COIN-OR::LEMON - Graph Library

source: lemon-1.2/lemon/cost_scaling.h @ 864:d3ea191c3412

Last change on this file since 864:d3ea191c3412 was 863:a93f1a27d831, checked in by Peter Kovacs <kpeter@…>, 10 years ago

Fix gcc 3.3 compilation error (#354)

gcc 3.3 requires that a class has a default constructor if it has
template named parameters. (That constructor can be protected.)

File size: 42.5 KB
Line 
1/* -*- C++ -*-
2 *
3 * This file is a part of LEMON, a generic C++ optimization library
4 *
5 * Copyright (C) 2003-2008
6 * Egervary Jeno Kombinatorikus Optimalizalasi Kutatocsoport
7 * (Egervary Research Group on Combinatorial Optimization, EGRES).
8 *
9 * Permission to use, modify and distribute this software is granted
10 * provided that this copyright notice appears in all copies. For
11 * precise terms see the accompanying LICENSE file.
12 *
13 * This software is provided "AS IS" with no warranty of any kind,
14 * express or implied, and with no claim as to its suitability for any
15 * purpose.
16 *
17 */
18
19#ifndef LEMON_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<Value> ValueVector;
205    typedef std::vector<Cost> CostVector;
206    typedef std::vector<LargeCost> LargeCostVector;
207    typedef std::vector<char> BoolVector;
208    // Note: vector<char> is used instead of vector<bool> for efficiency reasons
209
210  private:
211 
212    template <typename KT, typename VT>
213    class StaticVectorMap {
214    public:
215      typedef KT Key;
216      typedef VT Value;
217     
218      StaticVectorMap(std::vector<Value>& v) : _v(v) {}
219     
220      const Value& operator[](const Key& key) const {
221        return _v[StaticDigraph::id(key)];
222      }
223
224      Value& operator[](const Key& key) {
225        return _v[StaticDigraph::id(key)];
226      }
227     
228      void set(const Key& key, const Value& val) {
229        _v[StaticDigraph::id(key)] = val;
230      }
231
232    private:
233      std::vector<Value>& _v;
234    };
235
236    typedef StaticVectorMap<StaticDigraph::Node, LargeCost> LargeCostNodeMap;
237    typedef StaticVectorMap<StaticDigraph::Arc, LargeCost> LargeCostArcMap;
238
239  private:
240
241    // Data related to the underlying digraph
242    const GR &_graph;
243    int _node_num;
244    int _arc_num;
245    int _res_node_num;
246    int _res_arc_num;
247    int _root;
248
249    // Parameters of the problem
250    bool _have_lower;
251    Value _sum_supply;
252    int _sup_node_num;
253
254    // Data structures for storing the digraph
255    IntNodeMap _node_id;
256    IntArcMap _arc_idf;
257    IntArcMap _arc_idb;
258    IntVector _first_out;
259    BoolVector _forward;
260    IntVector _source;
261    IntVector _target;
262    IntVector _reverse;
263
264    // Node and arc data
265    ValueVector _lower;
266    ValueVector _upper;
267    CostVector _scost;
268    ValueVector _supply;
269
270    ValueVector _res_cap;
271    LargeCostVector _cost;
272    LargeCostVector _pi;
273    ValueVector _excess;
274    IntVector _next_out;
275    std::deque<int> _active_nodes;
276
277    // Data for scaling
278    LargeCost _epsilon;
279    int _alpha;
280
281    IntVector _buckets;
282    IntVector _bucket_next;
283    IntVector _bucket_prev;
284    IntVector _rank;
285    int _max_rank;
286 
287    // Data for a StaticDigraph structure
288    typedef std::pair<int, int> IntPair;
289    StaticDigraph _sgr;
290    std::vector<IntPair> _arc_vec;
291    std::vector<LargeCost> _cost_vec;
292    LargeCostArcMap _cost_map;
293    LargeCostNodeMap _pi_map;
294 
295  public:
296 
297    /// \brief Constant for infinite upper bounds (capacities).
298    ///
299    /// Constant for infinite upper bounds (capacities).
300    /// It is \c std::numeric_limits<Value>::infinity() if available,
301    /// \c std::numeric_limits<Value>::max() otherwise.
302    const Value INF;
303
304  public:
305
306    /// \name Named Template Parameters
307    /// @{
308
309    template <typename T>
310    struct SetLargeCostTraits : public Traits {
311      typedef T LargeCost;
312    };
313
314    /// \brief \ref named-templ-param "Named parameter" for setting
315    /// \c LargeCost type.
316    ///
317    /// \ref named-templ-param "Named parameter" for setting \c LargeCost
318    /// type, which is used for internal computations in the algorithm.
319    /// \c Cost must be convertible to \c LargeCost.
320    template <typename T>
321    struct SetLargeCost
322      : public CostScaling<GR, V, C, SetLargeCostTraits<T> > {
323      typedef  CostScaling<GR, V, C, SetLargeCostTraits<T> > Create;
324    };
325
326    /// @}
327
328  protected:
329
330    CostScaling() {}
331
332  public:
333
334    /// \brief Constructor.
335    ///
336    /// The constructor of the class.
337    ///
338    /// \param graph The digraph the algorithm runs on.
339    CostScaling(const GR& graph) :
340      _graph(graph), _node_id(graph), _arc_idf(graph), _arc_idb(graph),
341      _cost_map(_cost_vec), _pi_map(_pi),
342      INF(std::numeric_limits<Value>::has_infinity ?
343          std::numeric_limits<Value>::infinity() :
344          std::numeric_limits<Value>::max())
345    {
346      // Check the number types
347      LEMON_ASSERT(std::numeric_limits<Value>::is_signed,
348        "The flow type of CostScaling must be signed");
349      LEMON_ASSERT(std::numeric_limits<Cost>::is_signed,
350        "The cost type of CostScaling must be signed");
351     
352      // Reset data structures
353      reset();
354    }
355
356    /// \name Parameters
357    /// The parameters of the algorithm can be specified using these
358    /// functions.
359
360    /// @{
361
362    /// \brief Set the lower bounds on the arcs.
363    ///
364    /// This function sets the lower bounds on the arcs.
365    /// If it is not used before calling \ref run(), the lower bounds
366    /// will be set to zero on all arcs.
367    ///
368    /// \param map An arc map storing the lower bounds.
369    /// Its \c Value type must be convertible to the \c Value type
370    /// of the algorithm.
371    ///
372    /// \return <tt>(*this)</tt>
373    template <typename LowerMap>
374    CostScaling& lowerMap(const LowerMap& map) {
375      _have_lower = true;
376      for (ArcIt a(_graph); a != INVALID; ++a) {
377        _lower[_arc_idf[a]] = map[a];
378        _lower[_arc_idb[a]] = map[a];
379      }
380      return *this;
381    }
382
383    /// \brief Set the upper bounds (capacities) on the arcs.
384    ///
385    /// This function sets the upper bounds (capacities) on the arcs.
386    /// If it is not used before calling \ref run(), the upper bounds
387    /// will be set to \ref INF on all arcs (i.e. the flow value will be
388    /// unbounded from above).
389    ///
390    /// \param map An arc map storing the upper bounds.
391    /// Its \c Value type must be convertible to the \c Value type
392    /// of the algorithm.
393    ///
394    /// \return <tt>(*this)</tt>
395    template<typename UpperMap>
396    CostScaling& upperMap(const UpperMap& map) {
397      for (ArcIt a(_graph); a != INVALID; ++a) {
398        _upper[_arc_idf[a]] = map[a];
399      }
400      return *this;
401    }
402
403    /// \brief Set the costs of the arcs.
404    ///
405    /// This function sets the costs of the arcs.
406    /// If it is not used before calling \ref run(), the costs
407    /// will be set to \c 1 on all arcs.
408    ///
409    /// \param map An arc map storing the costs.
410    /// Its \c Value type must be convertible to the \c Cost type
411    /// of the algorithm.
412    ///
413    /// \return <tt>(*this)</tt>
414    template<typename CostMap>
415    CostScaling& costMap(const CostMap& map) {
416      for (ArcIt a(_graph); a != INVALID; ++a) {
417        _scost[_arc_idf[a]] =  map[a];
418        _scost[_arc_idb[a]] = -map[a];
419      }
420      return *this;
421    }
422
423    /// \brief Set the supply values of the nodes.
424    ///
425    /// This function sets the supply values of the nodes.
426    /// If neither this function nor \ref stSupply() is used before
427    /// calling \ref run(), the supply of each node will be set to zero.
428    ///
429    /// \param map A node map storing the supply values.
430    /// Its \c Value type must be convertible to the \c Value type
431    /// of the algorithm.
432    ///
433    /// \return <tt>(*this)</tt>
434    template<typename SupplyMap>
435    CostScaling& supplyMap(const SupplyMap& map) {
436      for (NodeIt n(_graph); n != INVALID; ++n) {
437        _supply[_node_id[n]] = map[n];
438      }
439      return *this;
440    }
441
442    /// \brief Set single source and target nodes and a supply value.
443    ///
444    /// This function sets a single source node and a single target node
445    /// and the required flow value.
446    /// If neither this function nor \ref supplyMap() is used before
447    /// calling \ref run(), the supply of each node will be set to zero.
448    ///
449    /// Using this function has the same effect as using \ref supplyMap()
450    /// with such a map in which \c k is assigned to \c s, \c -k is
451    /// assigned to \c t and all other nodes have zero supply value.
452    ///
453    /// \param s The source node.
454    /// \param t The target node.
455    /// \param k The required amount of flow from node \c s to node \c t
456    /// (i.e. the supply of \c s and the demand of \c t).
457    ///
458    /// \return <tt>(*this)</tt>
459    CostScaling& stSupply(const Node& s, const Node& t, Value k) {
460      for (int i = 0; i != _res_node_num; ++i) {
461        _supply[i] = 0;
462      }
463      _supply[_node_id[s]] =  k;
464      _supply[_node_id[t]] = -k;
465      return *this;
466    }
467   
468    /// @}
469
470    /// \name Execution control
471    /// The algorithm can be executed using \ref run().
472
473    /// @{
474
475    /// \brief Run the algorithm.
476    ///
477    /// This function runs the algorithm.
478    /// The paramters can be specified using functions \ref lowerMap(),
479    /// \ref upperMap(), \ref costMap(), \ref supplyMap(), \ref stSupply().
480    /// For example,
481    /// \code
482    ///   CostScaling<ListDigraph> cs(graph);
483    ///   cs.lowerMap(lower).upperMap(upper).costMap(cost)
484    ///     .supplyMap(sup).run();
485    /// \endcode
486    ///
487    /// This function can be called more than once. All the given parameters
488    /// are kept for the next call, unless \ref resetParams() or \ref reset()
489    /// is used, thus only the modified parameters have to be set again.
490    /// If the underlying digraph was also modified after the construction
491    /// of the class (or the last \ref reset() call), then the \ref reset()
492    /// function must be called.
493    ///
494    /// \param method The internal method that will be used in the
495    /// algorithm. For more information, see \ref Method.
496    /// \param factor The cost scaling factor. It must be larger than one.
497    ///
498    /// \return \c INFEASIBLE if no feasible flow exists,
499    /// \n \c OPTIMAL if the problem has optimal solution
500    /// (i.e. it is feasible and bounded), and the algorithm has found
501    /// optimal flow and node potentials (primal and dual solutions),
502    /// \n \c UNBOUNDED if the digraph contains an arc of negative cost
503    /// and infinite upper bound. It means that the objective function
504    /// is unbounded on that arc, however, note that it could actually be
505    /// bounded over the feasible flows, but this algroithm cannot handle
506    /// these cases.
507    ///
508    /// \see ProblemType, Method
509    /// \see resetParams(), reset()
510    ProblemType run(Method method = PARTIAL_AUGMENT, int factor = 8) {
511      _alpha = factor;
512      ProblemType pt = init();
513      if (pt != OPTIMAL) return pt;
514      start(method);
515      return OPTIMAL;
516    }
517
518    /// \brief Reset all the parameters that have been given before.
519    ///
520    /// This function resets all the paramaters that have been given
521    /// before using functions \ref lowerMap(), \ref upperMap(),
522    /// \ref costMap(), \ref supplyMap(), \ref stSupply().
523    ///
524    /// It is useful for multiple \ref run() calls. Basically, all the given
525    /// parameters are kept for the next \ref run() call, unless
526    /// \ref resetParams() or \ref reset() is used.
527    /// If the underlying digraph was also modified after the construction
528    /// of the class or the last \ref reset() call, then the \ref reset()
529    /// function must be used, otherwise \ref resetParams() is sufficient.
530    ///
531    /// For example,
532    /// \code
533    ///   CostScaling<ListDigraph> cs(graph);
534    ///
535    ///   // First run
536    ///   cs.lowerMap(lower).upperMap(upper).costMap(cost)
537    ///     .supplyMap(sup).run();
538    ///
539    ///   // Run again with modified cost map (resetParams() is not called,
540    ///   // so only the cost map have to be set again)
541    ///   cost[e] += 100;
542    ///   cs.costMap(cost).run();
543    ///
544    ///   // Run again from scratch using resetParams()
545    ///   // (the lower bounds will be set to zero on all arcs)
546    ///   cs.resetParams();
547    ///   cs.upperMap(capacity).costMap(cost)
548    ///     .supplyMap(sup).run();
549    /// \endcode
550    ///
551    /// \return <tt>(*this)</tt>
552    ///
553    /// \see reset(), run()
554    CostScaling& resetParams() {
555      for (int i = 0; i != _res_node_num; ++i) {
556        _supply[i] = 0;
557      }
558      int limit = _first_out[_root];
559      for (int j = 0; j != limit; ++j) {
560        _lower[j] = 0;
561        _upper[j] = INF;
562        _scost[j] = _forward[j] ? 1 : -1;
563      }
564      for (int j = limit; j != _res_arc_num; ++j) {
565        _lower[j] = 0;
566        _upper[j] = INF;
567        _scost[j] = 0;
568        _scost[_reverse[j]] = 0;
569      }     
570      _have_lower = false;
571      return *this;
572    }
573
574    /// \brief Reset all the parameters that have been given before.
575    ///
576    /// This function resets all the paramaters that have been given
577    /// before using functions \ref lowerMap(), \ref upperMap(),
578    /// \ref costMap(), \ref supplyMap(), \ref stSupply().
579    ///
580    /// It is useful for multiple run() calls. If this function is not
581    /// used, all the parameters given before are kept for the next
582    /// \ref run() call.
583    /// However, the underlying digraph must not be modified after this
584    /// class have been constructed, since it copies and extends the graph.
585    /// \return <tt>(*this)</tt>
586    CostScaling& reset() {
587      // Resize vectors
588      _node_num = countNodes(_graph);
589      _arc_num = countArcs(_graph);
590      _res_node_num = _node_num + 1;
591      _res_arc_num = 2 * (_arc_num + _node_num);
592      _root = _node_num;
593
594      _first_out.resize(_res_node_num + 1);
595      _forward.resize(_res_arc_num);
596      _source.resize(_res_arc_num);
597      _target.resize(_res_arc_num);
598      _reverse.resize(_res_arc_num);
599
600      _lower.resize(_res_arc_num);
601      _upper.resize(_res_arc_num);
602      _scost.resize(_res_arc_num);
603      _supply.resize(_res_node_num);
604     
605      _res_cap.resize(_res_arc_num);
606      _cost.resize(_res_arc_num);
607      _pi.resize(_res_node_num);
608      _excess.resize(_res_node_num);
609      _next_out.resize(_res_node_num);
610
611      _arc_vec.reserve(_res_arc_num);
612      _cost_vec.reserve(_res_arc_num);
613
614      // Copy the graph
615      int i = 0, j = 0, k = 2 * _arc_num + _node_num;
616      for (NodeIt n(_graph); n != INVALID; ++n, ++i) {
617        _node_id[n] = i;
618      }
619      i = 0;
620      for (NodeIt n(_graph); n != INVALID; ++n, ++i) {
621        _first_out[i] = j;
622        for (OutArcIt a(_graph, n); a != INVALID; ++a, ++j) {
623          _arc_idf[a] = j;
624          _forward[j] = true;
625          _source[j] = i;
626          _target[j] = _node_id[_graph.runningNode(a)];
627        }
628        for (InArcIt a(_graph, n); a != INVALID; ++a, ++j) {
629          _arc_idb[a] = j;
630          _forward[j] = false;
631          _source[j] = i;
632          _target[j] = _node_id[_graph.runningNode(a)];
633        }
634        _forward[j] = false;
635        _source[j] = i;
636        _target[j] = _root;
637        _reverse[j] = k;
638        _forward[k] = true;
639        _source[k] = _root;
640        _target[k] = i;
641        _reverse[k] = j;
642        ++j; ++k;
643      }
644      _first_out[i] = j;
645      _first_out[_res_node_num] = k;
646      for (ArcIt a(_graph); a != INVALID; ++a) {
647        int fi = _arc_idf[a];
648        int bi = _arc_idb[a];
649        _reverse[fi] = bi;
650        _reverse[bi] = fi;
651      }
652     
653      // Reset parameters
654      resetParams();
655      return *this;
656    }
657
658    /// @}
659
660    /// \name Query Functions
661    /// The results of the algorithm can be obtained using these
662    /// functions.\n
663    /// The \ref run() function must be called before using them.
664
665    /// @{
666
667    /// \brief Return the total cost of the found flow.
668    ///
669    /// This function returns the total cost of the found flow.
670    /// Its complexity is O(e).
671    ///
672    /// \note The return type of the function can be specified as a
673    /// template parameter. For example,
674    /// \code
675    ///   cs.totalCost<double>();
676    /// \endcode
677    /// It is useful if the total cost cannot be stored in the \c Cost
678    /// type of the algorithm, which is the default return type of the
679    /// function.
680    ///
681    /// \pre \ref run() must be called before using this function.
682    template <typename Number>
683    Number totalCost() const {
684      Number c = 0;
685      for (ArcIt a(_graph); a != INVALID; ++a) {
686        int i = _arc_idb[a];
687        c += static_cast<Number>(_res_cap[i]) *
688             (-static_cast<Number>(_scost[i]));
689      }
690      return c;
691    }
692
693#ifndef DOXYGEN
694    Cost totalCost() const {
695      return totalCost<Cost>();
696    }
697#endif
698
699    /// \brief Return the flow on the given arc.
700    ///
701    /// This function returns the flow on the given arc.
702    ///
703    /// \pre \ref run() must be called before using this function.
704    Value flow(const Arc& a) const {
705      return _res_cap[_arc_idb[a]];
706    }
707
708    /// \brief Return the flow map (the primal solution).
709    ///
710    /// This function copies the flow value on each arc into the given
711    /// map. The \c Value type of the algorithm must be convertible to
712    /// the \c Value type of the map.
713    ///
714    /// \pre \ref run() must be called before using this function.
715    template <typename FlowMap>
716    void flowMap(FlowMap &map) const {
717      for (ArcIt a(_graph); a != INVALID; ++a) {
718        map.set(a, _res_cap[_arc_idb[a]]);
719      }
720    }
721
722    /// \brief Return the potential (dual value) of the given node.
723    ///
724    /// This function returns the potential (dual value) of the
725    /// given node.
726    ///
727    /// \pre \ref run() must be called before using this function.
728    Cost potential(const Node& n) const {
729      return static_cast<Cost>(_pi[_node_id[n]]);
730    }
731
732    /// \brief Return the potential map (the dual solution).
733    ///
734    /// This function copies the potential (dual value) of each node
735    /// into the given map.
736    /// The \c Cost type of the algorithm must be convertible to the
737    /// \c Value type of the map.
738    ///
739    /// \pre \ref run() must be called before using this function.
740    template <typename PotentialMap>
741    void potentialMap(PotentialMap &map) const {
742      for (NodeIt n(_graph); n != INVALID; ++n) {
743        map.set(n, static_cast<Cost>(_pi[_node_id[n]]));
744      }
745    }
746
747    /// @}
748
749  private:
750
751    // Initialize the algorithm
752    ProblemType init() {
753      if (_res_node_num <= 1) return INFEASIBLE;
754
755      // Check the sum of supply values
756      _sum_supply = 0;
757      for (int i = 0; i != _root; ++i) {
758        _sum_supply += _supply[i];
759      }
760      if (_sum_supply > 0) return INFEASIBLE;
761     
762
763      // Initialize vectors
764      for (int i = 0; i != _res_node_num; ++i) {
765        _pi[i] = 0;
766        _excess[i] = _supply[i];
767      }
768     
769      // Remove infinite upper bounds and check negative arcs
770      const Value MAX = std::numeric_limits<Value>::max();
771      int last_out;
772      if (_have_lower) {
773        for (int i = 0; i != _root; ++i) {
774          last_out = _first_out[i+1];
775          for (int j = _first_out[i]; j != last_out; ++j) {
776            if (_forward[j]) {
777              Value c = _scost[j] < 0 ? _upper[j] : _lower[j];
778              if (c >= MAX) return UNBOUNDED;
779              _excess[i] -= c;
780              _excess[_target[j]] += c;
781            }
782          }
783        }
784      } else {
785        for (int i = 0; i != _root; ++i) {
786          last_out = _first_out[i+1];
787          for (int j = _first_out[i]; j != last_out; ++j) {
788            if (_forward[j] && _scost[j] < 0) {
789              Value c = _upper[j];
790              if (c >= MAX) return UNBOUNDED;
791              _excess[i] -= c;
792              _excess[_target[j]] += c;
793            }
794          }
795        }
796      }
797      Value ex, max_cap = 0;
798      for (int i = 0; i != _res_node_num; ++i) {
799        ex = _excess[i];
800        _excess[i] = 0;
801        if (ex < 0) max_cap -= ex;
802      }
803      for (int j = 0; j != _res_arc_num; ++j) {
804        if (_upper[j] >= MAX) _upper[j] = max_cap;
805      }
806
807      // Initialize the large cost vector and the epsilon parameter
808      _epsilon = 0;
809      LargeCost lc;
810      for (int i = 0; i != _root; ++i) {
811        last_out = _first_out[i+1];
812        for (int j = _first_out[i]; j != last_out; ++j) {
813          lc = static_cast<LargeCost>(_scost[j]) * _res_node_num * _alpha;
814          _cost[j] = lc;
815          if (lc > _epsilon) _epsilon = lc;
816        }
817      }
818      _epsilon /= _alpha;
819
820      // Initialize maps for Circulation and remove non-zero lower bounds
821      ConstMap<Arc, Value> low(0);
822      typedef typename Digraph::template ArcMap<Value> ValueArcMap;
823      typedef typename Digraph::template NodeMap<Value> ValueNodeMap;
824      ValueArcMap cap(_graph), flow(_graph);
825      ValueNodeMap sup(_graph);
826      for (NodeIt n(_graph); n != INVALID; ++n) {
827        sup[n] = _supply[_node_id[n]];
828      }
829      if (_have_lower) {
830        for (ArcIt a(_graph); a != INVALID; ++a) {
831          int j = _arc_idf[a];
832          Value c = _lower[j];
833          cap[a] = _upper[j] - c;
834          sup[_graph.source(a)] -= c;
835          sup[_graph.target(a)] += c;
836        }
837      } else {
838        for (ArcIt a(_graph); a != INVALID; ++a) {
839          cap[a] = _upper[_arc_idf[a]];
840        }
841      }
842
843      _sup_node_num = 0;
844      for (NodeIt n(_graph); n != INVALID; ++n) {
845        if (sup[n] > 0) ++_sup_node_num;
846      }
847
848      // Find a feasible flow using Circulation
849      Circulation<Digraph, ConstMap<Arc, Value>, ValueArcMap, ValueNodeMap>
850        circ(_graph, low, cap, sup);
851      if (!circ.flowMap(flow).run()) return INFEASIBLE;
852
853      // Set residual capacities and handle GEQ supply type
854      if (_sum_supply < 0) {
855        for (ArcIt a(_graph); a != INVALID; ++a) {
856          Value fa = flow[a];
857          _res_cap[_arc_idf[a]] = cap[a] - fa;
858          _res_cap[_arc_idb[a]] = fa;
859          sup[_graph.source(a)] -= fa;
860          sup[_graph.target(a)] += fa;
861        }
862        for (NodeIt n(_graph); n != INVALID; ++n) {
863          _excess[_node_id[n]] = sup[n];
864        }
865        for (int a = _first_out[_root]; a != _res_arc_num; ++a) {
866          int u = _target[a];
867          int ra = _reverse[a];
868          _res_cap[a] = -_sum_supply + 1;
869          _res_cap[ra] = -_excess[u];
870          _cost[a] = 0;
871          _cost[ra] = 0;
872          _excess[u] = 0;
873        }
874      } else {
875        for (ArcIt a(_graph); a != INVALID; ++a) {
876          Value fa = flow[a];
877          _res_cap[_arc_idf[a]] = cap[a] - fa;
878          _res_cap[_arc_idb[a]] = fa;
879        }
880        for (int a = _first_out[_root]; a != _res_arc_num; ++a) {
881          int ra = _reverse[a];
882          _res_cap[a] = 0;
883          _res_cap[ra] = 0;
884          _cost[a] = 0;
885          _cost[ra] = 0;
886        }
887      }
888     
889      return OPTIMAL;
890    }
891
892    // Execute the algorithm and transform the results
893    void start(Method method) {
894      // Maximum path length for partial augment
895      const int MAX_PATH_LENGTH = 4;
896
897      // Initialize data structures for buckets     
898      _max_rank = _alpha * _res_node_num;
899      _buckets.resize(_max_rank);
900      _bucket_next.resize(_res_node_num + 1);
901      _bucket_prev.resize(_res_node_num + 1);
902      _rank.resize(_res_node_num + 1);
903 
904      // Execute the algorithm
905      switch (method) {
906        case PUSH:
907          startPush();
908          break;
909        case AUGMENT:
910          startAugment();
911          break;
912        case PARTIAL_AUGMENT:
913          startAugment(MAX_PATH_LENGTH);
914          break;
915      }
916
917      // Compute node potentials for the original costs
918      _arc_vec.clear();
919      _cost_vec.clear();
920      for (int j = 0; j != _res_arc_num; ++j) {
921        if (_res_cap[j] > 0) {
922          _arc_vec.push_back(IntPair(_source[j], _target[j]));
923          _cost_vec.push_back(_scost[j]);
924        }
925      }
926      _sgr.build(_res_node_num, _arc_vec.begin(), _arc_vec.end());
927
928      typename BellmanFord<StaticDigraph, LargeCostArcMap>
929        ::template SetDistMap<LargeCostNodeMap>::Create bf(_sgr, _cost_map);
930      bf.distMap(_pi_map);
931      bf.init(0);
932      bf.start();
933
934      // Handle non-zero lower bounds
935      if (_have_lower) {
936        int limit = _first_out[_root];
937        for (int j = 0; j != limit; ++j) {
938          if (!_forward[j]) _res_cap[j] += _lower[j];
939        }
940      }
941    }
942   
943    // Initialize a cost scaling phase
944    void initPhase() {
945      // Saturate arcs not satisfying the optimality condition
946      for (int u = 0; u != _res_node_num; ++u) {
947        int last_out = _first_out[u+1];
948        LargeCost pi_u = _pi[u];
949        for (int a = _first_out[u]; a != last_out; ++a) {
950          int v = _target[a];
951          if (_res_cap[a] > 0 && _cost[a] + pi_u - _pi[v] < 0) {
952            Value delta = _res_cap[a];
953            _excess[u] -= delta;
954            _excess[v] += delta;
955            _res_cap[a] = 0;
956            _res_cap[_reverse[a]] += delta;
957          }
958        }
959      }
960     
961      // Find active nodes (i.e. nodes with positive excess)
962      for (int u = 0; u != _res_node_num; ++u) {
963        if (_excess[u] > 0) _active_nodes.push_back(u);
964      }
965
966      // Initialize the next arcs
967      for (int u = 0; u != _res_node_num; ++u) {
968        _next_out[u] = _first_out[u];
969      }
970    }
971   
972    // Early termination heuristic
973    bool earlyTermination() {
974      const double EARLY_TERM_FACTOR = 3.0;
975
976      // Build a static residual graph
977      _arc_vec.clear();
978      _cost_vec.clear();
979      for (int j = 0; j != _res_arc_num; ++j) {
980        if (_res_cap[j] > 0) {
981          _arc_vec.push_back(IntPair(_source[j], _target[j]));
982          _cost_vec.push_back(_cost[j] + 1);
983        }
984      }
985      _sgr.build(_res_node_num, _arc_vec.begin(), _arc_vec.end());
986
987      // Run Bellman-Ford algorithm to check if the current flow is optimal
988      BellmanFord<StaticDigraph, LargeCostArcMap> bf(_sgr, _cost_map);
989      bf.init(0);
990      bool done = false;
991      int K = int(EARLY_TERM_FACTOR * std::sqrt(double(_res_node_num)));
992      for (int i = 0; i < K && !done; ++i) {
993        done = bf.processNextWeakRound();
994      }
995      return done;
996    }
997
998    // Global potential update heuristic
999    void globalUpdate() {
1000      int bucket_end = _root + 1;
1001   
1002      // Initialize buckets
1003      for (int r = 0; r != _max_rank; ++r) {
1004        _buckets[r] = bucket_end;
1005      }
1006      Value total_excess = 0;
1007      for (int i = 0; i != _res_node_num; ++i) {
1008        if (_excess[i] < 0) {
1009          _rank[i] = 0;
1010          _bucket_next[i] = _buckets[0];
1011          _bucket_prev[_buckets[0]] = i;
1012          _buckets[0] = i;
1013        } else {
1014          total_excess += _excess[i];
1015          _rank[i] = _max_rank;
1016        }
1017      }
1018      if (total_excess == 0) return;
1019
1020      // Search the buckets
1021      int r = 0;
1022      for ( ; r != _max_rank; ++r) {
1023        while (_buckets[r] != bucket_end) {
1024          // Remove the first node from the current bucket
1025          int u = _buckets[r];
1026          _buckets[r] = _bucket_next[u];
1027         
1028          // Search the incomming arcs of u
1029          LargeCost pi_u = _pi[u];
1030          int last_out = _first_out[u+1];
1031          for (int a = _first_out[u]; a != last_out; ++a) {
1032            int ra = _reverse[a];
1033            if (_res_cap[ra] > 0) {
1034              int v = _source[ra];
1035              int old_rank_v = _rank[v];
1036              if (r < old_rank_v) {
1037                // Compute the new rank of v
1038                LargeCost nrc = (_cost[ra] + _pi[v] - pi_u) / _epsilon;
1039                int new_rank_v = old_rank_v;
1040                if (nrc < LargeCost(_max_rank))
1041                  new_rank_v = r + 1 + int(nrc);
1042                 
1043                // Change the rank of v
1044                if (new_rank_v < old_rank_v) {
1045                  _rank[v] = new_rank_v;
1046                  _next_out[v] = _first_out[v];
1047                 
1048                  // Remove v from its old bucket
1049                  if (old_rank_v < _max_rank) {
1050                    if (_buckets[old_rank_v] == v) {
1051                      _buckets[old_rank_v] = _bucket_next[v];
1052                    } else {
1053                      _bucket_next[_bucket_prev[v]] = _bucket_next[v];
1054                      _bucket_prev[_bucket_next[v]] = _bucket_prev[v];
1055                    }
1056                  }
1057                 
1058                  // Insert v to its new bucket
1059                  _bucket_next[v] = _buckets[new_rank_v];
1060                  _bucket_prev[_buckets[new_rank_v]] = v;
1061                  _buckets[new_rank_v] = v;
1062                }
1063              }
1064            }
1065          }
1066
1067          // Finish search if there are no more active nodes
1068          if (_excess[u] > 0) {
1069            total_excess -= _excess[u];
1070            if (total_excess <= 0) break;
1071          }
1072        }
1073        if (total_excess <= 0) break;
1074      }
1075     
1076      // Relabel nodes
1077      for (int u = 0; u != _res_node_num; ++u) {
1078        int k = std::min(_rank[u], r);
1079        if (k > 0) {
1080          _pi[u] -= _epsilon * k;
1081          _next_out[u] = _first_out[u];
1082        }
1083      }
1084    }
1085
1086    /// Execute the algorithm performing augment and relabel operations
1087    void startAugment(int max_length = std::numeric_limits<int>::max()) {
1088      // Paramters for heuristics
1089      const int EARLY_TERM_EPSILON_LIMIT = 1000;
1090      const double GLOBAL_UPDATE_FACTOR = 3.0;
1091
1092      const int global_update_freq = int(GLOBAL_UPDATE_FACTOR *
1093        (_res_node_num + _sup_node_num * _sup_node_num));
1094      int next_update_limit = global_update_freq;
1095     
1096      int relabel_cnt = 0;
1097     
1098      // Perform cost scaling phases
1099      std::vector<int> path;
1100      for ( ; _epsilon >= 1; _epsilon = _epsilon < _alpha && _epsilon > 1 ?
1101                                        1 : _epsilon / _alpha )
1102      {
1103        // Early termination heuristic
1104        if (_epsilon <= EARLY_TERM_EPSILON_LIMIT) {
1105          if (earlyTermination()) break;
1106        }
1107       
1108        // Initialize current phase
1109        initPhase();
1110       
1111        // Perform partial augment and relabel operations
1112        while (true) {
1113          // Select an active node (FIFO selection)
1114          while (_active_nodes.size() > 0 &&
1115                 _excess[_active_nodes.front()] <= 0) {
1116            _active_nodes.pop_front();
1117          }
1118          if (_active_nodes.size() == 0) break;
1119          int start = _active_nodes.front();
1120
1121          // Find an augmenting path from the start node
1122          path.clear();
1123          int tip = start;
1124          while (_excess[tip] >= 0 && int(path.size()) < max_length) {
1125            int u;
1126            LargeCost min_red_cost, rc, pi_tip = _pi[tip];
1127            int last_out = _first_out[tip+1];
1128            for (int a = _next_out[tip]; a != last_out; ++a) {
1129              u = _target[a];
1130              if (_res_cap[a] > 0 && _cost[a] + pi_tip - _pi[u] < 0) {
1131                path.push_back(a);
1132                _next_out[tip] = a;
1133                tip = u;
1134                goto next_step;
1135              }
1136            }
1137
1138            // Relabel tip node
1139            min_red_cost = std::numeric_limits<LargeCost>::max();
1140            if (tip != start) {
1141              int ra = _reverse[path.back()];
1142              min_red_cost = _cost[ra] + pi_tip - _pi[_target[ra]];
1143            }
1144            for (int a = _first_out[tip]; a != last_out; ++a) {
1145              rc = _cost[a] + pi_tip - _pi[_target[a]];
1146              if (_res_cap[a] > 0 && rc < min_red_cost) {
1147                min_red_cost = rc;
1148              }
1149            }
1150            _pi[tip] -= min_red_cost + _epsilon;
1151            _next_out[tip] = _first_out[tip];
1152            ++relabel_cnt;
1153
1154            // Step back
1155            if (tip != start) {
1156              tip = _source[path.back()];
1157              path.pop_back();
1158            }
1159
1160          next_step: ;
1161          }
1162
1163          // Augment along the found path (as much flow as possible)
1164          Value delta;
1165          int pa, u, v = start;
1166          for (int i = 0; i != int(path.size()); ++i) {
1167            pa = path[i];
1168            u = v;
1169            v = _target[pa];
1170            delta = std::min(_res_cap[pa], _excess[u]);
1171            _res_cap[pa] -= delta;
1172            _res_cap[_reverse[pa]] += delta;
1173            _excess[u] -= delta;
1174            _excess[v] += delta;
1175            if (_excess[v] > 0 && _excess[v] <= delta)
1176              _active_nodes.push_back(v);
1177          }
1178
1179          // Global update heuristic
1180          if (relabel_cnt >= next_update_limit) {
1181            globalUpdate();
1182            next_update_limit += global_update_freq;
1183          }
1184        }
1185      }
1186    }
1187
1188    /// Execute the algorithm performing push and relabel operations
1189    void startPush() {
1190      // Paramters for heuristics
1191      const int EARLY_TERM_EPSILON_LIMIT = 1000;
1192      const double GLOBAL_UPDATE_FACTOR = 2.0;
1193
1194      const int global_update_freq = int(GLOBAL_UPDATE_FACTOR *
1195        (_res_node_num + _sup_node_num * _sup_node_num));
1196      int next_update_limit = global_update_freq;
1197
1198      int relabel_cnt = 0;
1199     
1200      // Perform cost scaling phases
1201      BoolVector hyper(_res_node_num, false);
1202      LargeCostVector hyper_cost(_res_node_num);
1203      for ( ; _epsilon >= 1; _epsilon = _epsilon < _alpha && _epsilon > 1 ?
1204                                        1 : _epsilon / _alpha )
1205      {
1206        // Early termination heuristic
1207        if (_epsilon <= EARLY_TERM_EPSILON_LIMIT) {
1208          if (earlyTermination()) break;
1209        }
1210       
1211        // Initialize current phase
1212        initPhase();
1213
1214        // Perform push and relabel operations
1215        while (_active_nodes.size() > 0) {
1216          LargeCost min_red_cost, rc, pi_n;
1217          Value delta;
1218          int n, t, a, last_out = _res_arc_num;
1219
1220        next_node:
1221          // Select an active node (FIFO selection)
1222          n = _active_nodes.front();
1223          last_out = _first_out[n+1];
1224          pi_n = _pi[n];
1225         
1226          // Perform push operations if there are admissible arcs
1227          if (_excess[n] > 0) {
1228            for (a = _next_out[n]; a != last_out; ++a) {
1229              if (_res_cap[a] > 0 &&
1230                  _cost[a] + pi_n - _pi[_target[a]] < 0) {
1231                delta = std::min(_res_cap[a], _excess[n]);
1232                t = _target[a];
1233
1234                // Push-look-ahead heuristic
1235                Value ahead = -_excess[t];
1236                int last_out_t = _first_out[t+1];
1237                LargeCost pi_t = _pi[t];
1238                for (int ta = _next_out[t]; ta != last_out_t; ++ta) {
1239                  if (_res_cap[ta] > 0 &&
1240                      _cost[ta] + pi_t - _pi[_target[ta]] < 0)
1241                    ahead += _res_cap[ta];
1242                  if (ahead >= delta) break;
1243                }
1244                if (ahead < 0) ahead = 0;
1245
1246                // Push flow along the arc
1247                if (ahead < delta && !hyper[t]) {
1248                  _res_cap[a] -= ahead;
1249                  _res_cap[_reverse[a]] += ahead;
1250                  _excess[n] -= ahead;
1251                  _excess[t] += ahead;
1252                  _active_nodes.push_front(t);
1253                  hyper[t] = true;
1254                  hyper_cost[t] = _cost[a] + pi_n - pi_t;
1255                  _next_out[n] = a;
1256                  goto next_node;
1257                } else {
1258                  _res_cap[a] -= delta;
1259                  _res_cap[_reverse[a]] += delta;
1260                  _excess[n] -= delta;
1261                  _excess[t] += delta;
1262                  if (_excess[t] > 0 && _excess[t] <= delta)
1263                    _active_nodes.push_back(t);
1264                }
1265
1266                if (_excess[n] == 0) {
1267                  _next_out[n] = a;
1268                  goto remove_nodes;
1269                }
1270              }
1271            }
1272            _next_out[n] = a;
1273          }
1274
1275          // Relabel the node if it is still active (or hyper)
1276          if (_excess[n] > 0 || hyper[n]) {
1277             min_red_cost = hyper[n] ? -hyper_cost[n] :
1278               std::numeric_limits<LargeCost>::max();
1279            for (int a = _first_out[n]; a != last_out; ++a) {
1280              rc = _cost[a] + pi_n - _pi[_target[a]];
1281              if (_res_cap[a] > 0 && rc < min_red_cost) {
1282                min_red_cost = rc;
1283              }
1284            }
1285            _pi[n] -= min_red_cost + _epsilon;
1286            _next_out[n] = _first_out[n];
1287            hyper[n] = false;
1288            ++relabel_cnt;
1289          }
1290       
1291          // Remove nodes that are not active nor hyper
1292        remove_nodes:
1293          while ( _active_nodes.size() > 0 &&
1294                  _excess[_active_nodes.front()] <= 0 &&
1295                  !hyper[_active_nodes.front()] ) {
1296            _active_nodes.pop_front();
1297          }
1298         
1299          // Global update heuristic
1300          if (relabel_cnt >= next_update_limit) {
1301            globalUpdate();
1302            for (int u = 0; u != _res_node_num; ++u)
1303              hyper[u] = false;
1304            next_update_limit += global_update_freq;
1305          }
1306        }
1307      }
1308    }
1309
1310  }; //class CostScaling
1311
1312  ///@}
1313
1314} //namespace lemon
1315
1316#endif //LEMON_COST_SCALING_H
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