COIN-OR::LEMON - Graph Library

source: lemon/lemon/cost_scaling.h @ 1046:6ea176638264

Last change on this file since 1046:6ea176638264 was 1046:6ea176638264, checked in by Peter Kovacs <kpeter@…>, 8 years ago

Fix and improve refine methods 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::Node, LargeCost> LargeCostNodeMap;
241    typedef StaticVectorMap<StaticDigraph::Arc, LargeCost> LargeCostArcMap;
242
243  private:
244
245    // Data related to the underlying digraph
246    const GR &_graph;
247    int _node_num;
248    int _arc_num;
249    int _res_node_num;
250    int _res_arc_num;
251    int _root;
252
253    // Parameters of the problem
254    bool _have_lower;
255    Value _sum_supply;
256    int _sup_node_num;
257
258    // Data structures for storing the digraph
259    IntNodeMap _node_id;
260    IntArcMap _arc_idf;
261    IntArcMap _arc_idb;
262    IntVector _first_out;
263    BoolVector _forward;
264    IntVector _source;
265    IntVector _target;
266    IntVector _reverse;
267
268    // Node and arc data
269    ValueVector _lower;
270    ValueVector _upper;
271    CostVector _scost;
272    ValueVector _supply;
273
274    ValueVector _res_cap;
275    LargeCostVector _cost;
276    LargeCostVector _pi;
277    ValueVector _excess;
278    IntVector _next_out;
279    std::deque<int> _active_nodes;
280
281    // Data for scaling
282    LargeCost _epsilon;
283    int _alpha;
284
285    IntVector _buckets;
286    IntVector _bucket_next;
287    IntVector _bucket_prev;
288    IntVector _rank;
289    int _max_rank;
290
291    // Data for a StaticDigraph structure
292    typedef std::pair<int, int> IntPair;
293    StaticDigraph _sgr;
294    std::vector<IntPair> _arc_vec;
295    std::vector<LargeCost> _cost_vec;
296    LargeCostArcMap _cost_map;
297    LargeCostNodeMap _pi_map;
298
299  public:
300
301    /// \brief Constant for infinite upper bounds (capacities).
302    ///
303    /// Constant for infinite upper bounds (capacities).
304    /// It is \c std::numeric_limits<Value>::infinity() if available,
305    /// \c std::numeric_limits<Value>::max() otherwise.
306    const Value INF;
307
308  public:
309
310    /// \name Named Template Parameters
311    /// @{
312
313    template <typename T>
314    struct SetLargeCostTraits : public Traits {
315      typedef T LargeCost;
316    };
317
318    /// \brief \ref named-templ-param "Named parameter" for setting
319    /// \c LargeCost type.
320    ///
321    /// \ref named-templ-param "Named parameter" for setting \c LargeCost
322    /// type, which is used for internal computations in the algorithm.
323    /// \c Cost must be convertible to \c LargeCost.
324    template <typename T>
325    struct SetLargeCost
326      : public CostScaling<GR, V, C, SetLargeCostTraits<T> > {
327      typedef  CostScaling<GR, V, C, SetLargeCostTraits<T> > Create;
328    };
329
330    /// @}
331
332  protected:
333
334    CostScaling() {}
335
336  public:
337
338    /// \brief Constructor.
339    ///
340    /// The constructor of the class.
341    ///
342    /// \param graph The digraph the algorithm runs on.
343    CostScaling(const GR& graph) :
344      _graph(graph), _node_id(graph), _arc_idf(graph), _arc_idb(graph),
345      _cost_map(_cost_vec), _pi_map(_pi),
346      INF(std::numeric_limits<Value>::has_infinity ?
347          std::numeric_limits<Value>::infinity() :
348          std::numeric_limits<Value>::max())
349    {
350      // Check the number types
351      LEMON_ASSERT(std::numeric_limits<Value>::is_signed,
352        "The flow type of CostScaling must be signed");
353      LEMON_ASSERT(std::numeric_limits<Cost>::is_signed,
354        "The cost type of CostScaling must be signed");
355
356      // Reset data structures
357      reset();
358    }
359
360    /// \name Parameters
361    /// The parameters of the algorithm can be specified using these
362    /// functions.
363
364    /// @{
365
366    /// \brief Set the lower bounds on the arcs.
367    ///
368    /// This function sets the lower bounds on the arcs.
369    /// If it is not used before calling \ref run(), the lower bounds
370    /// will be set to zero on all arcs.
371    ///
372    /// \param map An arc map storing the lower bounds.
373    /// Its \c Value type must be convertible to the \c Value type
374    /// of the algorithm.
375    ///
376    /// \return <tt>(*this)</tt>
377    template <typename LowerMap>
378    CostScaling& lowerMap(const LowerMap& map) {
379      _have_lower = true;
380      for (ArcIt a(_graph); a != INVALID; ++a) {
381        _lower[_arc_idf[a]] = map[a];
382        _lower[_arc_idb[a]] = map[a];
383      }
384      return *this;
385    }
386
387    /// \brief Set the upper bounds (capacities) on the arcs.
388    ///
389    /// This function sets the upper bounds (capacities) on the arcs.
390    /// If it is not used before calling \ref run(), the upper bounds
391    /// will be set to \ref INF on all arcs (i.e. the flow value will be
392    /// unbounded from above).
393    ///
394    /// \param map An arc map storing the upper bounds.
395    /// Its \c Value type must be convertible to the \c Value type
396    /// of the algorithm.
397    ///
398    /// \return <tt>(*this)</tt>
399    template<typename UpperMap>
400    CostScaling& upperMap(const UpperMap& map) {
401      for (ArcIt a(_graph); a != INVALID; ++a) {
402        _upper[_arc_idf[a]] = map[a];
403      }
404      return *this;
405    }
406
407    /// \brief Set the costs of the arcs.
408    ///
409    /// This function sets the costs of the arcs.
410    /// If it is not used before calling \ref run(), the costs
411    /// will be set to \c 1 on all arcs.
412    ///
413    /// \param map An arc map storing the costs.
414    /// Its \c Value type must be convertible to the \c Cost type
415    /// of the algorithm.
416    ///
417    /// \return <tt>(*this)</tt>
418    template<typename CostMap>
419    CostScaling& costMap(const CostMap& map) {
420      for (ArcIt a(_graph); a != INVALID; ++a) {
421        _scost[_arc_idf[a]] =  map[a];
422        _scost[_arc_idb[a]] = -map[a];
423      }
424      return *this;
425    }
426
427    /// \brief Set the supply values of the nodes.
428    ///
429    /// This function sets the supply values of the nodes.
430    /// If neither this function nor \ref stSupply() is used before
431    /// calling \ref run(), the supply of each node will be set to zero.
432    ///
433    /// \param map A node map storing the supply values.
434    /// Its \c Value type must be convertible to the \c Value type
435    /// of the algorithm.
436    ///
437    /// \return <tt>(*this)</tt>
438    template<typename SupplyMap>
439    CostScaling& supplyMap(const SupplyMap& map) {
440      for (NodeIt n(_graph); n != INVALID; ++n) {
441        _supply[_node_id[n]] = map[n];
442      }
443      return *this;
444    }
445
446    /// \brief Set single source and target nodes and a supply value.
447    ///
448    /// This function sets a single source node and a single target node
449    /// and the required flow value.
450    /// If neither this function nor \ref supplyMap() is used before
451    /// calling \ref run(), the supply of each node will be set to zero.
452    ///
453    /// Using this function has the same effect as using \ref supplyMap()
454    /// with a map in which \c k is assigned to \c s, \c -k is
455    /// assigned to \c t and all other nodes have zero supply value.
456    ///
457    /// \param s The source node.
458    /// \param t The target node.
459    /// \param k The required amount of flow from node \c s to node \c t
460    /// (i.e. the supply of \c s and the demand of \c t).
461    ///
462    /// \return <tt>(*this)</tt>
463    CostScaling& stSupply(const Node& s, const Node& t, Value k) {
464      for (int i = 0; i != _res_node_num; ++i) {
465        _supply[i] = 0;
466      }
467      _supply[_node_id[s]] =  k;
468      _supply[_node_id[t]] = -k;
469      return *this;
470    }
471
472    /// @}
473
474    /// \name Execution control
475    /// The algorithm can be executed using \ref run().
476
477    /// @{
478
479    /// \brief Run the algorithm.
480    ///
481    /// This function runs the algorithm.
482    /// The paramters can be specified using functions \ref lowerMap(),
483    /// \ref upperMap(), \ref costMap(), \ref supplyMap(), \ref stSupply().
484    /// For example,
485    /// \code
486    ///   CostScaling<ListDigraph> cs(graph);
487    ///   cs.lowerMap(lower).upperMap(upper).costMap(cost)
488    ///     .supplyMap(sup).run();
489    /// \endcode
490    ///
491    /// This function can be called more than once. All the given parameters
492    /// are kept for the next call, unless \ref resetParams() or \ref reset()
493    /// is used, thus only the modified parameters have to be set again.
494    /// If the underlying digraph was also modified after the construction
495    /// of the class (or the last \ref reset() call), then the \ref reset()
496    /// function must be called.
497    ///
498    /// \param method The internal method that will be used in the
499    /// algorithm. For more information, see \ref Method.
500    /// \param factor The cost scaling factor. It must be larger than one.
501    ///
502    /// \return \c INFEASIBLE if no feasible flow exists,
503    /// \n \c OPTIMAL if the problem has optimal solution
504    /// (i.e. it is feasible and bounded), and the algorithm has found
505    /// optimal flow and node potentials (primal and dual solutions),
506    /// \n \c UNBOUNDED if the digraph contains an arc of negative cost
507    /// and infinite upper bound. It means that the objective function
508    /// is unbounded on that arc, however, note that it could actually be
509    /// bounded over the feasible flows, but this algroithm cannot handle
510    /// these cases.
511    ///
512    /// \see ProblemType, Method
513    /// \see resetParams(), reset()
514    ProblemType run(Method method = PARTIAL_AUGMENT, int factor = 8) {
515      _alpha = factor;
516      ProblemType pt = init();
517      if (pt != OPTIMAL) return pt;
518      start(method);
519      return OPTIMAL;
520    }
521
522    /// \brief Reset all the parameters that have been given before.
523    ///
524    /// This function resets all the paramaters that have been given
525    /// before using functions \ref lowerMap(), \ref upperMap(),
526    /// \ref costMap(), \ref supplyMap(), \ref stSupply().
527    ///
528    /// It is useful for multiple \ref run() calls. Basically, all the given
529    /// parameters are kept for the next \ref run() call, unless
530    /// \ref resetParams() or \ref reset() is used.
531    /// If the underlying digraph was also modified after the construction
532    /// of the class or the last \ref reset() call, then the \ref reset()
533    /// function must be used, otherwise \ref resetParams() is sufficient.
534    ///
535    /// For example,
536    /// \code
537    ///   CostScaling<ListDigraph> cs(graph);
538    ///
539    ///   // First run
540    ///   cs.lowerMap(lower).upperMap(upper).costMap(cost)
541    ///     .supplyMap(sup).run();
542    ///
543    ///   // Run again with modified cost map (resetParams() is not called,
544    ///   // so only the cost map have to be set again)
545    ///   cost[e] += 100;
546    ///   cs.costMap(cost).run();
547    ///
548    ///   // Run again from scratch using resetParams()
549    ///   // (the lower bounds will be set to zero on all arcs)
550    ///   cs.resetParams();
551    ///   cs.upperMap(capacity).costMap(cost)
552    ///     .supplyMap(sup).run();
553    /// \endcode
554    ///
555    /// \return <tt>(*this)</tt>
556    ///
557    /// \see reset(), run()
558    CostScaling& resetParams() {
559      for (int i = 0; i != _res_node_num; ++i) {
560        _supply[i] = 0;
561      }
562      int limit = _first_out[_root];
563      for (int j = 0; j != limit; ++j) {
564        _lower[j] = 0;
565        _upper[j] = INF;
566        _scost[j] = _forward[j] ? 1 : -1;
567      }
568      for (int j = limit; j != _res_arc_num; ++j) {
569        _lower[j] = 0;
570        _upper[j] = INF;
571        _scost[j] = 0;
572        _scost[_reverse[j]] = 0;
573      }
574      _have_lower = false;
575      return *this;
576    }
577
578    /// \brief Reset the internal data structures and all the parameters
579    /// that have been given before.
580    ///
581    /// This function resets the internal data structures and all the
582    /// paramaters that have been given before using functions \ref lowerMap(),
583    /// \ref upperMap(), \ref costMap(), \ref supplyMap(), \ref stSupply().
584    ///
585    /// It is useful for multiple \ref run() calls. By default, all the given
586    /// parameters are kept for the next \ref run() call, unless
587    /// \ref resetParams() or \ref reset() is used.
588    /// If the underlying digraph was also modified after the construction
589    /// of the class or the last \ref reset() call, then the \ref reset()
590    /// function must be used, otherwise \ref resetParams() is sufficient.
591    ///
592    /// See \ref resetParams() for examples.
593    ///
594    /// \return <tt>(*this)</tt>
595    ///
596    /// \see resetParams(), run()
597    CostScaling& reset() {
598      // Resize vectors
599      _node_num = countNodes(_graph);
600      _arc_num = countArcs(_graph);
601      _res_node_num = _node_num + 1;
602      _res_arc_num = 2 * (_arc_num + _node_num);
603      _root = _node_num;
604
605      _first_out.resize(_res_node_num + 1);
606      _forward.resize(_res_arc_num);
607      _source.resize(_res_arc_num);
608      _target.resize(_res_arc_num);
609      _reverse.resize(_res_arc_num);
610
611      _lower.resize(_res_arc_num);
612      _upper.resize(_res_arc_num);
613      _scost.resize(_res_arc_num);
614      _supply.resize(_res_node_num);
615
616      _res_cap.resize(_res_arc_num);
617      _cost.resize(_res_arc_num);
618      _pi.resize(_res_node_num);
619      _excess.resize(_res_node_num);
620      _next_out.resize(_res_node_num);
621
622      _arc_vec.reserve(_res_arc_num);
623      _cost_vec.reserve(_res_arc_num);
624
625      // Copy the graph
626      int i = 0, j = 0, k = 2 * _arc_num + _node_num;
627      for (NodeIt n(_graph); n != INVALID; ++n, ++i) {
628        _node_id[n] = i;
629      }
630      i = 0;
631      for (NodeIt n(_graph); n != INVALID; ++n, ++i) {
632        _first_out[i] = j;
633        for (OutArcIt a(_graph, n); a != INVALID; ++a, ++j) {
634          _arc_idf[a] = j;
635          _forward[j] = true;
636          _source[j] = i;
637          _target[j] = _node_id[_graph.runningNode(a)];
638        }
639        for (InArcIt a(_graph, n); a != INVALID; ++a, ++j) {
640          _arc_idb[a] = j;
641          _forward[j] = false;
642          _source[j] = i;
643          _target[j] = _node_id[_graph.runningNode(a)];
644        }
645        _forward[j] = false;
646        _source[j] = i;
647        _target[j] = _root;
648        _reverse[j] = k;
649        _forward[k] = true;
650        _source[k] = _root;
651        _target[k] = i;
652        _reverse[k] = j;
653        ++j; ++k;
654      }
655      _first_out[i] = j;
656      _first_out[_res_node_num] = k;
657      for (ArcIt a(_graph); a != INVALID; ++a) {
658        int fi = _arc_idf[a];
659        int bi = _arc_idb[a];
660        _reverse[fi] = bi;
661        _reverse[bi] = fi;
662      }
663
664      // Reset parameters
665      resetParams();
666      return *this;
667    }
668
669    /// @}
670
671    /// \name Query Functions
672    /// The results of the algorithm can be obtained using these
673    /// functions.\n
674    /// The \ref run() function must be called before using them.
675
676    /// @{
677
678    /// \brief Return the total cost of the found flow.
679    ///
680    /// This function returns the total cost of the found flow.
681    /// Its complexity is O(e).
682    ///
683    /// \note The return type of the function can be specified as a
684    /// template parameter. For example,
685    /// \code
686    ///   cs.totalCost<double>();
687    /// \endcode
688    /// It is useful if the total cost cannot be stored in the \c Cost
689    /// type of the algorithm, which is the default return type of the
690    /// function.
691    ///
692    /// \pre \ref run() must be called before using this function.
693    template <typename Number>
694    Number totalCost() const {
695      Number c = 0;
696      for (ArcIt a(_graph); a != INVALID; ++a) {
697        int i = _arc_idb[a];
698        c += static_cast<Number>(_res_cap[i]) *
699             (-static_cast<Number>(_scost[i]));
700      }
701      return c;
702    }
703
704#ifndef DOXYGEN
705    Cost totalCost() const {
706      return totalCost<Cost>();
707    }
708#endif
709
710    /// \brief Return the flow on the given arc.
711    ///
712    /// This function returns the flow on the given arc.
713    ///
714    /// \pre \ref run() must be called before using this function.
715    Value flow(const Arc& a) const {
716      return _res_cap[_arc_idb[a]];
717    }
718
719    /// \brief Return the flow map (the primal solution).
720    ///
721    /// This function copies the flow value on each arc into the given
722    /// map. The \c Value type of the algorithm must be convertible to
723    /// the \c Value type of the map.
724    ///
725    /// \pre \ref run() must be called before using this function.
726    template <typename FlowMap>
727    void flowMap(FlowMap &map) const {
728      for (ArcIt a(_graph); a != INVALID; ++a) {
729        map.set(a, _res_cap[_arc_idb[a]]);
730      }
731    }
732
733    /// \brief Return the potential (dual value) of the given node.
734    ///
735    /// This function returns the potential (dual value) of the
736    /// given node.
737    ///
738    /// \pre \ref run() must be called before using this function.
739    Cost potential(const Node& n) const {
740      return static_cast<Cost>(_pi[_node_id[n]]);
741    }
742
743    /// \brief Return the potential map (the dual solution).
744    ///
745    /// This function copies the potential (dual value) of each node
746    /// into the given map.
747    /// The \c Cost type of the algorithm must be convertible to the
748    /// \c Value type of the map.
749    ///
750    /// \pre \ref run() must be called before using this function.
751    template <typename PotentialMap>
752    void potentialMap(PotentialMap &map) const {
753      for (NodeIt n(_graph); n != INVALID; ++n) {
754        map.set(n, static_cast<Cost>(_pi[_node_id[n]]));
755      }
756    }
757
758    /// @}
759
760  private:
761
762    // Initialize the algorithm
763    ProblemType init() {
764      if (_res_node_num <= 1) return INFEASIBLE;
765
766      // Check the sum of supply values
767      _sum_supply = 0;
768      for (int i = 0; i != _root; ++i) {
769        _sum_supply += _supply[i];
770      }
771      if (_sum_supply > 0) return INFEASIBLE;
772
773
774      // Initialize vectors
775      for (int i = 0; i != _res_node_num; ++i) {
776        _pi[i] = 0;
777        _excess[i] = _supply[i];
778      }
779
780      // Remove infinite upper bounds and check negative arcs
781      const Value MAX = std::numeric_limits<Value>::max();
782      int last_out;
783      if (_have_lower) {
784        for (int i = 0; i != _root; ++i) {
785          last_out = _first_out[i+1];
786          for (int j = _first_out[i]; j != last_out; ++j) {
787            if (_forward[j]) {
788              Value c = _scost[j] < 0 ? _upper[j] : _lower[j];
789              if (c >= MAX) return UNBOUNDED;
790              _excess[i] -= c;
791              _excess[_target[j]] += c;
792            }
793          }
794        }
795      } else {
796        for (int i = 0; i != _root; ++i) {
797          last_out = _first_out[i+1];
798          for (int j = _first_out[i]; j != last_out; ++j) {
799            if (_forward[j] && _scost[j] < 0) {
800              Value c = _upper[j];
801              if (c >= MAX) return UNBOUNDED;
802              _excess[i] -= c;
803              _excess[_target[j]] += c;
804            }
805          }
806        }
807      }
808      Value ex, max_cap = 0;
809      for (int i = 0; i != _res_node_num; ++i) {
810        ex = _excess[i];
811        _excess[i] = 0;
812        if (ex < 0) max_cap -= ex;
813      }
814      for (int j = 0; j != _res_arc_num; ++j) {
815        if (_upper[j] >= MAX) _upper[j] = max_cap;
816      }
817
818      // Initialize the large cost vector and the epsilon parameter
819      _epsilon = 0;
820      LargeCost lc;
821      for (int i = 0; i != _root; ++i) {
822        last_out = _first_out[i+1];
823        for (int j = _first_out[i]; j != last_out; ++j) {
824          lc = static_cast<LargeCost>(_scost[j]) * _res_node_num * _alpha;
825          _cost[j] = lc;
826          if (lc > _epsilon) _epsilon = lc;
827        }
828      }
829      _epsilon /= _alpha;
830
831      // Initialize maps for Circulation and remove non-zero lower bounds
832      ConstMap<Arc, Value> low(0);
833      typedef typename Digraph::template ArcMap<Value> ValueArcMap;
834      typedef typename Digraph::template NodeMap<Value> ValueNodeMap;
835      ValueArcMap cap(_graph), flow(_graph);
836      ValueNodeMap sup(_graph);
837      for (NodeIt n(_graph); n != INVALID; ++n) {
838        sup[n] = _supply[_node_id[n]];
839      }
840      if (_have_lower) {
841        for (ArcIt a(_graph); a != INVALID; ++a) {
842          int j = _arc_idf[a];
843          Value c = _lower[j];
844          cap[a] = _upper[j] - c;
845          sup[_graph.source(a)] -= c;
846          sup[_graph.target(a)] += c;
847        }
848      } else {
849        for (ArcIt a(_graph); a != INVALID; ++a) {
850          cap[a] = _upper[_arc_idf[a]];
851        }
852      }
853
854      _sup_node_num = 0;
855      for (NodeIt n(_graph); n != INVALID; ++n) {
856        if (sup[n] > 0) ++_sup_node_num;
857      }
858
859      // Find a feasible flow using Circulation
860      Circulation<Digraph, ConstMap<Arc, Value>, ValueArcMap, ValueNodeMap>
861        circ(_graph, low, cap, sup);
862      if (!circ.flowMap(flow).run()) return INFEASIBLE;
863
864      // Set residual capacities and handle GEQ supply type
865      if (_sum_supply < 0) {
866        for (ArcIt a(_graph); a != INVALID; ++a) {
867          Value fa = flow[a];
868          _res_cap[_arc_idf[a]] = cap[a] - fa;
869          _res_cap[_arc_idb[a]] = fa;
870          sup[_graph.source(a)] -= fa;
871          sup[_graph.target(a)] += fa;
872        }
873        for (NodeIt n(_graph); n != INVALID; ++n) {
874          _excess[_node_id[n]] = sup[n];
875        }
876        for (int a = _first_out[_root]; a != _res_arc_num; ++a) {
877          int u = _target[a];
878          int ra = _reverse[a];
879          _res_cap[a] = -_sum_supply + 1;
880          _res_cap[ra] = -_excess[u];
881          _cost[a] = 0;
882          _cost[ra] = 0;
883          _excess[u] = 0;
884        }
885      } else {
886        for (ArcIt a(_graph); a != INVALID; ++a) {
887          Value fa = flow[a];
888          _res_cap[_arc_idf[a]] = cap[a] - fa;
889          _res_cap[_arc_idb[a]] = fa;
890        }
891        for (int a = _first_out[_root]; a != _res_arc_num; ++a) {
892          int ra = _reverse[a];
893          _res_cap[a] = 0;
894          _res_cap[ra] = 0;
895          _cost[a] = 0;
896          _cost[ra] = 0;
897        }
898      }
899
900      // Initialize data structures for buckets
901      _max_rank = _alpha * _res_node_num;
902      _buckets.resize(_max_rank);
903      _bucket_next.resize(_res_node_num + 1);
904      _bucket_prev.resize(_res_node_num + 1);
905      _rank.resize(_res_node_num + 1);
906
907      return OPTIMAL;
908    }
909
910    // Execute the algorithm and transform the results
911    void start(Method method) {
912      const int MAX_PARTIAL_PATH_LENGTH = 4;
913
914      switch (method) {
915        case PUSH:
916          startPush();
917          break;
918        case AUGMENT:
919          startAugment(_res_node_num - 1);
920          break;
921        case PARTIAL_AUGMENT:
922          startAugment(MAX_PARTIAL_PATH_LENGTH);
923          break;
924      }
925
926      // Compute node potentials for the original costs
927      _arc_vec.clear();
928      _cost_vec.clear();
929      for (int j = 0; j != _res_arc_num; ++j) {
930        if (_res_cap[j] > 0) {
931          _arc_vec.push_back(IntPair(_source[j], _target[j]));
932          _cost_vec.push_back(_scost[j]);
933        }
934      }
935      _sgr.build(_res_node_num, _arc_vec.begin(), _arc_vec.end());
936
937      typename BellmanFord<StaticDigraph, LargeCostArcMap>
938        ::template SetDistMap<LargeCostNodeMap>::Create bf(_sgr, _cost_map);
939      bf.distMap(_pi_map);
940      bf.init(0);
941      bf.start();
942
943      // Handle non-zero lower bounds
944      if (_have_lower) {
945        int limit = _first_out[_root];
946        for (int j = 0; j != limit; ++j) {
947          if (!_forward[j]) _res_cap[j] += _lower[j];
948        }
949      }
950    }
951
952    // Initialize a cost scaling phase
953    void initPhase() {
954      // Saturate arcs not satisfying the optimality condition
955      for (int u = 0; u != _res_node_num; ++u) {
956        int last_out = _first_out[u+1];
957        LargeCost pi_u = _pi[u];
958        for (int a = _first_out[u]; a != last_out; ++a) {
959          Value delta = _res_cap[a];
960          if (delta > 0) {
961            int v = _target[a];
962            if (_cost[a] + pi_u - _pi[v] < 0) {
963              _excess[u] -= delta;
964              _excess[v] += delta;
965              _res_cap[a] = 0;
966              _res_cap[_reverse[a]] += delta;
967            }
968          }
969        }
970      }
971
972      // Find active nodes (i.e. nodes with positive excess)
973      for (int u = 0; u != _res_node_num; ++u) {
974        if (_excess[u] > 0) _active_nodes.push_back(u);
975      }
976
977      // Initialize the next arcs
978      for (int u = 0; u != _res_node_num; ++u) {
979        _next_out[u] = _first_out[u];
980      }
981    }
982
983    // Early termination heuristic
984    bool earlyTermination() {
985      const double EARLY_TERM_FACTOR = 3.0;
986
987      // Build a static residual graph
988      _arc_vec.clear();
989      _cost_vec.clear();
990      for (int j = 0; j != _res_arc_num; ++j) {
991        if (_res_cap[j] > 0) {
992          _arc_vec.push_back(IntPair(_source[j], _target[j]));
993          _cost_vec.push_back(_cost[j] + 1);
994        }
995      }
996      _sgr.build(_res_node_num, _arc_vec.begin(), _arc_vec.end());
997
998      // Run Bellman-Ford algorithm to check if the current flow is optimal
999      BellmanFord<StaticDigraph, LargeCostArcMap> bf(_sgr, _cost_map);
1000      bf.init(0);
1001      bool done = false;
1002      int K = int(EARLY_TERM_FACTOR * std::sqrt(double(_res_node_num)));
1003      for (int i = 0; i < K && !done; ++i) {
1004        done = bf.processNextWeakRound();
1005      }
1006      return done;
1007    }
1008
1009    // Global potential update heuristic
1010    void globalUpdate() {
1011      const int bucket_end = _root + 1;
1012
1013      // Initialize buckets
1014      for (int r = 0; r != _max_rank; ++r) {
1015        _buckets[r] = bucket_end;
1016      }
1017      Value total_excess = 0;
1018      int b0 = bucket_end;
1019      for (int i = 0; i != _res_node_num; ++i) {
1020        if (_excess[i] < 0) {
1021          _rank[i] = 0;
1022          _bucket_next[i] = b0;
1023          _bucket_prev[b0] = i;
1024          b0 = i;
1025        } else {
1026          total_excess += _excess[i];
1027          _rank[i] = _max_rank;
1028        }
1029      }
1030      if (total_excess == 0) return;
1031      _buckets[0] = b0;
1032
1033      // Search the buckets
1034      int r = 0;
1035      for ( ; r != _max_rank; ++r) {
1036        while (_buckets[r] != bucket_end) {
1037          // Remove the first node from the current bucket
1038          int u = _buckets[r];
1039          _buckets[r] = _bucket_next[u];
1040
1041          // Search the incomming arcs of u
1042          LargeCost pi_u = _pi[u];
1043          int last_out = _first_out[u+1];
1044          for (int a = _first_out[u]; a != last_out; ++a) {
1045            int ra = _reverse[a];
1046            if (_res_cap[ra] > 0) {
1047              int v = _source[ra];
1048              int old_rank_v = _rank[v];
1049              if (r < old_rank_v) {
1050                // Compute the new rank of v
1051                LargeCost nrc = (_cost[ra] + _pi[v] - pi_u) / _epsilon;
1052                int new_rank_v = old_rank_v;
1053                if (nrc < LargeCost(_max_rank)) {
1054                  new_rank_v = r + 1 + static_cast<int>(nrc);
1055                }
1056
1057                // Change the rank of v
1058                if (new_rank_v < old_rank_v) {
1059                  _rank[v] = new_rank_v;
1060                  _next_out[v] = _first_out[v];
1061
1062                  // Remove v from its old bucket
1063                  if (old_rank_v < _max_rank) {
1064                    if (_buckets[old_rank_v] == v) {
1065                      _buckets[old_rank_v] = _bucket_next[v];
1066                    } else {
1067                      int pv = _bucket_prev[v], nv = _bucket_next[v];
1068                      _bucket_next[pv] = nv;
1069                      _bucket_prev[nv] = pv;
1070                    }
1071                  }
1072
1073                  // Insert v into its new bucket
1074                  int nv = _buckets[new_rank_v];
1075                  _bucket_next[v] = nv;
1076                  _bucket_prev[nv] = v;
1077                  _buckets[new_rank_v] = v;
1078                }
1079              }
1080            }
1081          }
1082
1083          // Finish search if there are no more active nodes
1084          if (_excess[u] > 0) {
1085            total_excess -= _excess[u];
1086            if (total_excess <= 0) break;
1087          }
1088        }
1089        if (total_excess <= 0) break;
1090      }
1091
1092      // Relabel nodes
1093      for (int u = 0; u != _res_node_num; ++u) {
1094        int k = std::min(_rank[u], r);
1095        if (k > 0) {
1096          _pi[u] -= _epsilon * k;
1097          _next_out[u] = _first_out[u];
1098        }
1099      }
1100    }
1101
1102    /// Execute the algorithm performing augment and relabel operations
1103    void startAugment(int max_length) {
1104      // Paramters for heuristics
1105      const int EARLY_TERM_EPSILON_LIMIT = 1000;
1106      const double GLOBAL_UPDATE_FACTOR = 1.0;
1107      const int global_update_skip = static_cast<int>(GLOBAL_UPDATE_FACTOR *
1108        (_res_node_num + _sup_node_num * _sup_node_num));
1109      int next_global_update_limit = global_update_skip;
1110
1111      // Perform cost scaling phases
1112      IntVector path;
1113      BoolVector path_arc(_res_arc_num, false);
1114      int relabel_cnt = 0;
1115      for ( ; _epsilon >= 1; _epsilon = _epsilon < _alpha && _epsilon > 1 ?
1116                                        1 : _epsilon / _alpha )
1117      {
1118        // Early termination heuristic
1119        if (_epsilon <= EARLY_TERM_EPSILON_LIMIT) {
1120          if (earlyTermination()) break;
1121        }
1122
1123        // Initialize current phase
1124        initPhase();
1125
1126        // Perform partial augment and relabel operations
1127        while (true) {
1128          // Select an active node (FIFO selection)
1129          while (_active_nodes.size() > 0 &&
1130                 _excess[_active_nodes.front()] <= 0) {
1131            _active_nodes.pop_front();
1132          }
1133          if (_active_nodes.size() == 0) break;
1134          int start = _active_nodes.front();
1135
1136          // Find an augmenting path from the start node
1137          int tip = start;
1138          while (int(path.size()) < max_length && _excess[tip] >= 0) {
1139            int u;
1140            LargeCost rc, min_red_cost = std::numeric_limits<LargeCost>::max();
1141            LargeCost pi_tip = _pi[tip];
1142            int last_out = _first_out[tip+1];
1143            for (int a = _next_out[tip]; a != last_out; ++a) {
1144              if (_res_cap[a] > 0) {
1145                u = _target[a];
1146                rc = _cost[a] + pi_tip - _pi[u];
1147                if (rc < 0) {
1148                  path.push_back(a);
1149                  _next_out[tip] = a;
1150                  if (path_arc[a]) {
1151                    goto augment;   // a cycle is found, stop path search
1152                  }
1153                  tip = u;
1154                  path_arc[a] = true;
1155                  goto next_step;
1156                }
1157                else if (rc < min_red_cost) {
1158                  min_red_cost = rc;
1159                }
1160              }
1161            }
1162
1163            // Relabel tip node
1164            if (tip != start) {
1165              int ra = _reverse[path.back()];
1166              min_red_cost =
1167                std::min(min_red_cost, _cost[ra] + pi_tip - _pi[_target[ra]]);
1168            }
1169            last_out = _next_out[tip];
1170            for (int a = _first_out[tip]; a != last_out; ++a) {
1171              if (_res_cap[a] > 0) {
1172                rc = _cost[a] + pi_tip - _pi[_target[a]];
1173                if (rc < min_red_cost) {
1174                  min_red_cost = rc;
1175                }
1176              }
1177            }
1178            _pi[tip] -= min_red_cost + _epsilon;
1179            _next_out[tip] = _first_out[tip];
1180            ++relabel_cnt;
1181
1182            // Step back
1183            if (tip != start) {
1184              int pa = path.back();
1185              path_arc[pa] = false;
1186              tip = _source[pa];
1187              path.pop_back();
1188            }
1189
1190          next_step: ;
1191          }
1192
1193          // Augment along the found path (as much flow as possible)
1194        augment:
1195          Value delta;
1196          int pa, u, v = start;
1197          for (int i = 0; i != int(path.size()); ++i) {
1198            pa = path[i];
1199            u = v;
1200            v = _target[pa];
1201            path_arc[pa] = false;
1202            delta = std::min(_res_cap[pa], _excess[u]);
1203            _res_cap[pa] -= delta;
1204            _res_cap[_reverse[pa]] += delta;
1205            _excess[u] -= delta;
1206            _excess[v] += delta;
1207            if (_excess[v] > 0 && _excess[v] <= delta) {
1208              _active_nodes.push_back(v);
1209            }
1210          }
1211          path.clear();
1212
1213          // Global update heuristic
1214          if (relabel_cnt >= next_global_update_limit) {
1215            globalUpdate();
1216            next_global_update_limit += global_update_skip;
1217          }
1218        }
1219
1220      }
1221
1222    }
1223
1224    /// Execute the algorithm performing push and relabel operations
1225    void startPush() {
1226      // Paramters for heuristics
1227      const int EARLY_TERM_EPSILON_LIMIT = 1000;
1228      const double GLOBAL_UPDATE_FACTOR = 2.0;
1229
1230      const int global_update_skip = static_cast<int>(GLOBAL_UPDATE_FACTOR *
1231        (_res_node_num + _sup_node_num * _sup_node_num));
1232      int next_global_update_limit = global_update_skip;
1233
1234      // Perform cost scaling phases
1235      BoolVector hyper(_res_node_num, false);
1236      LargeCostVector hyper_cost(_res_node_num);
1237      int relabel_cnt = 0;
1238      for ( ; _epsilon >= 1; _epsilon = _epsilon < _alpha && _epsilon > 1 ?
1239                                        1 : _epsilon / _alpha )
1240      {
1241        // Early termination heuristic
1242        if (_epsilon <= EARLY_TERM_EPSILON_LIMIT) {
1243          if (earlyTermination()) break;
1244        }
1245
1246        // Initialize current phase
1247        initPhase();
1248
1249        // Perform push and relabel operations
1250        while (_active_nodes.size() > 0) {
1251          LargeCost min_red_cost, rc, pi_n;
1252          Value delta;
1253          int n, t, a, last_out = _res_arc_num;
1254
1255        next_node:
1256          // Select an active node (FIFO selection)
1257          n = _active_nodes.front();
1258          last_out = _first_out[n+1];
1259          pi_n = _pi[n];
1260
1261          // Perform push operations if there are admissible arcs
1262          if (_excess[n] > 0) {
1263            for (a = _next_out[n]; a != last_out; ++a) {
1264              if (_res_cap[a] > 0 &&
1265                  _cost[a] + pi_n - _pi[_target[a]] < 0) {
1266                delta = std::min(_res_cap[a], _excess[n]);
1267                t = _target[a];
1268
1269                // Push-look-ahead heuristic
1270                Value ahead = -_excess[t];
1271                int last_out_t = _first_out[t+1];
1272                LargeCost pi_t = _pi[t];
1273                for (int ta = _next_out[t]; ta != last_out_t; ++ta) {
1274                  if (_res_cap[ta] > 0 &&
1275                      _cost[ta] + pi_t - _pi[_target[ta]] < 0)
1276                    ahead += _res_cap[ta];
1277                  if (ahead >= delta) break;
1278                }
1279                if (ahead < 0) ahead = 0;
1280
1281                // Push flow along the arc
1282                if (ahead < delta && !hyper[t]) {
1283                  _res_cap[a] -= ahead;
1284                  _res_cap[_reverse[a]] += ahead;
1285                  _excess[n] -= ahead;
1286                  _excess[t] += ahead;
1287                  _active_nodes.push_front(t);
1288                  hyper[t] = true;
1289                  hyper_cost[t] = _cost[a] + pi_n - pi_t;
1290                  _next_out[n] = a;
1291                  goto next_node;
1292                } else {
1293                  _res_cap[a] -= delta;
1294                  _res_cap[_reverse[a]] += delta;
1295                  _excess[n] -= delta;
1296                  _excess[t] += delta;
1297                  if (_excess[t] > 0 && _excess[t] <= delta)
1298                    _active_nodes.push_back(t);
1299                }
1300
1301                if (_excess[n] == 0) {
1302                  _next_out[n] = a;
1303                  goto remove_nodes;
1304                }
1305              }
1306            }
1307            _next_out[n] = a;
1308          }
1309
1310          // Relabel the node if it is still active (or hyper)
1311          if (_excess[n] > 0 || hyper[n]) {
1312             min_red_cost = hyper[n] ? -hyper_cost[n] :
1313               std::numeric_limits<LargeCost>::max();
1314            for (int a = _first_out[n]; a != last_out; ++a) {
1315              if (_res_cap[a] > 0) {
1316                rc = _cost[a] + pi_n - _pi[_target[a]];
1317                if (rc < min_red_cost) {
1318                  min_red_cost = rc;
1319                }
1320              }
1321            }
1322            _pi[n] -= min_red_cost + _epsilon;
1323            _next_out[n] = _first_out[n];
1324            hyper[n] = false;
1325            ++relabel_cnt;
1326          }
1327
1328          // Remove nodes that are not active nor hyper
1329        remove_nodes:
1330          while ( _active_nodes.size() > 0 &&
1331                  _excess[_active_nodes.front()] <= 0 &&
1332                  !hyper[_active_nodes.front()] ) {
1333            _active_nodes.pop_front();
1334          }
1335
1336          // Global update heuristic
1337          if (relabel_cnt >= next_global_update_limit) {
1338            globalUpdate();
1339            for (int u = 0; u != _res_node_num; ++u)
1340              hyper[u] = false;
1341            next_global_update_limit += global_update_skip;
1342          }
1343        }
1344      }
1345    }
1346
1347  }; //class CostScaling
1348
1349  ///@}
1350
1351} //namespace lemon
1352
1353#endif //LEMON_COST_SCALING_H
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