COIN-OR::LEMON - Graph Library

source: lemon/lemon/cycle_canceling.h @ 1025:140c953ad5d1

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1/* -*- mode: C++; indent-tabs-mode: nil; -*-
2 *
3 * This file is a part of LEMON, a generic C++ optimization library.
4 *
5 * Copyright (C) 2003-2010
6 * Egervary Jeno Kombinatorikus Optimalizalasi Kutatocsoport
7 * (Egervary Research Group on Combinatorial Optimization, EGRES).
8 *
9 * Permission to use, modify and distribute this software is granted
10 * provided that this copyright notice appears in all copies. For
11 * precise terms see the accompanying LICENSE file.
12 *
13 * This software is provided "AS IS" with no warranty of any kind,
14 * express or implied, and with no claim as to its suitability for any
15 * purpose.
16 *
17 */
18
19#ifndef LEMON_CYCLE_CANCELING_H
20#define LEMON_CYCLE_CANCELING_H
21
22/// \ingroup min_cost_flow_algs
23/// \file
24/// \brief Cycle-canceling algorithms for finding a minimum cost flow.
25
26#include <vector>
27#include <limits>
28
29#include <lemon/core.h>
30#include <lemon/maps.h>
31#include <lemon/path.h>
32#include <lemon/math.h>
33#include <lemon/static_graph.h>
34#include <lemon/adaptors.h>
35#include <lemon/circulation.h>
36#include <lemon/bellman_ford.h>
37#include <lemon/howard_mmc.h>
38
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 \c V and \c C must be signed number types.
69  /// \warning All input data (capacities, supply values, and costs) must
70  /// be integer.
71  /// \warning This algorithm does not support negative costs for such
72  /// arcs that have infinite upper bound.
73  ///
74  /// \note For more information about the three available methods,
75  /// see \ref Method.
76#ifdef DOXYGEN
77  template <typename GR, typename V, typename C>
78#else
79  template <typename GR, typename V = int, typename C = V>
80#endif
81  class CycleCanceling
82  {
83  public:
84
85    /// The type of the digraph
86    typedef GR Digraph;
87    /// The type of the flow amounts, capacity bounds and supply values
88    typedef V Value;
89    /// The type of the arc costs
90    typedef C Cost;
91
92  public:
93
94    /// \brief Problem type constants for the \c run() function.
95    ///
96    /// Enum type containing the problem type constants that can be
97    /// returned by the \ref run() function of the algorithm.
98    enum ProblemType {
99      /// The problem has no feasible solution (flow).
100      INFEASIBLE,
101      /// The problem has optimal solution (i.e. it is feasible and
102      /// bounded), and the algorithm has found optimal flow and node
103      /// potentials (primal and dual solutions).
104      OPTIMAL,
105      /// The digraph contains an arc of negative cost and infinite
106      /// upper bound. It means that the objective function is unbounded
107      /// on that arc, however, note that it could actually be bounded
108      /// over the feasible flows, but this algroithm cannot handle
109      /// these cases.
110      UNBOUNDED
111    };
112
113    /// \brief Constants for selecting the used method.
114    ///
115    /// Enum type containing constants for selecting the used method
116    /// for the \ref run() function.
117    ///
118    /// \ref CycleCanceling provides three different cycle-canceling
119    /// methods. By default, \ref CANCEL_AND_TIGHTEN "Cancel and Tighten"
120    /// is used, which proved to be the most efficient and the most robust
121    /// on various test inputs.
122    /// However, the other methods can be selected using the \ref run()
123    /// function with the proper parameter.
124    enum Method {
125      /// A simple cycle-canceling method, which uses the
126      /// \ref BellmanFord "Bellman-Ford" algorithm with limited iteration
127      /// number for detecting negative cycles in the residual network.
128      SIMPLE_CYCLE_CANCELING,
129      /// The "Minimum Mean Cycle-Canceling" algorithm, which is a
130      /// well-known strongly polynomial method
131      /// \ref goldberg89cyclecanceling. It improves along a
132      /// \ref min_mean_cycle "minimum mean cycle" in each iteration.
133      /// Its running time complexity is O(n<sup>2</sup>m<sup>3</sup>log(n)).
134      MINIMUM_MEAN_CYCLE_CANCELING,
135      /// The "Cancel And Tighten" algorithm, which can be viewed as an
136      /// improved version of the previous method
137      /// \ref goldberg89cyclecanceling.
138      /// It is faster both in theory and in practice, its running time
139      /// complexity is O(n<sup>2</sup>m<sup>2</sup>log(n)).
140      CANCEL_AND_TIGHTEN
141    };
142
143  private:
144
145    TEMPLATE_DIGRAPH_TYPEDEFS(GR);
146
147    typedef std::vector<int> IntVector;
148    typedef std::vector<double> DoubleVector;
149    typedef std::vector<Value> ValueVector;
150    typedef std::vector<Cost> CostVector;
151    typedef std::vector<char> BoolVector;
152    // Note: vector<char> is used instead of vector<bool> for efficiency reasons
153
154  private:
155
156    template <typename KT, typename VT>
157    class StaticVectorMap {
158    public:
159      typedef KT Key;
160      typedef VT Value;
161
162      StaticVectorMap(std::vector<Value>& v) : _v(v) {}
163
164      const Value& operator[](const Key& key) const {
165        return _v[StaticDigraph::id(key)];
166      }
167
168      Value& operator[](const Key& key) {
169        return _v[StaticDigraph::id(key)];
170      }
171
172      void set(const Key& key, const Value& val) {
173        _v[StaticDigraph::id(key)] = val;
174      }
175
176    private:
177      std::vector<Value>& _v;
178    };
179
180    typedef StaticVectorMap<StaticDigraph::Node, Cost> CostNodeMap;
181    typedef StaticVectorMap<StaticDigraph::Arc, Cost> CostArcMap;
182
183  private:
184
185
186    // Data related to the underlying digraph
187    const GR &_graph;
188    int _node_num;
189    int _arc_num;
190    int _res_node_num;
191    int _res_arc_num;
192    int _root;
193
194    // Parameters of the problem
195    bool _have_lower;
196    Value _sum_supply;
197
198    // Data structures for storing the digraph
199    IntNodeMap _node_id;
200    IntArcMap _arc_idf;
201    IntArcMap _arc_idb;
202    IntVector _first_out;
203    BoolVector _forward;
204    IntVector _source;
205    IntVector _target;
206    IntVector _reverse;
207
208    // Node and arc data
209    ValueVector _lower;
210    ValueVector _upper;
211    CostVector _cost;
212    ValueVector _supply;
213
214    ValueVector _res_cap;
215    CostVector _pi;
216
217    // Data for a StaticDigraph structure
218    typedef std::pair<int, int> IntPair;
219    StaticDigraph _sgr;
220    std::vector<IntPair> _arc_vec;
221    std::vector<Cost> _cost_vec;
222    IntVector _id_vec;
223    CostArcMap _cost_map;
224    CostNodeMap _pi_map;
225
226  public:
227
228    /// \brief Constant for infinite upper bounds (capacities).
229    ///
230    /// Constant for infinite upper bounds (capacities).
231    /// It is \c std::numeric_limits<Value>::infinity() if available,
232    /// \c std::numeric_limits<Value>::max() otherwise.
233    const Value INF;
234
235  public:
236
237    /// \brief Constructor.
238    ///
239    /// The constructor of the class.
240    ///
241    /// \param graph The digraph the algorithm runs on.
242    CycleCanceling(const GR& graph) :
243      _graph(graph), _node_id(graph), _arc_idf(graph), _arc_idb(graph),
244      _cost_map(_cost_vec), _pi_map(_pi),
245      INF(std::numeric_limits<Value>::has_infinity ?
246          std::numeric_limits<Value>::infinity() :
247          std::numeric_limits<Value>::max())
248    {
249      // Check the number types
250      LEMON_ASSERT(std::numeric_limits<Value>::is_signed,
251        "The flow type of CycleCanceling must be signed");
252      LEMON_ASSERT(std::numeric_limits<Cost>::is_signed,
253        "The cost type of CycleCanceling must be signed");
254
255      // Reset data structures
256      reset();
257    }
258
259    /// \name Parameters
260    /// The parameters of the algorithm can be specified using these
261    /// functions.
262
263    /// @{
264
265    /// \brief Set the lower bounds on the arcs.
266    ///
267    /// This function sets the lower bounds on the arcs.
268    /// If it is not used before calling \ref run(), the lower bounds
269    /// will be set to zero on all arcs.
270    ///
271    /// \param map An arc map storing the lower bounds.
272    /// Its \c Value type must be convertible to the \c Value type
273    /// of the algorithm.
274    ///
275    /// \return <tt>(*this)</tt>
276    template <typename LowerMap>
277    CycleCanceling& lowerMap(const LowerMap& map) {
278      _have_lower = true;
279      for (ArcIt a(_graph); a != INVALID; ++a) {
280        _lower[_arc_idf[a]] = map[a];
281        _lower[_arc_idb[a]] = map[a];
282      }
283      return *this;
284    }
285
286    /// \brief Set the upper bounds (capacities) on the arcs.
287    ///
288    /// This function sets the upper bounds (capacities) on the arcs.
289    /// If it is not used before calling \ref run(), the upper bounds
290    /// will be set to \ref INF on all arcs (i.e. the flow value will be
291    /// unbounded from above).
292    ///
293    /// \param map An arc map storing the upper bounds.
294    /// Its \c Value type must be convertible to the \c Value type
295    /// of the algorithm.
296    ///
297    /// \return <tt>(*this)</tt>
298    template<typename UpperMap>
299    CycleCanceling& upperMap(const UpperMap& map) {
300      for (ArcIt a(_graph); a != INVALID; ++a) {
301        _upper[_arc_idf[a]] = map[a];
302      }
303      return *this;
304    }
305
306    /// \brief Set the costs of the arcs.
307    ///
308    /// This function sets the costs of the arcs.
309    /// If it is not used before calling \ref run(), the costs
310    /// will be set to \c 1 on all arcs.
311    ///
312    /// \param map An arc map storing the costs.
313    /// Its \c Value type must be convertible to the \c Cost type
314    /// of the algorithm.
315    ///
316    /// \return <tt>(*this)</tt>
317    template<typename CostMap>
318    CycleCanceling& costMap(const CostMap& map) {
319      for (ArcIt a(_graph); a != INVALID; ++a) {
320        _cost[_arc_idf[a]] =  map[a];
321        _cost[_arc_idb[a]] = -map[a];
322      }
323      return *this;
324    }
325
326    /// \brief Set the supply values of the nodes.
327    ///
328    /// This function sets the supply values of the nodes.
329    /// If neither this function nor \ref stSupply() is used before
330    /// calling \ref run(), the supply of each node will be set to zero.
331    ///
332    /// \param map A node map storing the supply values.
333    /// Its \c Value type must be convertible to the \c Value type
334    /// of the algorithm.
335    ///
336    /// \return <tt>(*this)</tt>
337    template<typename SupplyMap>
338    CycleCanceling& supplyMap(const SupplyMap& map) {
339      for (NodeIt n(_graph); n != INVALID; ++n) {
340        _supply[_node_id[n]] = map[n];
341      }
342      return *this;
343    }
344
345    /// \brief Set single source and target nodes and a supply value.
346    ///
347    /// This function sets a single source node and a single target node
348    /// and the required flow value.
349    /// If neither this function nor \ref supplyMap() is used before
350    /// calling \ref run(), the supply of each node will be set to zero.
351    ///
352    /// Using this function has the same effect as using \ref supplyMap()
353    /// with such a map in which \c k is assigned to \c s, \c -k is
354    /// assigned to \c t and all other nodes have zero supply value.
355    ///
356    /// \param s The source node.
357    /// \param t The target node.
358    /// \param k The required amount of flow from node \c s to node \c t
359    /// (i.e. the supply of \c s and the demand of \c t).
360    ///
361    /// \return <tt>(*this)</tt>
362    CycleCanceling& stSupply(const Node& s, const Node& t, Value k) {
363      for (int i = 0; i != _res_node_num; ++i) {
364        _supply[i] = 0;
365      }
366      _supply[_node_id[s]] =  k;
367      _supply[_node_id[t]] = -k;
368      return *this;
369    }
370
371    /// @}
372
373    /// \name Execution control
374    /// The algorithm can be executed using \ref run().
375
376    /// @{
377
378    /// \brief Run the algorithm.
379    ///
380    /// This function runs the algorithm.
381    /// The paramters can be specified using functions \ref lowerMap(),
382    /// \ref upperMap(), \ref costMap(), \ref supplyMap(), \ref stSupply().
383    /// For example,
384    /// \code
385    ///   CycleCanceling<ListDigraph> cc(graph);
386    ///   cc.lowerMap(lower).upperMap(upper).costMap(cost)
387    ///     .supplyMap(sup).run();
388    /// \endcode
389    ///
390    /// This function can be called more than once. All the given parameters
391    /// are kept for the next call, unless \ref resetParams() or \ref reset()
392    /// is used, thus only the modified parameters have to be set again.
393    /// If the underlying digraph was also modified after the construction
394    /// of the class (or the last \ref reset() call), then the \ref reset()
395    /// function must be called.
396    ///
397    /// \param method The cycle-canceling method that will be used.
398    /// For more information, see \ref Method.
399    ///
400    /// \return \c INFEASIBLE if no feasible flow exists,
401    /// \n \c OPTIMAL if the problem has optimal solution
402    /// (i.e. it is feasible and bounded), and the algorithm has found
403    /// optimal flow and node potentials (primal and dual solutions),
404    /// \n \c UNBOUNDED if the digraph contains an arc of negative cost
405    /// and infinite upper bound. It means that the objective function
406    /// is unbounded on that arc, however, note that it could actually be
407    /// bounded over the feasible flows, but this algroithm cannot handle
408    /// these cases.
409    ///
410    /// \see ProblemType, Method
411    /// \see resetParams(), reset()
412    ProblemType run(Method method = CANCEL_AND_TIGHTEN) {
413      ProblemType pt = init();
414      if (pt != OPTIMAL) return pt;
415      start(method);
416      return OPTIMAL;
417    }
418
419    /// \brief Reset all the parameters that have been given before.
420    ///
421    /// This function resets all the paramaters that have been given
422    /// before using functions \ref lowerMap(), \ref upperMap(),
423    /// \ref costMap(), \ref supplyMap(), \ref stSupply().
424    ///
425    /// It is useful for multiple \ref run() calls. Basically, all the given
426    /// parameters are kept for the next \ref run() call, unless
427    /// \ref resetParams() or \ref reset() is used.
428    /// If the underlying digraph was also modified after the construction
429    /// of the class or the last \ref reset() call, then the \ref reset()
430    /// function must be used, otherwise \ref resetParams() is sufficient.
431    ///
432    /// For example,
433    /// \code
434    ///   CycleCanceling<ListDigraph> cs(graph);
435    ///
436    ///   // First run
437    ///   cc.lowerMap(lower).upperMap(upper).costMap(cost)
438    ///     .supplyMap(sup).run();
439    ///
440    ///   // Run again with modified cost map (resetParams() is not called,
441    ///   // so only the cost map have to be set again)
442    ///   cost[e] += 100;
443    ///   cc.costMap(cost).run();
444    ///
445    ///   // Run again from scratch using resetParams()
446    ///   // (the lower bounds will be set to zero on all arcs)
447    ///   cc.resetParams();
448    ///   cc.upperMap(capacity).costMap(cost)
449    ///     .supplyMap(sup).run();
450    /// \endcode
451    ///
452    /// \return <tt>(*this)</tt>
453    ///
454    /// \see reset(), run()
455    CycleCanceling& resetParams() {
456      for (int i = 0; i != _res_node_num; ++i) {
457        _supply[i] = 0;
458      }
459      int limit = _first_out[_root];
460      for (int j = 0; j != limit; ++j) {
461        _lower[j] = 0;
462        _upper[j] = INF;
463        _cost[j] = _forward[j] ? 1 : -1;
464      }
465      for (int j = limit; j != _res_arc_num; ++j) {
466        _lower[j] = 0;
467        _upper[j] = INF;
468        _cost[j] = 0;
469        _cost[_reverse[j]] = 0;
470      }
471      _have_lower = false;
472      return *this;
473    }
474
475    /// \brief Reset the internal data structures and all the parameters
476    /// that have been given before.
477    ///
478    /// This function resets the internal data structures and all the
479    /// paramaters that have been given before using functions \ref lowerMap(),
480    /// \ref upperMap(), \ref costMap(), \ref supplyMap(), \ref stSupply().
481    ///
482    /// It is useful for multiple \ref run() calls. Basically, all the given
483    /// parameters are kept for the next \ref run() call, unless
484    /// \ref resetParams() or \ref reset() is used.
485    /// If the underlying digraph was also modified after the construction
486    /// of the class or the last \ref reset() call, then the \ref reset()
487    /// function must be used, otherwise \ref resetParams() is sufficient.
488    ///
489    /// See \ref resetParams() for examples.
490    ///
491    /// \return <tt>(*this)</tt>
492    ///
493    /// \see resetParams(), run()
494    CycleCanceling& reset() {
495      // Resize vectors
496      _node_num = countNodes(_graph);
497      _arc_num = countArcs(_graph);
498      _res_node_num = _node_num + 1;
499      _res_arc_num = 2 * (_arc_num + _node_num);
500      _root = _node_num;
501
502      _first_out.resize(_res_node_num + 1);
503      _forward.resize(_res_arc_num);
504      _source.resize(_res_arc_num);
505      _target.resize(_res_arc_num);
506      _reverse.resize(_res_arc_num);
507
508      _lower.resize(_res_arc_num);
509      _upper.resize(_res_arc_num);
510      _cost.resize(_res_arc_num);
511      _supply.resize(_res_node_num);
512
513      _res_cap.resize(_res_arc_num);
514      _pi.resize(_res_node_num);
515
516      _arc_vec.reserve(_res_arc_num);
517      _cost_vec.reserve(_res_arc_num);
518      _id_vec.reserve(_res_arc_num);
519
520      // Copy the graph
521      int i = 0, j = 0, k = 2 * _arc_num + _node_num;
522      for (NodeIt n(_graph); n != INVALID; ++n, ++i) {
523        _node_id[n] = i;
524      }
525      i = 0;
526      for (NodeIt n(_graph); n != INVALID; ++n, ++i) {
527        _first_out[i] = j;
528        for (OutArcIt a(_graph, n); a != INVALID; ++a, ++j) {
529          _arc_idf[a] = j;
530          _forward[j] = true;
531          _source[j] = i;
532          _target[j] = _node_id[_graph.runningNode(a)];
533        }
534        for (InArcIt a(_graph, n); a != INVALID; ++a, ++j) {
535          _arc_idb[a] = j;
536          _forward[j] = false;
537          _source[j] = i;
538          _target[j] = _node_id[_graph.runningNode(a)];
539        }
540        _forward[j] = false;
541        _source[j] = i;
542        _target[j] = _root;
543        _reverse[j] = k;
544        _forward[k] = true;
545        _source[k] = _root;
546        _target[k] = i;
547        _reverse[k] = j;
548        ++j; ++k;
549      }
550      _first_out[i] = j;
551      _first_out[_res_node_num] = k;
552      for (ArcIt a(_graph); a != INVALID; ++a) {
553        int fi = _arc_idf[a];
554        int bi = _arc_idb[a];
555        _reverse[fi] = bi;
556        _reverse[bi] = fi;
557      }
558
559      // Reset parameters
560      resetParams();
561      return *this;
562    }
563
564    /// @}
565
566    /// \name Query Functions
567    /// The results of the algorithm can be obtained using these
568    /// functions.\n
569    /// The \ref run() function must be called before using them.
570
571    /// @{
572
573    /// \brief Return the total cost of the found flow.
574    ///
575    /// This function returns the total cost of the found flow.
576    /// Its complexity is O(e).
577    ///
578    /// \note The return type of the function can be specified as a
579    /// template parameter. For example,
580    /// \code
581    ///   cc.totalCost<double>();
582    /// \endcode
583    /// It is useful if the total cost cannot be stored in the \c Cost
584    /// type of the algorithm, which is the default return type of the
585    /// function.
586    ///
587    /// \pre \ref run() must be called before using this function.
588    template <typename Number>
589    Number totalCost() const {
590      Number c = 0;
591      for (ArcIt a(_graph); a != INVALID; ++a) {
592        int i = _arc_idb[a];
593        c += static_cast<Number>(_res_cap[i]) *
594             (-static_cast<Number>(_cost[i]));
595      }
596      return c;
597    }
598
599#ifndef DOXYGEN
600    Cost totalCost() const {
601      return totalCost<Cost>();
602    }
603#endif
604
605    /// \brief Return the flow on the given arc.
606    ///
607    /// This function returns the flow on the given arc.
608    ///
609    /// \pre \ref run() must be called before using this function.
610    Value flow(const Arc& a) const {
611      return _res_cap[_arc_idb[a]];
612    }
613
614    /// \brief Return the flow map (the primal solution).
615    ///
616    /// This function copies the flow value on each arc into the given
617    /// map. The \c Value type of the algorithm must be convertible to
618    /// the \c Value type of the map.
619    ///
620    /// \pre \ref run() must be called before using this function.
621    template <typename FlowMap>
622    void flowMap(FlowMap &map) const {
623      for (ArcIt a(_graph); a != INVALID; ++a) {
624        map.set(a, _res_cap[_arc_idb[a]]);
625      }
626    }
627
628    /// \brief Return the potential (dual value) of the given node.
629    ///
630    /// This function returns the potential (dual value) of the
631    /// given node.
632    ///
633    /// \pre \ref run() must be called before using this function.
634    Cost potential(const Node& n) const {
635      return static_cast<Cost>(_pi[_node_id[n]]);
636    }
637
638    /// \brief Return the potential map (the dual solution).
639    ///
640    /// This function copies the potential (dual value) of each node
641    /// into the given map.
642    /// The \c Cost type of the algorithm must be convertible to the
643    /// \c Value type of the map.
644    ///
645    /// \pre \ref run() must be called before using this function.
646    template <typename PotentialMap>
647    void potentialMap(PotentialMap &map) const {
648      for (NodeIt n(_graph); n != INVALID; ++n) {
649        map.set(n, static_cast<Cost>(_pi[_node_id[n]]));
650      }
651    }
652
653    /// @}
654
655  private:
656
657    // Initialize the algorithm
658    ProblemType init() {
659      if (_res_node_num <= 1) return INFEASIBLE;
660
661      // Check the sum of supply values
662      _sum_supply = 0;
663      for (int i = 0; i != _root; ++i) {
664        _sum_supply += _supply[i];
665      }
666      if (_sum_supply > 0) return INFEASIBLE;
667
668
669      // Initialize vectors
670      for (int i = 0; i != _res_node_num; ++i) {
671        _pi[i] = 0;
672      }
673      ValueVector excess(_supply);
674
675      // Remove infinite upper bounds and check negative arcs
676      const Value MAX = std::numeric_limits<Value>::max();
677      int last_out;
678      if (_have_lower) {
679        for (int i = 0; i != _root; ++i) {
680          last_out = _first_out[i+1];
681          for (int j = _first_out[i]; j != last_out; ++j) {
682            if (_forward[j]) {
683              Value c = _cost[j] < 0 ? _upper[j] : _lower[j];
684              if (c >= MAX) return UNBOUNDED;
685              excess[i] -= c;
686              excess[_target[j]] += c;
687            }
688          }
689        }
690      } else {
691        for (int i = 0; i != _root; ++i) {
692          last_out = _first_out[i+1];
693          for (int j = _first_out[i]; j != last_out; ++j) {
694            if (_forward[j] && _cost[j] < 0) {
695              Value c = _upper[j];
696              if (c >= MAX) return UNBOUNDED;
697              excess[i] -= c;
698              excess[_target[j]] += c;
699            }
700          }
701        }
702      }
703      Value ex, max_cap = 0;
704      for (int i = 0; i != _res_node_num; ++i) {
705        ex = excess[i];
706        if (ex < 0) max_cap -= ex;
707      }
708      for (int j = 0; j != _res_arc_num; ++j) {
709        if (_upper[j] >= MAX) _upper[j] = max_cap;
710      }
711
712      // Initialize maps for Circulation and remove non-zero lower bounds
713      ConstMap<Arc, Value> low(0);
714      typedef typename Digraph::template ArcMap<Value> ValueArcMap;
715      typedef typename Digraph::template NodeMap<Value> ValueNodeMap;
716      ValueArcMap cap(_graph), flow(_graph);
717      ValueNodeMap sup(_graph);
718      for (NodeIt n(_graph); n != INVALID; ++n) {
719        sup[n] = _supply[_node_id[n]];
720      }
721      if (_have_lower) {
722        for (ArcIt a(_graph); a != INVALID; ++a) {
723          int j = _arc_idf[a];
724          Value c = _lower[j];
725          cap[a] = _upper[j] - c;
726          sup[_graph.source(a)] -= c;
727          sup[_graph.target(a)] += c;
728        }
729      } else {
730        for (ArcIt a(_graph); a != INVALID; ++a) {
731          cap[a] = _upper[_arc_idf[a]];
732        }
733      }
734
735      // Find a feasible flow using Circulation
736      Circulation<Digraph, ConstMap<Arc, Value>, ValueArcMap, ValueNodeMap>
737        circ(_graph, low, cap, sup);
738      if (!circ.flowMap(flow).run()) return INFEASIBLE;
739
740      // Set residual capacities and handle GEQ supply type
741      if (_sum_supply < 0) {
742        for (ArcIt a(_graph); a != INVALID; ++a) {
743          Value fa = flow[a];
744          _res_cap[_arc_idf[a]] = cap[a] - fa;
745          _res_cap[_arc_idb[a]] = fa;
746          sup[_graph.source(a)] -= fa;
747          sup[_graph.target(a)] += fa;
748        }
749        for (NodeIt n(_graph); n != INVALID; ++n) {
750          excess[_node_id[n]] = sup[n];
751        }
752        for (int a = _first_out[_root]; a != _res_arc_num; ++a) {
753          int u = _target[a];
754          int ra = _reverse[a];
755          _res_cap[a] = -_sum_supply + 1;
756          _res_cap[ra] = -excess[u];
757          _cost[a] = 0;
758          _cost[ra] = 0;
759        }
760      } else {
761        for (ArcIt a(_graph); a != INVALID; ++a) {
762          Value fa = flow[a];
763          _res_cap[_arc_idf[a]] = cap[a] - fa;
764          _res_cap[_arc_idb[a]] = fa;
765        }
766        for (int a = _first_out[_root]; a != _res_arc_num; ++a) {
767          int ra = _reverse[a];
768          _res_cap[a] = 1;
769          _res_cap[ra] = 0;
770          _cost[a] = 0;
771          _cost[ra] = 0;
772        }
773      }
774
775      return OPTIMAL;
776    }
777
778    // Build a StaticDigraph structure containing the current
779    // residual network
780    void buildResidualNetwork() {
781      _arc_vec.clear();
782      _cost_vec.clear();
783      _id_vec.clear();
784      for (int j = 0; j != _res_arc_num; ++j) {
785        if (_res_cap[j] > 0) {
786          _arc_vec.push_back(IntPair(_source[j], _target[j]));
787          _cost_vec.push_back(_cost[j]);
788          _id_vec.push_back(j);
789        }
790      }
791      _sgr.build(_res_node_num, _arc_vec.begin(), _arc_vec.end());
792    }
793
794    // Execute the algorithm and transform the results
795    void start(Method method) {
796      // Execute the algorithm
797      switch (method) {
798        case SIMPLE_CYCLE_CANCELING:
799          startSimpleCycleCanceling();
800          break;
801        case MINIMUM_MEAN_CYCLE_CANCELING:
802          startMinMeanCycleCanceling();
803          break;
804        case CANCEL_AND_TIGHTEN:
805          startCancelAndTighten();
806          break;
807      }
808
809      // Compute node potentials
810      if (method != SIMPLE_CYCLE_CANCELING) {
811        buildResidualNetwork();
812        typename BellmanFord<StaticDigraph, CostArcMap>
813          ::template SetDistMap<CostNodeMap>::Create bf(_sgr, _cost_map);
814        bf.distMap(_pi_map);
815        bf.init(0);
816        bf.start();
817      }
818
819      // Handle non-zero lower bounds
820      if (_have_lower) {
821        int limit = _first_out[_root];
822        for (int j = 0; j != limit; ++j) {
823          if (!_forward[j]) _res_cap[j] += _lower[j];
824        }
825      }
826    }
827
828    // Execute the "Simple Cycle Canceling" method
829    void startSimpleCycleCanceling() {
830      // Constants for computing the iteration limits
831      const int BF_FIRST_LIMIT  = 2;
832      const double BF_LIMIT_FACTOR = 1.5;
833
834      typedef StaticVectorMap<StaticDigraph::Arc, Value> FilterMap;
835      typedef FilterArcs<StaticDigraph, FilterMap> ResDigraph;
836      typedef StaticVectorMap<StaticDigraph::Node, StaticDigraph::Arc> PredMap;
837      typedef typename BellmanFord<ResDigraph, CostArcMap>
838        ::template SetDistMap<CostNodeMap>
839        ::template SetPredMap<PredMap>::Create BF;
840
841      // Build the residual network
842      _arc_vec.clear();
843      _cost_vec.clear();
844      for (int j = 0; j != _res_arc_num; ++j) {
845        _arc_vec.push_back(IntPair(_source[j], _target[j]));
846        _cost_vec.push_back(_cost[j]);
847      }
848      _sgr.build(_res_node_num, _arc_vec.begin(), _arc_vec.end());
849
850      FilterMap filter_map(_res_cap);
851      ResDigraph rgr(_sgr, filter_map);
852      std::vector<int> cycle;
853      std::vector<StaticDigraph::Arc> pred(_res_arc_num);
854      PredMap pred_map(pred);
855      BF bf(rgr, _cost_map);
856      bf.distMap(_pi_map).predMap(pred_map);
857
858      int length_bound = BF_FIRST_LIMIT;
859      bool optimal = false;
860      while (!optimal) {
861        bf.init(0);
862        int iter_num = 0;
863        bool cycle_found = false;
864        while (!cycle_found) {
865          // Perform some iterations of the Bellman-Ford algorithm
866          int curr_iter_num = iter_num + length_bound <= _node_num ?
867            length_bound : _node_num - iter_num;
868          iter_num += curr_iter_num;
869          int real_iter_num = curr_iter_num;
870          for (int i = 0; i < curr_iter_num; ++i) {
871            if (bf.processNextWeakRound()) {
872              real_iter_num = i;
873              break;
874            }
875          }
876          if (real_iter_num < curr_iter_num) {
877            // Optimal flow is found
878            optimal = true;
879            break;
880          } else {
881            // Search for node disjoint negative cycles
882            std::vector<int> state(_res_node_num, 0);
883            int id = 0;
884            for (int u = 0; u != _res_node_num; ++u) {
885              if (state[u] != 0) continue;
886              ++id;
887              int v = u;
888              for (; v != -1 && state[v] == 0; v = pred[v] == INVALID ?
889                   -1 : rgr.id(rgr.source(pred[v]))) {
890                state[v] = id;
891              }
892              if (v != -1 && state[v] == id) {
893                // A negative cycle is found
894                cycle_found = true;
895                cycle.clear();
896                StaticDigraph::Arc a = pred[v];
897                Value d, delta = _res_cap[rgr.id(a)];
898                cycle.push_back(rgr.id(a));
899                while (rgr.id(rgr.source(a)) != v) {
900                  a = pred_map[rgr.source(a)];
901                  d = _res_cap[rgr.id(a)];
902                  if (d < delta) delta = d;
903                  cycle.push_back(rgr.id(a));
904                }
905
906                // Augment along the cycle
907                for (int i = 0; i < int(cycle.size()); ++i) {
908                  int j = cycle[i];
909                  _res_cap[j] -= delta;
910                  _res_cap[_reverse[j]] += delta;
911                }
912              }
913            }
914          }
915
916          // Increase iteration limit if no cycle is found
917          if (!cycle_found) {
918            length_bound = static_cast<int>(length_bound * BF_LIMIT_FACTOR);
919          }
920        }
921      }
922    }
923
924    // Execute the "Minimum Mean Cycle Canceling" method
925    void startMinMeanCycleCanceling() {
926      typedef SimplePath<StaticDigraph> SPath;
927      typedef typename SPath::ArcIt SPathArcIt;
928      typedef typename HowardMmc<StaticDigraph, CostArcMap>
929        ::template SetPath<SPath>::Create MMC;
930
931      SPath cycle;
932      MMC mmc(_sgr, _cost_map);
933      mmc.cycle(cycle);
934      buildResidualNetwork();
935      while (mmc.findCycleMean() && mmc.cycleCost() < 0) {
936        // Find the cycle
937        mmc.findCycle();
938
939        // Compute delta value
940        Value delta = INF;
941        for (SPathArcIt a(cycle); a != INVALID; ++a) {
942          Value d = _res_cap[_id_vec[_sgr.id(a)]];
943          if (d < delta) delta = d;
944        }
945
946        // Augment along the cycle
947        for (SPathArcIt a(cycle); a != INVALID; ++a) {
948          int j = _id_vec[_sgr.id(a)];
949          _res_cap[j] -= delta;
950          _res_cap[_reverse[j]] += delta;
951        }
952
953        // Rebuild the residual network
954        buildResidualNetwork();
955      }
956    }
957
958    // Execute the "Cancel And Tighten" method
959    void startCancelAndTighten() {
960      // Constants for the min mean cycle computations
961      const double LIMIT_FACTOR = 1.0;
962      const int MIN_LIMIT = 5;
963
964      // Contruct auxiliary data vectors
965      DoubleVector pi(_res_node_num, 0.0);
966      IntVector level(_res_node_num);
967      BoolVector reached(_res_node_num);
968      BoolVector processed(_res_node_num);
969      IntVector pred_node(_res_node_num);
970      IntVector pred_arc(_res_node_num);
971      std::vector<int> stack(_res_node_num);
972      std::vector<int> proc_vector(_res_node_num);
973
974      // Initialize epsilon
975      double epsilon = 0;
976      for (int a = 0; a != _res_arc_num; ++a) {
977        if (_res_cap[a] > 0 && -_cost[a] > epsilon)
978          epsilon = -_cost[a];
979      }
980
981      // Start phases
982      Tolerance<double> tol;
983      tol.epsilon(1e-6);
984      int limit = int(LIMIT_FACTOR * std::sqrt(double(_res_node_num)));
985      if (limit < MIN_LIMIT) limit = MIN_LIMIT;
986      int iter = limit;
987      while (epsilon * _res_node_num >= 1) {
988        // Find and cancel cycles in the admissible network using DFS
989        for (int u = 0; u != _res_node_num; ++u) {
990          reached[u] = false;
991          processed[u] = false;
992        }
993        int stack_head = -1;
994        int proc_head = -1;
995        for (int start = 0; start != _res_node_num; ++start) {
996          if (reached[start]) continue;
997
998          // New start node
999          reached[start] = true;
1000          pred_arc[start] = -1;
1001          pred_node[start] = -1;
1002
1003          // Find the first admissible outgoing arc
1004          double p = pi[start];
1005          int a = _first_out[start];
1006          int last_out = _first_out[start+1];
1007          for (; a != last_out && (_res_cap[a] == 0 ||
1008               !tol.negative(_cost[a] + p - pi[_target[a]])); ++a) ;
1009          if (a == last_out) {
1010            processed[start] = true;
1011            proc_vector[++proc_head] = start;
1012            continue;
1013          }
1014          stack[++stack_head] = a;
1015
1016          while (stack_head >= 0) {
1017            int sa = stack[stack_head];
1018            int u = _source[sa];
1019            int v = _target[sa];
1020
1021            if (!reached[v]) {
1022              // A new node is reached
1023              reached[v] = true;
1024              pred_node[v] = u;
1025              pred_arc[v] = sa;
1026              p = pi[v];
1027              a = _first_out[v];
1028              last_out = _first_out[v+1];
1029              for (; a != last_out && (_res_cap[a] == 0 ||
1030                   !tol.negative(_cost[a] + p - pi[_target[a]])); ++a) ;
1031              stack[++stack_head] = a == last_out ? -1 : a;
1032            } else {
1033              if (!processed[v]) {
1034                // A cycle is found
1035                int n, w = u;
1036                Value d, delta = _res_cap[sa];
1037                for (n = u; n != v; n = pred_node[n]) {
1038                  d = _res_cap[pred_arc[n]];
1039                  if (d <= delta) {
1040                    delta = d;
1041                    w = pred_node[n];
1042                  }
1043                }
1044
1045                // Augment along the cycle
1046                _res_cap[sa] -= delta;
1047                _res_cap[_reverse[sa]] += delta;
1048                for (n = u; n != v; n = pred_node[n]) {
1049                  int pa = pred_arc[n];
1050                  _res_cap[pa] -= delta;
1051                  _res_cap[_reverse[pa]] += delta;
1052                }
1053                for (n = u; stack_head > 0 && n != w; n = pred_node[n]) {
1054                  --stack_head;
1055                  reached[n] = false;
1056                }
1057                u = w;
1058              }
1059              v = u;
1060
1061              // Find the next admissible outgoing arc
1062              p = pi[v];
1063              a = stack[stack_head] + 1;
1064              last_out = _first_out[v+1];
1065              for (; a != last_out && (_res_cap[a] == 0 ||
1066                   !tol.negative(_cost[a] + p - pi[_target[a]])); ++a) ;
1067              stack[stack_head] = a == last_out ? -1 : a;
1068            }
1069
1070            while (stack_head >= 0 && stack[stack_head] == -1) {
1071              processed[v] = true;
1072              proc_vector[++proc_head] = v;
1073              if (--stack_head >= 0) {
1074                // Find the next admissible outgoing arc
1075                v = _source[stack[stack_head]];
1076                p = pi[v];
1077                a = stack[stack_head] + 1;
1078                last_out = _first_out[v+1];
1079                for (; a != last_out && (_res_cap[a] == 0 ||
1080                     !tol.negative(_cost[a] + p - pi[_target[a]])); ++a) ;
1081                stack[stack_head] = a == last_out ? -1 : a;
1082              }
1083            }
1084          }
1085        }
1086
1087        // Tighten potentials and epsilon
1088        if (--iter > 0) {
1089          for (int u = 0; u != _res_node_num; ++u) {
1090            level[u] = 0;
1091          }
1092          for (int i = proc_head; i > 0; --i) {
1093            int u = proc_vector[i];
1094            double p = pi[u];
1095            int l = level[u] + 1;
1096            int last_out = _first_out[u+1];
1097            for (int a = _first_out[u]; a != last_out; ++a) {
1098              int v = _target[a];
1099              if (_res_cap[a] > 0 && tol.negative(_cost[a] + p - pi[v]) &&
1100                  l > level[v]) level[v] = l;
1101            }
1102          }
1103
1104          // Modify potentials
1105          double q = std::numeric_limits<double>::max();
1106          for (int u = 0; u != _res_node_num; ++u) {
1107            int lu = level[u];
1108            double p, pu = pi[u];
1109            int last_out = _first_out[u+1];
1110            for (int a = _first_out[u]; a != last_out; ++a) {
1111              if (_res_cap[a] == 0) continue;
1112              int v = _target[a];
1113              int ld = lu - level[v];
1114              if (ld > 0) {
1115                p = (_cost[a] + pu - pi[v] + epsilon) / (ld + 1);
1116                if (p < q) q = p;
1117              }
1118            }
1119          }
1120          for (int u = 0; u != _res_node_num; ++u) {
1121            pi[u] -= q * level[u];
1122          }
1123
1124          // Modify epsilon
1125          epsilon = 0;
1126          for (int u = 0; u != _res_node_num; ++u) {
1127            double curr, pu = pi[u];
1128            int last_out = _first_out[u+1];
1129            for (int a = _first_out[u]; a != last_out; ++a) {
1130              if (_res_cap[a] == 0) continue;
1131              curr = _cost[a] + pu - pi[_target[a]];
1132              if (-curr > epsilon) epsilon = -curr;
1133            }
1134          }
1135        } else {
1136          typedef HowardMmc<StaticDigraph, CostArcMap> MMC;
1137          typedef typename BellmanFord<StaticDigraph, CostArcMap>
1138            ::template SetDistMap<CostNodeMap>::Create BF;
1139
1140          // Set epsilon to the minimum cycle mean
1141          buildResidualNetwork();
1142          MMC mmc(_sgr, _cost_map);
1143          mmc.findCycleMean();
1144          epsilon = -mmc.cycleMean();
1145          Cost cycle_cost = mmc.cycleCost();
1146          int cycle_size = mmc.cycleSize();
1147
1148          // Compute feasible potentials for the current epsilon
1149          for (int i = 0; i != int(_cost_vec.size()); ++i) {
1150            _cost_vec[i] = cycle_size * _cost_vec[i] - cycle_cost;
1151          }
1152          BF bf(_sgr, _cost_map);
1153          bf.distMap(_pi_map);
1154          bf.init(0);
1155          bf.start();
1156          for (int u = 0; u != _res_node_num; ++u) {
1157            pi[u] = static_cast<double>(_pi[u]) / cycle_size;
1158          }
1159
1160          iter = limit;
1161        }
1162      }
1163    }
1164
1165  }; //class CycleCanceling
1166
1167  ///@}
1168
1169} //namespace lemon
1170
1171#endif //LEMON_CYCLE_CANCELING_H
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