COIN-OR::LEMON - Graph Library

source: lemon/lemon/cost_scaling.h @ 1048:1226290a9b7d

Last change on this file since 1048:1226290a9b7d was 1048:1226290a9b7d, checked in by Peter Kovacs <kpeter@…>, 9 years ago

Faster computation of the dual solution in CostScaling? (#417)

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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_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 larger than one.
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 = 8) {
505      _alpha = factor;
506      ProblemType pt = init();
507      if (pt != OPTIMAL) return pt;
508      start(method);
509      return OPTIMAL;
510    }
511
512    /// \brief Reset all the parameters that have been given before.
513    ///
514    /// This function resets all the paramaters that have been given
515    /// before using functions \ref lowerMap(), \ref upperMap(),
516    /// \ref costMap(), \ref supplyMap(), \ref stSupply().
517    ///
518    /// It is useful for multiple \ref run() calls. Basically, all the given
519    /// parameters are kept for the next \ref run() call, unless
520    /// \ref resetParams() or \ref reset() is used.
521    /// If the underlying digraph was also modified after the construction
522    /// of the class or the last \ref reset() call, then the \ref reset()
523    /// function must be used, otherwise \ref resetParams() is sufficient.
524    ///
525    /// For example,
526    /// \code
527    ///   CostScaling<ListDigraph> cs(graph);
528    ///
529    ///   // First run
530    ///   cs.lowerMap(lower).upperMap(upper).costMap(cost)
531    ///     .supplyMap(sup).run();
532    ///
533    ///   // Run again with modified cost map (resetParams() is not called,
534    ///   // so only the cost map have to be set again)
535    ///   cost[e] += 100;
536    ///   cs.costMap(cost).run();
537    ///
538    ///   // Run again from scratch using resetParams()
539    ///   // (the lower bounds will be set to zero on all arcs)
540    ///   cs.resetParams();
541    ///   cs.upperMap(capacity).costMap(cost)
542    ///     .supplyMap(sup).run();
543    /// \endcode
544    ///
545    /// \return <tt>(*this)</tt>
546    ///
547    /// \see reset(), run()
548    CostScaling& resetParams() {
549      for (int i = 0; i != _res_node_num; ++i) {
550        _supply[i] = 0;
551      }
552      int limit = _first_out[_root];
553      for (int j = 0; j != limit; ++j) {
554        _lower[j] = 0;
555        _upper[j] = INF;
556        _scost[j] = _forward[j] ? 1 : -1;
557      }
558      for (int j = limit; j != _res_arc_num; ++j) {
559        _lower[j] = 0;
560        _upper[j] = INF;
561        _scost[j] = 0;
562        _scost[_reverse[j]] = 0;
563      }
564      _have_lower = false;
565      return *this;
566    }
567
568    /// \brief Reset the internal data structures and all the parameters
569    /// that have been given before.
570    ///
571    /// This function resets the internal data structures and all the
572    /// paramaters that have been given before using functions \ref lowerMap(),
573    /// \ref upperMap(), \ref costMap(), \ref supplyMap(), \ref stSupply().
574    ///
575    /// It is useful for multiple \ref run() calls. By default, all the given
576    /// parameters are kept for the next \ref run() call, unless
577    /// \ref resetParams() or \ref reset() is used.
578    /// If the underlying digraph was also modified after the construction
579    /// of the class or the last \ref reset() call, then the \ref reset()
580    /// function must be used, otherwise \ref resetParams() is sufficient.
581    ///
582    /// See \ref resetParams() for examples.
583    ///
584    /// \return <tt>(*this)</tt>
585    ///
586    /// \see resetParams(), run()
587    CostScaling& reset() {
588      // Resize vectors
589      _node_num = countNodes(_graph);
590      _arc_num = countArcs(_graph);
591      _res_node_num = _node_num + 1;
592      _res_arc_num = 2 * (_arc_num + _node_num);
593      _root = _node_num;
594
595      _first_out.resize(_res_node_num + 1);
596      _forward.resize(_res_arc_num);
597      _source.resize(_res_arc_num);
598      _target.resize(_res_arc_num);
599      _reverse.resize(_res_arc_num);
600
601      _lower.resize(_res_arc_num);
602      _upper.resize(_res_arc_num);
603      _scost.resize(_res_arc_num);
604      _supply.resize(_res_node_num);
605
606      _res_cap.resize(_res_arc_num);
607      _cost.resize(_res_arc_num);
608      _pi.resize(_res_node_num);
609      _excess.resize(_res_node_num);
610      _next_out.resize(_res_node_num);
611
612      // Copy the graph
613      int i = 0, j = 0, k = 2 * _arc_num + _node_num;
614      for (NodeIt n(_graph); n != INVALID; ++n, ++i) {
615        _node_id[n] = i;
616      }
617      i = 0;
618      for (NodeIt n(_graph); n != INVALID; ++n, ++i) {
619        _first_out[i] = j;
620        for (OutArcIt a(_graph, n); a != INVALID; ++a, ++j) {
621          _arc_idf[a] = j;
622          _forward[j] = true;
623          _source[j] = i;
624          _target[j] = _node_id[_graph.runningNode(a)];
625        }
626        for (InArcIt a(_graph, n); a != INVALID; ++a, ++j) {
627          _arc_idb[a] = j;
628          _forward[j] = false;
629          _source[j] = i;
630          _target[j] = _node_id[_graph.runningNode(a)];
631        }
632        _forward[j] = false;
633        _source[j] = i;
634        _target[j] = _root;
635        _reverse[j] = k;
636        _forward[k] = true;
637        _source[k] = _root;
638        _target[k] = i;
639        _reverse[k] = j;
640        ++j; ++k;
641      }
642      _first_out[i] = j;
643      _first_out[_res_node_num] = k;
644      for (ArcIt a(_graph); a != INVALID; ++a) {
645        int fi = _arc_idf[a];
646        int bi = _arc_idb[a];
647        _reverse[fi] = bi;
648        _reverse[bi] = fi;
649      }
650
651      // Reset parameters
652      resetParams();
653      return *this;
654    }
655
656    /// @}
657
658    /// \name Query Functions
659    /// The results of the algorithm can be obtained using these
660    /// functions.\n
661    /// The \ref run() function must be called before using them.
662
663    /// @{
664
665    /// \brief Return the total cost of the found flow.
666    ///
667    /// This function returns the total cost of the found flow.
668    /// Its complexity is O(e).
669    ///
670    /// \note The return type of the function can be specified as a
671    /// template parameter. For example,
672    /// \code
673    ///   cs.totalCost<double>();
674    /// \endcode
675    /// It is useful if the total cost cannot be stored in the \c Cost
676    /// type of the algorithm, which is the default return type of the
677    /// function.
678    ///
679    /// \pre \ref run() must be called before using this function.
680    template <typename Number>
681    Number totalCost() const {
682      Number c = 0;
683      for (ArcIt a(_graph); a != INVALID; ++a) {
684        int i = _arc_idb[a];
685        c += static_cast<Number>(_res_cap[i]) *
686             (-static_cast<Number>(_scost[i]));
687      }
688      return c;
689    }
690
691#ifndef DOXYGEN
692    Cost totalCost() const {
693      return totalCost<Cost>();
694    }
695#endif
696
697    /// \brief Return the flow on the given arc.
698    ///
699    /// This function returns the flow on the given arc.
700    ///
701    /// \pre \ref run() must be called before using this function.
702    Value flow(const Arc& a) const {
703      return _res_cap[_arc_idb[a]];
704    }
705
706    /// \brief Return the flow map (the primal solution).
707    ///
708    /// This function copies the flow value on each arc into the given
709    /// map. The \c Value type of the algorithm must be convertible to
710    /// the \c Value type of the map.
711    ///
712    /// \pre \ref run() must be called before using this function.
713    template <typename FlowMap>
714    void flowMap(FlowMap &map) const {
715      for (ArcIt a(_graph); a != INVALID; ++a) {
716        map.set(a, _res_cap[_arc_idb[a]]);
717      }
718    }
719
720    /// \brief Return the potential (dual value) of the given node.
721    ///
722    /// This function returns the potential (dual value) of the
723    /// given node.
724    ///
725    /// \pre \ref run() must be called before using this function.
726    Cost potential(const Node& n) const {
727      return static_cast<Cost>(_pi[_node_id[n]]);
728    }
729
730    /// \brief Return the potential map (the dual solution).
731    ///
732    /// This function copies the potential (dual value) of each node
733    /// into the given map.
734    /// The \c Cost type of the algorithm must be convertible to the
735    /// \c Value type of the map.
736    ///
737    /// \pre \ref run() must be called before using this function.
738    template <typename PotentialMap>
739    void potentialMap(PotentialMap &map) const {
740      for (NodeIt n(_graph); n != INVALID; ++n) {
741        map.set(n, static_cast<Cost>(_pi[_node_id[n]]));
742      }
743    }
744
745    /// @}
746
747  private:
748
749    // Initialize the algorithm
750    ProblemType init() {
751      if (_res_node_num <= 1) return INFEASIBLE;
752
753      // Check the sum of supply values
754      _sum_supply = 0;
755      for (int i = 0; i != _root; ++i) {
756        _sum_supply += _supply[i];
757      }
758      if (_sum_supply > 0) return INFEASIBLE;
759
760
761      // Initialize vectors
762      for (int i = 0; i != _res_node_num; ++i) {
763        _pi[i] = 0;
764        _excess[i] = _supply[i];
765      }
766
767      // Remove infinite upper bounds and check negative arcs
768      const Value MAX = std::numeric_limits<Value>::max();
769      int last_out;
770      if (_have_lower) {
771        for (int i = 0; i != _root; ++i) {
772          last_out = _first_out[i+1];
773          for (int j = _first_out[i]; j != last_out; ++j) {
774            if (_forward[j]) {
775              Value c = _scost[j] < 0 ? _upper[j] : _lower[j];
776              if (c >= MAX) return UNBOUNDED;
777              _excess[i] -= c;
778              _excess[_target[j]] += c;
779            }
780          }
781        }
782      } else {
783        for (int i = 0; i != _root; ++i) {
784          last_out = _first_out[i+1];
785          for (int j = _first_out[i]; j != last_out; ++j) {
786            if (_forward[j] && _scost[j] < 0) {
787              Value c = _upper[j];
788              if (c >= MAX) return UNBOUNDED;
789              _excess[i] -= c;
790              _excess[_target[j]] += c;
791            }
792          }
793        }
794      }
795      Value ex, max_cap = 0;
796      for (int i = 0; i != _res_node_num; ++i) {
797        ex = _excess[i];
798        _excess[i] = 0;
799        if (ex < 0) max_cap -= ex;
800      }
801      for (int j = 0; j != _res_arc_num; ++j) {
802        if (_upper[j] >= MAX) _upper[j] = max_cap;
803      }
804
805      // Initialize the large cost vector and the epsilon parameter
806      _epsilon = 0;
807      LargeCost lc;
808      for (int i = 0; i != _root; ++i) {
809        last_out = _first_out[i+1];
810        for (int j = _first_out[i]; j != last_out; ++j) {
811          lc = static_cast<LargeCost>(_scost[j]) * _res_node_num * _alpha;
812          _cost[j] = lc;
813          if (lc > _epsilon) _epsilon = lc;
814        }
815      }
816      _epsilon /= _alpha;
817
818      // Initialize maps for Circulation and remove non-zero lower bounds
819      ConstMap<Arc, Value> low(0);
820      typedef typename Digraph::template ArcMap<Value> ValueArcMap;
821      typedef typename Digraph::template NodeMap<Value> ValueNodeMap;
822      ValueArcMap cap(_graph), flow(_graph);
823      ValueNodeMap sup(_graph);
824      for (NodeIt n(_graph); n != INVALID; ++n) {
825        sup[n] = _supply[_node_id[n]];
826      }
827      if (_have_lower) {
828        for (ArcIt a(_graph); a != INVALID; ++a) {
829          int j = _arc_idf[a];
830          Value c = _lower[j];
831          cap[a] = _upper[j] - c;
832          sup[_graph.source(a)] -= c;
833          sup[_graph.target(a)] += c;
834        }
835      } else {
836        for (ArcIt a(_graph); a != INVALID; ++a) {
837          cap[a] = _upper[_arc_idf[a]];
838        }
839      }
840
841      _sup_node_num = 0;
842      for (NodeIt n(_graph); n != INVALID; ++n) {
843        if (sup[n] > 0) ++_sup_node_num;
844      }
845
846      // Find a feasible flow using Circulation
847      Circulation<Digraph, ConstMap<Arc, Value>, ValueArcMap, ValueNodeMap>
848        circ(_graph, low, cap, sup);
849      if (!circ.flowMap(flow).run()) return INFEASIBLE;
850
851      // Set residual capacities and handle GEQ supply type
852      if (_sum_supply < 0) {
853        for (ArcIt a(_graph); a != INVALID; ++a) {
854          Value fa = flow[a];
855          _res_cap[_arc_idf[a]] = cap[a] - fa;
856          _res_cap[_arc_idb[a]] = fa;
857          sup[_graph.source(a)] -= fa;
858          sup[_graph.target(a)] += fa;
859        }
860        for (NodeIt n(_graph); n != INVALID; ++n) {
861          _excess[_node_id[n]] = sup[n];
862        }
863        for (int a = _first_out[_root]; a != _res_arc_num; ++a) {
864          int u = _target[a];
865          int ra = _reverse[a];
866          _res_cap[a] = -_sum_supply + 1;
867          _res_cap[ra] = -_excess[u];
868          _cost[a] = 0;
869          _cost[ra] = 0;
870          _excess[u] = 0;
871        }
872      } else {
873        for (ArcIt a(_graph); a != INVALID; ++a) {
874          Value fa = flow[a];
875          _res_cap[_arc_idf[a]] = cap[a] - fa;
876          _res_cap[_arc_idb[a]] = fa;
877        }
878        for (int a = _first_out[_root]; a != _res_arc_num; ++a) {
879          int ra = _reverse[a];
880          _res_cap[a] = 0;
881          _res_cap[ra] = 0;
882          _cost[a] = 0;
883          _cost[ra] = 0;
884        }
885      }
886
887      // Initialize data structures for buckets
888      _max_rank = _alpha * _res_node_num;
889      _buckets.resize(_max_rank);
890      _bucket_next.resize(_res_node_num + 1);
891      _bucket_prev.resize(_res_node_num + 1);
892      _rank.resize(_res_node_num + 1);
893
894      return OPTIMAL;
895    }
896
897    // Execute the algorithm and transform the results
898    void start(Method method) {
899      const int MAX_PARTIAL_PATH_LENGTH = 4;
900
901      switch (method) {
902        case PUSH:
903          startPush();
904          break;
905        case AUGMENT:
906          startAugment(_res_node_num - 1);
907          break;
908        case PARTIAL_AUGMENT:
909          startAugment(MAX_PARTIAL_PATH_LENGTH);
910          break;
911      }
912
913      // Compute node potentials (dual solution)
914      for (int i = 0; i != _res_node_num; ++i) {
915        _pi[i] = static_cast<Cost>(_pi[i] / (_res_node_num * _alpha));
916      }
917      bool optimal = true;
918      for (int i = 0; optimal && i != _res_node_num; ++i) {
919        LargeCost pi_i = _pi[i];
920        int last_out = _first_out[i+1];
921        for (int j = _first_out[i]; j != last_out; ++j) {
922          if (_res_cap[j] > 0 && _scost[j] + pi_i - _pi[_target[j]] < 0) {
923            optimal = false;
924            break;
925          }
926        }
927      }
928
929      if (!optimal) {
930        // Compute node potentials for the original costs with BellmanFord
931        // (if it is necessary)
932        typedef std::pair<int, int> IntPair;
933        StaticDigraph sgr;
934        std::vector<IntPair> arc_vec;
935        std::vector<LargeCost> cost_vec;
936        LargeCostArcMap cost_map(cost_vec);
937
938        arc_vec.clear();
939        cost_vec.clear();
940        for (int j = 0; j != _res_arc_num; ++j) {
941          if (_res_cap[j] > 0) {
942            int u = _source[j], v = _target[j];
943            arc_vec.push_back(IntPair(u, v));
944            cost_vec.push_back(_scost[j] + _pi[u] - _pi[v]);
945          }
946        }
947        sgr.build(_res_node_num, arc_vec.begin(), arc_vec.end());
948
949        typename BellmanFord<StaticDigraph, LargeCostArcMap>::Create
950          bf(sgr, cost_map);
951        bf.init(0);
952        bf.start();
953
954        for (int i = 0; i != _res_node_num; ++i) {
955          _pi[i] += bf.dist(sgr.node(i));
956        }
957      }
958
959      // Shift potentials to meet the requirements of the GEQ type
960      // optimality conditions
961      LargeCost max_pot = _pi[_root];
962      for (int i = 0; i != _res_node_num; ++i) {
963        if (_pi[i] > max_pot) max_pot = _pi[i];
964      }
965      if (max_pot != 0) {
966        for (int i = 0; i != _res_node_num; ++i) {
967          _pi[i] -= max_pot;
968        }
969      }
970
971      // Handle non-zero lower bounds
972      if (_have_lower) {
973        int limit = _first_out[_root];
974        for (int j = 0; j != limit; ++j) {
975          if (!_forward[j]) _res_cap[j] += _lower[j];
976        }
977      }
978    }
979
980    // Initialize a cost scaling phase
981    void initPhase() {
982      // Saturate arcs not satisfying the optimality condition
983      for (int u = 0; u != _res_node_num; ++u) {
984        int last_out = _first_out[u+1];
985        LargeCost pi_u = _pi[u];
986        for (int a = _first_out[u]; a != last_out; ++a) {
987          Value delta = _res_cap[a];
988          if (delta > 0) {
989            int v = _target[a];
990            if (_cost[a] + pi_u - _pi[v] < 0) {
991              _excess[u] -= delta;
992              _excess[v] += delta;
993              _res_cap[a] = 0;
994              _res_cap[_reverse[a]] += delta;
995            }
996          }
997        }
998      }
999
1000      // Find active nodes (i.e. nodes with positive excess)
1001      for (int u = 0; u != _res_node_num; ++u) {
1002        if (_excess[u] > 0) _active_nodes.push_back(u);
1003      }
1004
1005      // Initialize the next arcs
1006      for (int u = 0; u != _res_node_num; ++u) {
1007        _next_out[u] = _first_out[u];
1008      }
1009    }
1010
1011    // Price (potential) refinement heuristic
1012    bool priceRefinement() {
1013
1014      // Stack for stroing the topological order
1015      IntVector stack(_res_node_num);
1016      int stack_top;
1017
1018      // Perform phases
1019      while (topologicalSort(stack, stack_top)) {
1020
1021        // Compute node ranks in the acyclic admissible network and
1022        // store the nodes in buckets
1023        for (int i = 0; i != _res_node_num; ++i) {
1024          _rank[i] = 0;
1025        }
1026        const int bucket_end = _root + 1;
1027        for (int r = 0; r != _max_rank; ++r) {
1028          _buckets[r] = bucket_end;
1029        }
1030        int top_rank = 0;
1031        for ( ; stack_top >= 0; --stack_top) {
1032          int u = stack[stack_top], v;
1033          int rank_u = _rank[u];
1034
1035          LargeCost rc, pi_u = _pi[u];
1036          int last_out = _first_out[u+1];
1037          for (int a = _first_out[u]; a != last_out; ++a) {
1038            if (_res_cap[a] > 0) {
1039              v = _target[a];
1040              rc = _cost[a] + pi_u - _pi[v];
1041              if (rc < 0) {
1042                LargeCost nrc = static_cast<LargeCost>((-rc - 0.5) / _epsilon);
1043                if (nrc < LargeCost(_max_rank)) {
1044                  int new_rank_v = rank_u + static_cast<int>(nrc);
1045                  if (new_rank_v > _rank[v]) {
1046                    _rank[v] = new_rank_v;
1047                  }
1048                }
1049              }
1050            }
1051          }
1052
1053          if (rank_u > 0) {
1054            top_rank = std::max(top_rank, rank_u);
1055            int bfirst = _buckets[rank_u];
1056            _bucket_next[u] = bfirst;
1057            _bucket_prev[bfirst] = u;
1058            _buckets[rank_u] = u;
1059          }
1060        }
1061
1062        // Check if the current flow is epsilon-optimal
1063        if (top_rank == 0) {
1064          return true;
1065        }
1066
1067        // Process buckets in top-down order
1068        for (int rank = top_rank; rank > 0; --rank) {
1069          while (_buckets[rank] != bucket_end) {
1070            // Remove the first node from the current bucket
1071            int u = _buckets[rank];
1072            _buckets[rank] = _bucket_next[u];
1073
1074            // Search the outgoing arcs of u
1075            LargeCost rc, pi_u = _pi[u];
1076            int last_out = _first_out[u+1];
1077            int v, old_rank_v, new_rank_v;
1078            for (int a = _first_out[u]; a != last_out; ++a) {
1079              if (_res_cap[a] > 0) {
1080                v = _target[a];
1081                old_rank_v = _rank[v];
1082
1083                if (old_rank_v < rank) {
1084
1085                  // Compute the new rank of node v
1086                  rc = _cost[a] + pi_u - _pi[v];
1087                  if (rc < 0) {
1088                    new_rank_v = rank;
1089                  } else {
1090                    LargeCost nrc = rc / _epsilon;
1091                    new_rank_v = 0;
1092                    if (nrc < LargeCost(_max_rank)) {
1093                      new_rank_v = rank - 1 - static_cast<int>(nrc);
1094                    }
1095                  }
1096
1097                  // Change the rank of node v
1098                  if (new_rank_v > old_rank_v) {
1099                    _rank[v] = new_rank_v;
1100
1101                    // Remove v from its old bucket
1102                    if (old_rank_v > 0) {
1103                      if (_buckets[old_rank_v] == v) {
1104                        _buckets[old_rank_v] = _bucket_next[v];
1105                      } else {
1106                        int pv = _bucket_prev[v], nv = _bucket_next[v];
1107                        _bucket_next[pv] = nv;
1108                        _bucket_prev[nv] = pv;
1109                      }
1110                    }
1111
1112                    // Insert v into its new bucket
1113                    int nv = _buckets[new_rank_v];
1114                    _bucket_next[v] = nv;
1115                    _bucket_prev[nv] = v;
1116                    _buckets[new_rank_v] = v;
1117                  }
1118                }
1119              }
1120            }
1121
1122            // Refine potential of node u
1123            _pi[u] -= rank * _epsilon;
1124          }
1125        }
1126
1127      }
1128
1129      return false;
1130    }
1131
1132    // Find and cancel cycles in the admissible network and
1133    // determine topological order using DFS
1134    bool topologicalSort(IntVector &stack, int &stack_top) {
1135      const int MAX_CYCLE_CANCEL = 1;
1136
1137      BoolVector reached(_res_node_num, false);
1138      BoolVector processed(_res_node_num, false);
1139      IntVector pred(_res_node_num);
1140      for (int i = 0; i != _res_node_num; ++i) {
1141        _next_out[i] = _first_out[i];
1142      }
1143      stack_top = -1;
1144
1145      int cycle_cnt = 0;
1146      for (int start = 0; start != _res_node_num; ++start) {
1147        if (reached[start]) continue;
1148
1149        // Start DFS search from this start node
1150        pred[start] = -1;
1151        int tip = start, v;
1152        while (true) {
1153          // Check the outgoing arcs of the current tip node
1154          reached[tip] = true;
1155          LargeCost pi_tip = _pi[tip];
1156          int a, last_out = _first_out[tip+1];
1157          for (a = _next_out[tip]; a != last_out; ++a) {
1158            if (_res_cap[a] > 0) {
1159              v = _target[a];
1160              if (_cost[a] + pi_tip - _pi[v] < 0) {
1161                if (!reached[v]) {
1162                  // A new node is reached
1163                  reached[v] = true;
1164                  pred[v] = tip;
1165                  _next_out[tip] = a;
1166                  tip = v;
1167                  a = _next_out[tip];
1168                  last_out = _first_out[tip+1];
1169                  break;
1170                }
1171                else if (!processed[v]) {
1172                  // A cycle is found
1173                  ++cycle_cnt;
1174                  _next_out[tip] = a;
1175
1176                  // Find the minimum residual capacity along the cycle
1177                  Value d, delta = _res_cap[a];
1178                  int u, delta_node = tip;
1179                  for (u = tip; u != v; ) {
1180                    u = pred[u];
1181                    d = _res_cap[_next_out[u]];
1182                    if (d <= delta) {
1183                      delta = d;
1184                      delta_node = u;
1185                    }
1186                  }
1187
1188                  // Augment along the cycle
1189                  _res_cap[a] -= delta;
1190                  _res_cap[_reverse[a]] += delta;
1191                  for (u = tip; u != v; ) {
1192                    u = pred[u];
1193                    int ca = _next_out[u];
1194                    _res_cap[ca] -= delta;
1195                    _res_cap[_reverse[ca]] += delta;
1196                  }
1197
1198                  // Check the maximum number of cycle canceling
1199                  if (cycle_cnt >= MAX_CYCLE_CANCEL) {
1200                    return false;
1201                  }
1202
1203                  // Roll back search to delta_node
1204                  if (delta_node != tip) {
1205                    for (u = tip; u != delta_node; u = pred[u]) {
1206                      reached[u] = false;
1207                    }
1208                    tip = delta_node;
1209                    a = _next_out[tip] + 1;
1210                    last_out = _first_out[tip+1];
1211                    break;
1212                  }
1213                }
1214              }
1215            }
1216          }
1217
1218          // Step back to the previous node
1219          if (a == last_out) {
1220            processed[tip] = true;
1221            stack[++stack_top] = tip;
1222            tip = pred[tip];
1223            if (tip < 0) {
1224              // Finish DFS from the current start node
1225              break;
1226            }
1227            ++_next_out[tip];
1228          }
1229        }
1230
1231      }
1232
1233      return (cycle_cnt == 0);
1234    }
1235
1236    // Global potential update heuristic
1237    void globalUpdate() {
1238      const int bucket_end = _root + 1;
1239
1240      // Initialize buckets
1241      for (int r = 0; r != _max_rank; ++r) {
1242        _buckets[r] = bucket_end;
1243      }
1244      Value total_excess = 0;
1245      int b0 = bucket_end;
1246      for (int i = 0; i != _res_node_num; ++i) {
1247        if (_excess[i] < 0) {
1248          _rank[i] = 0;
1249          _bucket_next[i] = b0;
1250          _bucket_prev[b0] = i;
1251          b0 = i;
1252        } else {
1253          total_excess += _excess[i];
1254          _rank[i] = _max_rank;
1255        }
1256      }
1257      if (total_excess == 0) return;
1258      _buckets[0] = b0;
1259
1260      // Search the buckets
1261      int r = 0;
1262      for ( ; r != _max_rank; ++r) {
1263        while (_buckets[r] != bucket_end) {
1264          // Remove the first node from the current bucket
1265          int u = _buckets[r];
1266          _buckets[r] = _bucket_next[u];
1267
1268          // Search the incomming arcs of u
1269          LargeCost pi_u = _pi[u];
1270          int last_out = _first_out[u+1];
1271          for (int a = _first_out[u]; a != last_out; ++a) {
1272            int ra = _reverse[a];
1273            if (_res_cap[ra] > 0) {
1274              int v = _source[ra];
1275              int old_rank_v = _rank[v];
1276              if (r < old_rank_v) {
1277                // Compute the new rank of v
1278                LargeCost nrc = (_cost[ra] + _pi[v] - pi_u) / _epsilon;
1279                int new_rank_v = old_rank_v;
1280                if (nrc < LargeCost(_max_rank)) {
1281                  new_rank_v = r + 1 + static_cast<int>(nrc);
1282                }
1283
1284                // Change the rank of v
1285                if (new_rank_v < old_rank_v) {
1286                  _rank[v] = new_rank_v;
1287                  _next_out[v] = _first_out[v];
1288
1289                  // Remove v from its old bucket
1290                  if (old_rank_v < _max_rank) {
1291                    if (_buckets[old_rank_v] == v) {
1292                      _buckets[old_rank_v] = _bucket_next[v];
1293                    } else {
1294                      int pv = _bucket_prev[v], nv = _bucket_next[v];
1295                      _bucket_next[pv] = nv;
1296                      _bucket_prev[nv] = pv;
1297                    }
1298                  }
1299
1300                  // Insert v into its new bucket
1301                  int nv = _buckets[new_rank_v];
1302                  _bucket_next[v] = nv;
1303                  _bucket_prev[nv] = v;
1304                  _buckets[new_rank_v] = v;
1305                }
1306              }
1307            }
1308          }
1309
1310          // Finish search if there are no more active nodes
1311          if (_excess[u] > 0) {
1312            total_excess -= _excess[u];
1313            if (total_excess <= 0) break;
1314          }
1315        }
1316        if (total_excess <= 0) break;
1317      }
1318
1319      // Relabel nodes
1320      for (int u = 0; u != _res_node_num; ++u) {
1321        int k = std::min(_rank[u], r);
1322        if (k > 0) {
1323          _pi[u] -= _epsilon * k;
1324          _next_out[u] = _first_out[u];
1325        }
1326      }
1327    }
1328
1329    /// Execute the algorithm performing augment and relabel operations
1330    void startAugment(int max_length) {
1331      // Paramters for heuristics
1332      const int PRICE_REFINEMENT_LIMIT = 2;
1333      const double GLOBAL_UPDATE_FACTOR = 1.0;
1334      const int global_update_skip = static_cast<int>(GLOBAL_UPDATE_FACTOR *
1335        (_res_node_num + _sup_node_num * _sup_node_num));
1336      int next_global_update_limit = global_update_skip;
1337
1338      // Perform cost scaling phases
1339      IntVector path;
1340      BoolVector path_arc(_res_arc_num, false);
1341      int relabel_cnt = 0;
1342      int eps_phase_cnt = 0;
1343      for ( ; _epsilon >= 1; _epsilon = _epsilon < _alpha && _epsilon > 1 ?
1344                                        1 : _epsilon / _alpha )
1345      {
1346        ++eps_phase_cnt;
1347
1348        // Price refinement heuristic
1349        if (eps_phase_cnt >= PRICE_REFINEMENT_LIMIT) {
1350          if (priceRefinement()) continue;
1351        }
1352
1353        // Initialize current phase
1354        initPhase();
1355
1356        // Perform partial augment and relabel operations
1357        while (true) {
1358          // Select an active node (FIFO selection)
1359          while (_active_nodes.size() > 0 &&
1360                 _excess[_active_nodes.front()] <= 0) {
1361            _active_nodes.pop_front();
1362          }
1363          if (_active_nodes.size() == 0) break;
1364          int start = _active_nodes.front();
1365
1366          // Find an augmenting path from the start node
1367          int tip = start;
1368          while (int(path.size()) < max_length && _excess[tip] >= 0) {
1369            int u;
1370            LargeCost rc, min_red_cost = std::numeric_limits<LargeCost>::max();
1371            LargeCost pi_tip = _pi[tip];
1372            int last_out = _first_out[tip+1];
1373            for (int a = _next_out[tip]; a != last_out; ++a) {
1374              if (_res_cap[a] > 0) {
1375                u = _target[a];
1376                rc = _cost[a] + pi_tip - _pi[u];
1377                if (rc < 0) {
1378                  path.push_back(a);
1379                  _next_out[tip] = a;
1380                  if (path_arc[a]) {
1381                    goto augment;   // a cycle is found, stop path search
1382                  }
1383                  tip = u;
1384                  path_arc[a] = true;
1385                  goto next_step;
1386                }
1387                else if (rc < min_red_cost) {
1388                  min_red_cost = rc;
1389                }
1390              }
1391            }
1392
1393            // Relabel tip node
1394            if (tip != start) {
1395              int ra = _reverse[path.back()];
1396              min_red_cost =
1397                std::min(min_red_cost, _cost[ra] + pi_tip - _pi[_target[ra]]);
1398            }
1399            last_out = _next_out[tip];
1400            for (int a = _first_out[tip]; a != last_out; ++a) {
1401              if (_res_cap[a] > 0) {
1402                rc = _cost[a] + pi_tip - _pi[_target[a]];
1403                if (rc < min_red_cost) {
1404                  min_red_cost = rc;
1405                }
1406              }
1407            }
1408            _pi[tip] -= min_red_cost + _epsilon;
1409            _next_out[tip] = _first_out[tip];
1410            ++relabel_cnt;
1411
1412            // Step back
1413            if (tip != start) {
1414              int pa = path.back();
1415              path_arc[pa] = false;
1416              tip = _source[pa];
1417              path.pop_back();
1418            }
1419
1420          next_step: ;
1421          }
1422
1423          // Augment along the found path (as much flow as possible)
1424        augment:
1425          Value delta;
1426          int pa, u, v = start;
1427          for (int i = 0; i != int(path.size()); ++i) {
1428            pa = path[i];
1429            u = v;
1430            v = _target[pa];
1431            path_arc[pa] = false;
1432            delta = std::min(_res_cap[pa], _excess[u]);
1433            _res_cap[pa] -= delta;
1434            _res_cap[_reverse[pa]] += delta;
1435            _excess[u] -= delta;
1436            _excess[v] += delta;
1437            if (_excess[v] > 0 && _excess[v] <= delta) {
1438              _active_nodes.push_back(v);
1439            }
1440          }
1441          path.clear();
1442
1443          // Global update heuristic
1444          if (relabel_cnt >= next_global_update_limit) {
1445            globalUpdate();
1446            next_global_update_limit += global_update_skip;
1447          }
1448        }
1449
1450      }
1451
1452    }
1453
1454    /// Execute the algorithm performing push and relabel operations
1455    void startPush() {
1456      // Paramters for heuristics
1457      const int PRICE_REFINEMENT_LIMIT = 2;
1458      const double GLOBAL_UPDATE_FACTOR = 2.0;
1459
1460      const int global_update_skip = static_cast<int>(GLOBAL_UPDATE_FACTOR *
1461        (_res_node_num + _sup_node_num * _sup_node_num));
1462      int next_global_update_limit = global_update_skip;
1463
1464      // Perform cost scaling phases
1465      BoolVector hyper(_res_node_num, false);
1466      LargeCostVector hyper_cost(_res_node_num);
1467      int relabel_cnt = 0;
1468      int eps_phase_cnt = 0;
1469      for ( ; _epsilon >= 1; _epsilon = _epsilon < _alpha && _epsilon > 1 ?
1470                                        1 : _epsilon / _alpha )
1471      {
1472        ++eps_phase_cnt;
1473
1474        // Price refinement heuristic
1475        if (eps_phase_cnt >= PRICE_REFINEMENT_LIMIT) {
1476          if (priceRefinement()) continue;
1477        }
1478
1479        // Initialize current phase
1480        initPhase();
1481
1482        // Perform push and relabel operations
1483        while (_active_nodes.size() > 0) {
1484          LargeCost min_red_cost, rc, pi_n;
1485          Value delta;
1486          int n, t, a, last_out = _res_arc_num;
1487
1488        next_node:
1489          // Select an active node (FIFO selection)
1490          n = _active_nodes.front();
1491          last_out = _first_out[n+1];
1492          pi_n = _pi[n];
1493
1494          // Perform push operations if there are admissible arcs
1495          if (_excess[n] > 0) {
1496            for (a = _next_out[n]; a != last_out; ++a) {
1497              if (_res_cap[a] > 0 &&
1498                  _cost[a] + pi_n - _pi[_target[a]] < 0) {
1499                delta = std::min(_res_cap[a], _excess[n]);
1500                t = _target[a];
1501
1502                // Push-look-ahead heuristic
1503                Value ahead = -_excess[t];
1504                int last_out_t = _first_out[t+1];
1505                LargeCost pi_t = _pi[t];
1506                for (int ta = _next_out[t]; ta != last_out_t; ++ta) {
1507                  if (_res_cap[ta] > 0 &&
1508                      _cost[ta] + pi_t - _pi[_target[ta]] < 0)
1509                    ahead += _res_cap[ta];
1510                  if (ahead >= delta) break;
1511                }
1512                if (ahead < 0) ahead = 0;
1513
1514                // Push flow along the arc
1515                if (ahead < delta && !hyper[t]) {
1516                  _res_cap[a] -= ahead;
1517                  _res_cap[_reverse[a]] += ahead;
1518                  _excess[n] -= ahead;
1519                  _excess[t] += ahead;
1520                  _active_nodes.push_front(t);
1521                  hyper[t] = true;
1522                  hyper_cost[t] = _cost[a] + pi_n - pi_t;
1523                  _next_out[n] = a;
1524                  goto next_node;
1525                } else {
1526                  _res_cap[a] -= delta;
1527                  _res_cap[_reverse[a]] += delta;
1528                  _excess[n] -= delta;
1529                  _excess[t] += delta;
1530                  if (_excess[t] > 0 && _excess[t] <= delta)
1531                    _active_nodes.push_back(t);
1532                }
1533
1534                if (_excess[n] == 0) {
1535                  _next_out[n] = a;
1536                  goto remove_nodes;
1537                }
1538              }
1539            }
1540            _next_out[n] = a;
1541          }
1542
1543          // Relabel the node if it is still active (or hyper)
1544          if (_excess[n] > 0 || hyper[n]) {
1545             min_red_cost = hyper[n] ? -hyper_cost[n] :
1546               std::numeric_limits<LargeCost>::max();
1547            for (int a = _first_out[n]; a != last_out; ++a) {
1548              if (_res_cap[a] > 0) {
1549                rc = _cost[a] + pi_n - _pi[_target[a]];
1550                if (rc < min_red_cost) {
1551                  min_red_cost = rc;
1552                }
1553              }
1554            }
1555            _pi[n] -= min_red_cost + _epsilon;
1556            _next_out[n] = _first_out[n];
1557            hyper[n] = false;
1558            ++relabel_cnt;
1559          }
1560
1561          // Remove nodes that are not active nor hyper
1562        remove_nodes:
1563          while ( _active_nodes.size() > 0 &&
1564                  _excess[_active_nodes.front()] <= 0 &&
1565                  !hyper[_active_nodes.front()] ) {
1566            _active_nodes.pop_front();
1567          }
1568
1569          // Global update heuristic
1570          if (relabel_cnt >= next_global_update_limit) {
1571            globalUpdate();
1572            for (int u = 0; u != _res_node_num; ++u)
1573              hyper[u] = false;
1574            next_global_update_limit += global_update_skip;
1575          }
1576        }
1577      }
1578    }
1579
1580  }; //class CostScaling
1581
1582  ///@}
1583
1584} //namespace lemon
1585
1586#endif //LEMON_COST_SCALING_H
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