COIN-OR::LEMON - Graph Library

source: lemon-1.2/lemon/cycle_canceling.h @ 830:75c97c3786d6

Last change on this file since 830:75c97c3786d6 was 830:75c97c3786d6, checked in by Peter Kovacs <kpeter@…>, 10 years ago

Handle graph changes in the MCF algorithms (#327)

The reset() functions are renamed to resetParams() and the new reset()
functions handle the graph chnages, as well.

File size: 38.2 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      // Reset data structures
254      reset();
255    }
256
257    /// \name Parameters
258    /// The parameters of the algorithm can be specified using these
259    /// functions.
260
261    /// @{
262
263    /// \brief Set the lower bounds on the arcs.
264    ///
265    /// This function sets the lower bounds on the arcs.
266    /// If it is not used before calling \ref run(), the lower bounds
267    /// will be set to zero on all arcs.
268    ///
269    /// \param map An arc map storing the lower bounds.
270    /// Its \c Value type must be convertible to the \c Value type
271    /// of the algorithm.
272    ///
273    /// \return <tt>(*this)</tt>
274    template <typename LowerMap>
275    CycleCanceling& lowerMap(const LowerMap& map) {
276      _have_lower = true;
277      for (ArcIt a(_graph); a != INVALID; ++a) {
278        _lower[_arc_idf[a]] = map[a];
279        _lower[_arc_idb[a]] = map[a];
280      }
281      return *this;
282    }
283
284    /// \brief Set the upper bounds (capacities) on the arcs.
285    ///
286    /// This function sets the upper bounds (capacities) on the arcs.
287    /// If it is not used before calling \ref run(), the upper bounds
288    /// will be set to \ref INF on all arcs (i.e. the flow value will be
289    /// unbounded from above).
290    ///
291    /// \param map An arc map storing the upper bounds.
292    /// Its \c Value type must be convertible to the \c Value type
293    /// of the algorithm.
294    ///
295    /// \return <tt>(*this)</tt>
296    template<typename UpperMap>
297    CycleCanceling& upperMap(const UpperMap& map) {
298      for (ArcIt a(_graph); a != INVALID; ++a) {
299        _upper[_arc_idf[a]] = map[a];
300      }
301      return *this;
302    }
303
304    /// \brief Set the costs of the arcs.
305    ///
306    /// This function sets the costs of the arcs.
307    /// If it is not used before calling \ref run(), the costs
308    /// will be set to \c 1 on all arcs.
309    ///
310    /// \param map An arc map storing the costs.
311    /// Its \c Value type must be convertible to the \c Cost type
312    /// of the algorithm.
313    ///
314    /// \return <tt>(*this)</tt>
315    template<typename CostMap>
316    CycleCanceling& costMap(const CostMap& map) {
317      for (ArcIt a(_graph); a != INVALID; ++a) {
318        _cost[_arc_idf[a]] =  map[a];
319        _cost[_arc_idb[a]] = -map[a];
320      }
321      return *this;
322    }
323
324    /// \brief Set the supply values of the nodes.
325    ///
326    /// This function sets the supply values of the nodes.
327    /// If neither this function nor \ref stSupply() is used before
328    /// calling \ref run(), the supply of each node will be set to zero.
329    ///
330    /// \param map A node map storing the supply values.
331    /// Its \c Value type must be convertible to the \c Value type
332    /// of the algorithm.
333    ///
334    /// \return <tt>(*this)</tt>
335    template<typename SupplyMap>
336    CycleCanceling& supplyMap(const SupplyMap& map) {
337      for (NodeIt n(_graph); n != INVALID; ++n) {
338        _supply[_node_id[n]] = map[n];
339      }
340      return *this;
341    }
342
343    /// \brief Set single source and target nodes and a supply value.
344    ///
345    /// This function sets a single source node and a single target node
346    /// and the required flow value.
347    /// If neither this function nor \ref supplyMap() is used before
348    /// calling \ref run(), the supply of each node will be set to zero.
349    ///
350    /// Using this function has the same effect as using \ref supplyMap()
351    /// with such a map in which \c k is assigned to \c s, \c -k is
352    /// assigned to \c t and all other nodes have zero supply value.
353    ///
354    /// \param s The source node.
355    /// \param t The target node.
356    /// \param k The required amount of flow from node \c s to node \c t
357    /// (i.e. the supply of \c s and the demand of \c t).
358    ///
359    /// \return <tt>(*this)</tt>
360    CycleCanceling& stSupply(const Node& s, const Node& t, Value k) {
361      for (int i = 0; i != _res_node_num; ++i) {
362        _supply[i] = 0;
363      }
364      _supply[_node_id[s]] =  k;
365      _supply[_node_id[t]] = -k;
366      return *this;
367    }
368   
369    /// @}
370
371    /// \name Execution control
372    /// The algorithm can be executed using \ref run().
373
374    /// @{
375
376    /// \brief Run the algorithm.
377    ///
378    /// This function runs the algorithm.
379    /// The paramters can be specified using functions \ref lowerMap(),
380    /// \ref upperMap(), \ref costMap(), \ref supplyMap(), \ref stSupply().
381    /// For example,
382    /// \code
383    ///   CycleCanceling<ListDigraph> cc(graph);
384    ///   cc.lowerMap(lower).upperMap(upper).costMap(cost)
385    ///     .supplyMap(sup).run();
386    /// \endcode
387    ///
388    /// This function can be called more than once. All the given parameters
389    /// are kept for the next call, unless \ref resetParams() or \ref reset()
390    /// is used, thus only the modified parameters have to be set again.
391    /// If the underlying digraph was also modified after the construction
392    /// of the class (or the last \ref reset() call), then the \ref reset()
393    /// function must be called.
394    ///
395    /// \param method The cycle-canceling method that will be used.
396    /// For more information, see \ref Method.
397    ///
398    /// \return \c INFEASIBLE if no feasible flow exists,
399    /// \n \c OPTIMAL if the problem has optimal solution
400    /// (i.e. it is feasible and bounded), and the algorithm has found
401    /// optimal flow and node potentials (primal and dual solutions),
402    /// \n \c UNBOUNDED if the digraph contains an arc of negative cost
403    /// and infinite upper bound. It means that the objective function
404    /// is unbounded on that arc, however, note that it could actually be
405    /// bounded over the feasible flows, but this algroithm cannot handle
406    /// these cases.
407    ///
408    /// \see ProblemType, Method
409    /// \see resetParams(), reset()
410    ProblemType run(Method method = CANCEL_AND_TIGHTEN) {
411      ProblemType pt = init();
412      if (pt != OPTIMAL) return pt;
413      start(method);
414      return OPTIMAL;
415    }
416
417    /// \brief Reset all the parameters that have been given before.
418    ///
419    /// This function resets all the paramaters that have been given
420    /// before using functions \ref lowerMap(), \ref upperMap(),
421    /// \ref costMap(), \ref supplyMap(), \ref stSupply().
422    ///
423    /// It is useful for multiple \ref run() calls. Basically, all the given
424    /// parameters are kept for the next \ref run() call, unless
425    /// \ref resetParams() or \ref reset() is used.
426    /// If the underlying digraph was also modified after the construction
427    /// of the class or the last \ref reset() call, then the \ref reset()
428    /// function must be used, otherwise \ref resetParams() is sufficient.
429    ///
430    /// For example,
431    /// \code
432    ///   CycleCanceling<ListDigraph> cs(graph);
433    ///
434    ///   // First run
435    ///   cc.lowerMap(lower).upperMap(upper).costMap(cost)
436    ///     .supplyMap(sup).run();
437    ///
438    ///   // Run again with modified cost map (resetParams() is not called,
439    ///   // so only the cost map have to be set again)
440    ///   cost[e] += 100;
441    ///   cc.costMap(cost).run();
442    ///
443    ///   // Run again from scratch using resetParams()
444    ///   // (the lower bounds will be set to zero on all arcs)
445    ///   cc.resetParams();
446    ///   cc.upperMap(capacity).costMap(cost)
447    ///     .supplyMap(sup).run();
448    /// \endcode
449    ///
450    /// \return <tt>(*this)</tt>
451    ///
452    /// \see reset(), run()
453    CycleCanceling& resetParams() {
454      for (int i = 0; i != _res_node_num; ++i) {
455        _supply[i] = 0;
456      }
457      int limit = _first_out[_root];
458      for (int j = 0; j != limit; ++j) {
459        _lower[j] = 0;
460        _upper[j] = INF;
461        _cost[j] = _forward[j] ? 1 : -1;
462      }
463      for (int j = limit; j != _res_arc_num; ++j) {
464        _lower[j] = 0;
465        _upper[j] = INF;
466        _cost[j] = 0;
467        _cost[_reverse[j]] = 0;
468      }     
469      _have_lower = false;
470      return *this;
471    }
472
473    /// \brief Reset the internal data structures and all the parameters
474    /// that have been given before.
475    ///
476    /// This function resets the internal data structures and all the
477    /// paramaters that have been given before using functions \ref lowerMap(),
478    /// \ref upperMap(), \ref costMap(), \ref supplyMap(), \ref stSupply().
479    ///
480    /// It is useful for multiple \ref run() calls. Basically, all the given
481    /// parameters are kept for the next \ref run() call, unless
482    /// \ref resetParams() or \ref reset() is used.
483    /// If the underlying digraph was also modified after the construction
484    /// of the class or the last \ref reset() call, then the \ref reset()
485    /// function must be used, otherwise \ref resetParams() is sufficient.
486    ///
487    /// See \ref resetParams() for examples.
488    ///
489    /// \return <tt>(*this)</tt>
490    ///
491    /// \see resetParams(), run()
492    CycleCanceling& reset() {
493      // Resize vectors
494      _node_num = countNodes(_graph);
495      _arc_num = countArcs(_graph);
496      _res_node_num = _node_num + 1;
497      _res_arc_num = 2 * (_arc_num + _node_num);
498      _root = _node_num;
499
500      _first_out.resize(_res_node_num + 1);
501      _forward.resize(_res_arc_num);
502      _source.resize(_res_arc_num);
503      _target.resize(_res_arc_num);
504      _reverse.resize(_res_arc_num);
505
506      _lower.resize(_res_arc_num);
507      _upper.resize(_res_arc_num);
508      _cost.resize(_res_arc_num);
509      _supply.resize(_res_node_num);
510     
511      _res_cap.resize(_res_arc_num);
512      _pi.resize(_res_node_num);
513
514      _arc_vec.reserve(_res_arc_num);
515      _cost_vec.reserve(_res_arc_num);
516      _id_vec.reserve(_res_arc_num);
517
518      // Copy the graph
519      int i = 0, j = 0, k = 2 * _arc_num + _node_num;
520      for (NodeIt n(_graph); n != INVALID; ++n, ++i) {
521        _node_id[n] = i;
522      }
523      i = 0;
524      for (NodeIt n(_graph); n != INVALID; ++n, ++i) {
525        _first_out[i] = j;
526        for (OutArcIt a(_graph, n); a != INVALID; ++a, ++j) {
527          _arc_idf[a] = j;
528          _forward[j] = true;
529          _source[j] = i;
530          _target[j] = _node_id[_graph.runningNode(a)];
531        }
532        for (InArcIt a(_graph, n); a != INVALID; ++a, ++j) {
533          _arc_idb[a] = j;
534          _forward[j] = false;
535          _source[j] = i;
536          _target[j] = _node_id[_graph.runningNode(a)];
537        }
538        _forward[j] = false;
539        _source[j] = i;
540        _target[j] = _root;
541        _reverse[j] = k;
542        _forward[k] = true;
543        _source[k] = _root;
544        _target[k] = i;
545        _reverse[k] = j;
546        ++j; ++k;
547      }
548      _first_out[i] = j;
549      _first_out[_res_node_num] = k;
550      for (ArcIt a(_graph); a != INVALID; ++a) {
551        int fi = _arc_idf[a];
552        int bi = _arc_idb[a];
553        _reverse[fi] = bi;
554        _reverse[bi] = fi;
555      }
556     
557      // Reset parameters
558      resetParams();
559      return *this;
560    }
561
562    /// @}
563
564    /// \name Query Functions
565    /// The results of the algorithm can be obtained using these
566    /// functions.\n
567    /// The \ref run() function must be called before using them.
568
569    /// @{
570
571    /// \brief Return the total cost of the found flow.
572    ///
573    /// This function returns the total cost of the found flow.
574    /// Its complexity is O(e).
575    ///
576    /// \note The return type of the function can be specified as a
577    /// template parameter. For example,
578    /// \code
579    ///   cc.totalCost<double>();
580    /// \endcode
581    /// It is useful if the total cost cannot be stored in the \c Cost
582    /// type of the algorithm, which is the default return type of the
583    /// function.
584    ///
585    /// \pre \ref run() must be called before using this function.
586    template <typename Number>
587    Number totalCost() const {
588      Number c = 0;
589      for (ArcIt a(_graph); a != INVALID; ++a) {
590        int i = _arc_idb[a];
591        c += static_cast<Number>(_res_cap[i]) *
592             (-static_cast<Number>(_cost[i]));
593      }
594      return c;
595    }
596
597#ifndef DOXYGEN
598    Cost totalCost() const {
599      return totalCost<Cost>();
600    }
601#endif
602
603    /// \brief Return the flow on the given arc.
604    ///
605    /// This function returns the flow on the given arc.
606    ///
607    /// \pre \ref run() must be called before using this function.
608    Value flow(const Arc& a) const {
609      return _res_cap[_arc_idb[a]];
610    }
611
612    /// \brief Return the flow map (the primal solution).
613    ///
614    /// This function copies the flow value on each arc into the given
615    /// map. The \c Value type of the algorithm must be convertible to
616    /// the \c Value type of the map.
617    ///
618    /// \pre \ref run() must be called before using this function.
619    template <typename FlowMap>
620    void flowMap(FlowMap &map) const {
621      for (ArcIt a(_graph); a != INVALID; ++a) {
622        map.set(a, _res_cap[_arc_idb[a]]);
623      }
624    }
625
626    /// \brief Return the potential (dual value) of the given node.
627    ///
628    /// This function returns the potential (dual value) of the
629    /// given node.
630    ///
631    /// \pre \ref run() must be called before using this function.
632    Cost potential(const Node& n) const {
633      return static_cast<Cost>(_pi[_node_id[n]]);
634    }
635
636    /// \brief Return the potential map (the dual solution).
637    ///
638    /// This function copies the potential (dual value) of each node
639    /// into the given map.
640    /// The \c Cost type of the algorithm must be convertible to the
641    /// \c Value type of the map.
642    ///
643    /// \pre \ref run() must be called before using this function.
644    template <typename PotentialMap>
645    void potentialMap(PotentialMap &map) const {
646      for (NodeIt n(_graph); n != INVALID; ++n) {
647        map.set(n, static_cast<Cost>(_pi[_node_id[n]]));
648      }
649    }
650
651    /// @}
652
653  private:
654
655    // Initialize the algorithm
656    ProblemType init() {
657      if (_res_node_num <= 1) return INFEASIBLE;
658
659      // Check the sum of supply values
660      _sum_supply = 0;
661      for (int i = 0; i != _root; ++i) {
662        _sum_supply += _supply[i];
663      }
664      if (_sum_supply > 0) return INFEASIBLE;
665     
666
667      // Initialize vectors
668      for (int i = 0; i != _res_node_num; ++i) {
669        _pi[i] = 0;
670      }
671      ValueVector excess(_supply);
672     
673      // Remove infinite upper bounds and check negative arcs
674      const Value MAX = std::numeric_limits<Value>::max();
675      int last_out;
676      if (_have_lower) {
677        for (int i = 0; i != _root; ++i) {
678          last_out = _first_out[i+1];
679          for (int j = _first_out[i]; j != last_out; ++j) {
680            if (_forward[j]) {
681              Value c = _cost[j] < 0 ? _upper[j] : _lower[j];
682              if (c >= MAX) return UNBOUNDED;
683              excess[i] -= c;
684              excess[_target[j]] += c;
685            }
686          }
687        }
688      } else {
689        for (int i = 0; i != _root; ++i) {
690          last_out = _first_out[i+1];
691          for (int j = _first_out[i]; j != last_out; ++j) {
692            if (_forward[j] && _cost[j] < 0) {
693              Value c = _upper[j];
694              if (c >= MAX) return UNBOUNDED;
695              excess[i] -= c;
696              excess[_target[j]] += c;
697            }
698          }
699        }
700      }
701      Value ex, max_cap = 0;
702      for (int i = 0; i != _res_node_num; ++i) {
703        ex = excess[i];
704        if (ex < 0) max_cap -= ex;
705      }
706      for (int j = 0; j != _res_arc_num; ++j) {
707        if (_upper[j] >= MAX) _upper[j] = max_cap;
708      }
709
710      // Initialize maps for Circulation and remove non-zero lower bounds
711      ConstMap<Arc, Value> low(0);
712      typedef typename Digraph::template ArcMap<Value> ValueArcMap;
713      typedef typename Digraph::template NodeMap<Value> ValueNodeMap;
714      ValueArcMap cap(_graph), flow(_graph);
715      ValueNodeMap sup(_graph);
716      for (NodeIt n(_graph); n != INVALID; ++n) {
717        sup[n] = _supply[_node_id[n]];
718      }
719      if (_have_lower) {
720        for (ArcIt a(_graph); a != INVALID; ++a) {
721          int j = _arc_idf[a];
722          Value c = _lower[j];
723          cap[a] = _upper[j] - c;
724          sup[_graph.source(a)] -= c;
725          sup[_graph.target(a)] += c;
726        }
727      } else {
728        for (ArcIt a(_graph); a != INVALID; ++a) {
729          cap[a] = _upper[_arc_idf[a]];
730        }
731      }
732
733      // Find a feasible flow using Circulation
734      Circulation<Digraph, ConstMap<Arc, Value>, ValueArcMap, ValueNodeMap>
735        circ(_graph, low, cap, sup);
736      if (!circ.flowMap(flow).run()) return INFEASIBLE;
737
738      // Set residual capacities and handle GEQ supply type
739      if (_sum_supply < 0) {
740        for (ArcIt a(_graph); a != INVALID; ++a) {
741          Value fa = flow[a];
742          _res_cap[_arc_idf[a]] = cap[a] - fa;
743          _res_cap[_arc_idb[a]] = fa;
744          sup[_graph.source(a)] -= fa;
745          sup[_graph.target(a)] += fa;
746        }
747        for (NodeIt n(_graph); n != INVALID; ++n) {
748          excess[_node_id[n]] = sup[n];
749        }
750        for (int a = _first_out[_root]; a != _res_arc_num; ++a) {
751          int u = _target[a];
752          int ra = _reverse[a];
753          _res_cap[a] = -_sum_supply + 1;
754          _res_cap[ra] = -excess[u];
755          _cost[a] = 0;
756          _cost[ra] = 0;
757        }
758      } else {
759        for (ArcIt a(_graph); a != INVALID; ++a) {
760          Value fa = flow[a];
761          _res_cap[_arc_idf[a]] = cap[a] - fa;
762          _res_cap[_arc_idb[a]] = fa;
763        }
764        for (int a = _first_out[_root]; a != _res_arc_num; ++a) {
765          int ra = _reverse[a];
766          _res_cap[a] = 1;
767          _res_cap[ra] = 0;
768          _cost[a] = 0;
769          _cost[ra] = 0;
770        }
771      }
772     
773      return OPTIMAL;
774    }
775   
776    // Build a StaticDigraph structure containing the current
777    // residual network
778    void buildResidualNetwork() {
779      _arc_vec.clear();
780      _cost_vec.clear();
781      _id_vec.clear();
782      for (int j = 0; j != _res_arc_num; ++j) {
783        if (_res_cap[j] > 0) {
784          _arc_vec.push_back(IntPair(_source[j], _target[j]));
785          _cost_vec.push_back(_cost[j]);
786          _id_vec.push_back(j);
787        }
788      }
789      _sgr.build(_res_node_num, _arc_vec.begin(), _arc_vec.end());
790    }
791
792    // Execute the algorithm and transform the results
793    void start(Method method) {
794      // Execute the algorithm
795      switch (method) {
796        case SIMPLE_CYCLE_CANCELING:
797          startSimpleCycleCanceling();
798          break;
799        case MINIMUM_MEAN_CYCLE_CANCELING:
800          startMinMeanCycleCanceling();
801          break;
802        case CANCEL_AND_TIGHTEN:
803          startCancelAndTighten();
804          break;
805      }
806
807      // Compute node potentials
808      if (method != SIMPLE_CYCLE_CANCELING) {
809        buildResidualNetwork();
810        typename BellmanFord<StaticDigraph, CostArcMap>
811          ::template SetDistMap<CostNodeMap>::Create bf(_sgr, _cost_map);
812        bf.distMap(_pi_map);
813        bf.init(0);
814        bf.start();
815      }
816
817      // Handle non-zero lower bounds
818      if (_have_lower) {
819        int limit = _first_out[_root];
820        for (int j = 0; j != limit; ++j) {
821          if (!_forward[j]) _res_cap[j] += _lower[j];
822        }
823      }
824    }
825
826    // Execute the "Simple Cycle Canceling" method
827    void startSimpleCycleCanceling() {
828      // Constants for computing the iteration limits
829      const int BF_FIRST_LIMIT  = 2;
830      const double BF_LIMIT_FACTOR = 1.5;
831     
832      typedef StaticVectorMap<StaticDigraph::Arc, Value> FilterMap;
833      typedef FilterArcs<StaticDigraph, FilterMap> ResDigraph;
834      typedef StaticVectorMap<StaticDigraph::Node, StaticDigraph::Arc> PredMap;
835      typedef typename BellmanFord<ResDigraph, CostArcMap>
836        ::template SetDistMap<CostNodeMap>
837        ::template SetPredMap<PredMap>::Create BF;
838     
839      // Build the residual network
840      _arc_vec.clear();
841      _cost_vec.clear();
842      for (int j = 0; j != _res_arc_num; ++j) {
843        _arc_vec.push_back(IntPair(_source[j], _target[j]));
844        _cost_vec.push_back(_cost[j]);
845      }
846      _sgr.build(_res_node_num, _arc_vec.begin(), _arc_vec.end());
847
848      FilterMap filter_map(_res_cap);
849      ResDigraph rgr(_sgr, filter_map);
850      std::vector<int> cycle;
851      std::vector<StaticDigraph::Arc> pred(_res_arc_num);
852      PredMap pred_map(pred);
853      BF bf(rgr, _cost_map);
854      bf.distMap(_pi_map).predMap(pred_map);
855
856      int length_bound = BF_FIRST_LIMIT;
857      bool optimal = false;
858      while (!optimal) {
859        bf.init(0);
860        int iter_num = 0;
861        bool cycle_found = false;
862        while (!cycle_found) {
863          // Perform some iterations of the Bellman-Ford algorithm
864          int curr_iter_num = iter_num + length_bound <= _node_num ?
865            length_bound : _node_num - iter_num;
866          iter_num += curr_iter_num;
867          int real_iter_num = curr_iter_num;
868          for (int i = 0; i < curr_iter_num; ++i) {
869            if (bf.processNextWeakRound()) {
870              real_iter_num = i;
871              break;
872            }
873          }
874          if (real_iter_num < curr_iter_num) {
875            // Optimal flow is found
876            optimal = true;
877            break;
878          } else {
879            // Search for node disjoint negative cycles
880            std::vector<int> state(_res_node_num, 0);
881            int id = 0;
882            for (int u = 0; u != _res_node_num; ++u) {
883              if (state[u] != 0) continue;
884              ++id;
885              int v = u;
886              for (; v != -1 && state[v] == 0; v = pred[v] == INVALID ?
887                   -1 : rgr.id(rgr.source(pred[v]))) {
888                state[v] = id;
889              }
890              if (v != -1 && state[v] == id) {
891                // A negative cycle is found
892                cycle_found = true;
893                cycle.clear();
894                StaticDigraph::Arc a = pred[v];
895                Value d, delta = _res_cap[rgr.id(a)];
896                cycle.push_back(rgr.id(a));
897                while (rgr.id(rgr.source(a)) != v) {
898                  a = pred_map[rgr.source(a)];
899                  d = _res_cap[rgr.id(a)];
900                  if (d < delta) delta = d;
901                  cycle.push_back(rgr.id(a));
902                }
903
904                // Augment along the cycle
905                for (int i = 0; i < int(cycle.size()); ++i) {
906                  int j = cycle[i];
907                  _res_cap[j] -= delta;
908                  _res_cap[_reverse[j]] += delta;
909                }
910              }
911            }
912          }
913
914          // Increase iteration limit if no cycle is found
915          if (!cycle_found) {
916            length_bound = static_cast<int>(length_bound * BF_LIMIT_FACTOR);
917          }
918        }
919      }
920    }
921
922    // Execute the "Minimum Mean Cycle Canceling" method
923    void startMinMeanCycleCanceling() {
924      typedef SimplePath<StaticDigraph> SPath;
925      typedef typename SPath::ArcIt SPathArcIt;
926      typedef typename Howard<StaticDigraph, CostArcMap>
927        ::template SetPath<SPath>::Create MMC;
928     
929      SPath cycle;
930      MMC mmc(_sgr, _cost_map);
931      mmc.cycle(cycle);
932      buildResidualNetwork();
933      while (mmc.findMinMean() && mmc.cycleLength() < 0) {
934        // Find the cycle
935        mmc.findCycle();
936
937        // Compute delta value
938        Value delta = INF;
939        for (SPathArcIt a(cycle); a != INVALID; ++a) {
940          Value d = _res_cap[_id_vec[_sgr.id(a)]];
941          if (d < delta) delta = d;
942        }
943
944        // Augment along the cycle
945        for (SPathArcIt a(cycle); a != INVALID; ++a) {
946          int j = _id_vec[_sgr.id(a)];
947          _res_cap[j] -= delta;
948          _res_cap[_reverse[j]] += delta;
949        }
950
951        // Rebuild the residual network       
952        buildResidualNetwork();
953      }
954    }
955
956    // Execute the "Cancel And Tighten" method
957    void startCancelAndTighten() {
958      // Constants for the min mean cycle computations
959      const double LIMIT_FACTOR = 1.0;
960      const int MIN_LIMIT = 5;
961
962      // Contruct auxiliary data vectors
963      DoubleVector pi(_res_node_num, 0.0);
964      IntVector level(_res_node_num);
965      CharVector reached(_res_node_num);
966      CharVector processed(_res_node_num);
967      IntVector pred_node(_res_node_num);
968      IntVector pred_arc(_res_node_num);
969      std::vector<int> stack(_res_node_num);
970      std::vector<int> proc_vector(_res_node_num);
971
972      // Initialize epsilon
973      double epsilon = 0;
974      for (int a = 0; a != _res_arc_num; ++a) {
975        if (_res_cap[a] > 0 && -_cost[a] > epsilon)
976          epsilon = -_cost[a];
977      }
978
979      // Start phases
980      Tolerance<double> tol;
981      tol.epsilon(1e-6);
982      int limit = int(LIMIT_FACTOR * std::sqrt(double(_res_node_num)));
983      if (limit < MIN_LIMIT) limit = MIN_LIMIT;
984      int iter = limit;
985      while (epsilon * _res_node_num >= 1) {
986        // Find and cancel cycles in the admissible network using DFS
987        for (int u = 0; u != _res_node_num; ++u) {
988          reached[u] = false;
989          processed[u] = false;
990        }
991        int stack_head = -1;
992        int proc_head = -1;
993        for (int start = 0; start != _res_node_num; ++start) {
994          if (reached[start]) continue;
995
996          // New start node
997          reached[start] = true;
998          pred_arc[start] = -1;
999          pred_node[start] = -1;
1000
1001          // Find the first admissible outgoing arc
1002          double p = pi[start];
1003          int a = _first_out[start];
1004          int last_out = _first_out[start+1];
1005          for (; a != last_out && (_res_cap[a] == 0 ||
1006               !tol.negative(_cost[a] + p - pi[_target[a]])); ++a) ;
1007          if (a == last_out) {
1008            processed[start] = true;
1009            proc_vector[++proc_head] = start;
1010            continue;
1011          }
1012          stack[++stack_head] = a;
1013
1014          while (stack_head >= 0) {
1015            int sa = stack[stack_head];
1016            int u = _source[sa];
1017            int v = _target[sa];
1018
1019            if (!reached[v]) {
1020              // A new node is reached
1021              reached[v] = true;
1022              pred_node[v] = u;
1023              pred_arc[v] = sa;
1024              p = pi[v];
1025              a = _first_out[v];
1026              last_out = _first_out[v+1];
1027              for (; a != last_out && (_res_cap[a] == 0 ||
1028                   !tol.negative(_cost[a] + p - pi[_target[a]])); ++a) ;
1029              stack[++stack_head] = a == last_out ? -1 : a;
1030            } else {
1031              if (!processed[v]) {
1032                // A cycle is found
1033                int n, w = u;
1034                Value d, delta = _res_cap[sa];
1035                for (n = u; n != v; n = pred_node[n]) {
1036                  d = _res_cap[pred_arc[n]];
1037                  if (d <= delta) {
1038                    delta = d;
1039                    w = pred_node[n];
1040                  }
1041                }
1042
1043                // Augment along the cycle
1044                _res_cap[sa] -= delta;
1045                _res_cap[_reverse[sa]] += delta;
1046                for (n = u; n != v; n = pred_node[n]) {
1047                  int pa = pred_arc[n];
1048                  _res_cap[pa] -= delta;
1049                  _res_cap[_reverse[pa]] += delta;
1050                }
1051                for (n = u; stack_head > 0 && n != w; n = pred_node[n]) {
1052                  --stack_head;
1053                  reached[n] = false;
1054                }
1055                u = w;
1056              }
1057              v = u;
1058
1059              // Find the next admissible outgoing arc
1060              p = pi[v];
1061              a = stack[stack_head] + 1;
1062              last_out = _first_out[v+1];
1063              for (; a != last_out && (_res_cap[a] == 0 ||
1064                   !tol.negative(_cost[a] + p - pi[_target[a]])); ++a) ;
1065              stack[stack_head] = a == last_out ? -1 : a;
1066            }
1067
1068            while (stack_head >= 0 && stack[stack_head] == -1) {
1069              processed[v] = true;
1070              proc_vector[++proc_head] = v;
1071              if (--stack_head >= 0) {
1072                // Find the next admissible outgoing arc
1073                v = _source[stack[stack_head]];
1074                p = pi[v];
1075                a = stack[stack_head] + 1;
1076                last_out = _first_out[v+1];
1077                for (; a != last_out && (_res_cap[a] == 0 ||
1078                     !tol.negative(_cost[a] + p - pi[_target[a]])); ++a) ;
1079                stack[stack_head] = a == last_out ? -1 : a;
1080              }
1081            }
1082          }
1083        }
1084
1085        // Tighten potentials and epsilon
1086        if (--iter > 0) {
1087          for (int u = 0; u != _res_node_num; ++u) {
1088            level[u] = 0;
1089          }
1090          for (int i = proc_head; i > 0; --i) {
1091            int u = proc_vector[i];
1092            double p = pi[u];
1093            int l = level[u] + 1;
1094            int last_out = _first_out[u+1];
1095            for (int a = _first_out[u]; a != last_out; ++a) {
1096              int v = _target[a];
1097              if (_res_cap[a] > 0 && tol.negative(_cost[a] + p - pi[v]) &&
1098                  l > level[v]) level[v] = l;
1099            }
1100          }
1101
1102          // Modify potentials
1103          double q = std::numeric_limits<double>::max();
1104          for (int u = 0; u != _res_node_num; ++u) {
1105            int lu = level[u];
1106            double p, pu = pi[u];
1107            int last_out = _first_out[u+1];
1108            for (int a = _first_out[u]; a != last_out; ++a) {
1109              if (_res_cap[a] == 0) continue;
1110              int v = _target[a];
1111              int ld = lu - level[v];
1112              if (ld > 0) {
1113                p = (_cost[a] + pu - pi[v] + epsilon) / (ld + 1);
1114                if (p < q) q = p;
1115              }
1116            }
1117          }
1118          for (int u = 0; u != _res_node_num; ++u) {
1119            pi[u] -= q * level[u];
1120          }
1121
1122          // Modify epsilon
1123          epsilon = 0;
1124          for (int u = 0; u != _res_node_num; ++u) {
1125            double curr, pu = pi[u];
1126            int last_out = _first_out[u+1];
1127            for (int a = _first_out[u]; a != last_out; ++a) {
1128              if (_res_cap[a] == 0) continue;
1129              curr = _cost[a] + pu - pi[_target[a]];
1130              if (-curr > epsilon) epsilon = -curr;
1131            }
1132          }
1133        } else {
1134          typedef Howard<StaticDigraph, CostArcMap> MMC;
1135          typedef typename BellmanFord<StaticDigraph, CostArcMap>
1136            ::template SetDistMap<CostNodeMap>::Create BF;
1137
1138          // Set epsilon to the minimum cycle mean
1139          buildResidualNetwork();
1140          MMC mmc(_sgr, _cost_map);
1141          mmc.findMinMean();
1142          epsilon = -mmc.cycleMean();
1143          Cost cycle_cost = mmc.cycleLength();
1144          int cycle_size = mmc.cycleArcNum();
1145         
1146          // Compute feasible potentials for the current epsilon
1147          for (int i = 0; i != int(_cost_vec.size()); ++i) {
1148            _cost_vec[i] = cycle_size * _cost_vec[i] - cycle_cost;
1149          }
1150          BF bf(_sgr, _cost_map);
1151          bf.distMap(_pi_map);
1152          bf.init(0);
1153          bf.start();
1154          for (int u = 0; u != _res_node_num; ++u) {
1155            pi[u] = static_cast<double>(_pi[u]) / cycle_size;
1156          }
1157       
1158          iter = limit;
1159        }
1160      }
1161    }
1162
1163  }; //class CycleCanceling
1164
1165  ///@}
1166
1167} //namespace lemon
1168
1169#endif //LEMON_CYCLE_CANCELING_H
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