COIN-OR::LEMON - Graph Library

source: lemon-1.2/lemon/cost_scaling.h @ 840:2914b6f0fde0

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1/* -*- C++ -*-
2 *
3 * This file is a part of LEMON, a generic C++ optimization library
4 *
5 * Copyright (C) 2003-2008
6 * Egervary Jeno Kombinatorikus Optimalizalasi Kutatocsoport
7 * (Egervary Research Group on Combinatorial Optimization, EGRES).
8 *
9 * Permission to use, modify and distribute this software is granted
10 * provided that this copyright notice appears in all copies. For
11 * precise terms see the accompanying LICENSE file.
12 *
13 * This software is provided "AS IS" with no warranty of any kind,
14 * express or implied, and with no claim as to its suitability for any
15 * purpose.
16 *
17 */
18
19#ifndef LEMON_COST_SCALING_H
20#define LEMON_COST_SCALING_H
21
22/// \ingroup min_cost_flow_algs
23/// \file
24/// \brief Cost scaling algorithm for finding a minimum cost flow.
25
26#include <vector>
27#include <deque>
28#include <limits>
29
30#include <lemon/core.h>
31#include <lemon/maps.h>
32#include <lemon/math.h>
33#include <lemon/static_graph.h>
34#include <lemon/circulation.h>
35#include <lemon/bellman_ford.h>
36
37namespace lemon {
38
39  /// \brief Default traits class of CostScaling algorithm.
40  ///
41  /// Default traits class of CostScaling algorithm.
42  /// \tparam GR Digraph type.
43  /// \tparam V The number type used for flow amounts, capacity bounds
44  /// and supply values. By default it is \c int.
45  /// \tparam C The number type used for costs and potentials.
46  /// By default it is the same as \c V.
47#ifdef DOXYGEN
48  template <typename GR, typename V = int, typename C = V>
49#else
50  template < typename GR, typename V = int, typename C = V,
51             bool integer = std::numeric_limits<C>::is_integer >
52#endif
53  struct CostScalingDefaultTraits
54  {
55    /// The type of the digraph
56    typedef GR Digraph;
57    /// The type of the flow amounts, capacity bounds and supply values
58    typedef V Value;
59    /// The type of the arc costs
60    typedef C Cost;
61
62    /// \brief The large cost type used for internal computations
63    ///
64    /// The large cost type used for internal computations.
65    /// It is \c long \c long if the \c Cost type is integer,
66    /// otherwise it is \c double.
67    /// \c Cost must be convertible to \c LargeCost.
68    typedef double LargeCost;
69  };
70
71  // Default traits class for integer cost types
72  template <typename GR, typename V, typename C>
73  struct CostScalingDefaultTraits<GR, V, C, true>
74  {
75    typedef GR Digraph;
76    typedef V Value;
77    typedef C Cost;
78#ifdef LEMON_HAVE_LONG_LONG
79    typedef long long LargeCost;
80#else
81    typedef long LargeCost;
82#endif
83  };
84
85
86  /// \addtogroup min_cost_flow_algs
87  /// @{
88
89  /// \brief Implementation of the Cost Scaling algorithm for
90  /// finding a \ref min_cost_flow "minimum cost flow".
91  ///
92  /// \ref CostScaling implements a cost scaling algorithm that performs
93  /// push/augment and relabel operations for finding a \ref min_cost_flow
94  /// "minimum cost flow" \ref amo93networkflows, \ref goldberg90approximation,
95  /// \ref goldberg97efficient, \ref bunnagel98efficient.
96  /// It is a highly efficient primal-dual solution method, which
97  /// can be viewed as the generalization of the \ref Preflow
98  /// "preflow push-relabel" algorithm for the maximum flow problem.
99  ///
100  /// Most of the parameters of the problem (except for the digraph)
101  /// can be given using separate functions, and the algorithm can be
102  /// executed using the \ref run() function. If some parameters are not
103  /// specified, then default values will be used.
104  ///
105  /// \tparam GR The digraph type the algorithm runs on.
106  /// \tparam V The number type used for flow amounts, capacity bounds
107  /// and supply values in the algorithm. By default, it is \c int.
108  /// \tparam C The number type used for costs and potentials in the
109  /// algorithm. By default, it is the same as \c V.
110  /// \tparam TR The traits class that defines various types used by the
111  /// algorithm. By default, it is \ref CostScalingDefaultTraits
112  /// "CostScalingDefaultTraits<GR, V, C>".
113  /// In most cases, this parameter should not be set directly,
114  /// consider to use the named template parameters instead.
115  ///
116  /// \warning Both number types must be signed and all input data must
117  /// be integer.
118  /// \warning This algorithm does not support negative costs for such
119  /// arcs that have infinite upper bound.
120  ///
121  /// \note %CostScaling provides three different internal methods,
122  /// from which the most efficient one is used by default.
123  /// For more information, see \ref Method.
124#ifdef DOXYGEN
125  template <typename GR, typename V, typename C, typename TR>
126#else
127  template < typename GR, typename V = int, typename C = V,
128             typename TR = CostScalingDefaultTraits<GR, V, C> >
129#endif
130  class CostScaling
131  {
132  public:
133
134    /// The type of the digraph
135    typedef typename TR::Digraph Digraph;
136    /// The type of the flow amounts, capacity bounds and supply values
137    typedef typename TR::Value Value;
138    /// The type of the arc costs
139    typedef typename TR::Cost Cost;
140
141    /// \brief The large cost type
142    ///
143    /// The large cost type used for internal computations.
144    /// By default, it is \c long \c long if the \c Cost type is integer,
145    /// otherwise it is \c double.
146    typedef typename TR::LargeCost LargeCost;
147
148    /// The \ref CostScalingDefaultTraits "traits class" of the algorithm
149    typedef TR Traits;
150
151  public:
152
153    /// \brief Problem type constants for the \c run() function.
154    ///
155    /// Enum type containing the problem type constants that can be
156    /// returned by the \ref run() function of the algorithm.
157    enum ProblemType {
158      /// The problem has no feasible solution (flow).
159      INFEASIBLE,
160      /// The problem has optimal solution (i.e. it is feasible and
161      /// bounded), and the algorithm has found optimal flow and node
162      /// potentials (primal and dual solutions).
163      OPTIMAL,
164      /// The digraph contains an arc of negative cost and infinite
165      /// upper bound. It means that the objective function is unbounded
166      /// on that arc, however, note that it could actually be bounded
167      /// over the feasible flows, but this algroithm cannot handle
168      /// these cases.
169      UNBOUNDED
170    };
171
172    /// \brief Constants for selecting the internal method.
173    ///
174    /// Enum type containing constants for selecting the internal method
175    /// for the \ref run() function.
176    ///
177    /// \ref CostScaling provides three internal methods that differ mainly
178    /// in their base operations, which are used in conjunction with the
179    /// relabel operation.
180    /// By default, the so called \ref PARTIAL_AUGMENT
181    /// "Partial Augment-Relabel" method is used, which proved to be
182    /// the most efficient and the most robust on various test inputs.
183    /// However, the other methods can be selected using the \ref run()
184    /// function with the proper parameter.
185    enum Method {
186      /// Local push operations are used, i.e. flow is moved only on one
187      /// admissible arc at once.
188      PUSH,
189      /// Augment operations are used, i.e. flow is moved on admissible
190      /// paths from a node with excess to a node with deficit.
191      AUGMENT,
192      /// Partial augment operations are used, i.e. flow is moved on
193      /// admissible paths started from a node with excess, but the
194      /// lengths of these paths are limited. This method can be viewed
195      /// as a combined version of the previous two operations.
196      PARTIAL_AUGMENT
197    };
198
199  private:
200
201    TEMPLATE_DIGRAPH_TYPEDEFS(GR);
202
203    typedef std::vector<int> IntVector;
204    typedef std::vector<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  public:
329
330    /// \brief Constructor.
331    ///
332    /// The constructor of the class.
333    ///
334    /// \param graph The digraph the algorithm runs on.
335    CostScaling(const GR& graph) :
336      _graph(graph), _node_id(graph), _arc_idf(graph), _arc_idb(graph),
337      _cost_map(_cost_vec), _pi_map(_pi),
338      INF(std::numeric_limits<Value>::has_infinity ?
339          std::numeric_limits<Value>::infinity() :
340          std::numeric_limits<Value>::max())
341    {
342      // Check the number types
343      LEMON_ASSERT(std::numeric_limits<Value>::is_signed,
344        "The flow type of CostScaling must be signed");
345      LEMON_ASSERT(std::numeric_limits<Cost>::is_signed,
346        "The cost type of CostScaling must be signed");
347     
348      // Reset data structures
349      reset();
350    }
351
352    /// \name Parameters
353    /// The parameters of the algorithm can be specified using these
354    /// functions.
355
356    /// @{
357
358    /// \brief Set the lower bounds on the arcs.
359    ///
360    /// This function sets the lower bounds on the arcs.
361    /// If it is not used before calling \ref run(), the lower bounds
362    /// will be set to zero on all arcs.
363    ///
364    /// \param map An arc map storing the lower bounds.
365    /// Its \c Value type must be convertible to the \c Value type
366    /// of the algorithm.
367    ///
368    /// \return <tt>(*this)</tt>
369    template <typename LowerMap>
370    CostScaling& lowerMap(const LowerMap& map) {
371      _have_lower = true;
372      for (ArcIt a(_graph); a != INVALID; ++a) {
373        _lower[_arc_idf[a]] = map[a];
374        _lower[_arc_idb[a]] = map[a];
375      }
376      return *this;
377    }
378
379    /// \brief Set the upper bounds (capacities) on the arcs.
380    ///
381    /// This function sets the upper bounds (capacities) on the arcs.
382    /// If it is not used before calling \ref run(), the upper bounds
383    /// will be set to \ref INF on all arcs (i.e. the flow value will be
384    /// unbounded from above).
385    ///
386    /// \param map An arc map storing the upper bounds.
387    /// Its \c Value type must be convertible to the \c Value type
388    /// of the algorithm.
389    ///
390    /// \return <tt>(*this)</tt>
391    template<typename UpperMap>
392    CostScaling& upperMap(const UpperMap& map) {
393      for (ArcIt a(_graph); a != INVALID; ++a) {
394        _upper[_arc_idf[a]] = map[a];
395      }
396      return *this;
397    }
398
399    /// \brief Set the costs of the arcs.
400    ///
401    /// This function sets the costs of the arcs.
402    /// If it is not used before calling \ref run(), the costs
403    /// will be set to \c 1 on all arcs.
404    ///
405    /// \param map An arc map storing the costs.
406    /// Its \c Value type must be convertible to the \c Cost type
407    /// of the algorithm.
408    ///
409    /// \return <tt>(*this)</tt>
410    template<typename CostMap>
411    CostScaling& costMap(const CostMap& map) {
412      for (ArcIt a(_graph); a != INVALID; ++a) {
413        _scost[_arc_idf[a]] =  map[a];
414        _scost[_arc_idb[a]] = -map[a];
415      }
416      return *this;
417    }
418
419    /// \brief Set the supply values of the nodes.
420    ///
421    /// This function sets the supply values of the nodes.
422    /// If neither this function nor \ref stSupply() is used before
423    /// calling \ref run(), the supply of each node will be set to zero.
424    ///
425    /// \param map A node map storing the supply values.
426    /// Its \c Value type must be convertible to the \c Value type
427    /// of the algorithm.
428    ///
429    /// \return <tt>(*this)</tt>
430    template<typename SupplyMap>
431    CostScaling& supplyMap(const SupplyMap& map) {
432      for (NodeIt n(_graph); n != INVALID; ++n) {
433        _supply[_node_id[n]] = map[n];
434      }
435      return *this;
436    }
437
438    /// \brief Set single source and target nodes and a supply value.
439    ///
440    /// This function sets a single source node and a single target node
441    /// and the required flow value.
442    /// If neither this function nor \ref supplyMap() is used before
443    /// calling \ref run(), the supply of each node will be set to zero.
444    ///
445    /// Using this function has the same effect as using \ref supplyMap()
446    /// with such a map in which \c k is assigned to \c s, \c -k is
447    /// assigned to \c t and all other nodes have zero supply value.
448    ///
449    /// \param s The source node.
450    /// \param t The target node.
451    /// \param k The required amount of flow from node \c s to node \c t
452    /// (i.e. the supply of \c s and the demand of \c t).
453    ///
454    /// \return <tt>(*this)</tt>
455    CostScaling& stSupply(const Node& s, const Node& t, Value k) {
456      for (int i = 0; i != _res_node_num; ++i) {
457        _supply[i] = 0;
458      }
459      _supply[_node_id[s]] =  k;
460      _supply[_node_id[t]] = -k;
461      return *this;
462    }
463   
464    /// @}
465
466    /// \name Execution control
467    /// The algorithm can be executed using \ref run().
468
469    /// @{
470
471    /// \brief Run the algorithm.
472    ///
473    /// This function runs the algorithm.
474    /// The paramters can be specified using functions \ref lowerMap(),
475    /// \ref upperMap(), \ref costMap(), \ref supplyMap(), \ref stSupply().
476    /// For example,
477    /// \code
478    ///   CostScaling<ListDigraph> cs(graph);
479    ///   cs.lowerMap(lower).upperMap(upper).costMap(cost)
480    ///     .supplyMap(sup).run();
481    /// \endcode
482    ///
483    /// This function can be called more than once. All the given parameters
484    /// are kept for the next call, unless \ref resetParams() or \ref reset()
485    /// is used, thus only the modified parameters have to be set again.
486    /// If the underlying digraph was also modified after the construction
487    /// of the class (or the last \ref reset() call), then the \ref reset()
488    /// function must be called.
489    ///
490    /// \param method The internal method that will be used in the
491    /// algorithm. For more information, see \ref Method.
492    /// \param factor The cost scaling factor. It must be larger than one.
493    ///
494    /// \return \c INFEASIBLE if no feasible flow exists,
495    /// \n \c OPTIMAL if the problem has optimal solution
496    /// (i.e. it is feasible and bounded), and the algorithm has found
497    /// optimal flow and node potentials (primal and dual solutions),
498    /// \n \c UNBOUNDED if the digraph contains an arc of negative cost
499    /// and infinite upper bound. It means that the objective function
500    /// is unbounded on that arc, however, note that it could actually be
501    /// bounded over the feasible flows, but this algroithm cannot handle
502    /// these cases.
503    ///
504    /// \see ProblemType, Method
505    /// \see resetParams(), reset()
506    ProblemType run(Method method = PARTIAL_AUGMENT, int factor = 8) {
507      _alpha = factor;
508      ProblemType pt = init();
509      if (pt != OPTIMAL) return pt;
510      start(method);
511      return OPTIMAL;
512    }
513
514    /// \brief Reset all the parameters that have been given before.
515    ///
516    /// This function resets all the paramaters that have been given
517    /// before using functions \ref lowerMap(), \ref upperMap(),
518    /// \ref costMap(), \ref supplyMap(), \ref stSupply().
519    ///
520    /// It is useful for multiple \ref run() calls. Basically, all the given
521    /// parameters are kept for the next \ref run() call, unless
522    /// \ref resetParams() or \ref reset() is used.
523    /// If the underlying digraph was also modified after the construction
524    /// of the class or the last \ref reset() call, then the \ref reset()
525    /// function must be used, otherwise \ref resetParams() is sufficient.
526    ///
527    /// For example,
528    /// \code
529    ///   CostScaling<ListDigraph> cs(graph);
530    ///
531    ///   // First run
532    ///   cs.lowerMap(lower).upperMap(upper).costMap(cost)
533    ///     .supplyMap(sup).run();
534    ///
535    ///   // Run again with modified cost map (resetParams() is not called,
536    ///   // so only the cost map have to be set again)
537    ///   cost[e] += 100;
538    ///   cs.costMap(cost).run();
539    ///
540    ///   // Run again from scratch using resetParams()
541    ///   // (the lower bounds will be set to zero on all arcs)
542    ///   cs.resetParams();
543    ///   cs.upperMap(capacity).costMap(cost)
544    ///     .supplyMap(sup).run();
545    /// \endcode
546    ///
547    /// \return <tt>(*this)</tt>
548    ///
549    /// \see reset(), run()
550    CostScaling& resetParams() {
551      for (int i = 0; i != _res_node_num; ++i) {
552        _supply[i] = 0;
553      }
554      int limit = _first_out[_root];
555      for (int j = 0; j != limit; ++j) {
556        _lower[j] = 0;
557        _upper[j] = INF;
558        _scost[j] = _forward[j] ? 1 : -1;
559      }
560      for (int j = limit; j != _res_arc_num; ++j) {
561        _lower[j] = 0;
562        _upper[j] = INF;
563        _scost[j] = 0;
564        _scost[_reverse[j]] = 0;
565      }     
566      _have_lower = false;
567      return *this;
568    }
569
570    /// \brief Reset all the parameters that have been given before.
571    ///
572    /// This function resets all the paramaters that have been given
573    /// before using functions \ref lowerMap(), \ref upperMap(),
574    /// \ref costMap(), \ref supplyMap(), \ref stSupply().
575    ///
576    /// It is useful for multiple run() calls. If this function is not
577    /// used, all the parameters given before are kept for the next
578    /// \ref run() call.
579    /// However, the underlying digraph must not be modified after this
580    /// class have been constructed, since it copies and extends the graph.
581    /// \return <tt>(*this)</tt>
582    CostScaling& reset() {
583      // Resize vectors
584      _node_num = countNodes(_graph);
585      _arc_num = countArcs(_graph);
586      _res_node_num = _node_num + 1;
587      _res_arc_num = 2 * (_arc_num + _node_num);
588      _root = _node_num;
589
590      _first_out.resize(_res_node_num + 1);
591      _forward.resize(_res_arc_num);
592      _source.resize(_res_arc_num);
593      _target.resize(_res_arc_num);
594      _reverse.resize(_res_arc_num);
595
596      _lower.resize(_res_arc_num);
597      _upper.resize(_res_arc_num);
598      _scost.resize(_res_arc_num);
599      _supply.resize(_res_node_num);
600     
601      _res_cap.resize(_res_arc_num);
602      _cost.resize(_res_arc_num);
603      _pi.resize(_res_node_num);
604      _excess.resize(_res_node_num);
605      _next_out.resize(_res_node_num);
606
607      _arc_vec.reserve(_res_arc_num);
608      _cost_vec.reserve(_res_arc_num);
609
610      // Copy the graph
611      int i = 0, j = 0, k = 2 * _arc_num + _node_num;
612      for (NodeIt n(_graph); n != INVALID; ++n, ++i) {
613        _node_id[n] = i;
614      }
615      i = 0;
616      for (NodeIt n(_graph); n != INVALID; ++n, ++i) {
617        _first_out[i] = j;
618        for (OutArcIt a(_graph, n); a != INVALID; ++a, ++j) {
619          _arc_idf[a] = j;
620          _forward[j] = true;
621          _source[j] = i;
622          _target[j] = _node_id[_graph.runningNode(a)];
623        }
624        for (InArcIt a(_graph, n); a != INVALID; ++a, ++j) {
625          _arc_idb[a] = j;
626          _forward[j] = false;
627          _source[j] = i;
628          _target[j] = _node_id[_graph.runningNode(a)];
629        }
630        _forward[j] = false;
631        _source[j] = i;
632        _target[j] = _root;
633        _reverse[j] = k;
634        _forward[k] = true;
635        _source[k] = _root;
636        _target[k] = i;
637        _reverse[k] = j;
638        ++j; ++k;
639      }
640      _first_out[i] = j;
641      _first_out[_res_node_num] = k;
642      for (ArcIt a(_graph); a != INVALID; ++a) {
643        int fi = _arc_idf[a];
644        int bi = _arc_idb[a];
645        _reverse[fi] = bi;
646        _reverse[bi] = fi;
647      }
648     
649      // Reset parameters
650      resetParams();
651      return *this;
652    }
653
654    /// @}
655
656    /// \name Query Functions
657    /// The results of the algorithm can be obtained using these
658    /// functions.\n
659    /// The \ref run() function must be called before using them.
660
661    /// @{
662
663    /// \brief Return the total cost of the found flow.
664    ///
665    /// This function returns the total cost of the found flow.
666    /// Its complexity is O(e).
667    ///
668    /// \note The return type of the function can be specified as a
669    /// template parameter. For example,
670    /// \code
671    ///   cs.totalCost<double>();
672    /// \endcode
673    /// It is useful if the total cost cannot be stored in the \c Cost
674    /// type of the algorithm, which is the default return type of the
675    /// function.
676    ///
677    /// \pre \ref run() must be called before using this function.
678    template <typename Number>
679    Number totalCost() const {
680      Number c = 0;
681      for (ArcIt a(_graph); a != INVALID; ++a) {
682        int i = _arc_idb[a];
683        c += static_cast<Number>(_res_cap[i]) *
684             (-static_cast<Number>(_scost[i]));
685      }
686      return c;
687    }
688
689#ifndef DOXYGEN
690    Cost totalCost() const {
691      return totalCost<Cost>();
692    }
693#endif
694
695    /// \brief Return the flow on the given arc.
696    ///
697    /// This function returns the flow on the given arc.
698    ///
699    /// \pre \ref run() must be called before using this function.
700    Value flow(const Arc& a) const {
701      return _res_cap[_arc_idb[a]];
702    }
703
704    /// \brief Return the flow map (the primal solution).
705    ///
706    /// This function copies the flow value on each arc into the given
707    /// map. The \c Value type of the algorithm must be convertible to
708    /// the \c Value type of the map.
709    ///
710    /// \pre \ref run() must be called before using this function.
711    template <typename FlowMap>
712    void flowMap(FlowMap &map) const {
713      for (ArcIt a(_graph); a != INVALID; ++a) {
714        map.set(a, _res_cap[_arc_idb[a]]);
715      }
716    }
717
718    /// \brief Return the potential (dual value) of the given node.
719    ///
720    /// This function returns the potential (dual value) of the
721    /// given node.
722    ///
723    /// \pre \ref run() must be called before using this function.
724    Cost potential(const Node& n) const {
725      return static_cast<Cost>(_pi[_node_id[n]]);
726    }
727
728    /// \brief Return the potential map (the dual solution).
729    ///
730    /// This function copies the potential (dual value) of each node
731    /// into the given map.
732    /// The \c Cost type of the algorithm must be convertible to the
733    /// \c Value type of the map.
734    ///
735    /// \pre \ref run() must be called before using this function.
736    template <typename PotentialMap>
737    void potentialMap(PotentialMap &map) const {
738      for (NodeIt n(_graph); n != INVALID; ++n) {
739        map.set(n, static_cast<Cost>(_pi[_node_id[n]]));
740      }
741    }
742
743    /// @}
744
745  private:
746
747    // Initialize the algorithm
748    ProblemType init() {
749      if (_res_node_num <= 1) return INFEASIBLE;
750
751      // Check the sum of supply values
752      _sum_supply = 0;
753      for (int i = 0; i != _root; ++i) {
754        _sum_supply += _supply[i];
755      }
756      if (_sum_supply > 0) return INFEASIBLE;
757     
758
759      // Initialize vectors
760      for (int i = 0; i != _res_node_num; ++i) {
761        _pi[i] = 0;
762        _excess[i] = _supply[i];
763      }
764     
765      // Remove infinite upper bounds and check negative arcs
766      const Value MAX = std::numeric_limits<Value>::max();
767      int last_out;
768      if (_have_lower) {
769        for (int i = 0; i != _root; ++i) {
770          last_out = _first_out[i+1];
771          for (int j = _first_out[i]; j != last_out; ++j) {
772            if (_forward[j]) {
773              Value c = _scost[j] < 0 ? _upper[j] : _lower[j];
774              if (c >= MAX) return UNBOUNDED;
775              _excess[i] -= c;
776              _excess[_target[j]] += c;
777            }
778          }
779        }
780      } else {
781        for (int i = 0; i != _root; ++i) {
782          last_out = _first_out[i+1];
783          for (int j = _first_out[i]; j != last_out; ++j) {
784            if (_forward[j] && _scost[j] < 0) {
785              Value c = _upper[j];
786              if (c >= MAX) return UNBOUNDED;
787              _excess[i] -= c;
788              _excess[_target[j]] += c;
789            }
790          }
791        }
792      }
793      Value ex, max_cap = 0;
794      for (int i = 0; i != _res_node_num; ++i) {
795        ex = _excess[i];
796        _excess[i] = 0;
797        if (ex < 0) max_cap -= ex;
798      }
799      for (int j = 0; j != _res_arc_num; ++j) {
800        if (_upper[j] >= MAX) _upper[j] = max_cap;
801      }
802
803      // Initialize the large cost vector and the epsilon parameter
804      _epsilon = 0;
805      LargeCost lc;
806      for (int i = 0; i != _root; ++i) {
807        last_out = _first_out[i+1];
808        for (int j = _first_out[i]; j != last_out; ++j) {
809          lc = static_cast<LargeCost>(_scost[j]) * _res_node_num * _alpha;
810          _cost[j] = lc;
811          if (lc > _epsilon) _epsilon = lc;
812        }
813      }
814      _epsilon /= _alpha;
815
816      // Initialize maps for Circulation and remove non-zero lower bounds
817      ConstMap<Arc, Value> low(0);
818      typedef typename Digraph::template ArcMap<Value> ValueArcMap;
819      typedef typename Digraph::template NodeMap<Value> ValueNodeMap;
820      ValueArcMap cap(_graph), flow(_graph);
821      ValueNodeMap sup(_graph);
822      for (NodeIt n(_graph); n != INVALID; ++n) {
823        sup[n] = _supply[_node_id[n]];
824      }
825      if (_have_lower) {
826        for (ArcIt a(_graph); a != INVALID; ++a) {
827          int j = _arc_idf[a];
828          Value c = _lower[j];
829          cap[a] = _upper[j] - c;
830          sup[_graph.source(a)] -= c;
831          sup[_graph.target(a)] += c;
832        }
833      } else {
834        for (ArcIt a(_graph); a != INVALID; ++a) {
835          cap[a] = _upper[_arc_idf[a]];
836        }
837      }
838
839      _sup_node_num = 0;
840      for (NodeIt n(_graph); n != INVALID; ++n) {
841        if (sup[n] > 0) ++_sup_node_num;
842      }
843
844      // Find a feasible flow using Circulation
845      Circulation<Digraph, ConstMap<Arc, Value>, ValueArcMap, ValueNodeMap>
846        circ(_graph, low, cap, sup);
847      if (!circ.flowMap(flow).run()) return INFEASIBLE;
848
849      // Set residual capacities and handle GEQ supply type
850      if (_sum_supply < 0) {
851        for (ArcIt a(_graph); a != INVALID; ++a) {
852          Value fa = flow[a];
853          _res_cap[_arc_idf[a]] = cap[a] - fa;
854          _res_cap[_arc_idb[a]] = fa;
855          sup[_graph.source(a)] -= fa;
856          sup[_graph.target(a)] += fa;
857        }
858        for (NodeIt n(_graph); n != INVALID; ++n) {
859          _excess[_node_id[n]] = sup[n];
860        }
861        for (int a = _first_out[_root]; a != _res_arc_num; ++a) {
862          int u = _target[a];
863          int ra = _reverse[a];
864          _res_cap[a] = -_sum_supply + 1;
865          _res_cap[ra] = -_excess[u];
866          _cost[a] = 0;
867          _cost[ra] = 0;
868          _excess[u] = 0;
869        }
870      } else {
871        for (ArcIt a(_graph); a != INVALID; ++a) {
872          Value fa = flow[a];
873          _res_cap[_arc_idf[a]] = cap[a] - fa;
874          _res_cap[_arc_idb[a]] = fa;
875        }
876        for (int a = _first_out[_root]; a != _res_arc_num; ++a) {
877          int ra = _reverse[a];
878          _res_cap[a] = 0;
879          _res_cap[ra] = 0;
880          _cost[a] = 0;
881          _cost[ra] = 0;
882        }
883      }
884     
885      return OPTIMAL;
886    }
887
888    // Execute the algorithm and transform the results
889    void start(Method method) {
890      // Maximum path length for partial augment
891      const int MAX_PATH_LENGTH = 4;
892
893      // Initialize data structures for buckets     
894      _max_rank = _alpha * _res_node_num;
895      _buckets.resize(_max_rank);
896      _bucket_next.resize(_res_node_num + 1);
897      _bucket_prev.resize(_res_node_num + 1);
898      _rank.resize(_res_node_num + 1);
899 
900      // Execute the algorithm
901      switch (method) {
902        case PUSH:
903          startPush();
904          break;
905        case AUGMENT:
906          startAugment();
907          break;
908        case PARTIAL_AUGMENT:
909          startAugment(MAX_PATH_LENGTH);
910          break;
911      }
912
913      // Compute node potentials for the original costs
914      _arc_vec.clear();
915      _cost_vec.clear();
916      for (int j = 0; j != _res_arc_num; ++j) {
917        if (_res_cap[j] > 0) {
918          _arc_vec.push_back(IntPair(_source[j], _target[j]));
919          _cost_vec.push_back(_scost[j]);
920        }
921      }
922      _sgr.build(_res_node_num, _arc_vec.begin(), _arc_vec.end());
923
924      typename BellmanFord<StaticDigraph, LargeCostArcMap>
925        ::template SetDistMap<LargeCostNodeMap>::Create bf(_sgr, _cost_map);
926      bf.distMap(_pi_map);
927      bf.init(0);
928      bf.start();
929
930      // Handle non-zero lower bounds
931      if (_have_lower) {
932        int limit = _first_out[_root];
933        for (int j = 0; j != limit; ++j) {
934          if (!_forward[j]) _res_cap[j] += _lower[j];
935        }
936      }
937    }
938   
939    // Initialize a cost scaling phase
940    void initPhase() {
941      // Saturate arcs not satisfying the optimality condition
942      for (int u = 0; u != _res_node_num; ++u) {
943        int last_out = _first_out[u+1];
944        LargeCost pi_u = _pi[u];
945        for (int a = _first_out[u]; a != last_out; ++a) {
946          int v = _target[a];
947          if (_res_cap[a] > 0 && _cost[a] + pi_u - _pi[v] < 0) {
948            Value delta = _res_cap[a];
949            _excess[u] -= delta;
950            _excess[v] += delta;
951            _res_cap[a] = 0;
952            _res_cap[_reverse[a]] += delta;
953          }
954        }
955      }
956     
957      // Find active nodes (i.e. nodes with positive excess)
958      for (int u = 0; u != _res_node_num; ++u) {
959        if (_excess[u] > 0) _active_nodes.push_back(u);
960      }
961
962      // Initialize the next arcs
963      for (int u = 0; u != _res_node_num; ++u) {
964        _next_out[u] = _first_out[u];
965      }
966    }
967   
968    // Early termination heuristic
969    bool earlyTermination() {
970      const double EARLY_TERM_FACTOR = 3.0;
971
972      // Build a static residual graph
973      _arc_vec.clear();
974      _cost_vec.clear();
975      for (int j = 0; j != _res_arc_num; ++j) {
976        if (_res_cap[j] > 0) {
977          _arc_vec.push_back(IntPair(_source[j], _target[j]));
978          _cost_vec.push_back(_cost[j] + 1);
979        }
980      }
981      _sgr.build(_res_node_num, _arc_vec.begin(), _arc_vec.end());
982
983      // Run Bellman-Ford algorithm to check if the current flow is optimal
984      BellmanFord<StaticDigraph, LargeCostArcMap> bf(_sgr, _cost_map);
985      bf.init(0);
986      bool done = false;
987      int K = int(EARLY_TERM_FACTOR * std::sqrt(double(_res_node_num)));
988      for (int i = 0; i < K && !done; ++i) {
989        done = bf.processNextWeakRound();
990      }
991      return done;
992    }
993
994    // Global potential update heuristic
995    void globalUpdate() {
996      int bucket_end = _root + 1;
997   
998      // Initialize buckets
999      for (int r = 0; r != _max_rank; ++r) {
1000        _buckets[r] = bucket_end;
1001      }
1002      Value total_excess = 0;
1003      for (int i = 0; i != _res_node_num; ++i) {
1004        if (_excess[i] < 0) {
1005          _rank[i] = 0;
1006          _bucket_next[i] = _buckets[0];
1007          _bucket_prev[_buckets[0]] = i;
1008          _buckets[0] = i;
1009        } else {
1010          total_excess += _excess[i];
1011          _rank[i] = _max_rank;
1012        }
1013      }
1014      if (total_excess == 0) return;
1015
1016      // Search the buckets
1017      int r = 0;
1018      for ( ; r != _max_rank; ++r) {
1019        while (_buckets[r] != bucket_end) {
1020          // Remove the first node from the current bucket
1021          int u = _buckets[r];
1022          _buckets[r] = _bucket_next[u];
1023         
1024          // Search the incomming arcs of u
1025          LargeCost pi_u = _pi[u];
1026          int last_out = _first_out[u+1];
1027          for (int a = _first_out[u]; a != last_out; ++a) {
1028            int ra = _reverse[a];
1029            if (_res_cap[ra] > 0) {
1030              int v = _source[ra];
1031              int old_rank_v = _rank[v];
1032              if (r < old_rank_v) {
1033                // Compute the new rank of v
1034                LargeCost nrc = (_cost[ra] + _pi[v] - pi_u) / _epsilon;
1035                int new_rank_v = old_rank_v;
1036                if (nrc < LargeCost(_max_rank))
1037                  new_rank_v = r + 1 + int(nrc);
1038                 
1039                // Change the rank of v
1040                if (new_rank_v < old_rank_v) {
1041                  _rank[v] = new_rank_v;
1042                  _next_out[v] = _first_out[v];
1043                 
1044                  // Remove v from its old bucket
1045                  if (old_rank_v < _max_rank) {
1046                    if (_buckets[old_rank_v] == v) {
1047                      _buckets[old_rank_v] = _bucket_next[v];
1048                    } else {
1049                      _bucket_next[_bucket_prev[v]] = _bucket_next[v];
1050                      _bucket_prev[_bucket_next[v]] = _bucket_prev[v];
1051                    }
1052                  }
1053                 
1054                  // Insert v to its new bucket
1055                  _bucket_next[v] = _buckets[new_rank_v];
1056                  _bucket_prev[_buckets[new_rank_v]] = v;
1057                  _buckets[new_rank_v] = v;
1058                }
1059              }
1060            }
1061          }
1062
1063          // Finish search if there are no more active nodes
1064          if (_excess[u] > 0) {
1065            total_excess -= _excess[u];
1066            if (total_excess <= 0) break;
1067          }
1068        }
1069        if (total_excess <= 0) break;
1070      }
1071     
1072      // Relabel nodes
1073      for (int u = 0; u != _res_node_num; ++u) {
1074        int k = std::min(_rank[u], r);
1075        if (k > 0) {
1076          _pi[u] -= _epsilon * k;
1077          _next_out[u] = _first_out[u];
1078        }
1079      }
1080    }
1081
1082    /// Execute the algorithm performing augment and relabel operations
1083    void startAugment(int max_length = std::numeric_limits<int>::max()) {
1084      // Paramters for heuristics
1085      const int EARLY_TERM_EPSILON_LIMIT = 1000;
1086      const double GLOBAL_UPDATE_FACTOR = 3.0;
1087
1088      const int global_update_freq = int(GLOBAL_UPDATE_FACTOR *
1089        (_res_node_num + _sup_node_num * _sup_node_num));
1090      int next_update_limit = global_update_freq;
1091     
1092      int relabel_cnt = 0;
1093     
1094      // Perform cost scaling phases
1095      std::vector<int> path;
1096      for ( ; _epsilon >= 1; _epsilon = _epsilon < _alpha && _epsilon > 1 ?
1097                                        1 : _epsilon / _alpha )
1098      {
1099        // Early termination heuristic
1100        if (_epsilon <= EARLY_TERM_EPSILON_LIMIT) {
1101          if (earlyTermination()) break;
1102        }
1103       
1104        // Initialize current phase
1105        initPhase();
1106       
1107        // Perform partial augment and relabel operations
1108        while (true) {
1109          // Select an active node (FIFO selection)
1110          while (_active_nodes.size() > 0 &&
1111                 _excess[_active_nodes.front()] <= 0) {
1112            _active_nodes.pop_front();
1113          }
1114          if (_active_nodes.size() == 0) break;
1115          int start = _active_nodes.front();
1116
1117          // Find an augmenting path from the start node
1118          path.clear();
1119          int tip = start;
1120          while (_excess[tip] >= 0 && int(path.size()) < max_length) {
1121            int u;
1122            LargeCost min_red_cost, rc, pi_tip = _pi[tip];
1123            int last_out = _first_out[tip+1];
1124            for (int a = _next_out[tip]; a != last_out; ++a) {
1125              u = _target[a];
1126              if (_res_cap[a] > 0 && _cost[a] + pi_tip - _pi[u] < 0) {
1127                path.push_back(a);
1128                _next_out[tip] = a;
1129                tip = u;
1130                goto next_step;
1131              }
1132            }
1133
1134            // Relabel tip node
1135            min_red_cost = std::numeric_limits<LargeCost>::max();
1136            if (tip != start) {
1137              int ra = _reverse[path.back()];
1138              min_red_cost = _cost[ra] + pi_tip - _pi[_target[ra]];
1139            }
1140            for (int a = _first_out[tip]; a != last_out; ++a) {
1141              rc = _cost[a] + pi_tip - _pi[_target[a]];
1142              if (_res_cap[a] > 0 && rc < min_red_cost) {
1143                min_red_cost = rc;
1144              }
1145            }
1146            _pi[tip] -= min_red_cost + _epsilon;
1147            _next_out[tip] = _first_out[tip];
1148            ++relabel_cnt;
1149
1150            // Step back
1151            if (tip != start) {
1152              tip = _source[path.back()];
1153              path.pop_back();
1154            }
1155
1156          next_step: ;
1157          }
1158
1159          // Augment along the found path (as much flow as possible)
1160          Value delta;
1161          int pa, u, v = start;
1162          for (int i = 0; i != int(path.size()); ++i) {
1163            pa = path[i];
1164            u = v;
1165            v = _target[pa];
1166            delta = std::min(_res_cap[pa], _excess[u]);
1167            _res_cap[pa] -= delta;
1168            _res_cap[_reverse[pa]] += delta;
1169            _excess[u] -= delta;
1170            _excess[v] += delta;
1171            if (_excess[v] > 0 && _excess[v] <= delta)
1172              _active_nodes.push_back(v);
1173          }
1174
1175          // Global update heuristic
1176          if (relabel_cnt >= next_update_limit) {
1177            globalUpdate();
1178            next_update_limit += global_update_freq;
1179          }
1180        }
1181      }
1182    }
1183
1184    /// Execute the algorithm performing push and relabel operations
1185    void startPush() {
1186      // Paramters for heuristics
1187      const int EARLY_TERM_EPSILON_LIMIT = 1000;
1188      const double GLOBAL_UPDATE_FACTOR = 2.0;
1189
1190      const int global_update_freq = int(GLOBAL_UPDATE_FACTOR *
1191        (_res_node_num + _sup_node_num * _sup_node_num));
1192      int next_update_limit = global_update_freq;
1193
1194      int relabel_cnt = 0;
1195     
1196      // Perform cost scaling phases
1197      BoolVector hyper(_res_node_num, false);
1198      LargeCostVector hyper_cost(_res_node_num);
1199      for ( ; _epsilon >= 1; _epsilon = _epsilon < _alpha && _epsilon > 1 ?
1200                                        1 : _epsilon / _alpha )
1201      {
1202        // Early termination heuristic
1203        if (_epsilon <= EARLY_TERM_EPSILON_LIMIT) {
1204          if (earlyTermination()) break;
1205        }
1206       
1207        // Initialize current phase
1208        initPhase();
1209
1210        // Perform push and relabel operations
1211        while (_active_nodes.size() > 0) {
1212          LargeCost min_red_cost, rc, pi_n;
1213          Value delta;
1214          int n, t, a, last_out = _res_arc_num;
1215
1216        next_node:
1217          // Select an active node (FIFO selection)
1218          n = _active_nodes.front();
1219          last_out = _first_out[n+1];
1220          pi_n = _pi[n];
1221         
1222          // Perform push operations if there are admissible arcs
1223          if (_excess[n] > 0) {
1224            for (a = _next_out[n]; a != last_out; ++a) {
1225              if (_res_cap[a] > 0 &&
1226                  _cost[a] + pi_n - _pi[_target[a]] < 0) {
1227                delta = std::min(_res_cap[a], _excess[n]);
1228                t = _target[a];
1229
1230                // Push-look-ahead heuristic
1231                Value ahead = -_excess[t];
1232                int last_out_t = _first_out[t+1];
1233                LargeCost pi_t = _pi[t];
1234                for (int ta = _next_out[t]; ta != last_out_t; ++ta) {
1235                  if (_res_cap[ta] > 0 &&
1236                      _cost[ta] + pi_t - _pi[_target[ta]] < 0)
1237                    ahead += _res_cap[ta];
1238                  if (ahead >= delta) break;
1239                }
1240                if (ahead < 0) ahead = 0;
1241
1242                // Push flow along the arc
1243                if (ahead < delta && !hyper[t]) {
1244                  _res_cap[a] -= ahead;
1245                  _res_cap[_reverse[a]] += ahead;
1246                  _excess[n] -= ahead;
1247                  _excess[t] += ahead;
1248                  _active_nodes.push_front(t);
1249                  hyper[t] = true;
1250                  hyper_cost[t] = _cost[a] + pi_n - pi_t;
1251                  _next_out[n] = a;
1252                  goto next_node;
1253                } else {
1254                  _res_cap[a] -= delta;
1255                  _res_cap[_reverse[a]] += delta;
1256                  _excess[n] -= delta;
1257                  _excess[t] += delta;
1258                  if (_excess[t] > 0 && _excess[t] <= delta)
1259                    _active_nodes.push_back(t);
1260                }
1261
1262                if (_excess[n] == 0) {
1263                  _next_out[n] = a;
1264                  goto remove_nodes;
1265                }
1266              }
1267            }
1268            _next_out[n] = a;
1269          }
1270
1271          // Relabel the node if it is still active (or hyper)
1272          if (_excess[n] > 0 || hyper[n]) {
1273             min_red_cost = hyper[n] ? -hyper_cost[n] :
1274               std::numeric_limits<LargeCost>::max();
1275            for (int a = _first_out[n]; a != last_out; ++a) {
1276              rc = _cost[a] + pi_n - _pi[_target[a]];
1277              if (_res_cap[a] > 0 && rc < min_red_cost) {
1278                min_red_cost = rc;
1279              }
1280            }
1281            _pi[n] -= min_red_cost + _epsilon;
1282            _next_out[n] = _first_out[n];
1283            hyper[n] = false;
1284            ++relabel_cnt;
1285          }
1286       
1287          // Remove nodes that are not active nor hyper
1288        remove_nodes:
1289          while ( _active_nodes.size() > 0 &&
1290                  _excess[_active_nodes.front()] <= 0 &&
1291                  !hyper[_active_nodes.front()] ) {
1292            _active_nodes.pop_front();
1293          }
1294         
1295          // Global update heuristic
1296          if (relabel_cnt >= next_update_limit) {
1297            globalUpdate();
1298            for (int u = 0; u != _res_node_num; ++u)
1299              hyper[u] = false;
1300            next_update_limit += global_update_freq;
1301          }
1302        }
1303      }
1304    }
1305
1306  }; //class CostScaling
1307
1308  ///@}
1309
1310} //namespace lemon
1311
1312#endif //LEMON_COST_SCALING_H
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