COIN-OR::LEMON - Graph Library

source: lemon/lemon/cycle_canceling.h @ 881:aef153f430e1

Last change on this file since 881:aef153f430e1 was 881:aef153f430e1, checked in by Peter Kovacs <kpeter@…>, 10 years ago

Entirely rework cycle canceling algorithms (#180)

  • Move the cycle canceling algorithms (CycleCanceling?, CancelAndTighten?) into one class (CycleCanceling?).
  • Add a Method parameter to the run() function to be able to select the used cycle canceling method.
  • Use the new interface similarly to NetworkSimplex?.
  • Rework the implementations using an efficient internal structure for handling the residual network. This improvement made the codes much faster.
  • Handle GEQ supply type (LEQ is not supported).
  • Handle infinite upper bounds.
  • Handle negative costs (for arcs of finite upper bound).
  • Extend the documentation.
File size: 36.8 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  /// The most efficent one (both theoretically and practically)
50  /// is the \ref CANCEL_AND_TIGHTEN "Cancel and Tighten" algorithm,
51  /// thus it is the default method.
52  /// It is strongly polynomial, but in practice, it is typically much
53  /// slower than the scaling algorithms and NetworkSimplex.
54  ///
55  /// Most of the parameters of the problem (except for the digraph)
56  /// can be given using separate functions, and the algorithm can be
57  /// executed using the \ref run() function. If some parameters are not
58  /// specified, then default values will be used.
59  ///
60  /// \tparam GR The digraph type the algorithm runs on.
61  /// \tparam V The number type used for flow amounts, capacity bounds
62  /// and supply values in the algorithm. By default, it is \c int.
63  /// \tparam C The number type used for costs and potentials in the
64  /// algorithm. By default, it is the same as \c V.
65  ///
66  /// \warning Both number types must be signed and all input data must
67  /// be integer.
68  /// \warning This algorithm does not support negative costs for such
69  /// arcs that have infinite upper bound.
70  ///
71  /// \note For more information about the three available methods,
72  /// see \ref Method.
73#ifdef DOXYGEN
74  template <typename GR, typename V, typename C>
75#else
76  template <typename GR, typename V = int, typename C = V>
77#endif
78  class CycleCanceling
79  {
80  public:
81
82    /// The type of the digraph
83    typedef GR Digraph;
84    /// The type of the flow amounts, capacity bounds and supply values
85    typedef V Value;
86    /// The type of the arc costs
87    typedef C Cost;
88
89  public:
90
91    /// \brief Problem type constants for the \c run() function.
92    ///
93    /// Enum type containing the problem type constants that can be
94    /// returned by the \ref run() function of the algorithm.
95    enum ProblemType {
96      /// The problem has no feasible solution (flow).
97      INFEASIBLE,
98      /// The problem has optimal solution (i.e. it is feasible and
99      /// bounded), and the algorithm has found optimal flow and node
100      /// potentials (primal and dual solutions).
101      OPTIMAL,
102      /// The digraph contains an arc of negative cost and infinite
103      /// upper bound. It means that the objective function is unbounded
104      /// on that arc, however, note that it could actually be bounded
105      /// over the feasible flows, but this algroithm cannot handle
106      /// these cases.
107      UNBOUNDED
108    };
109
110    /// \brief Constants for selecting the used method.
111    ///
112    /// Enum type containing constants for selecting the used method
113    /// for the \ref run() function.
114    ///
115    /// \ref CycleCanceling provides three different cycle-canceling
116    /// methods. By default, \ref CANCEL_AND_TIGHTEN "Cancel and Tighten"
117    /// is used, which proved to be the most efficient and the most robust
118    /// on various test inputs.
119    /// However, the other methods can be selected using the \ref run()
120    /// function with the proper parameter.
121    enum Method {
122      /// A simple cycle-canceling method, which uses the
123      /// \ref BellmanFord "Bellman-Ford" algorithm with limited iteration
124      /// number for detecting negative cycles in the residual network.
125      SIMPLE_CYCLE_CANCELING,
126      /// The "Minimum Mean Cycle-Canceling" algorithm, which is a
127      /// well-known strongly polynomial method. It improves along a
128      /// \ref min_mean_cycle "minimum mean cycle" in each iteration.
129      /// Its running time complexity is O(n<sup>2</sup>m<sup>3</sup>log(n)).
130      MINIMUM_MEAN_CYCLE_CANCELING,
131      /// The "Cancel And Tighten" algorithm, which can be viewed as an
132      /// improved version of the previous method.
133      /// It is faster both in theory and in practice, its running time
134      /// complexity is O(n<sup>2</sup>m<sup>2</sup>log(n)).
135      CANCEL_AND_TIGHTEN
136    };
137
138  private:
139
140    TEMPLATE_DIGRAPH_TYPEDEFS(GR);
141   
142    typedef std::vector<int> IntVector;
143    typedef std::vector<char> CharVector;
144    typedef std::vector<double> DoubleVector;
145    typedef std::vector<Value> ValueVector;
146    typedef std::vector<Cost> CostVector;
147
148  private:
149 
150    template <typename KT, typename VT>
151    class VectorMap {
152    public:
153      typedef KT Key;
154      typedef VT Value;
155     
156      VectorMap(std::vector<Value>& v) : _v(v) {}
157     
158      const Value& operator[](const Key& key) const {
159        return _v[StaticDigraph::id(key)];
160      }
161
162      Value& operator[](const Key& key) {
163        return _v[StaticDigraph::id(key)];
164      }
165     
166      void set(const Key& key, const Value& val) {
167        _v[StaticDigraph::id(key)] = val;
168      }
169
170    private:
171      std::vector<Value>& _v;
172    };
173
174    typedef VectorMap<StaticDigraph::Node, Cost> CostNodeMap;
175    typedef VectorMap<StaticDigraph::Arc, Cost> CostArcMap;
176
177  private:
178
179
180    // Data related to the underlying digraph
181    const GR &_graph;
182    int _node_num;
183    int _arc_num;
184    int _res_node_num;
185    int _res_arc_num;
186    int _root;
187
188    // Parameters of the problem
189    bool _have_lower;
190    Value _sum_supply;
191
192    // Data structures for storing the digraph
193    IntNodeMap _node_id;
194    IntArcMap _arc_idf;
195    IntArcMap _arc_idb;
196    IntVector _first_out;
197    CharVector _forward;
198    IntVector _source;
199    IntVector _target;
200    IntVector _reverse;
201
202    // Node and arc data
203    ValueVector _lower;
204    ValueVector _upper;
205    CostVector _cost;
206    ValueVector _supply;
207
208    ValueVector _res_cap;
209    CostVector _pi;
210
211    // Data for a StaticDigraph structure
212    typedef std::pair<int, int> IntPair;
213    StaticDigraph _sgr;
214    std::vector<IntPair> _arc_vec;
215    std::vector<Cost> _cost_vec;
216    IntVector _id_vec;
217    CostArcMap _cost_map;
218    CostNodeMap _pi_map;
219 
220  public:
221 
222    /// \brief Constant for infinite upper bounds (capacities).
223    ///
224    /// Constant for infinite upper bounds (capacities).
225    /// It is \c std::numeric_limits<Value>::infinity() if available,
226    /// \c std::numeric_limits<Value>::max() otherwise.
227    const Value INF;
228
229  public:
230
231    /// \brief Constructor.
232    ///
233    /// The constructor of the class.
234    ///
235    /// \param graph The digraph the algorithm runs on.
236    CycleCanceling(const GR& graph) :
237      _graph(graph), _node_id(graph), _arc_idf(graph), _arc_idb(graph),
238      _cost_map(_cost_vec), _pi_map(_pi),
239      INF(std::numeric_limits<Value>::has_infinity ?
240          std::numeric_limits<Value>::infinity() :
241          std::numeric_limits<Value>::max())
242    {
243      // Check the number types
244      LEMON_ASSERT(std::numeric_limits<Value>::is_signed,
245        "The flow type of CycleCanceling must be signed");
246      LEMON_ASSERT(std::numeric_limits<Cost>::is_signed,
247        "The cost type of CycleCanceling must be signed");
248
249      // Resize vectors
250      _node_num = countNodes(_graph);
251      _arc_num = countArcs(_graph);
252      _res_node_num = _node_num + 1;
253      _res_arc_num = 2 * (_arc_num + _node_num);
254      _root = _node_num;
255
256      _first_out.resize(_res_node_num + 1);
257      _forward.resize(_res_arc_num);
258      _source.resize(_res_arc_num);
259      _target.resize(_res_arc_num);
260      _reverse.resize(_res_arc_num);
261
262      _lower.resize(_res_arc_num);
263      _upper.resize(_res_arc_num);
264      _cost.resize(_res_arc_num);
265      _supply.resize(_res_node_num);
266     
267      _res_cap.resize(_res_arc_num);
268      _pi.resize(_res_node_num);
269
270      _arc_vec.reserve(_res_arc_num);
271      _cost_vec.reserve(_res_arc_num);
272      _id_vec.reserve(_res_arc_num);
273
274      // Copy the graph
275      int i = 0, j = 0, k = 2 * _arc_num + _node_num;
276      for (NodeIt n(_graph); n != INVALID; ++n, ++i) {
277        _node_id[n] = i;
278      }
279      i = 0;
280      for (NodeIt n(_graph); n != INVALID; ++n, ++i) {
281        _first_out[i] = j;
282        for (OutArcIt a(_graph, n); a != INVALID; ++a, ++j) {
283          _arc_idf[a] = j;
284          _forward[j] = true;
285          _source[j] = i;
286          _target[j] = _node_id[_graph.runningNode(a)];
287        }
288        for (InArcIt a(_graph, n); a != INVALID; ++a, ++j) {
289          _arc_idb[a] = j;
290          _forward[j] = false;
291          _source[j] = i;
292          _target[j] = _node_id[_graph.runningNode(a)];
293        }
294        _forward[j] = false;
295        _source[j] = i;
296        _target[j] = _root;
297        _reverse[j] = k;
298        _forward[k] = true;
299        _source[k] = _root;
300        _target[k] = i;
301        _reverse[k] = j;
302        ++j; ++k;
303      }
304      _first_out[i] = j;
305      _first_out[_res_node_num] = k;
306      for (ArcIt a(_graph); a != INVALID; ++a) {
307        int fi = _arc_idf[a];
308        int bi = _arc_idb[a];
309        _reverse[fi] = bi;
310        _reverse[bi] = fi;
311      }
312     
313      // Reset parameters
314      reset();
315    }
316
317    /// \name Parameters
318    /// The parameters of the algorithm can be specified using these
319    /// functions.
320
321    /// @{
322
323    /// \brief Set the lower bounds on the arcs.
324    ///
325    /// This function sets the lower bounds on the arcs.
326    /// If it is not used before calling \ref run(), the lower bounds
327    /// will be set to zero on all arcs.
328    ///
329    /// \param map An arc map storing the lower bounds.
330    /// Its \c Value type must be convertible to the \c Value type
331    /// of the algorithm.
332    ///
333    /// \return <tt>(*this)</tt>
334    template <typename LowerMap>
335    CycleCanceling& lowerMap(const LowerMap& map) {
336      _have_lower = true;
337      for (ArcIt a(_graph); a != INVALID; ++a) {
338        _lower[_arc_idf[a]] = map[a];
339        _lower[_arc_idb[a]] = map[a];
340      }
341      return *this;
342    }
343
344    /// \brief Set the upper bounds (capacities) on the arcs.
345    ///
346    /// This function sets the upper bounds (capacities) on the arcs.
347    /// If it is not used before calling \ref run(), the upper bounds
348    /// will be set to \ref INF on all arcs (i.e. the flow value will be
349    /// unbounded from above).
350    ///
351    /// \param map An arc map storing the upper bounds.
352    /// Its \c Value type must be convertible to the \c Value type
353    /// of the algorithm.
354    ///
355    /// \return <tt>(*this)</tt>
356    template<typename UpperMap>
357    CycleCanceling& upperMap(const UpperMap& map) {
358      for (ArcIt a(_graph); a != INVALID; ++a) {
359        _upper[_arc_idf[a]] = map[a];
360      }
361      return *this;
362    }
363
364    /// \brief Set the costs of the arcs.
365    ///
366    /// This function sets the costs of the arcs.
367    /// If it is not used before calling \ref run(), the costs
368    /// will be set to \c 1 on all arcs.
369    ///
370    /// \param map An arc map storing the costs.
371    /// Its \c Value type must be convertible to the \c Cost type
372    /// of the algorithm.
373    ///
374    /// \return <tt>(*this)</tt>
375    template<typename CostMap>
376    CycleCanceling& costMap(const CostMap& map) {
377      for (ArcIt a(_graph); a != INVALID; ++a) {
378        _cost[_arc_idf[a]] =  map[a];
379        _cost[_arc_idb[a]] = -map[a];
380      }
381      return *this;
382    }
383
384    /// \brief Set the supply values of the nodes.
385    ///
386    /// This function sets the supply values of the nodes.
387    /// If neither this function nor \ref stSupply() is used before
388    /// calling \ref run(), the supply of each node will be set to zero.
389    ///
390    /// \param map A node map storing the supply values.
391    /// Its \c Value type must be convertible to the \c Value type
392    /// of the algorithm.
393    ///
394    /// \return <tt>(*this)</tt>
395    template<typename SupplyMap>
396    CycleCanceling& supplyMap(const SupplyMap& map) {
397      for (NodeIt n(_graph); n != INVALID; ++n) {
398        _supply[_node_id[n]] = map[n];
399      }
400      return *this;
401    }
402
403    /// \brief Set single source and target nodes and a supply value.
404    ///
405    /// This function sets a single source node and a single target node
406    /// and the required flow value.
407    /// If neither this function nor \ref supplyMap() is used before
408    /// calling \ref run(), the supply of each node will be set to zero.
409    ///
410    /// Using this function has the same effect as using \ref supplyMap()
411    /// with such a map in which \c k is assigned to \c s, \c -k is
412    /// assigned to \c t and all other nodes have zero supply value.
413    ///
414    /// \param s The source node.
415    /// \param t The target node.
416    /// \param k The required amount of flow from node \c s to node \c t
417    /// (i.e. the supply of \c s and the demand of \c t).
418    ///
419    /// \return <tt>(*this)</tt>
420    CycleCanceling& stSupply(const Node& s, const Node& t, Value k) {
421      for (int i = 0; i != _res_node_num; ++i) {
422        _supply[i] = 0;
423      }
424      _supply[_node_id[s]] =  k;
425      _supply[_node_id[t]] = -k;
426      return *this;
427    }
428   
429    /// @}
430
431    /// \name Execution control
432    /// The algorithm can be executed using \ref run().
433
434    /// @{
435
436    /// \brief Run the algorithm.
437    ///
438    /// This function runs the algorithm.
439    /// The paramters can be specified using functions \ref lowerMap(),
440    /// \ref upperMap(), \ref costMap(), \ref supplyMap(), \ref stSupply().
441    /// For example,
442    /// \code
443    ///   CycleCanceling<ListDigraph> cc(graph);
444    ///   cc.lowerMap(lower).upperMap(upper).costMap(cost)
445    ///     .supplyMap(sup).run();
446    /// \endcode
447    ///
448    /// This function can be called more than once. All the parameters
449    /// that have been given are kept for the next call, unless
450    /// \ref reset() is called, thus only the modified parameters
451    /// have to be set again. See \ref reset() for examples.
452    /// However, the underlying digraph must not be modified after this
453    /// class have been constructed, since it copies and extends the graph.
454    ///
455    /// \param method The cycle-canceling method that will be used.
456    /// For more information, see \ref Method.
457    ///
458    /// \return \c INFEASIBLE if no feasible flow exists,
459    /// \n \c OPTIMAL if the problem has optimal solution
460    /// (i.e. it is feasible and bounded), and the algorithm has found
461    /// optimal flow and node potentials (primal and dual solutions),
462    /// \n \c UNBOUNDED if the digraph contains an arc of negative cost
463    /// and infinite upper bound. It means that the objective function
464    /// is unbounded on that arc, however, note that it could actually be
465    /// bounded over the feasible flows, but this algroithm cannot handle
466    /// these cases.
467    ///
468    /// \see ProblemType, Method
469    ProblemType run(Method method = CANCEL_AND_TIGHTEN) {
470      ProblemType pt = init();
471      if (pt != OPTIMAL) return pt;
472      start(method);
473      return OPTIMAL;
474    }
475
476    /// \brief Reset all the parameters that have been given before.
477    ///
478    /// This function resets all the paramaters that have been given
479    /// before using functions \ref lowerMap(), \ref upperMap(),
480    /// \ref costMap(), \ref supplyMap(), \ref stSupply().
481    ///
482    /// It is useful for multiple run() calls. If this function is not
483    /// used, all the parameters given before are kept for the next
484    /// \ref run() call.
485    /// However, the underlying digraph must not be modified after this
486    /// class have been constructed, since it copies and extends the graph.
487    ///
488    /// For example,
489    /// \code
490    ///   CycleCanceling<ListDigraph> cs(graph);
491    ///
492    ///   // First run
493    ///   cc.lowerMap(lower).upperMap(upper).costMap(cost)
494    ///     .supplyMap(sup).run();
495    ///
496    ///   // Run again with modified cost map (reset() is not called,
497    ///   // so only the cost map have to be set again)
498    ///   cost[e] += 100;
499    ///   cc.costMap(cost).run();
500    ///
501    ///   // Run again from scratch using reset()
502    ///   // (the lower bounds will be set to zero on all arcs)
503    ///   cc.reset();
504    ///   cc.upperMap(capacity).costMap(cost)
505    ///     .supplyMap(sup).run();
506    /// \endcode
507    ///
508    /// \return <tt>(*this)</tt>
509    CycleCanceling& reset() {
510      for (int i = 0; i != _res_node_num; ++i) {
511        _supply[i] = 0;
512      }
513      int limit = _first_out[_root];
514      for (int j = 0; j != limit; ++j) {
515        _lower[j] = 0;
516        _upper[j] = INF;
517        _cost[j] = _forward[j] ? 1 : -1;
518      }
519      for (int j = limit; j != _res_arc_num; ++j) {
520        _lower[j] = 0;
521        _upper[j] = INF;
522        _cost[j] = 0;
523        _cost[_reverse[j]] = 0;
524      }     
525      _have_lower = false;
526      return *this;
527    }
528
529    /// @}
530
531    /// \name Query Functions
532    /// The results of the algorithm can be obtained using these
533    /// functions.\n
534    /// The \ref run() function must be called before using them.
535
536    /// @{
537
538    /// \brief Return the total cost of the found flow.
539    ///
540    /// This function returns the total cost of the found flow.
541    /// Its complexity is O(e).
542    ///
543    /// \note The return type of the function can be specified as a
544    /// template parameter. For example,
545    /// \code
546    ///   cc.totalCost<double>();
547    /// \endcode
548    /// It is useful if the total cost cannot be stored in the \c Cost
549    /// type of the algorithm, which is the default return type of the
550    /// function.
551    ///
552    /// \pre \ref run() must be called before using this function.
553    template <typename Number>
554    Number totalCost() const {
555      Number c = 0;
556      for (ArcIt a(_graph); a != INVALID; ++a) {
557        int i = _arc_idb[a];
558        c += static_cast<Number>(_res_cap[i]) *
559             (-static_cast<Number>(_cost[i]));
560      }
561      return c;
562    }
563
564#ifndef DOXYGEN
565    Cost totalCost() const {
566      return totalCost<Cost>();
567    }
568#endif
569
570    /// \brief Return the flow on the given arc.
571    ///
572    /// This function returns the flow on the given arc.
573    ///
574    /// \pre \ref run() must be called before using this function.
575    Value flow(const Arc& a) const {
576      return _res_cap[_arc_idb[a]];
577    }
578
579    /// \brief Return the flow map (the primal solution).
580    ///
581    /// This function copies the flow value on each arc into the given
582    /// map. The \c Value type of the algorithm must be convertible to
583    /// the \c Value type of the map.
584    ///
585    /// \pre \ref run() must be called before using this function.
586    template <typename FlowMap>
587    void flowMap(FlowMap &map) const {
588      for (ArcIt a(_graph); a != INVALID; ++a) {
589        map.set(a, _res_cap[_arc_idb[a]]);
590      }
591    }
592
593    /// \brief Return the potential (dual value) of the given node.
594    ///
595    /// This function returns the potential (dual value) of the
596    /// given node.
597    ///
598    /// \pre \ref run() must be called before using this function.
599    Cost potential(const Node& n) const {
600      return static_cast<Cost>(_pi[_node_id[n]]);
601    }
602
603    /// \brief Return the potential map (the dual solution).
604    ///
605    /// This function copies the potential (dual value) of each node
606    /// into the given map.
607    /// The \c Cost type of the algorithm must be convertible to the
608    /// \c Value type of the map.
609    ///
610    /// \pre \ref run() must be called before using this function.
611    template <typename PotentialMap>
612    void potentialMap(PotentialMap &map) const {
613      for (NodeIt n(_graph); n != INVALID; ++n) {
614        map.set(n, static_cast<Cost>(_pi[_node_id[n]]));
615      }
616    }
617
618    /// @}
619
620  private:
621
622    // Initialize the algorithm
623    ProblemType init() {
624      if (_res_node_num <= 1) return INFEASIBLE;
625
626      // Check the sum of supply values
627      _sum_supply = 0;
628      for (int i = 0; i != _root; ++i) {
629        _sum_supply += _supply[i];
630      }
631      if (_sum_supply > 0) return INFEASIBLE;
632     
633
634      // Initialize vectors
635      for (int i = 0; i != _res_node_num; ++i) {
636        _pi[i] = 0;
637      }
638      ValueVector excess(_supply);
639     
640      // Remove infinite upper bounds and check negative arcs
641      const Value MAX = std::numeric_limits<Value>::max();
642      int last_out;
643      if (_have_lower) {
644        for (int i = 0; i != _root; ++i) {
645          last_out = _first_out[i+1];
646          for (int j = _first_out[i]; j != last_out; ++j) {
647            if (_forward[j]) {
648              Value c = _cost[j] < 0 ? _upper[j] : _lower[j];
649              if (c >= MAX) return UNBOUNDED;
650              excess[i] -= c;
651              excess[_target[j]] += c;
652            }
653          }
654        }
655      } else {
656        for (int i = 0; i != _root; ++i) {
657          last_out = _first_out[i+1];
658          for (int j = _first_out[i]; j != last_out; ++j) {
659            if (_forward[j] && _cost[j] < 0) {
660              Value c = _upper[j];
661              if (c >= MAX) return UNBOUNDED;
662              excess[i] -= c;
663              excess[_target[j]] += c;
664            }
665          }
666        }
667      }
668      Value ex, max_cap = 0;
669      for (int i = 0; i != _res_node_num; ++i) {
670        ex = excess[i];
671        if (ex < 0) max_cap -= ex;
672      }
673      for (int j = 0; j != _res_arc_num; ++j) {
674        if (_upper[j] >= MAX) _upper[j] = max_cap;
675      }
676
677      // Initialize maps for Circulation and remove non-zero lower bounds
678      ConstMap<Arc, Value> low(0);
679      typedef typename Digraph::template ArcMap<Value> ValueArcMap;
680      typedef typename Digraph::template NodeMap<Value> ValueNodeMap;
681      ValueArcMap cap(_graph), flow(_graph);
682      ValueNodeMap sup(_graph);
683      for (NodeIt n(_graph); n != INVALID; ++n) {
684        sup[n] = _supply[_node_id[n]];
685      }
686      if (_have_lower) {
687        for (ArcIt a(_graph); a != INVALID; ++a) {
688          int j = _arc_idf[a];
689          Value c = _lower[j];
690          cap[a] = _upper[j] - c;
691          sup[_graph.source(a)] -= c;
692          sup[_graph.target(a)] += c;
693        }
694      } else {
695        for (ArcIt a(_graph); a != INVALID; ++a) {
696          cap[a] = _upper[_arc_idf[a]];
697        }
698      }
699
700      // Find a feasible flow using Circulation
701      Circulation<Digraph, ConstMap<Arc, Value>, ValueArcMap, ValueNodeMap>
702        circ(_graph, low, cap, sup);
703      if (!circ.flowMap(flow).run()) return INFEASIBLE;
704
705      // Set residual capacities and handle GEQ supply type
706      if (_sum_supply < 0) {
707        for (ArcIt a(_graph); a != INVALID; ++a) {
708          Value fa = flow[a];
709          _res_cap[_arc_idf[a]] = cap[a] - fa;
710          _res_cap[_arc_idb[a]] = fa;
711          sup[_graph.source(a)] -= fa;
712          sup[_graph.target(a)] += fa;
713        }
714        for (NodeIt n(_graph); n != INVALID; ++n) {
715          excess[_node_id[n]] = sup[n];
716        }
717        for (int a = _first_out[_root]; a != _res_arc_num; ++a) {
718          int u = _target[a];
719          int ra = _reverse[a];
720          _res_cap[a] = -_sum_supply + 1;
721          _res_cap[ra] = -excess[u];
722          _cost[a] = 0;
723          _cost[ra] = 0;
724        }
725      } else {
726        for (ArcIt a(_graph); a != INVALID; ++a) {
727          Value fa = flow[a];
728          _res_cap[_arc_idf[a]] = cap[a] - fa;
729          _res_cap[_arc_idb[a]] = fa;
730        }
731        for (int a = _first_out[_root]; a != _res_arc_num; ++a) {
732          int ra = _reverse[a];
733          _res_cap[a] = 1;
734          _res_cap[ra] = 0;
735          _cost[a] = 0;
736          _cost[ra] = 0;
737        }
738      }
739     
740      return OPTIMAL;
741    }
742   
743    // Build a StaticDigraph structure containing the current
744    // residual network
745    void buildResidualNetwork() {
746      _arc_vec.clear();
747      _cost_vec.clear();
748      _id_vec.clear();
749      for (int j = 0; j != _res_arc_num; ++j) {
750        if (_res_cap[j] > 0) {
751          _arc_vec.push_back(IntPair(_source[j], _target[j]));
752          _cost_vec.push_back(_cost[j]);
753          _id_vec.push_back(j);
754        }
755      }
756      _sgr.build(_res_node_num, _arc_vec.begin(), _arc_vec.end());
757    }
758
759    // Execute the algorithm and transform the results
760    void start(Method method) {
761      // Execute the algorithm
762      switch (method) {
763        case SIMPLE_CYCLE_CANCELING:
764          startSimpleCycleCanceling();
765          break;
766        case MINIMUM_MEAN_CYCLE_CANCELING:
767          startMinMeanCycleCanceling();
768          break;
769        case CANCEL_AND_TIGHTEN:
770          startCancelAndTighten();
771          break;
772      }
773
774      // Compute node potentials
775      if (method != SIMPLE_CYCLE_CANCELING) {
776        buildResidualNetwork();
777        typename BellmanFord<StaticDigraph, CostArcMap>
778          ::template SetDistMap<CostNodeMap>::Create bf(_sgr, _cost_map);
779        bf.distMap(_pi_map);
780        bf.init(0);
781        bf.start();
782      }
783
784      // Handle non-zero lower bounds
785      if (_have_lower) {
786        int limit = _first_out[_root];
787        for (int j = 0; j != limit; ++j) {
788          if (!_forward[j]) _res_cap[j] += _lower[j];
789        }
790      }
791    }
792
793    // Execute the "Simple Cycle Canceling" method
794    void startSimpleCycleCanceling() {
795      // Constants for computing the iteration limits
796      const int BF_FIRST_LIMIT  = 2;
797      const double BF_LIMIT_FACTOR = 1.5;
798     
799      typedef VectorMap<StaticDigraph::Arc, Value> FilterMap;
800      typedef FilterArcs<StaticDigraph, FilterMap> ResDigraph;
801      typedef VectorMap<StaticDigraph::Node, StaticDigraph::Arc> PredMap;
802      typedef typename BellmanFord<ResDigraph, CostArcMap>
803        ::template SetDistMap<CostNodeMap>
804        ::template SetPredMap<PredMap>::Create BF;
805     
806      // Build the residual network
807      _arc_vec.clear();
808      _cost_vec.clear();
809      for (int j = 0; j != _res_arc_num; ++j) {
810        _arc_vec.push_back(IntPair(_source[j], _target[j]));
811        _cost_vec.push_back(_cost[j]);
812      }
813      _sgr.build(_res_node_num, _arc_vec.begin(), _arc_vec.end());
814
815      FilterMap filter_map(_res_cap);
816      ResDigraph rgr(_sgr, filter_map);
817      std::vector<int> cycle;
818      std::vector<StaticDigraph::Arc> pred(_res_arc_num);
819      PredMap pred_map(pred);
820      BF bf(rgr, _cost_map);
821      bf.distMap(_pi_map).predMap(pred_map);
822
823      int length_bound = BF_FIRST_LIMIT;
824      bool optimal = false;
825      while (!optimal) {
826        bf.init(0);
827        int iter_num = 0;
828        bool cycle_found = false;
829        while (!cycle_found) {
830          // Perform some iterations of the Bellman-Ford algorithm
831          int curr_iter_num = iter_num + length_bound <= _node_num ?
832            length_bound : _node_num - iter_num;
833          iter_num += curr_iter_num;
834          int real_iter_num = curr_iter_num;
835          for (int i = 0; i < curr_iter_num; ++i) {
836            if (bf.processNextWeakRound()) {
837              real_iter_num = i;
838              break;
839            }
840          }
841          if (real_iter_num < curr_iter_num) {
842            // Optimal flow is found
843            optimal = true;
844            break;
845          } else {
846            // Search for node disjoint negative cycles
847            std::vector<int> state(_res_node_num, 0);
848            int id = 0;
849            for (int u = 0; u != _res_node_num; ++u) {
850              if (state[u] != 0) continue;
851              ++id;
852              int v = u;
853              for (; v != -1 && state[v] == 0; v = pred[v] == INVALID ?
854                   -1 : rgr.id(rgr.source(pred[v]))) {
855                state[v] = id;
856              }
857              if (v != -1 && state[v] == id) {
858                // A negative cycle is found
859                cycle_found = true;
860                cycle.clear();
861                StaticDigraph::Arc a = pred[v];
862                Value d, delta = _res_cap[rgr.id(a)];
863                cycle.push_back(rgr.id(a));
864                while (rgr.id(rgr.source(a)) != v) {
865                  a = pred_map[rgr.source(a)];
866                  d = _res_cap[rgr.id(a)];
867                  if (d < delta) delta = d;
868                  cycle.push_back(rgr.id(a));
869                }
870
871                // Augment along the cycle
872                for (int i = 0; i < int(cycle.size()); ++i) {
873                  int j = cycle[i];
874                  _res_cap[j] -= delta;
875                  _res_cap[_reverse[j]] += delta;
876                }
877              }
878            }
879          }
880
881          // Increase iteration limit if no cycle is found
882          if (!cycle_found) {
883            length_bound = static_cast<int>(length_bound * BF_LIMIT_FACTOR);
884          }
885        }
886      }
887    }
888
889    // Execute the "Minimum Mean Cycle Canceling" method
890    void startMinMeanCycleCanceling() {
891      typedef SimplePath<StaticDigraph> SPath;
892      typedef typename SPath::ArcIt SPathArcIt;
893      typedef typename Howard<StaticDigraph, CostArcMap>
894        ::template SetPath<SPath>::Create MMC;
895     
896      SPath cycle;
897      MMC mmc(_sgr, _cost_map);
898      mmc.cycle(cycle);
899      buildResidualNetwork();
900      while (mmc.findMinMean() && mmc.cycleLength() < 0) {
901        // Find the cycle
902        mmc.findCycle();
903
904        // Compute delta value
905        Value delta = INF;
906        for (SPathArcIt a(cycle); a != INVALID; ++a) {
907          Value d = _res_cap[_id_vec[_sgr.id(a)]];
908          if (d < delta) delta = d;
909        }
910
911        // Augment along the cycle
912        for (SPathArcIt a(cycle); a != INVALID; ++a) {
913          int j = _id_vec[_sgr.id(a)];
914          _res_cap[j] -= delta;
915          _res_cap[_reverse[j]] += delta;
916        }
917
918        // Rebuild the residual network       
919        buildResidualNetwork();
920      }
921    }
922
923    // Execute the "Cancel And Tighten" method
924    void startCancelAndTighten() {
925      // Constants for the min mean cycle computations
926      const double LIMIT_FACTOR = 1.0;
927      const int MIN_LIMIT = 5;
928
929      // Contruct auxiliary data vectors
930      DoubleVector pi(_res_node_num, 0.0);
931      IntVector level(_res_node_num);
932      CharVector reached(_res_node_num);
933      CharVector processed(_res_node_num);
934      IntVector pred_node(_res_node_num);
935      IntVector pred_arc(_res_node_num);
936      std::vector<int> stack(_res_node_num);
937      std::vector<int> proc_vector(_res_node_num);
938
939      // Initialize epsilon
940      double epsilon = 0;
941      for (int a = 0; a != _res_arc_num; ++a) {
942        if (_res_cap[a] > 0 && -_cost[a] > epsilon)
943          epsilon = -_cost[a];
944      }
945
946      // Start phases
947      Tolerance<double> tol;
948      tol.epsilon(1e-6);
949      int limit = int(LIMIT_FACTOR * std::sqrt(double(_res_node_num)));
950      if (limit < MIN_LIMIT) limit = MIN_LIMIT;
951      int iter = limit;
952      while (epsilon * _res_node_num >= 1) {
953        // Find and cancel cycles in the admissible network using DFS
954        for (int u = 0; u != _res_node_num; ++u) {
955          reached[u] = false;
956          processed[u] = false;
957        }
958        int stack_head = -1;
959        int proc_head = -1;
960        for (int start = 0; start != _res_node_num; ++start) {
961          if (reached[start]) continue;
962
963          // New start node
964          reached[start] = true;
965          pred_arc[start] = -1;
966          pred_node[start] = -1;
967
968          // Find the first admissible outgoing arc
969          double p = pi[start];
970          int a = _first_out[start];
971          int last_out = _first_out[start+1];
972          for (; a != last_out && (_res_cap[a] == 0 ||
973               !tol.negative(_cost[a] + p - pi[_target[a]])); ++a) ;
974          if (a == last_out) {
975            processed[start] = true;
976            proc_vector[++proc_head] = start;
977            continue;
978          }
979          stack[++stack_head] = a;
980
981          while (stack_head >= 0) {
982            int sa = stack[stack_head];
983            int u = _source[sa];
984            int v = _target[sa];
985
986            if (!reached[v]) {
987              // A new node is reached
988              reached[v] = true;
989              pred_node[v] = u;
990              pred_arc[v] = sa;
991              p = pi[v];
992              a = _first_out[v];
993              last_out = _first_out[v+1];
994              for (; a != last_out && (_res_cap[a] == 0 ||
995                   !tol.negative(_cost[a] + p - pi[_target[a]])); ++a) ;
996              stack[++stack_head] = a == last_out ? -1 : a;
997            } else {
998              if (!processed[v]) {
999                // A cycle is found
1000                int n, w = u;
1001                Value d, delta = _res_cap[sa];
1002                for (n = u; n != v; n = pred_node[n]) {
1003                  d = _res_cap[pred_arc[n]];
1004                  if (d <= delta) {
1005                    delta = d;
1006                    w = pred_node[n];
1007                  }
1008                }
1009
1010                // Augment along the cycle
1011                _res_cap[sa] -= delta;
1012                _res_cap[_reverse[sa]] += delta;
1013                for (n = u; n != v; n = pred_node[n]) {
1014                  int pa = pred_arc[n];
1015                  _res_cap[pa] -= delta;
1016                  _res_cap[_reverse[pa]] += delta;
1017                }
1018                for (n = u; stack_head > 0 && n != w; n = pred_node[n]) {
1019                  --stack_head;
1020                  reached[n] = false;
1021                }
1022                u = w;
1023              }
1024              v = u;
1025
1026              // Find the next admissible outgoing arc
1027              p = pi[v];
1028              a = stack[stack_head] + 1;
1029              last_out = _first_out[v+1];
1030              for (; a != last_out && (_res_cap[a] == 0 ||
1031                   !tol.negative(_cost[a] + p - pi[_target[a]])); ++a) ;
1032              stack[stack_head] = a == last_out ? -1 : a;
1033            }
1034
1035            while (stack_head >= 0 && stack[stack_head] == -1) {
1036              processed[v] = true;
1037              proc_vector[++proc_head] = v;
1038              if (--stack_head >= 0) {
1039                // Find the next admissible outgoing arc
1040                v = _source[stack[stack_head]];
1041                p = pi[v];
1042                a = stack[stack_head] + 1;
1043                last_out = _first_out[v+1];
1044                for (; a != last_out && (_res_cap[a] == 0 ||
1045                     !tol.negative(_cost[a] + p - pi[_target[a]])); ++a) ;
1046                stack[stack_head] = a == last_out ? -1 : a;
1047              }
1048            }
1049          }
1050        }
1051
1052        // Tighten potentials and epsilon
1053        if (--iter > 0) {
1054          for (int u = 0; u != _res_node_num; ++u) {
1055            level[u] = 0;
1056          }
1057          for (int i = proc_head; i > 0; --i) {
1058            int u = proc_vector[i];
1059            double p = pi[u];
1060            int l = level[u] + 1;
1061            int last_out = _first_out[u+1];
1062            for (int a = _first_out[u]; a != last_out; ++a) {
1063              int v = _target[a];
1064              if (_res_cap[a] > 0 && tol.negative(_cost[a] + p - pi[v]) &&
1065                  l > level[v]) level[v] = l;
1066            }
1067          }
1068
1069          // Modify potentials
1070          double q = std::numeric_limits<double>::max();
1071          for (int u = 0; u != _res_node_num; ++u) {
1072            int lu = level[u];
1073            double p, pu = pi[u];
1074            int last_out = _first_out[u+1];
1075            for (int a = _first_out[u]; a != last_out; ++a) {
1076              if (_res_cap[a] == 0) continue;
1077              int v = _target[a];
1078              int ld = lu - level[v];
1079              if (ld > 0) {
1080                p = (_cost[a] + pu - pi[v] + epsilon) / (ld + 1);
1081                if (p < q) q = p;
1082              }
1083            }
1084          }
1085          for (int u = 0; u != _res_node_num; ++u) {
1086            pi[u] -= q * level[u];
1087          }
1088
1089          // Modify epsilon
1090          epsilon = 0;
1091          for (int u = 0; u != _res_node_num; ++u) {
1092            double curr, pu = pi[u];
1093            int last_out = _first_out[u+1];
1094            for (int a = _first_out[u]; a != last_out; ++a) {
1095              if (_res_cap[a] == 0) continue;
1096              curr = _cost[a] + pu - pi[_target[a]];
1097              if (-curr > epsilon) epsilon = -curr;
1098            }
1099          }
1100        } else {
1101          typedef Howard<StaticDigraph, CostArcMap> MMC;
1102          typedef typename BellmanFord<StaticDigraph, CostArcMap>
1103            ::template SetDistMap<CostNodeMap>::Create BF;
1104
1105          // Set epsilon to the minimum cycle mean
1106          buildResidualNetwork();
1107          MMC mmc(_sgr, _cost_map);
1108          mmc.findMinMean();
1109          epsilon = -mmc.cycleMean();
1110          Cost cycle_cost = mmc.cycleLength();
1111          int cycle_size = mmc.cycleArcNum();
1112         
1113          // Compute feasible potentials for the current epsilon
1114          for (int i = 0; i != int(_cost_vec.size()); ++i) {
1115            _cost_vec[i] = cycle_size * _cost_vec[i] - cycle_cost;
1116          }
1117          BF bf(_sgr, _cost_map);
1118          bf.distMap(_pi_map);
1119          bf.init(0);
1120          bf.start();
1121          for (int u = 0; u != _res_node_num; ++u) {
1122            pi[u] = static_cast<double>(_pi[u]) / cycle_size;
1123          }
1124       
1125          iter = limit;
1126        }
1127      }
1128    }
1129
1130  }; //class CycleCanceling
1131
1132  ///@}
1133
1134} //namespace lemon
1135
1136#endif //LEMON_CYCLE_CANCELING_H
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