COIN-OR::LEMON - Graph Library

source: lemon/lemon/cost_scaling.h @ 1071:2d583da4ba40

Last change on this file since 1071:2d583da4ba40 was 1049:a07b6b27fe69, checked in by Peter Kovacs <kpeter@…>, 13 years ago

Change the default scaling factor in CostScaling? (#417)

File size: 50.8 KB
Line 
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_COST_SCALING_H
20#define LEMON_COST_SCALING_H
21
22/// \ingroup min_cost_flow_algs
23/// \file
24/// \brief Cost scaling algorithm for finding a minimum cost flow.
25
26#include <vector>
27#include <deque>
28#include <limits>
29
30#include <lemon/core.h>
31#include <lemon/maps.h>
32#include <lemon/math.h>
33#include <lemon/static_graph.h>
34#include <lemon/circulation.h>
35#include <lemon/bellman_ford.h>
36
37namespace lemon {
38
39  /// \brief Default traits class of CostScaling algorithm.
40  ///
41  /// Default traits class of CostScaling algorithm.
42  /// \tparam GR Digraph type.
43  /// \tparam V The number type used for flow amounts, capacity bounds
44  /// and supply values. By default it is \c int.
45  /// \tparam C The number type used for costs and potentials.
46  /// By default it is the same as \c V.
47#ifdef DOXYGEN
48  template <typename GR, typename V = int, typename C = V>
49#else
50  template < typename GR, typename V = int, typename C = V,
51             bool integer = std::numeric_limits<C>::is_integer >
52#endif
53  struct CostScalingDefaultTraits
54  {
55    /// The type of the digraph
56    typedef GR Digraph;
57    /// The type of the flow amounts, capacity bounds and supply values
58    typedef V Value;
59    /// The type of the arc costs
60    typedef C Cost;
61
62    /// \brief The large cost type used for internal computations
63    ///
64    /// The large cost type used for internal computations.
65    /// It is \c long \c long if the \c Cost type is integer,
66    /// otherwise it is \c double.
67    /// \c Cost must be convertible to \c LargeCost.
68    typedef double LargeCost;
69  };
70
71  // Default traits class for integer cost types
72  template <typename GR, typename V, typename C>
73  struct CostScalingDefaultTraits<GR, V, C, true>
74  {
75    typedef GR Digraph;
76    typedef V Value;
77    typedef C Cost;
78#ifdef LEMON_HAVE_LONG_LONG
79    typedef long long LargeCost;
80#else
81    typedef long LargeCost;
82#endif
83  };
84
85
86  /// \addtogroup min_cost_flow_algs
87  /// @{
88
89  /// \brief Implementation of the Cost Scaling algorithm for
90  /// finding a \ref min_cost_flow "minimum cost flow".
91  ///
92  /// \ref CostScaling implements a cost scaling algorithm that performs
93  /// push/augment and relabel operations for finding a \ref min_cost_flow
94  /// "minimum cost flow" \ref amo93networkflows, \ref goldberg90approximation,
95  /// \ref goldberg97efficient, \ref bunnagel98efficient.
96  /// It is a highly efficient primal-dual solution method, which
97  /// can be viewed as the generalization of the \ref Preflow
98  /// "preflow push-relabel" algorithm for the maximum flow problem.
99  ///
100  /// In general, \ref NetworkSimplex and \ref CostScaling are the fastest
101  /// implementations available in LEMON for this problem.
102  ///
103  /// Most of the parameters of the problem (except for the digraph)
104  /// can be given using separate functions, and the algorithm can be
105  /// executed using the \ref run() function. If some parameters are not
106  /// specified, then default values will be used.
107  ///
108  /// \tparam GR The digraph type the algorithm runs on.
109  /// \tparam V The number type used for flow amounts, capacity bounds
110  /// and supply values in the algorithm. By default, it is \c int.
111  /// \tparam C The number type used for costs and potentials in the
112  /// algorithm. By default, it is the same as \c V.
113  /// \tparam TR The traits class that defines various types used by the
114  /// algorithm. By default, it is \ref CostScalingDefaultTraits
115  /// "CostScalingDefaultTraits<GR, V, C>".
116  /// In most cases, this parameter should not be set directly,
117  /// consider to use the named template parameters instead.
118  ///
119  /// \warning Both \c V and \c C must be signed number types.
120  /// \warning All input data (capacities, supply values, and costs) must
121  /// be integer.
122  /// \warning This algorithm does not support negative costs for
123  /// arcs having infinite upper bound.
124  ///
125  /// \note %CostScaling provides three different internal methods,
126  /// from which the most efficient one is used by default.
127  /// For more information, see \ref Method.
128#ifdef DOXYGEN
129  template <typename GR, typename V, typename C, typename TR>
130#else
131  template < typename GR, typename V = int, typename C = V,
132             typename TR = CostScalingDefaultTraits<GR, V, C> >
133#endif
134  class CostScaling
135  {
136  public:
137
138    /// The type of the digraph
139    typedef typename TR::Digraph Digraph;
140    /// The type of the flow amounts, capacity bounds and supply values
141    typedef typename TR::Value Value;
142    /// The type of the arc costs
143    typedef typename TR::Cost Cost;
144
145    /// \brief The large cost type
146    ///
147    /// The large cost type used for internal computations.
148    /// By default, it is \c long \c long if the \c Cost type is integer,
149    /// otherwise it is \c double.
150    typedef typename TR::LargeCost LargeCost;
151
152    /// The \ref CostScalingDefaultTraits "traits class" of the algorithm
153    typedef TR Traits;
154
155  public:
156
157    /// \brief Problem type constants for the \c run() function.
158    ///
159    /// Enum type containing the problem type constants that can be
160    /// returned by the \ref run() function of the algorithm.
161    enum ProblemType {
162      /// The problem has no feasible solution (flow).
163      INFEASIBLE,
164      /// The problem has optimal solution (i.e. it is feasible and
165      /// bounded), and the algorithm has found optimal flow and node
166      /// potentials (primal and dual solutions).
167      OPTIMAL,
168      /// The digraph contains an arc of negative cost and infinite
169      /// upper bound. It means that the objective function is unbounded
170      /// on that arc, however, note that it could actually be bounded
171      /// over the feasible flows, but this algroithm cannot handle
172      /// these cases.
173      UNBOUNDED
174    };
175
176    /// \brief Constants for selecting the internal method.
177    ///
178    /// Enum type containing constants for selecting the internal method
179    /// for the \ref run() function.
180    ///
181    /// \ref CostScaling provides three internal methods that differ mainly
182    /// in their base operations, which are used in conjunction with the
183    /// relabel operation.
184    /// By default, the so called \ref PARTIAL_AUGMENT
185    /// "Partial Augment-Relabel" method is used, which turned out to be
186    /// the most efficient and the most robust on various test inputs.
187    /// However, the other methods can be selected using the \ref run()
188    /// function with the proper parameter.
189    enum Method {
190      /// Local push operations are used, i.e. flow is moved only on one
191      /// admissible arc at once.
192      PUSH,
193      /// Augment operations are used, i.e. flow is moved on admissible
194      /// paths from a node with excess to a node with deficit.
195      AUGMENT,
196      /// Partial augment operations are used, i.e. flow is moved on
197      /// admissible paths started from a node with excess, but the
198      /// lengths of these paths are limited. This method can be viewed
199      /// as a combined version of the previous two operations.
200      PARTIAL_AUGMENT
201    };
202
203  private:
204
205    TEMPLATE_DIGRAPH_TYPEDEFS(GR);
206
207    typedef std::vector<int> IntVector;
208    typedef std::vector<Value> ValueVector;
209    typedef std::vector<Cost> CostVector;
210    typedef std::vector<LargeCost> LargeCostVector;
211    typedef std::vector<char> BoolVector;
212    // Note: vector<char> is used instead of vector<bool> for efficiency reasons
213
214  private:
215
216    template <typename KT, typename VT>
217    class StaticVectorMap {
218    public:
219      typedef KT Key;
220      typedef VT Value;
221
222      StaticVectorMap(std::vector<Value>& v) : _v(v) {}
223
224      const Value& operator[](const Key& key) const {
225        return _v[StaticDigraph::id(key)];
226      }
227
228      Value& operator[](const Key& key) {
229        return _v[StaticDigraph::id(key)];
230      }
231
232      void set(const Key& key, const Value& val) {
233        _v[StaticDigraph::id(key)] = val;
234      }
235
236    private:
237      std::vector<Value>& _v;
238    };
239
240    typedef StaticVectorMap<StaticDigraph::Arc, LargeCost> LargeCostArcMap;
241
242  private:
243
244    // Data related to the underlying digraph
245    const GR &_graph;
246    int _node_num;
247    int _arc_num;
248    int _res_node_num;
249    int _res_arc_num;
250    int _root;
251
252    // Parameters of the problem
253    bool _have_lower;
254    Value _sum_supply;
255    int _sup_node_num;
256
257    // Data structures for storing the digraph
258    IntNodeMap _node_id;
259    IntArcMap _arc_idf;
260    IntArcMap _arc_idb;
261    IntVector _first_out;
262    BoolVector _forward;
263    IntVector _source;
264    IntVector _target;
265    IntVector _reverse;
266
267    // Node and arc data
268    ValueVector _lower;
269    ValueVector _upper;
270    CostVector _scost;
271    ValueVector _supply;
272
273    ValueVector _res_cap;
274    LargeCostVector _cost;
275    LargeCostVector _pi;
276    ValueVector _excess;
277    IntVector _next_out;
278    std::deque<int> _active_nodes;
279
280    // Data for scaling
281    LargeCost _epsilon;
282    int _alpha;
283
284    IntVector _buckets;
285    IntVector _bucket_next;
286    IntVector _bucket_prev;
287    IntVector _rank;
288    int _max_rank;
289
290  public:
291
292    /// \brief Constant for infinite upper bounds (capacities).
293    ///
294    /// Constant for infinite upper bounds (capacities).
295    /// It is \c std::numeric_limits<Value>::infinity() if available,
296    /// \c std::numeric_limits<Value>::max() otherwise.
297    const Value INF;
298
299  public:
300
301    /// \name Named Template Parameters
302    /// @{
303
304    template <typename T>
305    struct SetLargeCostTraits : public Traits {
306      typedef T LargeCost;
307    };
308
309    /// \brief \ref named-templ-param "Named parameter" for setting
310    /// \c LargeCost type.
311    ///
312    /// \ref named-templ-param "Named parameter" for setting \c LargeCost
313    /// type, which is used for internal computations in the algorithm.
314    /// \c Cost must be convertible to \c LargeCost.
315    template <typename T>
316    struct SetLargeCost
317      : public CostScaling<GR, V, C, SetLargeCostTraits<T> > {
318      typedef  CostScaling<GR, V, C, SetLargeCostTraits<T> > Create;
319    };
320
321    /// @}
322
323  protected:
324
325    CostScaling() {}
326
327  public:
328
329    /// \brief Constructor.
330    ///
331    /// The constructor of the class.
332    ///
333    /// \param graph The digraph the algorithm runs on.
334    CostScaling(const GR& graph) :
335      _graph(graph), _node_id(graph), _arc_idf(graph), _arc_idb(graph),
336      INF(std::numeric_limits<Value>::has_infinity ?
337          std::numeric_limits<Value>::infinity() :
338          std::numeric_limits<Value>::max())
339    {
340      // Check the number types
341      LEMON_ASSERT(std::numeric_limits<Value>::is_signed,
342        "The flow type of CostScaling must be signed");
343      LEMON_ASSERT(std::numeric_limits<Cost>::is_signed,
344        "The cost type of CostScaling must be signed");
345
346      // Reset data structures
347      reset();
348    }
349
350    /// \name Parameters
351    /// The parameters of the algorithm can be specified using these
352    /// functions.
353
354    /// @{
355
356    /// \brief Set the lower bounds on the arcs.
357    ///
358    /// This function sets the lower bounds on the arcs.
359    /// If it is not used before calling \ref run(), the lower bounds
360    /// will be set to zero on all arcs.
361    ///
362    /// \param map An arc map storing the lower bounds.
363    /// Its \c Value type must be convertible to the \c Value type
364    /// of the algorithm.
365    ///
366    /// \return <tt>(*this)</tt>
367    template <typename LowerMap>
368    CostScaling& lowerMap(const LowerMap& map) {
369      _have_lower = true;
370      for (ArcIt a(_graph); a != INVALID; ++a) {
371        _lower[_arc_idf[a]] = map[a];
372        _lower[_arc_idb[a]] = map[a];
373      }
374      return *this;
375    }
376
377    /// \brief Set the upper bounds (capacities) on the arcs.
378    ///
379    /// This function sets the upper bounds (capacities) on the arcs.
380    /// If it is not used before calling \ref run(), the upper bounds
381    /// will be set to \ref INF on all arcs (i.e. the flow value will be
382    /// unbounded from above).
383    ///
384    /// \param map An arc map storing the upper bounds.
385    /// Its \c Value type must be convertible to the \c Value type
386    /// of the algorithm.
387    ///
388    /// \return <tt>(*this)</tt>
389    template<typename UpperMap>
390    CostScaling& upperMap(const UpperMap& map) {
391      for (ArcIt a(_graph); a != INVALID; ++a) {
392        _upper[_arc_idf[a]] = map[a];
393      }
394      return *this;
395    }
396
397    /// \brief Set the costs of the arcs.
398    ///
399    /// This function sets the costs of the arcs.
400    /// If it is not used before calling \ref run(), the costs
401    /// will be set to \c 1 on all arcs.
402    ///
403    /// \param map An arc map storing the costs.
404    /// Its \c Value type must be convertible to the \c Cost type
405    /// of the algorithm.
406    ///
407    /// \return <tt>(*this)</tt>
408    template<typename CostMap>
409    CostScaling& costMap(const CostMap& map) {
410      for (ArcIt a(_graph); a != INVALID; ++a) {
411        _scost[_arc_idf[a]] =  map[a];
412        _scost[_arc_idb[a]] = -map[a];
413      }
414      return *this;
415    }
416
417    /// \brief Set the supply values of the nodes.
418    ///
419    /// This function sets the supply values of the nodes.
420    /// If neither this function nor \ref stSupply() is used before
421    /// calling \ref run(), the supply of each node will be set to zero.
422    ///
423    /// \param map A node map storing the supply values.
424    /// Its \c Value type must be convertible to the \c Value type
425    /// of the algorithm.
426    ///
427    /// \return <tt>(*this)</tt>
428    template<typename SupplyMap>
429    CostScaling& supplyMap(const SupplyMap& map) {
430      for (NodeIt n(_graph); n != INVALID; ++n) {
431        _supply[_node_id[n]] = map[n];
432      }
433      return *this;
434    }
435
436    /// \brief Set single source and target nodes and a supply value.
437    ///
438    /// This function sets a single source node and a single target node
439    /// and the required flow value.
440    /// If neither this function nor \ref supplyMap() is used before
441    /// calling \ref run(), the supply of each node will be set to zero.
442    ///
443    /// Using this function has the same effect as using \ref supplyMap()
444    /// with a map in which \c k is assigned to \c s, \c -k is
445    /// assigned to \c t and all other nodes have zero supply value.
446    ///
447    /// \param s The source node.
448    /// \param t The target node.
449    /// \param k The required amount of flow from node \c s to node \c t
450    /// (i.e. the supply of \c s and the demand of \c t).
451    ///
452    /// \return <tt>(*this)</tt>
453    CostScaling& stSupply(const Node& s, const Node& t, Value k) {
454      for (int i = 0; i != _res_node_num; ++i) {
455        _supply[i] = 0;
456      }
457      _supply[_node_id[s]] =  k;
458      _supply[_node_id[t]] = -k;
459      return *this;
460    }
461
462    /// @}
463
464    /// \name Execution control
465    /// The algorithm can be executed using \ref run().
466
467    /// @{
468
469    /// \brief Run the algorithm.
470    ///
471    /// This function runs the algorithm.
472    /// The paramters can be specified using functions \ref lowerMap(),
473    /// \ref upperMap(), \ref costMap(), \ref supplyMap(), \ref stSupply().
474    /// For example,
475    /// \code
476    ///   CostScaling<ListDigraph> cs(graph);
477    ///   cs.lowerMap(lower).upperMap(upper).costMap(cost)
478    ///     .supplyMap(sup).run();
479    /// \endcode
480    ///
481    /// This function can be called more than once. All the given parameters
482    /// are kept for the next call, unless \ref resetParams() or \ref reset()
483    /// is used, thus only the modified parameters have to be set again.
484    /// If the underlying digraph was also modified after the construction
485    /// of the class (or the last \ref reset() call), then the \ref reset()
486    /// function must be called.
487    ///
488    /// \param method The internal method that will be used in the
489    /// algorithm. For more information, see \ref Method.
490    /// \param factor The cost scaling factor. It must be at least two.
491    ///
492    /// \return \c INFEASIBLE if no feasible flow exists,
493    /// \n \c OPTIMAL if the problem has optimal solution
494    /// (i.e. it is feasible and bounded), and the algorithm has found
495    /// optimal flow and node potentials (primal and dual solutions),
496    /// \n \c UNBOUNDED if the digraph contains an arc of negative cost
497    /// and infinite upper bound. It means that the objective function
498    /// is unbounded on that arc, however, note that it could actually be
499    /// bounded over the feasible flows, but this algroithm cannot handle
500    /// these cases.
501    ///
502    /// \see ProblemType, Method
503    /// \see resetParams(), reset()
504    ProblemType run(Method method = PARTIAL_AUGMENT, int factor = 16) {
505      LEMON_ASSERT(factor >= 2, "The scaling factor must be at least 2");
506      _alpha = factor;
507      ProblemType pt = init();
508      if (pt != OPTIMAL) return pt;
509      start(method);
510      return OPTIMAL;
511    }
512
513    /// \brief Reset all the parameters that have been given before.
514    ///
515    /// This function resets all the paramaters that have been given
516    /// before using functions \ref lowerMap(), \ref upperMap(),
517    /// \ref costMap(), \ref supplyMap(), \ref stSupply().
518    ///
519    /// It is useful for multiple \ref run() calls. Basically, all the given
520    /// parameters are kept for the next \ref run() call, unless
521    /// \ref resetParams() or \ref reset() is used.
522    /// If the underlying digraph was also modified after the construction
523    /// of the class or the last \ref reset() call, then the \ref reset()
524    /// function must be used, otherwise \ref resetParams() is sufficient.
525    ///
526    /// For example,
527    /// \code
528    ///   CostScaling<ListDigraph> cs(graph);
529    ///
530    ///   // First run
531    ///   cs.lowerMap(lower).upperMap(upper).costMap(cost)
532    ///     .supplyMap(sup).run();
533    ///
534    ///   // Run again with modified cost map (resetParams() is not called,
535    ///   // so only the cost map have to be set again)
536    ///   cost[e] += 100;
537    ///   cs.costMap(cost).run();
538    ///
539    ///   // Run again from scratch using resetParams()
540    ///   // (the lower bounds will be set to zero on all arcs)
541    ///   cs.resetParams();
542    ///   cs.upperMap(capacity).costMap(cost)
543    ///     .supplyMap(sup).run();
544    /// \endcode
545    ///
546    /// \return <tt>(*this)</tt>
547    ///
548    /// \see reset(), run()
549    CostScaling& resetParams() {
550      for (int i = 0; i != _res_node_num; ++i) {
551        _supply[i] = 0;
552      }
553      int limit = _first_out[_root];
554      for (int j = 0; j != limit; ++j) {
555        _lower[j] = 0;
556        _upper[j] = INF;
557        _scost[j] = _forward[j] ? 1 : -1;
558      }
559      for (int j = limit; j != _res_arc_num; ++j) {
560        _lower[j] = 0;
561        _upper[j] = INF;
562        _scost[j] = 0;
563        _scost[_reverse[j]] = 0;
564      }
565      _have_lower = false;
566      return *this;
567    }
568
569    /// \brief Reset the internal data structures and all the parameters
570    /// that have been given before.
571    ///
572    /// This function resets the internal data structures and all the
573    /// paramaters that have been given before using functions \ref lowerMap(),
574    /// \ref upperMap(), \ref costMap(), \ref supplyMap(), \ref stSupply().
575    ///
576    /// It is useful for multiple \ref run() calls. By default, all the given
577    /// parameters are kept for the next \ref run() call, unless
578    /// \ref resetParams() or \ref reset() is used.
579    /// If the underlying digraph was also modified after the construction
580    /// of the class or the last \ref reset() call, then the \ref reset()
581    /// function must be used, otherwise \ref resetParams() is sufficient.
582    ///
583    /// See \ref resetParams() for examples.
584    ///
585    /// \return <tt>(*this)</tt>
586    ///
587    /// \see resetParams(), run()
588    CostScaling& reset() {
589      // Resize vectors
590      _node_num = countNodes(_graph);
591      _arc_num = countArcs(_graph);
592      _res_node_num = _node_num + 1;
593      _res_arc_num = 2 * (_arc_num + _node_num);
594      _root = _node_num;
595
596      _first_out.resize(_res_node_num + 1);
597      _forward.resize(_res_arc_num);
598      _source.resize(_res_arc_num);
599      _target.resize(_res_arc_num);
600      _reverse.resize(_res_arc_num);
601
602      _lower.resize(_res_arc_num);
603      _upper.resize(_res_arc_num);
604      _scost.resize(_res_arc_num);
605      _supply.resize(_res_node_num);
606
607      _res_cap.resize(_res_arc_num);
608      _cost.resize(_res_arc_num);
609      _pi.resize(_res_node_num);
610      _excess.resize(_res_node_num);
611      _next_out.resize(_res_node_num);
612
613      // Copy the graph
614      int i = 0, j = 0, k = 2 * _arc_num + _node_num;
615      for (NodeIt n(_graph); n != INVALID; ++n, ++i) {
616        _node_id[n] = i;
617      }
618      i = 0;
619      for (NodeIt n(_graph); n != INVALID; ++n, ++i) {
620        _first_out[i] = j;
621        for (OutArcIt a(_graph, n); a != INVALID; ++a, ++j) {
622          _arc_idf[a] = j;
623          _forward[j] = true;
624          _source[j] = i;
625          _target[j] = _node_id[_graph.runningNode(a)];
626        }
627        for (InArcIt a(_graph, n); a != INVALID; ++a, ++j) {
628          _arc_idb[a] = j;
629          _forward[j] = false;
630          _source[j] = i;
631          _target[j] = _node_id[_graph.runningNode(a)];
632        }
633        _forward[j] = false;
634        _source[j] = i;
635        _target[j] = _root;
636        _reverse[j] = k;
637        _forward[k] = true;
638        _source[k] = _root;
639        _target[k] = i;
640        _reverse[k] = j;
641        ++j; ++k;
642      }
643      _first_out[i] = j;
644      _first_out[_res_node_num] = k;
645      for (ArcIt a(_graph); a != INVALID; ++a) {
646        int fi = _arc_idf[a];
647        int bi = _arc_idb[a];
648        _reverse[fi] = bi;
649        _reverse[bi] = fi;
650      }
651
652      // Reset parameters
653      resetParams();
654      return *this;
655    }
656
657    /// @}
658
659    /// \name Query Functions
660    /// The results of the algorithm can be obtained using these
661    /// functions.\n
662    /// The \ref run() function must be called before using them.
663
664    /// @{
665
666    /// \brief Return the total cost of the found flow.
667    ///
668    /// This function returns the total cost of the found flow.
669    /// Its complexity is O(e).
670    ///
671    /// \note The return type of the function can be specified as a
672    /// template parameter. For example,
673    /// \code
674    ///   cs.totalCost<double>();
675    /// \endcode
676    /// It is useful if the total cost cannot be stored in the \c Cost
677    /// type of the algorithm, which is the default return type of the
678    /// function.
679    ///
680    /// \pre \ref run() must be called before using this function.
681    template <typename Number>
682    Number totalCost() const {
683      Number c = 0;
684      for (ArcIt a(_graph); a != INVALID; ++a) {
685        int i = _arc_idb[a];
686        c += static_cast<Number>(_res_cap[i]) *
687             (-static_cast<Number>(_scost[i]));
688      }
689      return c;
690    }
691
692#ifndef DOXYGEN
693    Cost totalCost() const {
694      return totalCost<Cost>();
695    }
696#endif
697
698    /// \brief Return the flow on the given arc.
699    ///
700    /// This function returns the flow on the given arc.
701    ///
702    /// \pre \ref run() must be called before using this function.
703    Value flow(const Arc& a) const {
704      return _res_cap[_arc_idb[a]];
705    }
706
707    /// \brief Return the flow map (the primal solution).
708    ///
709    /// This function copies the flow value on each arc into the given
710    /// map. The \c Value type of the algorithm must be convertible to
711    /// the \c Value type of the map.
712    ///
713    /// \pre \ref run() must be called before using this function.
714    template <typename FlowMap>
715    void flowMap(FlowMap &map) const {
716      for (ArcIt a(_graph); a != INVALID; ++a) {
717        map.set(a, _res_cap[_arc_idb[a]]);
718      }
719    }
720
721    /// \brief Return the potential (dual value) of the given node.
722    ///
723    /// This function returns the potential (dual value) of the
724    /// given node.
725    ///
726    /// \pre \ref run() must be called before using this function.
727    Cost potential(const Node& n) const {
728      return static_cast<Cost>(_pi[_node_id[n]]);
729    }
730
731    /// \brief Return the potential map (the dual solution).
732    ///
733    /// This function copies the potential (dual value) of each node
734    /// into the given map.
735    /// The \c Cost type of the algorithm must be convertible to the
736    /// \c Value type of the map.
737    ///
738    /// \pre \ref run() must be called before using this function.
739    template <typename PotentialMap>
740    void potentialMap(PotentialMap &map) const {
741      for (NodeIt n(_graph); n != INVALID; ++n) {
742        map.set(n, static_cast<Cost>(_pi[_node_id[n]]));
743      }
744    }
745
746    /// @}
747
748  private:
749
750    // Initialize the algorithm
751    ProblemType init() {
752      if (_res_node_num <= 1) return INFEASIBLE;
753
754      // Check the sum of supply values
755      _sum_supply = 0;
756      for (int i = 0; i != _root; ++i) {
757        _sum_supply += _supply[i];
758      }
759      if (_sum_supply > 0) return INFEASIBLE;
760
761
762      // Initialize vectors
763      for (int i = 0; i != _res_node_num; ++i) {
764        _pi[i] = 0;
765        _excess[i] = _supply[i];
766      }
767
768      // Remove infinite upper bounds and check negative arcs
769      const Value MAX = std::numeric_limits<Value>::max();
770      int last_out;
771      if (_have_lower) {
772        for (int i = 0; i != _root; ++i) {
773          last_out = _first_out[i+1];
774          for (int j = _first_out[i]; j != last_out; ++j) {
775            if (_forward[j]) {
776              Value c = _scost[j] < 0 ? _upper[j] : _lower[j];
777              if (c >= MAX) return UNBOUNDED;
778              _excess[i] -= c;
779              _excess[_target[j]] += c;
780            }
781          }
782        }
783      } else {
784        for (int i = 0; i != _root; ++i) {
785          last_out = _first_out[i+1];
786          for (int j = _first_out[i]; j != last_out; ++j) {
787            if (_forward[j] && _scost[j] < 0) {
788              Value c = _upper[j];
789              if (c >= MAX) return UNBOUNDED;
790              _excess[i] -= c;
791              _excess[_target[j]] += c;
792            }
793          }
794        }
795      }
796      Value ex, max_cap = 0;
797      for (int i = 0; i != _res_node_num; ++i) {
798        ex = _excess[i];
799        _excess[i] = 0;
800        if (ex < 0) max_cap -= ex;
801      }
802      for (int j = 0; j != _res_arc_num; ++j) {
803        if (_upper[j] >= MAX) _upper[j] = max_cap;
804      }
805
806      // Initialize the large cost vector and the epsilon parameter
807      _epsilon = 0;
808      LargeCost lc;
809      for (int i = 0; i != _root; ++i) {
810        last_out = _first_out[i+1];
811        for (int j = _first_out[i]; j != last_out; ++j) {
812          lc = static_cast<LargeCost>(_scost[j]) * _res_node_num * _alpha;
813          _cost[j] = lc;
814          if (lc > _epsilon) _epsilon = lc;
815        }
816      }
817      _epsilon /= _alpha;
818
819      // Initialize maps for Circulation and remove non-zero lower bounds
820      ConstMap<Arc, Value> low(0);
821      typedef typename Digraph::template ArcMap<Value> ValueArcMap;
822      typedef typename Digraph::template NodeMap<Value> ValueNodeMap;
823      ValueArcMap cap(_graph), flow(_graph);
824      ValueNodeMap sup(_graph);
825      for (NodeIt n(_graph); n != INVALID; ++n) {
826        sup[n] = _supply[_node_id[n]];
827      }
828      if (_have_lower) {
829        for (ArcIt a(_graph); a != INVALID; ++a) {
830          int j = _arc_idf[a];
831          Value c = _lower[j];
832          cap[a] = _upper[j] - c;
833          sup[_graph.source(a)] -= c;
834          sup[_graph.target(a)] += c;
835        }
836      } else {
837        for (ArcIt a(_graph); a != INVALID; ++a) {
838          cap[a] = _upper[_arc_idf[a]];
839        }
840      }
841
842      _sup_node_num = 0;
843      for (NodeIt n(_graph); n != INVALID; ++n) {
844        if (sup[n] > 0) ++_sup_node_num;
845      }
846
847      // Find a feasible flow using Circulation
848      Circulation<Digraph, ConstMap<Arc, Value>, ValueArcMap, ValueNodeMap>
849        circ(_graph, low, cap, sup);
850      if (!circ.flowMap(flow).run()) return INFEASIBLE;
851
852      // Set residual capacities and handle GEQ supply type
853      if (_sum_supply < 0) {
854        for (ArcIt a(_graph); a != INVALID; ++a) {
855          Value fa = flow[a];
856          _res_cap[_arc_idf[a]] = cap[a] - fa;
857          _res_cap[_arc_idb[a]] = fa;
858          sup[_graph.source(a)] -= fa;
859          sup[_graph.target(a)] += fa;
860        }
861        for (NodeIt n(_graph); n != INVALID; ++n) {
862          _excess[_node_id[n]] = sup[n];
863        }
864        for (int a = _first_out[_root]; a != _res_arc_num; ++a) {
865          int u = _target[a];
866          int ra = _reverse[a];
867          _res_cap[a] = -_sum_supply + 1;
868          _res_cap[ra] = -_excess[u];
869          _cost[a] = 0;
870          _cost[ra] = 0;
871          _excess[u] = 0;
872        }
873      } else {
874        for (ArcIt a(_graph); a != INVALID; ++a) {
875          Value fa = flow[a];
876          _res_cap[_arc_idf[a]] = cap[a] - fa;
877          _res_cap[_arc_idb[a]] = fa;
878        }
879        for (int a = _first_out[_root]; a != _res_arc_num; ++a) {
880          int ra = _reverse[a];
881          _res_cap[a] = 0;
882          _res_cap[ra] = 0;
883          _cost[a] = 0;
884          _cost[ra] = 0;
885        }
886      }
887
888      // Initialize data structures for buckets
889      _max_rank = _alpha * _res_node_num;
890      _buckets.resize(_max_rank);
891      _bucket_next.resize(_res_node_num + 1);
892      _bucket_prev.resize(_res_node_num + 1);
893      _rank.resize(_res_node_num + 1);
894
895      return OPTIMAL;
896    }
897
898    // Execute the algorithm and transform the results
899    void start(Method method) {
900      const int MAX_PARTIAL_PATH_LENGTH = 4;
901
902      switch (method) {
903        case PUSH:
904          startPush();
905          break;
906        case AUGMENT:
907          startAugment(_res_node_num - 1);
908          break;
909        case PARTIAL_AUGMENT:
910          startAugment(MAX_PARTIAL_PATH_LENGTH);
911          break;
912      }
913
914      // Compute node potentials (dual solution)
915      for (int i = 0; i != _res_node_num; ++i) {
916        _pi[i] = static_cast<Cost>(_pi[i] / (_res_node_num * _alpha));
917      }
918      bool optimal = true;
919      for (int i = 0; optimal && i != _res_node_num; ++i) {
920        LargeCost pi_i = _pi[i];
921        int last_out = _first_out[i+1];
922        for (int j = _first_out[i]; j != last_out; ++j) {
923          if (_res_cap[j] > 0 && _scost[j] + pi_i - _pi[_target[j]] < 0) {
924            optimal = false;
925            break;
926          }
927        }
928      }
929
930      if (!optimal) {
931        // Compute node potentials for the original costs with BellmanFord
932        // (if it is necessary)
933        typedef std::pair<int, int> IntPair;
934        StaticDigraph sgr;
935        std::vector<IntPair> arc_vec;
936        std::vector<LargeCost> cost_vec;
937        LargeCostArcMap cost_map(cost_vec);
938
939        arc_vec.clear();
940        cost_vec.clear();
941        for (int j = 0; j != _res_arc_num; ++j) {
942          if (_res_cap[j] > 0) {
943            int u = _source[j], v = _target[j];
944            arc_vec.push_back(IntPair(u, v));
945            cost_vec.push_back(_scost[j] + _pi[u] - _pi[v]);
946          }
947        }
948        sgr.build(_res_node_num, arc_vec.begin(), arc_vec.end());
949
950        typename BellmanFord<StaticDigraph, LargeCostArcMap>::Create
951          bf(sgr, cost_map);
952        bf.init(0);
953        bf.start();
954
955        for (int i = 0; i != _res_node_num; ++i) {
956          _pi[i] += bf.dist(sgr.node(i));
957        }
958      }
959
960      // Shift potentials to meet the requirements of the GEQ type
961      // optimality conditions
962      LargeCost max_pot = _pi[_root];
963      for (int i = 0; i != _res_node_num; ++i) {
964        if (_pi[i] > max_pot) max_pot = _pi[i];
965      }
966      if (max_pot != 0) {
967        for (int i = 0; i != _res_node_num; ++i) {
968          _pi[i] -= max_pot;
969        }
970      }
971
972      // Handle non-zero lower bounds
973      if (_have_lower) {
974        int limit = _first_out[_root];
975        for (int j = 0; j != limit; ++j) {
976          if (!_forward[j]) _res_cap[j] += _lower[j];
977        }
978      }
979    }
980
981    // Initialize a cost scaling phase
982    void initPhase() {
983      // Saturate arcs not satisfying the optimality condition
984      for (int u = 0; u != _res_node_num; ++u) {
985        int last_out = _first_out[u+1];
986        LargeCost pi_u = _pi[u];
987        for (int a = _first_out[u]; a != last_out; ++a) {
988          Value delta = _res_cap[a];
989          if (delta > 0) {
990            int v = _target[a];
991            if (_cost[a] + pi_u - _pi[v] < 0) {
992              _excess[u] -= delta;
993              _excess[v] += delta;
994              _res_cap[a] = 0;
995              _res_cap[_reverse[a]] += delta;
996            }
997          }
998        }
999      }
1000
1001      // Find active nodes (i.e. nodes with positive excess)
1002      for (int u = 0; u != _res_node_num; ++u) {
1003        if (_excess[u] > 0) _active_nodes.push_back(u);
1004      }
1005
1006      // Initialize the next arcs
1007      for (int u = 0; u != _res_node_num; ++u) {
1008        _next_out[u] = _first_out[u];
1009      }
1010    }
1011
1012    // Price (potential) refinement heuristic
1013    bool priceRefinement() {
1014
1015      // Stack for stroing the topological order
1016      IntVector stack(_res_node_num);
1017      int stack_top;
1018
1019      // Perform phases
1020      while (topologicalSort(stack, stack_top)) {
1021
1022        // Compute node ranks in the acyclic admissible network and
1023        // store the nodes in buckets
1024        for (int i = 0; i != _res_node_num; ++i) {
1025          _rank[i] = 0;
1026        }
1027        const int bucket_end = _root + 1;
1028        for (int r = 0; r != _max_rank; ++r) {
1029          _buckets[r] = bucket_end;
1030        }
1031        int top_rank = 0;
1032        for ( ; stack_top >= 0; --stack_top) {
1033          int u = stack[stack_top], v;
1034          int rank_u = _rank[u];
1035
1036          LargeCost rc, pi_u = _pi[u];
1037          int last_out = _first_out[u+1];
1038          for (int a = _first_out[u]; a != last_out; ++a) {
1039            if (_res_cap[a] > 0) {
1040              v = _target[a];
1041              rc = _cost[a] + pi_u - _pi[v];
1042              if (rc < 0) {
1043                LargeCost nrc = static_cast<LargeCost>((-rc - 0.5) / _epsilon);
1044                if (nrc < LargeCost(_max_rank)) {
1045                  int new_rank_v = rank_u + static_cast<int>(nrc);
1046                  if (new_rank_v > _rank[v]) {
1047                    _rank[v] = new_rank_v;
1048                  }
1049                }
1050              }
1051            }
1052          }
1053
1054          if (rank_u > 0) {
1055            top_rank = std::max(top_rank, rank_u);
1056            int bfirst = _buckets[rank_u];
1057            _bucket_next[u] = bfirst;
1058            _bucket_prev[bfirst] = u;
1059            _buckets[rank_u] = u;
1060          }
1061        }
1062
1063        // Check if the current flow is epsilon-optimal
1064        if (top_rank == 0) {
1065          return true;
1066        }
1067
1068        // Process buckets in top-down order
1069        for (int rank = top_rank; rank > 0; --rank) {
1070          while (_buckets[rank] != bucket_end) {
1071            // Remove the first node from the current bucket
1072            int u = _buckets[rank];
1073            _buckets[rank] = _bucket_next[u];
1074
1075            // Search the outgoing arcs of u
1076            LargeCost rc, pi_u = _pi[u];
1077            int last_out = _first_out[u+1];
1078            int v, old_rank_v, new_rank_v;
1079            for (int a = _first_out[u]; a != last_out; ++a) {
1080              if (_res_cap[a] > 0) {
1081                v = _target[a];
1082                old_rank_v = _rank[v];
1083
1084                if (old_rank_v < rank) {
1085
1086                  // Compute the new rank of node v
1087                  rc = _cost[a] + pi_u - _pi[v];
1088                  if (rc < 0) {
1089                    new_rank_v = rank;
1090                  } else {
1091                    LargeCost nrc = rc / _epsilon;
1092                    new_rank_v = 0;
1093                    if (nrc < LargeCost(_max_rank)) {
1094                      new_rank_v = rank - 1 - static_cast<int>(nrc);
1095                    }
1096                  }
1097
1098                  // Change the rank of node v
1099                  if (new_rank_v > old_rank_v) {
1100                    _rank[v] = new_rank_v;
1101
1102                    // Remove v from its old bucket
1103                    if (old_rank_v > 0) {
1104                      if (_buckets[old_rank_v] == v) {
1105                        _buckets[old_rank_v] = _bucket_next[v];
1106                      } else {
1107                        int pv = _bucket_prev[v], nv = _bucket_next[v];
1108                        _bucket_next[pv] = nv;
1109                        _bucket_prev[nv] = pv;
1110                      }
1111                    }
1112
1113                    // Insert v into its new bucket
1114                    int nv = _buckets[new_rank_v];
1115                    _bucket_next[v] = nv;
1116                    _bucket_prev[nv] = v;
1117                    _buckets[new_rank_v] = v;
1118                  }
1119                }
1120              }
1121            }
1122
1123            // Refine potential of node u
1124            _pi[u] -= rank * _epsilon;
1125          }
1126        }
1127
1128      }
1129
1130      return false;
1131    }
1132
1133    // Find and cancel cycles in the admissible network and
1134    // determine topological order using DFS
1135    bool topologicalSort(IntVector &stack, int &stack_top) {
1136      const int MAX_CYCLE_CANCEL = 1;
1137
1138      BoolVector reached(_res_node_num, false);
1139      BoolVector processed(_res_node_num, false);
1140      IntVector pred(_res_node_num);
1141      for (int i = 0; i != _res_node_num; ++i) {
1142        _next_out[i] = _first_out[i];
1143      }
1144      stack_top = -1;
1145
1146      int cycle_cnt = 0;
1147      for (int start = 0; start != _res_node_num; ++start) {
1148        if (reached[start]) continue;
1149
1150        // Start DFS search from this start node
1151        pred[start] = -1;
1152        int tip = start, v;
1153        while (true) {
1154          // Check the outgoing arcs of the current tip node
1155          reached[tip] = true;
1156          LargeCost pi_tip = _pi[tip];
1157          int a, last_out = _first_out[tip+1];
1158          for (a = _next_out[tip]; a != last_out; ++a) {
1159            if (_res_cap[a] > 0) {
1160              v = _target[a];
1161              if (_cost[a] + pi_tip - _pi[v] < 0) {
1162                if (!reached[v]) {
1163                  // A new node is reached
1164                  reached[v] = true;
1165                  pred[v] = tip;
1166                  _next_out[tip] = a;
1167                  tip = v;
1168                  a = _next_out[tip];
1169                  last_out = _first_out[tip+1];
1170                  break;
1171                }
1172                else if (!processed[v]) {
1173                  // A cycle is found
1174                  ++cycle_cnt;
1175                  _next_out[tip] = a;
1176
1177                  // Find the minimum residual capacity along the cycle
1178                  Value d, delta = _res_cap[a];
1179                  int u, delta_node = tip;
1180                  for (u = tip; u != v; ) {
1181                    u = pred[u];
1182                    d = _res_cap[_next_out[u]];
1183                    if (d <= delta) {
1184                      delta = d;
1185                      delta_node = u;
1186                    }
1187                  }
1188
1189                  // Augment along the cycle
1190                  _res_cap[a] -= delta;
1191                  _res_cap[_reverse[a]] += delta;
1192                  for (u = tip; u != v; ) {
1193                    u = pred[u];
1194                    int ca = _next_out[u];
1195                    _res_cap[ca] -= delta;
1196                    _res_cap[_reverse[ca]] += delta;
1197                  }
1198
1199                  // Check the maximum number of cycle canceling
1200                  if (cycle_cnt >= MAX_CYCLE_CANCEL) {
1201                    return false;
1202                  }
1203
1204                  // Roll back search to delta_node
1205                  if (delta_node != tip) {
1206                    for (u = tip; u != delta_node; u = pred[u]) {
1207                      reached[u] = false;
1208                    }
1209                    tip = delta_node;
1210                    a = _next_out[tip] + 1;
1211                    last_out = _first_out[tip+1];
1212                    break;
1213                  }
1214                }
1215              }
1216            }
1217          }
1218
1219          // Step back to the previous node
1220          if (a == last_out) {
1221            processed[tip] = true;
1222            stack[++stack_top] = tip;
1223            tip = pred[tip];
1224            if (tip < 0) {
1225              // Finish DFS from the current start node
1226              break;
1227            }
1228            ++_next_out[tip];
1229          }
1230        }
1231
1232      }
1233
1234      return (cycle_cnt == 0);
1235    }
1236
1237    // Global potential update heuristic
1238    void globalUpdate() {
1239      const int bucket_end = _root + 1;
1240
1241      // Initialize buckets
1242      for (int r = 0; r != _max_rank; ++r) {
1243        _buckets[r] = bucket_end;
1244      }
1245      Value total_excess = 0;
1246      int b0 = bucket_end;
1247      for (int i = 0; i != _res_node_num; ++i) {
1248        if (_excess[i] < 0) {
1249          _rank[i] = 0;
1250          _bucket_next[i] = b0;
1251          _bucket_prev[b0] = i;
1252          b0 = i;
1253        } else {
1254          total_excess += _excess[i];
1255          _rank[i] = _max_rank;
1256        }
1257      }
1258      if (total_excess == 0) return;
1259      _buckets[0] = b0;
1260
1261      // Search the buckets
1262      int r = 0;
1263      for ( ; r != _max_rank; ++r) {
1264        while (_buckets[r] != bucket_end) {
1265          // Remove the first node from the current bucket
1266          int u = _buckets[r];
1267          _buckets[r] = _bucket_next[u];
1268
1269          // Search the incomming arcs of u
1270          LargeCost pi_u = _pi[u];
1271          int last_out = _first_out[u+1];
1272          for (int a = _first_out[u]; a != last_out; ++a) {
1273            int ra = _reverse[a];
1274            if (_res_cap[ra] > 0) {
1275              int v = _source[ra];
1276              int old_rank_v = _rank[v];
1277              if (r < old_rank_v) {
1278                // Compute the new rank of v
1279                LargeCost nrc = (_cost[ra] + _pi[v] - pi_u) / _epsilon;
1280                int new_rank_v = old_rank_v;
1281                if (nrc < LargeCost(_max_rank)) {
1282                  new_rank_v = r + 1 + static_cast<int>(nrc);
1283                }
1284
1285                // Change the rank of v
1286                if (new_rank_v < old_rank_v) {
1287                  _rank[v] = new_rank_v;
1288                  _next_out[v] = _first_out[v];
1289
1290                  // Remove v from its old bucket
1291                  if (old_rank_v < _max_rank) {
1292                    if (_buckets[old_rank_v] == v) {
1293                      _buckets[old_rank_v] = _bucket_next[v];
1294                    } else {
1295                      int pv = _bucket_prev[v], nv = _bucket_next[v];
1296                      _bucket_next[pv] = nv;
1297                      _bucket_prev[nv] = pv;
1298                    }
1299                  }
1300
1301                  // Insert v into its new bucket
1302                  int nv = _buckets[new_rank_v];
1303                  _bucket_next[v] = nv;
1304                  _bucket_prev[nv] = v;
1305                  _buckets[new_rank_v] = v;
1306                }
1307              }
1308            }
1309          }
1310
1311          // Finish search if there are no more active nodes
1312          if (_excess[u] > 0) {
1313            total_excess -= _excess[u];
1314            if (total_excess <= 0) break;
1315          }
1316        }
1317        if (total_excess <= 0) break;
1318      }
1319
1320      // Relabel nodes
1321      for (int u = 0; u != _res_node_num; ++u) {
1322        int k = std::min(_rank[u], r);
1323        if (k > 0) {
1324          _pi[u] -= _epsilon * k;
1325          _next_out[u] = _first_out[u];
1326        }
1327      }
1328    }
1329
1330    /// Execute the algorithm performing augment and relabel operations
1331    void startAugment(int max_length) {
1332      // Paramters for heuristics
1333      const int PRICE_REFINEMENT_LIMIT = 2;
1334      const double GLOBAL_UPDATE_FACTOR = 1.0;
1335      const int global_update_skip = static_cast<int>(GLOBAL_UPDATE_FACTOR *
1336        (_res_node_num + _sup_node_num * _sup_node_num));
1337      int next_global_update_limit = global_update_skip;
1338
1339      // Perform cost scaling phases
1340      IntVector path;
1341      BoolVector path_arc(_res_arc_num, false);
1342      int relabel_cnt = 0;
1343      int eps_phase_cnt = 0;
1344      for ( ; _epsilon >= 1; _epsilon = _epsilon < _alpha && _epsilon > 1 ?
1345                                        1 : _epsilon / _alpha )
1346      {
1347        ++eps_phase_cnt;
1348
1349        // Price refinement heuristic
1350        if (eps_phase_cnt >= PRICE_REFINEMENT_LIMIT) {
1351          if (priceRefinement()) continue;
1352        }
1353
1354        // Initialize current phase
1355        initPhase();
1356
1357        // Perform partial augment and relabel operations
1358        while (true) {
1359          // Select an active node (FIFO selection)
1360          while (_active_nodes.size() > 0 &&
1361                 _excess[_active_nodes.front()] <= 0) {
1362            _active_nodes.pop_front();
1363          }
1364          if (_active_nodes.size() == 0) break;
1365          int start = _active_nodes.front();
1366
1367          // Find an augmenting path from the start node
1368          int tip = start;
1369          while (int(path.size()) < max_length && _excess[tip] >= 0) {
1370            int u;
1371            LargeCost rc, min_red_cost = std::numeric_limits<LargeCost>::max();
1372            LargeCost pi_tip = _pi[tip];
1373            int last_out = _first_out[tip+1];
1374            for (int a = _next_out[tip]; a != last_out; ++a) {
1375              if (_res_cap[a] > 0) {
1376                u = _target[a];
1377                rc = _cost[a] + pi_tip - _pi[u];
1378                if (rc < 0) {
1379                  path.push_back(a);
1380                  _next_out[tip] = a;
1381                  if (path_arc[a]) {
1382                    goto augment;   // a cycle is found, stop path search
1383                  }
1384                  tip = u;
1385                  path_arc[a] = true;
1386                  goto next_step;
1387                }
1388                else if (rc < min_red_cost) {
1389                  min_red_cost = rc;
1390                }
1391              }
1392            }
1393
1394            // Relabel tip node
1395            if (tip != start) {
1396              int ra = _reverse[path.back()];
1397              min_red_cost =
1398                std::min(min_red_cost, _cost[ra] + pi_tip - _pi[_target[ra]]);
1399            }
1400            last_out = _next_out[tip];
1401            for (int a = _first_out[tip]; a != last_out; ++a) {
1402              if (_res_cap[a] > 0) {
1403                rc = _cost[a] + pi_tip - _pi[_target[a]];
1404                if (rc < min_red_cost) {
1405                  min_red_cost = rc;
1406                }
1407              }
1408            }
1409            _pi[tip] -= min_red_cost + _epsilon;
1410            _next_out[tip] = _first_out[tip];
1411            ++relabel_cnt;
1412
1413            // Step back
1414            if (tip != start) {
1415              int pa = path.back();
1416              path_arc[pa] = false;
1417              tip = _source[pa];
1418              path.pop_back();
1419            }
1420
1421          next_step: ;
1422          }
1423
1424          // Augment along the found path (as much flow as possible)
1425        augment:
1426          Value delta;
1427          int pa, u, v = start;
1428          for (int i = 0; i != int(path.size()); ++i) {
1429            pa = path[i];
1430            u = v;
1431            v = _target[pa];
1432            path_arc[pa] = false;
1433            delta = std::min(_res_cap[pa], _excess[u]);
1434            _res_cap[pa] -= delta;
1435            _res_cap[_reverse[pa]] += delta;
1436            _excess[u] -= delta;
1437            _excess[v] += delta;
1438            if (_excess[v] > 0 && _excess[v] <= delta) {
1439              _active_nodes.push_back(v);
1440            }
1441          }
1442          path.clear();
1443
1444          // Global update heuristic
1445          if (relabel_cnt >= next_global_update_limit) {
1446            globalUpdate();
1447            next_global_update_limit += global_update_skip;
1448          }
1449        }
1450
1451      }
1452
1453    }
1454
1455    /// Execute the algorithm performing push and relabel operations
1456    void startPush() {
1457      // Paramters for heuristics
1458      const int PRICE_REFINEMENT_LIMIT = 2;
1459      const double GLOBAL_UPDATE_FACTOR = 2.0;
1460
1461      const int global_update_skip = static_cast<int>(GLOBAL_UPDATE_FACTOR *
1462        (_res_node_num + _sup_node_num * _sup_node_num));
1463      int next_global_update_limit = global_update_skip;
1464
1465      // Perform cost scaling phases
1466      BoolVector hyper(_res_node_num, false);
1467      LargeCostVector hyper_cost(_res_node_num);
1468      int relabel_cnt = 0;
1469      int eps_phase_cnt = 0;
1470      for ( ; _epsilon >= 1; _epsilon = _epsilon < _alpha && _epsilon > 1 ?
1471                                        1 : _epsilon / _alpha )
1472      {
1473        ++eps_phase_cnt;
1474
1475        // Price refinement heuristic
1476        if (eps_phase_cnt >= PRICE_REFINEMENT_LIMIT) {
1477          if (priceRefinement()) continue;
1478        }
1479
1480        // Initialize current phase
1481        initPhase();
1482
1483        // Perform push and relabel operations
1484        while (_active_nodes.size() > 0) {
1485          LargeCost min_red_cost, rc, pi_n;
1486          Value delta;
1487          int n, t, a, last_out = _res_arc_num;
1488
1489        next_node:
1490          // Select an active node (FIFO selection)
1491          n = _active_nodes.front();
1492          last_out = _first_out[n+1];
1493          pi_n = _pi[n];
1494
1495          // Perform push operations if there are admissible arcs
1496          if (_excess[n] > 0) {
1497            for (a = _next_out[n]; a != last_out; ++a) {
1498              if (_res_cap[a] > 0 &&
1499                  _cost[a] + pi_n - _pi[_target[a]] < 0) {
1500                delta = std::min(_res_cap[a], _excess[n]);
1501                t = _target[a];
1502
1503                // Push-look-ahead heuristic
1504                Value ahead = -_excess[t];
1505                int last_out_t = _first_out[t+1];
1506                LargeCost pi_t = _pi[t];
1507                for (int ta = _next_out[t]; ta != last_out_t; ++ta) {
1508                  if (_res_cap[ta] > 0 &&
1509                      _cost[ta] + pi_t - _pi[_target[ta]] < 0)
1510                    ahead += _res_cap[ta];
1511                  if (ahead >= delta) break;
1512                }
1513                if (ahead < 0) ahead = 0;
1514
1515                // Push flow along the arc
1516                if (ahead < delta && !hyper[t]) {
1517                  _res_cap[a] -= ahead;
1518                  _res_cap[_reverse[a]] += ahead;
1519                  _excess[n] -= ahead;
1520                  _excess[t] += ahead;
1521                  _active_nodes.push_front(t);
1522                  hyper[t] = true;
1523                  hyper_cost[t] = _cost[a] + pi_n - pi_t;
1524                  _next_out[n] = a;
1525                  goto next_node;
1526                } else {
1527                  _res_cap[a] -= delta;
1528                  _res_cap[_reverse[a]] += delta;
1529                  _excess[n] -= delta;
1530                  _excess[t] += delta;
1531                  if (_excess[t] > 0 && _excess[t] <= delta)
1532                    _active_nodes.push_back(t);
1533                }
1534
1535                if (_excess[n] == 0) {
1536                  _next_out[n] = a;
1537                  goto remove_nodes;
1538                }
1539              }
1540            }
1541            _next_out[n] = a;
1542          }
1543
1544          // Relabel the node if it is still active (or hyper)
1545          if (_excess[n] > 0 || hyper[n]) {
1546             min_red_cost = hyper[n] ? -hyper_cost[n] :
1547               std::numeric_limits<LargeCost>::max();
1548            for (int a = _first_out[n]; a != last_out; ++a) {
1549              if (_res_cap[a] > 0) {
1550                rc = _cost[a] + pi_n - _pi[_target[a]];
1551                if (rc < min_red_cost) {
1552                  min_red_cost = rc;
1553                }
1554              }
1555            }
1556            _pi[n] -= min_red_cost + _epsilon;
1557            _next_out[n] = _first_out[n];
1558            hyper[n] = false;
1559            ++relabel_cnt;
1560          }
1561
1562          // Remove nodes that are not active nor hyper
1563        remove_nodes:
1564          while ( _active_nodes.size() > 0 &&
1565                  _excess[_active_nodes.front()] <= 0 &&
1566                  !hyper[_active_nodes.front()] ) {
1567            _active_nodes.pop_front();
1568          }
1569
1570          // Global update heuristic
1571          if (relabel_cnt >= next_global_update_limit) {
1572            globalUpdate();
1573            for (int u = 0; u != _res_node_num; ++u)
1574              hyper[u] = false;
1575            next_global_update_limit += global_update_skip;
1576          }
1577        }
1578      }
1579    }
1580
1581  }; //class CostScaling
1582
1583  ///@}
1584
1585} //namespace lemon
1586
1587#endif //LEMON_COST_SCALING_H
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