COIN-OR::LEMON - Graph Library

source: lemon-1.2/lemon/cost_scaling.h @ 825:75e6020b19b1

Last change on this file since 825:75e6020b19b1 was 825:75e6020b19b1, checked in by Peter Kovacs <kpeter@…>, 10 years ago

Add doc for the traits class parameters (#315)

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1/* -*- C++ -*-
2 *
3 * This file is a part of LEMON, a generic C++ optimization library
4 *
5 * Copyright (C) 2003-2008
6 * Egervary Jeno Kombinatorikus Optimalizalasi Kutatocsoport
7 * (Egervary Research Group on Combinatorial Optimization, EGRES).
8 *
9 * Permission to use, modify and distribute this software is granted
10 * provided that this copyright notice appears in all copies. For
11 * precise terms see the accompanying LICENSE file.
12 *
13 * This software is provided "AS IS" with no warranty of any kind,
14 * express or implied, and with no claim as to its suitability for any
15 * purpose.
16 *
17 */
18
19#ifndef LEMON_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  /// Most of the parameters of the problem (except for the digraph)
101  /// can be given using separate functions, and the algorithm can be
102  /// executed using the \ref run() function. If some parameters are not
103  /// specified, then default values will be used.
104  ///
105  /// \tparam GR The digraph type the algorithm runs on.
106  /// \tparam V The number type used for flow amounts, capacity bounds
107  /// and supply values in the algorithm. By default, it is \c int.
108  /// \tparam C The number type used for costs and potentials in the
109  /// algorithm. By default, it is the same as \c V.
110  /// \tparam TR The traits class that defines various types used by the
111  /// algorithm. By default, it is \ref CostScalingDefaultTraits
112  /// "CostScalingDefaultTraits<GR, V, C>".
113  /// In most cases, this parameter should not be set directly,
114  /// consider to use the named template parameters instead.
115  ///
116  /// \warning Both number types must be signed and all input data must
117  /// be integer.
118  /// \warning This algorithm does not support negative costs for such
119  /// arcs that have infinite upper bound.
120  ///
121  /// \note %CostScaling provides three different internal methods,
122  /// from which the most efficient one is used by default.
123  /// For more information, see \ref Method.
124#ifdef DOXYGEN
125  template <typename GR, typename V, typename C, typename TR>
126#else
127  template < typename GR, typename V = int, typename C = V,
128             typename TR = CostScalingDefaultTraits<GR, V, C> >
129#endif
130  class CostScaling
131  {
132  public:
133
134    /// The type of the digraph
135    typedef typename TR::Digraph Digraph;
136    /// The type of the flow amounts, capacity bounds and supply values
137    typedef typename TR::Value Value;
138    /// The type of the arc costs
139    typedef typename TR::Cost Cost;
140
141    /// \brief The large cost type
142    ///
143    /// The large cost type used for internal computations.
144    /// By default, it is \c long \c long if the \c Cost type is integer,
145    /// otherwise it is \c double.
146    typedef typename TR::LargeCost LargeCost;
147
148    /// The \ref CostScalingDefaultTraits "traits class" of the algorithm
149    typedef TR Traits;
150
151  public:
152
153    /// \brief Problem type constants for the \c run() function.
154    ///
155    /// Enum type containing the problem type constants that can be
156    /// returned by the \ref run() function of the algorithm.
157    enum ProblemType {
158      /// The problem has no feasible solution (flow).
159      INFEASIBLE,
160      /// The problem has optimal solution (i.e. it is feasible and
161      /// bounded), and the algorithm has found optimal flow and node
162      /// potentials (primal and dual solutions).
163      OPTIMAL,
164      /// The digraph contains an arc of negative cost and infinite
165      /// upper bound. It means that the objective function is unbounded
166      /// on that arc, however, note that it could actually be bounded
167      /// over the feasible flows, but this algroithm cannot handle
168      /// these cases.
169      UNBOUNDED
170    };
171
172    /// \brief Constants for selecting the internal method.
173    ///
174    /// Enum type containing constants for selecting the internal method
175    /// for the \ref run() function.
176    ///
177    /// \ref CostScaling provides three internal methods that differ mainly
178    /// in their base operations, which are used in conjunction with the
179    /// relabel operation.
180    /// By default, the so called \ref PARTIAL_AUGMENT
181    /// "Partial Augment-Relabel" method is used, which proved to be
182    /// the most efficient and the most robust on various test inputs.
183    /// However, the other methods can be selected using the \ref run()
184    /// function with the proper parameter.
185    enum Method {
186      /// Local push operations are used, i.e. flow is moved only on one
187      /// admissible arc at once.
188      PUSH,
189      /// Augment operations are used, i.e. flow is moved on admissible
190      /// paths from a node with excess to a node with deficit.
191      AUGMENT,
192      /// Partial augment operations are used, i.e. flow is moved on
193      /// admissible paths started from a node with excess, but the
194      /// lengths of these paths are limited. This method can be viewed
195      /// as a combined version of the previous two operations.
196      PARTIAL_AUGMENT
197    };
198
199  private:
200
201    TEMPLATE_DIGRAPH_TYPEDEFS(GR);
202
203    typedef std::vector<int> IntVector;
204    typedef std::vector<char> BoolVector;
205    typedef std::vector<Value> ValueVector;
206    typedef std::vector<Cost> CostVector;
207    typedef std::vector<LargeCost> LargeCostVector;
208
209  private:
210 
211    template <typename KT, typename VT>
212    class StaticVectorMap {
213    public:
214      typedef KT Key;
215      typedef VT Value;
216     
217      StaticVectorMap(std::vector<Value>& v) : _v(v) {}
218     
219      const Value& operator[](const Key& key) const {
220        return _v[StaticDigraph::id(key)];
221      }
222
223      Value& operator[](const Key& key) {
224        return _v[StaticDigraph::id(key)];
225      }
226     
227      void set(const Key& key, const Value& val) {
228        _v[StaticDigraph::id(key)] = val;
229      }
230
231    private:
232      std::vector<Value>& _v;
233    };
234
235    typedef StaticVectorMap<StaticDigraph::Node, LargeCost> LargeCostNodeMap;
236    typedef StaticVectorMap<StaticDigraph::Arc, LargeCost> LargeCostArcMap;
237
238  private:
239
240    // Data related to the underlying digraph
241    const GR &_graph;
242    int _node_num;
243    int _arc_num;
244    int _res_node_num;
245    int _res_arc_num;
246    int _root;
247
248    // Parameters of the problem
249    bool _have_lower;
250    Value _sum_supply;
251
252    // Data structures for storing the digraph
253    IntNodeMap _node_id;
254    IntArcMap _arc_idf;
255    IntArcMap _arc_idb;
256    IntVector _first_out;
257    BoolVector _forward;
258    IntVector _source;
259    IntVector _target;
260    IntVector _reverse;
261
262    // Node and arc data
263    ValueVector _lower;
264    ValueVector _upper;
265    CostVector _scost;
266    ValueVector _supply;
267
268    ValueVector _res_cap;
269    LargeCostVector _cost;
270    LargeCostVector _pi;
271    ValueVector _excess;
272    IntVector _next_out;
273    std::deque<int> _active_nodes;
274
275    // Data for scaling
276    LargeCost _epsilon;
277    int _alpha;
278
279    // Data for a StaticDigraph structure
280    typedef std::pair<int, int> IntPair;
281    StaticDigraph _sgr;
282    std::vector<IntPair> _arc_vec;
283    std::vector<LargeCost> _cost_vec;
284    LargeCostArcMap _cost_map;
285    LargeCostNodeMap _pi_map;
286 
287  public:
288 
289    /// \brief Constant for infinite upper bounds (capacities).
290    ///
291    /// Constant for infinite upper bounds (capacities).
292    /// It is \c std::numeric_limits<Value>::infinity() if available,
293    /// \c std::numeric_limits<Value>::max() otherwise.
294    const Value INF;
295
296  public:
297
298    /// \name Named Template Parameters
299    /// @{
300
301    template <typename T>
302    struct SetLargeCostTraits : public Traits {
303      typedef T LargeCost;
304    };
305
306    /// \brief \ref named-templ-param "Named parameter" for setting
307    /// \c LargeCost type.
308    ///
309    /// \ref named-templ-param "Named parameter" for setting \c LargeCost
310    /// type, which is used for internal computations in the algorithm.
311    /// \c Cost must be convertible to \c LargeCost.
312    template <typename T>
313    struct SetLargeCost
314      : public CostScaling<GR, V, C, SetLargeCostTraits<T> > {
315      typedef  CostScaling<GR, V, C, SetLargeCostTraits<T> > Create;
316    };
317
318    /// @}
319
320  public:
321
322    /// \brief Constructor.
323    ///
324    /// The constructor of the class.
325    ///
326    /// \param graph The digraph the algorithm runs on.
327    CostScaling(const GR& graph) :
328      _graph(graph), _node_id(graph), _arc_idf(graph), _arc_idb(graph),
329      _cost_map(_cost_vec), _pi_map(_pi),
330      INF(std::numeric_limits<Value>::has_infinity ?
331          std::numeric_limits<Value>::infinity() :
332          std::numeric_limits<Value>::max())
333    {
334      // Check the number types
335      LEMON_ASSERT(std::numeric_limits<Value>::is_signed,
336        "The flow type of CostScaling must be signed");
337      LEMON_ASSERT(std::numeric_limits<Cost>::is_signed,
338        "The cost type of CostScaling must be signed");
339
340      // Resize vectors
341      _node_num = countNodes(_graph);
342      _arc_num = countArcs(_graph);
343      _res_node_num = _node_num + 1;
344      _res_arc_num = 2 * (_arc_num + _node_num);
345      _root = _node_num;
346
347      _first_out.resize(_res_node_num + 1);
348      _forward.resize(_res_arc_num);
349      _source.resize(_res_arc_num);
350      _target.resize(_res_arc_num);
351      _reverse.resize(_res_arc_num);
352
353      _lower.resize(_res_arc_num);
354      _upper.resize(_res_arc_num);
355      _scost.resize(_res_arc_num);
356      _supply.resize(_res_node_num);
357     
358      _res_cap.resize(_res_arc_num);
359      _cost.resize(_res_arc_num);
360      _pi.resize(_res_node_num);
361      _excess.resize(_res_node_num);
362      _next_out.resize(_res_node_num);
363
364      _arc_vec.reserve(_res_arc_num);
365      _cost_vec.reserve(_res_arc_num);
366
367      // Copy the graph
368      int i = 0, j = 0, k = 2 * _arc_num + _node_num;
369      for (NodeIt n(_graph); n != INVALID; ++n, ++i) {
370        _node_id[n] = i;
371      }
372      i = 0;
373      for (NodeIt n(_graph); n != INVALID; ++n, ++i) {
374        _first_out[i] = j;
375        for (OutArcIt a(_graph, n); a != INVALID; ++a, ++j) {
376          _arc_idf[a] = j;
377          _forward[j] = true;
378          _source[j] = i;
379          _target[j] = _node_id[_graph.runningNode(a)];
380        }
381        for (InArcIt a(_graph, n); a != INVALID; ++a, ++j) {
382          _arc_idb[a] = j;
383          _forward[j] = false;
384          _source[j] = i;
385          _target[j] = _node_id[_graph.runningNode(a)];
386        }
387        _forward[j] = false;
388        _source[j] = i;
389        _target[j] = _root;
390        _reverse[j] = k;
391        _forward[k] = true;
392        _source[k] = _root;
393        _target[k] = i;
394        _reverse[k] = j;
395        ++j; ++k;
396      }
397      _first_out[i] = j;
398      _first_out[_res_node_num] = k;
399      for (ArcIt a(_graph); a != INVALID; ++a) {
400        int fi = _arc_idf[a];
401        int bi = _arc_idb[a];
402        _reverse[fi] = bi;
403        _reverse[bi] = fi;
404      }
405     
406      // Reset parameters
407      reset();
408    }
409
410    /// \name Parameters
411    /// The parameters of the algorithm can be specified using these
412    /// functions.
413
414    /// @{
415
416    /// \brief Set the lower bounds on the arcs.
417    ///
418    /// This function sets the lower bounds on the arcs.
419    /// If it is not used before calling \ref run(), the lower bounds
420    /// will be set to zero on all arcs.
421    ///
422    /// \param map An arc map storing the lower bounds.
423    /// Its \c Value type must be convertible to the \c Value type
424    /// of the algorithm.
425    ///
426    /// \return <tt>(*this)</tt>
427    template <typename LowerMap>
428    CostScaling& lowerMap(const LowerMap& map) {
429      _have_lower = true;
430      for (ArcIt a(_graph); a != INVALID; ++a) {
431        _lower[_arc_idf[a]] = map[a];
432        _lower[_arc_idb[a]] = map[a];
433      }
434      return *this;
435    }
436
437    /// \brief Set the upper bounds (capacities) on the arcs.
438    ///
439    /// This function sets the upper bounds (capacities) on the arcs.
440    /// If it is not used before calling \ref run(), the upper bounds
441    /// will be set to \ref INF on all arcs (i.e. the flow value will be
442    /// unbounded from above).
443    ///
444    /// \param map An arc map storing the upper bounds.
445    /// Its \c Value type must be convertible to the \c Value type
446    /// of the algorithm.
447    ///
448    /// \return <tt>(*this)</tt>
449    template<typename UpperMap>
450    CostScaling& upperMap(const UpperMap& map) {
451      for (ArcIt a(_graph); a != INVALID; ++a) {
452        _upper[_arc_idf[a]] = map[a];
453      }
454      return *this;
455    }
456
457    /// \brief Set the costs of the arcs.
458    ///
459    /// This function sets the costs of the arcs.
460    /// If it is not used before calling \ref run(), the costs
461    /// will be set to \c 1 on all arcs.
462    ///
463    /// \param map An arc map storing the costs.
464    /// Its \c Value type must be convertible to the \c Cost type
465    /// of the algorithm.
466    ///
467    /// \return <tt>(*this)</tt>
468    template<typename CostMap>
469    CostScaling& costMap(const CostMap& map) {
470      for (ArcIt a(_graph); a != INVALID; ++a) {
471        _scost[_arc_idf[a]] =  map[a];
472        _scost[_arc_idb[a]] = -map[a];
473      }
474      return *this;
475    }
476
477    /// \brief Set the supply values of the nodes.
478    ///
479    /// This function sets the supply values of the nodes.
480    /// If neither this function nor \ref stSupply() is used before
481    /// calling \ref run(), the supply of each node will be set to zero.
482    ///
483    /// \param map A node map storing the supply values.
484    /// Its \c Value type must be convertible to the \c Value type
485    /// of the algorithm.
486    ///
487    /// \return <tt>(*this)</tt>
488    template<typename SupplyMap>
489    CostScaling& supplyMap(const SupplyMap& map) {
490      for (NodeIt n(_graph); n != INVALID; ++n) {
491        _supply[_node_id[n]] = map[n];
492      }
493      return *this;
494    }
495
496    /// \brief Set single source and target nodes and a supply value.
497    ///
498    /// This function sets a single source node and a single target node
499    /// and the required flow value.
500    /// If neither this function nor \ref supplyMap() is used before
501    /// calling \ref run(), the supply of each node will be set to zero.
502    ///
503    /// Using this function has the same effect as using \ref supplyMap()
504    /// with such a map in which \c k is assigned to \c s, \c -k is
505    /// assigned to \c t and all other nodes have zero supply value.
506    ///
507    /// \param s The source node.
508    /// \param t The target node.
509    /// \param k The required amount of flow from node \c s to node \c t
510    /// (i.e. the supply of \c s and the demand of \c t).
511    ///
512    /// \return <tt>(*this)</tt>
513    CostScaling& stSupply(const Node& s, const Node& t, Value k) {
514      for (int i = 0; i != _res_node_num; ++i) {
515        _supply[i] = 0;
516      }
517      _supply[_node_id[s]] =  k;
518      _supply[_node_id[t]] = -k;
519      return *this;
520    }
521   
522    /// @}
523
524    /// \name Execution control
525    /// The algorithm can be executed using \ref run().
526
527    /// @{
528
529    /// \brief Run the algorithm.
530    ///
531    /// This function runs the algorithm.
532    /// The paramters can be specified using functions \ref lowerMap(),
533    /// \ref upperMap(), \ref costMap(), \ref supplyMap(), \ref stSupply().
534    /// For example,
535    /// \code
536    ///   CostScaling<ListDigraph> cs(graph);
537    ///   cs.lowerMap(lower).upperMap(upper).costMap(cost)
538    ///     .supplyMap(sup).run();
539    /// \endcode
540    ///
541    /// This function can be called more than once. All the parameters
542    /// that have been given are kept for the next call, unless
543    /// \ref reset() is called, thus only the modified parameters
544    /// have to be set again. See \ref reset() for examples.
545    /// However, the underlying digraph must not be modified after this
546    /// class have been constructed, since it copies and extends the graph.
547    ///
548    /// \param method The internal method that will be used in the
549    /// algorithm. For more information, see \ref Method.
550    /// \param factor The cost scaling factor. It must be larger than one.
551    ///
552    /// \return \c INFEASIBLE if no feasible flow exists,
553    /// \n \c OPTIMAL if the problem has optimal solution
554    /// (i.e. it is feasible and bounded), and the algorithm has found
555    /// optimal flow and node potentials (primal and dual solutions),
556    /// \n \c UNBOUNDED if the digraph contains an arc of negative cost
557    /// and infinite upper bound. It means that the objective function
558    /// is unbounded on that arc, however, note that it could actually be
559    /// bounded over the feasible flows, but this algroithm cannot handle
560    /// these cases.
561    ///
562    /// \see ProblemType, Method
563    ProblemType run(Method method = PARTIAL_AUGMENT, int factor = 8) {
564      _alpha = factor;
565      ProblemType pt = init();
566      if (pt != OPTIMAL) return pt;
567      start(method);
568      return OPTIMAL;
569    }
570
571    /// \brief Reset all the parameters that have been given before.
572    ///
573    /// This function resets all the paramaters that have been given
574    /// before using functions \ref lowerMap(), \ref upperMap(),
575    /// \ref costMap(), \ref supplyMap(), \ref stSupply().
576    ///
577    /// It is useful for multiple run() calls. If this function is not
578    /// used, all the parameters given before are kept for the next
579    /// \ref run() call.
580    /// However, the underlying digraph must not be modified after this
581    /// class have been constructed, since it copies and extends the graph.
582    ///
583    /// For example,
584    /// \code
585    ///   CostScaling<ListDigraph> cs(graph);
586    ///
587    ///   // First run
588    ///   cs.lowerMap(lower).upperMap(upper).costMap(cost)
589    ///     .supplyMap(sup).run();
590    ///
591    ///   // Run again with modified cost map (reset() is not called,
592    ///   // so only the cost map have to be set again)
593    ///   cost[e] += 100;
594    ///   cs.costMap(cost).run();
595    ///
596    ///   // Run again from scratch using reset()
597    ///   // (the lower bounds will be set to zero on all arcs)
598    ///   cs.reset();
599    ///   cs.upperMap(capacity).costMap(cost)
600    ///     .supplyMap(sup).run();
601    /// \endcode
602    ///
603    /// \return <tt>(*this)</tt>
604    CostScaling& reset() {
605      for (int i = 0; i != _res_node_num; ++i) {
606        _supply[i] = 0;
607      }
608      int limit = _first_out[_root];
609      for (int j = 0; j != limit; ++j) {
610        _lower[j] = 0;
611        _upper[j] = INF;
612        _scost[j] = _forward[j] ? 1 : -1;
613      }
614      for (int j = limit; j != _res_arc_num; ++j) {
615        _lower[j] = 0;
616        _upper[j] = INF;
617        _scost[j] = 0;
618        _scost[_reverse[j]] = 0;
619      }     
620      _have_lower = false;
621      return *this;
622    }
623
624    /// @}
625
626    /// \name Query Functions
627    /// The results of the algorithm can be obtained using these
628    /// functions.\n
629    /// The \ref run() function must be called before using them.
630
631    /// @{
632
633    /// \brief Return the total cost of the found flow.
634    ///
635    /// This function returns the total cost of the found flow.
636    /// Its complexity is O(e).
637    ///
638    /// \note The return type of the function can be specified as a
639    /// template parameter. For example,
640    /// \code
641    ///   cs.totalCost<double>();
642    /// \endcode
643    /// It is useful if the total cost cannot be stored in the \c Cost
644    /// type of the algorithm, which is the default return type of the
645    /// function.
646    ///
647    /// \pre \ref run() must be called before using this function.
648    template <typename Number>
649    Number totalCost() const {
650      Number c = 0;
651      for (ArcIt a(_graph); a != INVALID; ++a) {
652        int i = _arc_idb[a];
653        c += static_cast<Number>(_res_cap[i]) *
654             (-static_cast<Number>(_scost[i]));
655      }
656      return c;
657    }
658
659#ifndef DOXYGEN
660    Cost totalCost() const {
661      return totalCost<Cost>();
662    }
663#endif
664
665    /// \brief Return the flow on the given arc.
666    ///
667    /// This function returns the flow on the given arc.
668    ///
669    /// \pre \ref run() must be called before using this function.
670    Value flow(const Arc& a) const {
671      return _res_cap[_arc_idb[a]];
672    }
673
674    /// \brief Return the flow map (the primal solution).
675    ///
676    /// This function copies the flow value on each arc into the given
677    /// map. The \c Value type of the algorithm must be convertible to
678    /// the \c Value type of the map.
679    ///
680    /// \pre \ref run() must be called before using this function.
681    template <typename FlowMap>
682    void flowMap(FlowMap &map) const {
683      for (ArcIt a(_graph); a != INVALID; ++a) {
684        map.set(a, _res_cap[_arc_idb[a]]);
685      }
686    }
687
688    /// \brief Return the potential (dual value) of the given node.
689    ///
690    /// This function returns the potential (dual value) of the
691    /// given node.
692    ///
693    /// \pre \ref run() must be called before using this function.
694    Cost potential(const Node& n) const {
695      return static_cast<Cost>(_pi[_node_id[n]]);
696    }
697
698    /// \brief Return the potential map (the dual solution).
699    ///
700    /// This function copies the potential (dual value) of each node
701    /// into the given map.
702    /// The \c Cost type of the algorithm must be convertible to the
703    /// \c Value type of the map.
704    ///
705    /// \pre \ref run() must be called before using this function.
706    template <typename PotentialMap>
707    void potentialMap(PotentialMap &map) const {
708      for (NodeIt n(_graph); n != INVALID; ++n) {
709        map.set(n, static_cast<Cost>(_pi[_node_id[n]]));
710      }
711    }
712
713    /// @}
714
715  private:
716
717    // Initialize the algorithm
718    ProblemType init() {
719      if (_res_node_num <= 1) return INFEASIBLE;
720
721      // Check the sum of supply values
722      _sum_supply = 0;
723      for (int i = 0; i != _root; ++i) {
724        _sum_supply += _supply[i];
725      }
726      if (_sum_supply > 0) return INFEASIBLE;
727     
728
729      // Initialize vectors
730      for (int i = 0; i != _res_node_num; ++i) {
731        _pi[i] = 0;
732        _excess[i] = _supply[i];
733      }
734     
735      // Remove infinite upper bounds and check negative arcs
736      const Value MAX = std::numeric_limits<Value>::max();
737      int last_out;
738      if (_have_lower) {
739        for (int i = 0; i != _root; ++i) {
740          last_out = _first_out[i+1];
741          for (int j = _first_out[i]; j != last_out; ++j) {
742            if (_forward[j]) {
743              Value c = _scost[j] < 0 ? _upper[j] : _lower[j];
744              if (c >= MAX) return UNBOUNDED;
745              _excess[i] -= c;
746              _excess[_target[j]] += c;
747            }
748          }
749        }
750      } else {
751        for (int i = 0; i != _root; ++i) {
752          last_out = _first_out[i+1];
753          for (int j = _first_out[i]; j != last_out; ++j) {
754            if (_forward[j] && _scost[j] < 0) {
755              Value c = _upper[j];
756              if (c >= MAX) return UNBOUNDED;
757              _excess[i] -= c;
758              _excess[_target[j]] += c;
759            }
760          }
761        }
762      }
763      Value ex, max_cap = 0;
764      for (int i = 0; i != _res_node_num; ++i) {
765        ex = _excess[i];
766        _excess[i] = 0;
767        if (ex < 0) max_cap -= ex;
768      }
769      for (int j = 0; j != _res_arc_num; ++j) {
770        if (_upper[j] >= MAX) _upper[j] = max_cap;
771      }
772
773      // Initialize the large cost vector and the epsilon parameter
774      _epsilon = 0;
775      LargeCost lc;
776      for (int i = 0; i != _root; ++i) {
777        last_out = _first_out[i+1];
778        for (int j = _first_out[i]; j != last_out; ++j) {
779          lc = static_cast<LargeCost>(_scost[j]) * _res_node_num * _alpha;
780          _cost[j] = lc;
781          if (lc > _epsilon) _epsilon = lc;
782        }
783      }
784      _epsilon /= _alpha;
785
786      // Initialize maps for Circulation and remove non-zero lower bounds
787      ConstMap<Arc, Value> low(0);
788      typedef typename Digraph::template ArcMap<Value> ValueArcMap;
789      typedef typename Digraph::template NodeMap<Value> ValueNodeMap;
790      ValueArcMap cap(_graph), flow(_graph);
791      ValueNodeMap sup(_graph);
792      for (NodeIt n(_graph); n != INVALID; ++n) {
793        sup[n] = _supply[_node_id[n]];
794      }
795      if (_have_lower) {
796        for (ArcIt a(_graph); a != INVALID; ++a) {
797          int j = _arc_idf[a];
798          Value c = _lower[j];
799          cap[a] = _upper[j] - c;
800          sup[_graph.source(a)] -= c;
801          sup[_graph.target(a)] += c;
802        }
803      } else {
804        for (ArcIt a(_graph); a != INVALID; ++a) {
805          cap[a] = _upper[_arc_idf[a]];
806        }
807      }
808
809      // Find a feasible flow using Circulation
810      Circulation<Digraph, ConstMap<Arc, Value>, ValueArcMap, ValueNodeMap>
811        circ(_graph, low, cap, sup);
812      if (!circ.flowMap(flow).run()) return INFEASIBLE;
813
814      // Set residual capacities and handle GEQ supply type
815      if (_sum_supply < 0) {
816        for (ArcIt a(_graph); a != INVALID; ++a) {
817          Value fa = flow[a];
818          _res_cap[_arc_idf[a]] = cap[a] - fa;
819          _res_cap[_arc_idb[a]] = fa;
820          sup[_graph.source(a)] -= fa;
821          sup[_graph.target(a)] += fa;
822        }
823        for (NodeIt n(_graph); n != INVALID; ++n) {
824          _excess[_node_id[n]] = sup[n];
825        }
826        for (int a = _first_out[_root]; a != _res_arc_num; ++a) {
827          int u = _target[a];
828          int ra = _reverse[a];
829          _res_cap[a] = -_sum_supply + 1;
830          _res_cap[ra] = -_excess[u];
831          _cost[a] = 0;
832          _cost[ra] = 0;
833          _excess[u] = 0;
834        }
835      } else {
836        for (ArcIt a(_graph); a != INVALID; ++a) {
837          Value fa = flow[a];
838          _res_cap[_arc_idf[a]] = cap[a] - fa;
839          _res_cap[_arc_idb[a]] = fa;
840        }
841        for (int a = _first_out[_root]; a != _res_arc_num; ++a) {
842          int ra = _reverse[a];
843          _res_cap[a] = 1;
844          _res_cap[ra] = 0;
845          _cost[a] = 0;
846          _cost[ra] = 0;
847        }
848      }
849     
850      return OPTIMAL;
851    }
852
853    // Execute the algorithm and transform the results
854    void start(Method method) {
855      // Maximum path length for partial augment
856      const int MAX_PATH_LENGTH = 4;
857     
858      // Execute the algorithm
859      switch (method) {
860        case PUSH:
861          startPush();
862          break;
863        case AUGMENT:
864          startAugment();
865          break;
866        case PARTIAL_AUGMENT:
867          startAugment(MAX_PATH_LENGTH);
868          break;
869      }
870
871      // Compute node potentials for the original costs
872      _arc_vec.clear();
873      _cost_vec.clear();
874      for (int j = 0; j != _res_arc_num; ++j) {
875        if (_res_cap[j] > 0) {
876          _arc_vec.push_back(IntPair(_source[j], _target[j]));
877          _cost_vec.push_back(_scost[j]);
878        }
879      }
880      _sgr.build(_res_node_num, _arc_vec.begin(), _arc_vec.end());
881
882      typename BellmanFord<StaticDigraph, LargeCostArcMap>
883        ::template SetDistMap<LargeCostNodeMap>::Create bf(_sgr, _cost_map);
884      bf.distMap(_pi_map);
885      bf.init(0);
886      bf.start();
887
888      // Handle non-zero lower bounds
889      if (_have_lower) {
890        int limit = _first_out[_root];
891        for (int j = 0; j != limit; ++j) {
892          if (!_forward[j]) _res_cap[j] += _lower[j];
893        }
894      }
895    }
896
897    /// Execute the algorithm performing augment and relabel operations
898    void startAugment(int max_length = std::numeric_limits<int>::max()) {
899      // Paramters for heuristics
900      const int BF_HEURISTIC_EPSILON_BOUND = 1000;
901      const int BF_HEURISTIC_BOUND_FACTOR  = 3;
902
903      // Perform cost scaling phases
904      IntVector pred_arc(_res_node_num);
905      std::vector<int> path_nodes;
906      for ( ; _epsilon >= 1; _epsilon = _epsilon < _alpha && _epsilon > 1 ?
907                                        1 : _epsilon / _alpha )
908      {
909        // "Early Termination" heuristic: use Bellman-Ford algorithm
910        // to check if the current flow is optimal
911        if (_epsilon <= BF_HEURISTIC_EPSILON_BOUND) {
912          _arc_vec.clear();
913          _cost_vec.clear();
914          for (int j = 0; j != _res_arc_num; ++j) {
915            if (_res_cap[j] > 0) {
916              _arc_vec.push_back(IntPair(_source[j], _target[j]));
917              _cost_vec.push_back(_cost[j] + 1);
918            }
919          }
920          _sgr.build(_res_node_num, _arc_vec.begin(), _arc_vec.end());
921
922          BellmanFord<StaticDigraph, LargeCostArcMap> bf(_sgr, _cost_map);
923          bf.init(0);
924          bool done = false;
925          int K = int(BF_HEURISTIC_BOUND_FACTOR * sqrt(_res_node_num));
926          for (int i = 0; i < K && !done; ++i)
927            done = bf.processNextWeakRound();
928          if (done) break;
929        }
930
931        // Saturate arcs not satisfying the optimality condition
932        for (int a = 0; a != _res_arc_num; ++a) {
933          if (_res_cap[a] > 0 &&
934              _cost[a] + _pi[_source[a]] - _pi[_target[a]] < 0) {
935            Value delta = _res_cap[a];
936            _excess[_source[a]] -= delta;
937            _excess[_target[a]] += delta;
938            _res_cap[a] = 0;
939            _res_cap[_reverse[a]] += delta;
940          }
941        }
942       
943        // Find active nodes (i.e. nodes with positive excess)
944        for (int u = 0; u != _res_node_num; ++u) {
945          if (_excess[u] > 0) _active_nodes.push_back(u);
946        }
947
948        // Initialize the next arcs
949        for (int u = 0; u != _res_node_num; ++u) {
950          _next_out[u] = _first_out[u];
951        }
952
953        // Perform partial augment and relabel operations
954        while (true) {
955          // Select an active node (FIFO selection)
956          while (_active_nodes.size() > 0 &&
957                 _excess[_active_nodes.front()] <= 0) {
958            _active_nodes.pop_front();
959          }
960          if (_active_nodes.size() == 0) break;
961          int start = _active_nodes.front();
962          path_nodes.clear();
963          path_nodes.push_back(start);
964
965          // Find an augmenting path from the start node
966          int tip = start;
967          while (_excess[tip] >= 0 &&
968                 int(path_nodes.size()) <= max_length) {
969            int u;
970            LargeCost min_red_cost, rc;
971            int last_out = _sum_supply < 0 ?
972              _first_out[tip+1] : _first_out[tip+1] - 1;
973            for (int a = _next_out[tip]; a != last_out; ++a) {
974              if (_res_cap[a] > 0 &&
975                  _cost[a] + _pi[_source[a]] - _pi[_target[a]] < 0) {
976                u = _target[a];
977                pred_arc[u] = a;
978                _next_out[tip] = a;
979                tip = u;
980                path_nodes.push_back(tip);
981                goto next_step;
982              }
983            }
984
985            // Relabel tip node
986            min_red_cost = std::numeric_limits<LargeCost>::max() / 2;
987            for (int a = _first_out[tip]; a != last_out; ++a) {
988              rc = _cost[a] + _pi[_source[a]] - _pi[_target[a]];
989              if (_res_cap[a] > 0 && rc < min_red_cost) {
990                min_red_cost = rc;
991              }
992            }
993            _pi[tip] -= min_red_cost + _epsilon;
994
995            // Reset the next arc of tip
996            _next_out[tip] = _first_out[tip];
997
998            // Step back
999            if (tip != start) {
1000              path_nodes.pop_back();
1001              tip = path_nodes.back();
1002            }
1003
1004          next_step: ;
1005          }
1006
1007          // Augment along the found path (as much flow as possible)
1008          Value delta;
1009          int u, v = path_nodes.front(), pa;
1010          for (int i = 1; i < int(path_nodes.size()); ++i) {
1011            u = v;
1012            v = path_nodes[i];
1013            pa = pred_arc[v];
1014            delta = std::min(_res_cap[pa], _excess[u]);
1015            _res_cap[pa] -= delta;
1016            _res_cap[_reverse[pa]] += delta;
1017            _excess[u] -= delta;
1018            _excess[v] += delta;
1019            if (_excess[v] > 0 && _excess[v] <= delta)
1020              _active_nodes.push_back(v);
1021          }
1022        }
1023      }
1024    }
1025
1026    /// Execute the algorithm performing push and relabel operations
1027    void startPush() {
1028      // Paramters for heuristics
1029      const int BF_HEURISTIC_EPSILON_BOUND = 1000;
1030      const int BF_HEURISTIC_BOUND_FACTOR  = 3;
1031
1032      // Perform cost scaling phases
1033      BoolVector hyper(_res_node_num, false);
1034      for ( ; _epsilon >= 1; _epsilon = _epsilon < _alpha && _epsilon > 1 ?
1035                                        1 : _epsilon / _alpha )
1036      {
1037        // "Early Termination" heuristic: use Bellman-Ford algorithm
1038        // to check if the current flow is optimal
1039        if (_epsilon <= BF_HEURISTIC_EPSILON_BOUND) {
1040          _arc_vec.clear();
1041          _cost_vec.clear();
1042          for (int j = 0; j != _res_arc_num; ++j) {
1043            if (_res_cap[j] > 0) {
1044              _arc_vec.push_back(IntPair(_source[j], _target[j]));
1045              _cost_vec.push_back(_cost[j] + 1);
1046            }
1047          }
1048          _sgr.build(_res_node_num, _arc_vec.begin(), _arc_vec.end());
1049
1050          BellmanFord<StaticDigraph, LargeCostArcMap> bf(_sgr, _cost_map);
1051          bf.init(0);
1052          bool done = false;
1053          int K = int(BF_HEURISTIC_BOUND_FACTOR * sqrt(_res_node_num));
1054          for (int i = 0; i < K && !done; ++i)
1055            done = bf.processNextWeakRound();
1056          if (done) break;
1057        }
1058
1059        // Saturate arcs not satisfying the optimality condition
1060        for (int a = 0; a != _res_arc_num; ++a) {
1061          if (_res_cap[a] > 0 &&
1062              _cost[a] + _pi[_source[a]] - _pi[_target[a]] < 0) {
1063            Value delta = _res_cap[a];
1064            _excess[_source[a]] -= delta;
1065            _excess[_target[a]] += delta;
1066            _res_cap[a] = 0;
1067            _res_cap[_reverse[a]] += delta;
1068          }
1069        }
1070
1071        // Find active nodes (i.e. nodes with positive excess)
1072        for (int u = 0; u != _res_node_num; ++u) {
1073          if (_excess[u] > 0) _active_nodes.push_back(u);
1074        }
1075
1076        // Initialize the next arcs
1077        for (int u = 0; u != _res_node_num; ++u) {
1078          _next_out[u] = _first_out[u];
1079        }
1080
1081        // Perform push and relabel operations
1082        while (_active_nodes.size() > 0) {
1083          LargeCost min_red_cost, rc;
1084          Value delta;
1085          int n, t, a, last_out = _res_arc_num;
1086
1087          // Select an active node (FIFO selection)
1088        next_node:
1089          n = _active_nodes.front();
1090          last_out = _sum_supply < 0 ?
1091            _first_out[n+1] : _first_out[n+1] - 1;
1092
1093          // Perform push operations if there are admissible arcs
1094          if (_excess[n] > 0) {
1095            for (a = _next_out[n]; a != last_out; ++a) {
1096              if (_res_cap[a] > 0 &&
1097                  _cost[a] + _pi[_source[a]] - _pi[_target[a]] < 0) {
1098                delta = std::min(_res_cap[a], _excess[n]);
1099                t = _target[a];
1100
1101                // Push-look-ahead heuristic
1102                Value ahead = -_excess[t];
1103                int last_out_t = _sum_supply < 0 ?
1104                  _first_out[t+1] : _first_out[t+1] - 1;
1105                for (int ta = _next_out[t]; ta != last_out_t; ++ta) {
1106                  if (_res_cap[ta] > 0 &&
1107                      _cost[ta] + _pi[_source[ta]] - _pi[_target[ta]] < 0)
1108                    ahead += _res_cap[ta];
1109                  if (ahead >= delta) break;
1110                }
1111                if (ahead < 0) ahead = 0;
1112
1113                // Push flow along the arc
1114                if (ahead < delta) {
1115                  _res_cap[a] -= ahead;
1116                  _res_cap[_reverse[a]] += ahead;
1117                  _excess[n] -= ahead;
1118                  _excess[t] += ahead;
1119                  _active_nodes.push_front(t);
1120                  hyper[t] = true;
1121                  _next_out[n] = a;
1122                  goto next_node;
1123                } else {
1124                  _res_cap[a] -= delta;
1125                  _res_cap[_reverse[a]] += delta;
1126                  _excess[n] -= delta;
1127                  _excess[t] += delta;
1128                  if (_excess[t] > 0 && _excess[t] <= delta)
1129                    _active_nodes.push_back(t);
1130                }
1131
1132                if (_excess[n] == 0) {
1133                  _next_out[n] = a;
1134                  goto remove_nodes;
1135                }
1136              }
1137            }
1138            _next_out[n] = a;
1139          }
1140
1141          // Relabel the node if it is still active (or hyper)
1142          if (_excess[n] > 0 || hyper[n]) {
1143            min_red_cost = std::numeric_limits<LargeCost>::max() / 2;
1144            for (int a = _first_out[n]; a != last_out; ++a) {
1145              rc = _cost[a] + _pi[_source[a]] - _pi[_target[a]];
1146              if (_res_cap[a] > 0 && rc < min_red_cost) {
1147                min_red_cost = rc;
1148              }
1149            }
1150            _pi[n] -= min_red_cost + _epsilon;
1151            hyper[n] = false;
1152
1153            // Reset the next arc
1154            _next_out[n] = _first_out[n];
1155          }
1156       
1157          // Remove nodes that are not active nor hyper
1158        remove_nodes:
1159          while ( _active_nodes.size() > 0 &&
1160                  _excess[_active_nodes.front()] <= 0 &&
1161                  !hyper[_active_nodes.front()] ) {
1162            _active_nodes.pop_front();
1163          }
1164        }
1165      }
1166    }
1167
1168  }; //class CostScaling
1169
1170  ///@}
1171
1172} //namespace lemon
1173
1174#endif //LEMON_COST_SCALING_H
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