 0;
832        bool cycle_found = false;
833        while (!cycle_found) {
834          // Perform some iterations of the Bellman-Ford algorithm
835          int curr_iter_num = iter_num + length_bound <= _node_num ?
836            length_bound : _node_num - iter_num;
837          iter_num += curr_iter_num;
838          int real_iter_num = curr_iter_num;
839          for (int i = 0; i < curr_iter_num; ++i) {
840            if (bf.processNextWeakRound()) {
841              real_iter_num = i;
842              break;
843            }
844          }
845          if (real_iter_num < curr_iter_num) {
846            // Optimal flow is found
847            optimal = true;
848            break;
849          } else {
850            // Search for node disjoint negative cycles
851            std::vector<int> state(_res_node_num, 0);
852            int id = 0;
853            for (int u = 0; u != _res_node_num; ++u) {
854              if (state[u] != 0) continue;
855              ++id;
856              int v = u;
857              for (; v != -1 && state[v] == 0; v = pred[v] == INVALID ?
858                   -1 : rgr.id(rgr.source(pred[v]))) {
859                state[v] = id;
860              }
861              if (v != -1 && state[v] == id) {
862                // A negative cycle is found
863                cycle_found = true;
864                cycle.clear();
865                StaticDigraph::Arc a = pred[v];
866                Value d, delta = _res_cap[rgr.id(a)];
867                cycle.push_back(rgr.id(a));
868                while (rgr.id(rgr.source(a)) != v) {
869                  a = pred_map[rgr.source(a)];
870                  d = _res_cap[rgr.id(a)];
871                  if (d < delta) delta = d;
872                  cycle.push_back(rgr.id(a));
873                }
874
875                // Augment along the cycle
876                for (int i = 0; i < int(cycle.size()); ++i) {
877                  int j = cycle[i];
878                  _res_cap[j] -= delta;
879                  _res_cap[_reverse[j]] += delta;
880                }
881              }
882            }
883          }
884
885          // Increase iteration limit if no cycle is found
886          if (!cycle_found) {
887            length_bound = static_cast<int>(length_bound * BF_LIMIT_FACTOR);
888          }
889        }
890      }
891    }
892
893    // Execute the "Minimum Mean Cycle Canceling" method
894    void startMinMeanCycleCanceling() {
895      typedef SimplePath<StaticDigraph> SPath;
896      typedef typename SPath::ArcIt SPathArcIt;
897      typedef typename Howard<StaticDigraph, CostArcMap>
898        ::template SetPath<SPath>::Create MMC;
899     
900      SPath cycle;
901      MMC mmc(_sgr, _cost_map);
902      mmc.cycle(cycle);
903      buildResidualNetwork();
904      while (mmc.findMinMean() && mmc.cycleLength() < 0) {
905        // Find the cycle
906        mmc.findCycle();
907
908        // Compute delta value
909        Value delta = INF;
910        for (SPathArcIt a(cycle); a != INVALID; ++a) {
911          Value d = _res_cap[_id_vec[_sgr.id(a)]];
912          if (d < delta) delta = d;
913        }
914
915        // Augment along the cycle
916        for (SPathArcIt a(cycle); a != INVALID; ++a) {
917          int j = _id_vec[_sgr.id(a)];
918          _res_cap[j] -= delta;
919          _res_cap[_reverse[j]] += delta;
920        }
921
922        // Rebuild the residual network       
923        buildResidualNetwork();
924      }
925    }
926
927    // Execute the "Cancel And Tighten" method
928    void startCancelAndTighten() {
929      // Constants for the min mean cycle computations
930      const double LIMIT_FACTOR = 1.0;
931      const int MIN_LIMIT = 5;
932
933      // Contruct auxiliary data vectors
934      DoubleVector pi(_res_node_num, 0.0);
935      IntVector level(_res_node_num);
936      CharVector reached(_res_node_num);
937      CharVector processed(_res_node_num);
938      IntVector pred_node(_res_node_num);
939      IntVector pred_arc(_res_node_num);
940      std::vector<int> stack(_res_node_num);
941      std::vector<int> proc_vector(_res_node_num);
942
943      // Initialize epsilon
944      double epsilon = 0;
945      for (int a = 0; a != _res_arc_num; ++a) {
946        if (_res_cap[a] > 0 && -_cost[a] > epsilon)
947          epsilon = -_cost[a];
948      }
949
950      // Start phases
951      Tolerance<double> tol;
952      tol.epsilon(1e-6);
953      int limit = int(LIMIT_FACTOR * std::sqrt(double(_res_node_num)));
954      if (limit < MIN_LIMIT) limit = MIN_LIMIT;
955      int iter = limit;
956      while (epsilon * _res_node_num >= 1) {
957        // Find and cancel cycles in the admissible network using DFS
958        for (int u = 0; u != _res_node_num; ++u) {
959          reached[u] = false;
960          processed[u] = false;
961        }
962        int stack_head = -1;
963        int proc_head = -1;
964        for (int start = 0; start != _res_node_num; ++start) {
965          if (reached[start]) continue;
966
967          // New start node
968          reached[start] = true;
969          pred_arc[start] = -1;
970          pred_node[start] = -1;
971
972          // Find the first admissible outgoing arc
973          double p = pi[start];
974          int a = _first_out[start];
975          int last_out = _first_out[start+1];
976          for (; a != last_out && (_res_cap[a] == 0 ||
977               !tol.negative(_cost[a] + p - pi[_target[a]])); ++a) ;
978          if (a == last_out) {
979            processed[start] = true;
980            proc_vector[++proc_head] = start;
981            continue;
982          }
983          stack[++stack_head] = a;
984
985          while (stack_head >= 0) {
986            int sa = stack[stack_head];
987            int u = _source[sa];
988            int v = _target[sa];
989
990            if (!reached[v]) {
991              // A new node is reached
992              reached[v] = true;
993              pred_node[v] = u;
994              pred_arc[v] = sa;
995              p = pi[v];
996              a = _first_out[v];
997              last_out = _first_out[v+1];
998              for (; a != last_out && (_res_cap[a] == 0 ||
999                   !tol.negative(_cost[a] + p - pi[_target[a]])); ++a) ;
1000              stack[++stack_head] = a == last_out ? -1 : a;
1001            } else {
1002              if (!processed[v]) {
1003                // A cycle is found
1004                int n, w = u;
1005                Value d, delta = _res_cap[sa];
1006                for (n = u; n != v; n = pred_node[n]) {
1007                  d = _res_cap[pred_arc[n]];
1008                  if (d <= delta) {
1009                    delta = d;
1010                    w = pred_node[n];
1011                  }
1012                }
1013
1014                // Augment along the cycle
1015                _res_cap[sa] -= delta;
1016                _res_cap[_reverse[sa]] += delta;
1017                for (n = u; n != v; n = pred_node[n]) {
1018                  int pa = pred_arc[n];
1019                  _res_cap[pa] -= delta;
1020                  _res_cap[_reverse[pa]] += delta;
1021                }
1022                for (n = u; stack_head > 0 && n != w; n = pred_node[n]) {
1023                  --stack_head;
1024                  reached[n] = false;
1025                }
1026                u = w;
1027              }
1028              v = u;
1029
1030              // Find the next admissible outgoing arc
1031              p = pi[v];
1032              a = stack[stack_head] + 1;
1033              last_out = _first_out[v+1];
1034              for (; a != last_out && (_res_cap[a] == 0 ||
1035                   !tol.negative(_cost[a] + p - pi[_target[a]])); ++a) ;
1036              stack[stack_head] = a == last_out ? -1 : a;
1037            }
1038
1039            while (stack_head >= 0 && stack[stack_head] == -1) {
1040              processed[v] = true;
1041              proc_vector[++proc_head] = v;
1042              if (--stack_head >= 0) {
1043                // Find the next admissible outgoing arc
1044                v = _source[stack[stack_head]];
1045                p = pi[v];
1046                a = stack[stack_head] + 1;
1047                last_out = _first_out[v+1];
1048                for (; a != last_out && (_res_cap[a] == 0 ||
1049                     !tol.negative(_cost[a] + p - pi[_target[a]])); ++a) ;
1050                stack[stack_head] = a == last_out ? -1 : a;
1051              }
1052            }
1053          }
1054        }
1055
1056        // Tighten potentials and epsilon
1057        if (--iter > 0) {
1058          for (int u = 0; u != _res_node_num; ++u) {
1059            level[u] = 0;
1060          }
1061          for (int i = proc_head; i > 0; --i) {
1062            int u = proc_vector[i];
1063            double p = pi[u];
1064            int l = level[u] + 1;
1065            int last_out = _first_out[u+1];
1066            for (int a = _first_out[u]; a != last_out; ++a) {
1067              int v = _target[a];
1068              if (_res_cap[a] > 0 && tol.negative(_cost[a] + p - pi[v]) &&
1069                  l > level[v]) level[v] = l;
1070            }
1071          }
1072
1073          // Modify potentials
1074          double q = std::numeric_limits<double>::max();
1075          for (int u = 0; u != _res_node_num; ++u) {
1076            int lu = level[u];
1077            double p, pu = pi[u];
1078            int last_out = _first_out[u+1];
1079            for (int a = _first_out[u]; a != last_out; ++a) {
1080              if (_res_cap[a] == 0) continue;
1081              int v = _target[a];
1082              int ld = lu - level[v];
1083              if (ld > 0) {
1084                p = (_cost[a] + pu - pi[v] + epsilon) / (ld + 1);
1085                if (p < q) q = p;
1086              }
1087            }
1088          }
1089          for (int u = 0; u != _res_node_num; ++u) {
1090            pi[u] -= q * level[u];
1091          }
1092
1093          // Modify epsilon
1094          epsilon = 0;
1095          for (int u = 0; u != _res_node_num; ++u) {
1096            double curr, pu = pi[u];
1097            int last_out = _first_out[u+1];
1098            for (int a = _first_out[u]; a != last_out; ++a) {
1099              if (_res_cap[a] == 0) continue;
1100              curr = _cost[a] + pu - pi[_target[a]];
1101              if (-curr > epsilon) epsilon = -curr;
1102            }
1103          }
1104        } else {
1105          typedef Howard<StaticDigraph, CostArcMap> MMC;
1106          typedef typename BellmanFord<StaticDigraph, CostArcMap>
1107            ::template SetDistMap<CostNodeMap>::Create BF;
1108
1109          // Set epsilon to the minimum cycle mean
1110          buildResidualNetwork();
1111          MMC mmc(_sgr, _cost_map);
1112          mmc.findMinMean();
1113          epsilon = -mmc.cycleMean();
1114          Cost cycle_cost = mmc.cycleLength();
1115          int cycle_size = mmc.cycleArcNum();
1116         
1117          // Compute feasible potentials for the current epsilon
1118          for (int i = 0; i != int(_cost_vec.size()); ++i) {
1119            _cost_vec[i] = cycle_size * _cost_vec[i] - cycle_cost;
1120          }
1121          BF bf(_sgr, _cost_map);
1122          bf.distMap(_pi_map);
1123          bf.init(0);
1124          bf.start();
1125          for (int u = 0; u != _res_node_num; ++u) {
1126            pi[u] = static_cast<double>(_pi[u]) / cycle_size;
1127          }
1128       
1129          iter = limit;
1130        }
1131      }
1132    }
1133
1134  }; //class CycleCanceling
1135
1136  ///@}
1137
1138} //namespace lemon
1139
1140#endif //LEMON_CYCLE_CANCELING_H
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