COIN-OR::LEMON - Graph Library

source: lemon-1.2/lemon/cost_scaling.h @ 831:cc9e0c15d747

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1/* -*- C++ -*-
2 *
3 * This file is a part of LEMON, a generic C++ optimization library
4 *
5 * Copyright (C) 2003-2008
6 * Egervary Jeno Kombinatorikus Optimalizalasi Kutatocsoport
7 * (Egervary Research Group on Combinatorial Optimization, EGRES).
8 *
9 * Permission to use, modify and distribute this software is granted
10 * provided that this copyright notice appears in all copies. For
11 * precise terms see the accompanying LICENSE file.
12 *
13 * This software is provided "AS IS" with no warranty of any kind,
14 * express or implied, and with no claim as to its suitability for any
15 * purpose.
16 *
17 */
18
19#ifndef LEMON_COST_SCALING_H
20#define LEMON_COST_SCALING_H
21
22/// \ingroup min_cost_flow_algs
23/// \file
24/// \brief Cost scaling algorithm for finding a minimum cost flow.
25
26#include <vector>
27#include <deque>
28#include <limits>
29
30#include <lemon/core.h>
31#include <lemon/maps.h>
32#include <lemon/math.h>
33#include <lemon/static_graph.h>
34#include <lemon/circulation.h>
35#include <lemon/bellman_ford.h>
36
37namespace lemon {
38
39  /// \brief Default traits class of CostScaling algorithm.
40  ///
41  /// Default traits class of CostScaling algorithm.
42  /// \tparam GR Digraph type.
43  /// \tparam V The number type used for flow amounts, capacity bounds
44  /// and supply values. By default it is \c int.
45  /// \tparam C The number type used for costs and potentials.
46  /// By default it is the same as \c V.
47#ifdef DOXYGEN
48  template <typename GR, typename V = int, typename C = V>
49#else
50  template < typename GR, typename V = int, typename C = V,
51             bool integer = std::numeric_limits<C>::is_integer >
52#endif
53  struct CostScalingDefaultTraits
54  {
55    /// The type of the digraph
56    typedef GR Digraph;
57    /// The type of the flow amounts, capacity bounds and supply values
58    typedef V Value;
59    /// The type of the arc costs
60    typedef C Cost;
61
62    /// \brief The large cost type used for internal computations
63    ///
64    /// The large cost type used for internal computations.
65    /// It is \c long \c long if the \c Cost type is integer,
66    /// otherwise it is \c double.
67    /// \c Cost must be convertible to \c LargeCost.
68    typedef double LargeCost;
69  };
70
71  // Default traits class for integer cost types
72  template <typename GR, typename V, typename C>
73  struct CostScalingDefaultTraits<GR, V, C, true>
74  {
75    typedef GR Digraph;
76    typedef V Value;
77    typedef C Cost;
78#ifdef LEMON_HAVE_LONG_LONG
79    typedef long long LargeCost;
80#else
81    typedef long LargeCost;
82#endif
83  };
84
85
86  /// \addtogroup min_cost_flow_algs
87  /// @{
88
89  /// \brief Implementation of the Cost Scaling algorithm for
90  /// finding a \ref min_cost_flow "minimum cost flow".
91  ///
92  /// \ref CostScaling implements a cost scaling algorithm that performs
93  /// push/augment and relabel operations for finding a \ref min_cost_flow
94  /// "minimum cost flow" \ref amo93networkflows, \ref goldberg90approximation,
95  /// \ref goldberg97efficient, \ref bunnagel98efficient.
96  /// It is a highly efficient primal-dual solution method, which
97  /// can be viewed as the generalization of the \ref Preflow
98  /// "preflow push-relabel" algorithm for the maximum flow problem.
99  ///
100  /// Most of the parameters of the problem (except for the digraph)
101  /// can be given using separate functions, and the algorithm can be
102  /// executed using the \ref run() function. If some parameters are not
103  /// specified, then default values will be used.
104  ///
105  /// \tparam GR The digraph type the algorithm runs on.
106  /// \tparam V The number type used for flow amounts, capacity bounds
107  /// and supply values in the algorithm. By default, it is \c int.
108  /// \tparam C The number type used for costs and potentials in the
109  /// algorithm. By default, it is the same as \c V.
110  /// \tparam TR The traits class that defines various types used by the
111  /// algorithm. By default, it is \ref CostScalingDefaultTraits
112  /// "CostScalingDefaultTraits<GR, V, C>".
113  /// In most cases, this parameter should not be set directly,
114  /// consider to use the named template parameters instead.
115  ///
116  /// \warning Both number types must be signed and all input data must
117  /// be integer.
118  /// \warning This algorithm does not support negative costs for such
119  /// arcs that have infinite upper bound.
120  ///
121  /// \note %CostScaling provides three different internal methods,
122  /// from which the most efficient one is used by default.
123  /// For more information, see \ref Method.
124#ifdef DOXYGEN
125  template <typename GR, typename V, typename C, typename TR>
126#else
127  template < typename GR, typename V = int, typename C = V,
128             typename TR = CostScalingDefaultTraits<GR, V, C> >
129#endif
130  class CostScaling
131  {
132  public:
133
134    /// The type of the digraph
135    typedef typename TR::Digraph Digraph;
136    /// The type of the flow amounts, capacity bounds and supply values
137    typedef typename TR::Value Value;
138    /// The type of the arc costs
139    typedef typename TR::Cost Cost;
140
141    /// \brief The large cost type
142    ///
143    /// The large cost type used for internal computations.
144    /// By default, it is \c long \c long if the \c Cost type is integer,
145    /// otherwise it is \c double.
146    typedef typename TR::LargeCost LargeCost;
147
148    /// The \ref CostScalingDefaultTraits "traits class" of the algorithm
149    typedef TR Traits;
150
151  public:
152
153    /// \brief Problem type constants for the \c run() function.
154    ///
155    /// Enum type containing the problem type constants that can be
156    /// returned by the \ref run() function of the algorithm.
157    enum ProblemType {
158      /// The problem has no feasible solution (flow).
159      INFEASIBLE,
160      /// The problem has optimal solution (i.e. it is feasible and
161      /// bounded), and the algorithm has found optimal flow and node
162      /// potentials (primal and dual solutions).
163      OPTIMAL,
164      /// The digraph contains an arc of negative cost and infinite
165      /// upper bound. It means that the objective function is unbounded
166      /// on that arc, however, note that it could actually be bounded
167      /// over the feasible flows, but this algroithm cannot handle
168      /// these cases.
169      UNBOUNDED
170    };
171
172    /// \brief Constants for selecting the internal method.
173    ///
174    /// Enum type containing constants for selecting the internal method
175    /// for the \ref run() function.
176    ///
177    /// \ref CostScaling provides three internal methods that differ mainly
178    /// in their base operations, which are used in conjunction with the
179    /// relabel operation.
180    /// By default, the so called \ref PARTIAL_AUGMENT
181    /// "Partial Augment-Relabel" method is used, which proved to be
182    /// the most efficient and the most robust on various test inputs.
183    /// However, the other methods can be selected using the \ref run()
184    /// function with the proper parameter.
185    enum Method {
186      /// Local push operations are used, i.e. flow is moved only on one
187      /// admissible arc at once.
188      PUSH,
189      /// Augment operations are used, i.e. flow is moved on admissible
190      /// paths from a node with excess to a node with deficit.
191      AUGMENT,
192      /// Partial augment operations are used, i.e. flow is moved on
193      /// admissible paths started from a node with excess, but the
194      /// lengths of these paths are limited. This method can be viewed
195      /// as a combined version of the previous two operations.
196      PARTIAL_AUGMENT
197    };
198
199  private:
200
201    TEMPLATE_DIGRAPH_TYPEDEFS(GR);
202
203    typedef std::vector<int> IntVector;
204    typedef std::vector<char> BoolVector;
205    typedef std::vector<Value> ValueVector;
206    typedef std::vector<Cost> CostVector;
207    typedef std::vector<LargeCost> LargeCostVector;
208
209  private:
210 
211    template <typename KT, typename VT>
212    class StaticVectorMap {
213    public:
214      typedef KT Key;
215      typedef VT Value;
216     
217      StaticVectorMap(std::vector<Value>& v) : _v(v) {}
218     
219      const Value& operator[](const Key& key) const {
220        return _v[StaticDigraph::id(key)];
221      }
222
223      Value& operator[](const Key& key) {
224        return _v[StaticDigraph::id(key)];
225      }
226     
227      void set(const Key& key, const Value& val) {
228        _v[StaticDigraph::id(key)] = val;
229      }
230
231    private:
232      std::vector<Value>& _v;
233    };
234
235    typedef StaticVectorMap<StaticDigraph::Node, LargeCost> LargeCostNodeMap;
236    typedef StaticVectorMap<StaticDigraph::Arc, LargeCost> LargeCostArcMap;
237
238  private:
239
240    // Data related to the underlying digraph
241    const GR &_graph;
242    int _node_num;
243    int _arc_num;
244    int _res_node_num;
245    int _res_arc_num;
246    int _root;
247
248    // Parameters of the problem
249    bool _have_lower;
250    Value _sum_supply;
251
252    // Data structures for storing the digraph
253    IntNodeMap _node_id;
254    IntArcMap _arc_idf;
255    IntArcMap _arc_idb;
256    IntVector _first_out;
257    BoolVector _forward;
258    IntVector _source;
259    IntVector _target;
260    IntVector _reverse;
261
262    // Node and arc data
263    ValueVector _lower;
264    ValueVector _upper;
265    CostVector _scost;
266    ValueVector _supply;
267
268    ValueVector _res_cap;
269    LargeCostVector _cost;
270    LargeCostVector _pi;
271    ValueVector _excess;
272    IntVector _next_out;
273    std::deque<int> _active_nodes;
274
275    // Data for scaling
276    LargeCost _epsilon;
277    int _alpha;
278
279    // Data for a StaticDigraph structure
280    typedef std::pair<int, int> IntPair;
281    StaticDigraph _sgr;
282    std::vector<IntPair> _arc_vec;
283    std::vector<LargeCost> _cost_vec;
284    LargeCostArcMap _cost_map;
285    LargeCostNodeMap _pi_map;
286 
287  public:
288 
289    /// \brief Constant for infinite upper bounds (capacities).
290    ///
291    /// Constant for infinite upper bounds (capacities).
292    /// It is \c std::numeric_limits<Value>::infinity() if available,
293    /// \c std::numeric_limits<Value>::max() otherwise.
294    const Value INF;
295
296  public:
297
298    /// \name Named Template Parameters
299    /// @{
300
301    template <typename T>
302    struct SetLargeCostTraits : public Traits {
303      typedef T LargeCost;
304    };
305
306    /// \brief \ref named-templ-param "Named parameter" for setting
307    /// \c LargeCost type.
308    ///
309    /// \ref named-templ-param "Named parameter" for setting \c LargeCost
310    /// type, which is used for internal computations in the algorithm.
311    /// \c Cost must be convertible to \c LargeCost.
312    template <typename T>
313    struct SetLargeCost
314      : public CostScaling<GR, V, C, SetLargeCostTraits<T> > {
315      typedef  CostScaling<GR, V, C, SetLargeCostTraits<T> > Create;
316    };
317
318    /// @}
319
320  public:
321
322    /// \brief Constructor.
323    ///
324    /// The constructor of the class.
325    ///
326    /// \param graph The digraph the algorithm runs on.
327    CostScaling(const GR& graph) :
328      _graph(graph), _node_id(graph), _arc_idf(graph), _arc_idb(graph),
329      _cost_map(_cost_vec), _pi_map(_pi),
330      INF(std::numeric_limits<Value>::has_infinity ?
331          std::numeric_limits<Value>::infinity() :
332          std::numeric_limits<Value>::max())
333    {
334      // Check the number types
335      LEMON_ASSERT(std::numeric_limits<Value>::is_signed,
336        "The flow type of CostScaling must be signed");
337      LEMON_ASSERT(std::numeric_limits<Cost>::is_signed,
338        "The cost type of CostScaling must be signed");
339     
340      // Reset data structures
341      reset();
342    }
343
344    /// \name Parameters
345    /// The parameters of the algorithm can be specified using these
346    /// functions.
347
348    /// @{
349
350    /// \brief Set the lower bounds on the arcs.
351    ///
352    /// This function sets the lower bounds on the arcs.
353    /// If it is not used before calling \ref run(), the lower bounds
354    /// will be set to zero on all arcs.
355    ///
356    /// \param map An arc map storing the lower bounds.
357    /// Its \c Value type must be convertible to the \c Value type
358    /// of the algorithm.
359    ///
360    /// \return <tt>(*this)</tt>
361    template <typename LowerMap>
362    CostScaling& lowerMap(const LowerMap& map) {
363      _have_lower = true;
364      for (ArcIt a(_graph); a != INVALID; ++a) {
365        _lower[_arc_idf[a]] = map[a];
366        _lower[_arc_idb[a]] = map[a];
367      }
368      return *this;
369    }
370
371    /// \brief Set the upper bounds (capacities) on the arcs.
372    ///
373    /// This function sets the upper bounds (capacities) on the arcs.
374    /// If it is not used before calling \ref run(), the upper bounds
375    /// will be set to \ref INF on all arcs (i.e. the flow value will be
376    /// unbounded from above).
377    ///
378    /// \param map An arc map storing the upper bounds.
379    /// Its \c Value type must be convertible to the \c Value type
380    /// of the algorithm.
381    ///
382    /// \return <tt>(*this)</tt>
383    template<typename UpperMap>
384    CostScaling& upperMap(const UpperMap& map) {
385      for (ArcIt a(_graph); a != INVALID; ++a) {
386        _upper[_arc_idf[a]] = map[a];
387      }
388      return *this;
389    }
390
391    /// \brief Set the costs of the arcs.
392    ///
393    /// This function sets the costs of the arcs.
394    /// If it is not used before calling \ref run(), the costs
395    /// will be set to \c 1 on all arcs.
396    ///
397    /// \param map An arc map storing the costs.
398    /// Its \c Value type must be convertible to the \c Cost type
399    /// of the algorithm.
400    ///
401    /// \return <tt>(*this)</tt>
402    template<typename CostMap>
403    CostScaling& costMap(const CostMap& map) {
404      for (ArcIt a(_graph); a != INVALID; ++a) {
405        _scost[_arc_idf[a]] =  map[a];
406        _scost[_arc_idb[a]] = -map[a];
407      }
408      return *this;
409    }
410
411    /// \brief Set the supply values of the nodes.
412    ///
413    /// This function sets the supply values of the nodes.
414    /// If neither this function nor \ref stSupply() is used before
415    /// calling \ref run(), the supply of each node will be set to zero.
416    ///
417    /// \param map A node map storing the supply values.
418    /// Its \c Value type must be convertible to the \c Value type
419    /// of the algorithm.
420    ///
421    /// \return <tt>(*this)</tt>
422    template<typename SupplyMap>
423    CostScaling& supplyMap(const SupplyMap& map) {
424      for (NodeIt n(_graph); n != INVALID; ++n) {
425        _supply[_node_id[n]] = map[n];
426      }
427      return *this;
428    }
429
430    /// \brief Set single source and target nodes and a supply value.
431    ///
432    /// This function sets a single source node and a single target node
433    /// and the required flow value.
434    /// If neither this function nor \ref supplyMap() is used before
435    /// calling \ref run(), the supply of each node will be set to zero.
436    ///
437    /// Using this function has the same effect as using \ref supplyMap()
438    /// with such a map in which \c k is assigned to \c s, \c -k is
439    /// assigned to \c t and all other nodes have zero supply value.
440    ///
441    /// \param s The source node.
442    /// \param t The target node.
443    /// \param k The required amount of flow from node \c s to node \c t
444    /// (i.e. the supply of \c s and the demand of \c t).
445    ///
446    /// \return <tt>(*this)</tt>
447    CostScaling& stSupply(const Node& s, const Node& t, Value k) {
448      for (int i = 0; i != _res_node_num; ++i) {
449        _supply[i] = 0;
450      }
451      _supply[_node_id[s]] =  k;
452      _supply[_node_id[t]] = -k;
453      return *this;
454    }
455   
456    /// @}
457
458    /// \name Execution control
459    /// The algorithm can be executed using \ref run().
460
461    /// @{
462
463    /// \brief Run the algorithm.
464    ///
465    /// This function runs the algorithm.
466    /// The paramters can be specified using functions \ref lowerMap(),
467    /// \ref upperMap(), \ref costMap(), \ref supplyMap(), \ref stSupply().
468    /// For example,
469    /// \code
470    ///   CostScaling<ListDigraph> cs(graph);
471    ///   cs.lowerMap(lower).upperMap(upper).costMap(cost)
472    ///     .supplyMap(sup).run();
473    /// \endcode
474    ///
475    /// This function can be called more than once. All the given parameters
476    /// are kept for the next call, unless \ref resetParams() or \ref reset()
477    /// is used, thus only the modified parameters have to be set again.
478    /// If the underlying digraph was also modified after the construction
479    /// of the class (or the last \ref reset() call), then the \ref reset()
480    /// function must be called.
481    ///
482    /// \param method The internal method that will be used in the
483    /// algorithm. For more information, see \ref Method.
484    /// \param factor The cost scaling factor. It must be larger than one.
485    ///
486    /// \return \c INFEASIBLE if no feasible flow exists,
487    /// \n \c OPTIMAL if the problem has optimal solution
488    /// (i.e. it is feasible and bounded), and the algorithm has found
489    /// optimal flow and node potentials (primal and dual solutions),
490    /// \n \c UNBOUNDED if the digraph contains an arc of negative cost
491    /// and infinite upper bound. It means that the objective function
492    /// is unbounded on that arc, however, note that it could actually be
493    /// bounded over the feasible flows, but this algroithm cannot handle
494    /// these cases.
495    ///
496    /// \see ProblemType, Method
497    /// \see resetParams(), reset()
498    ProblemType run(Method method = PARTIAL_AUGMENT, int factor = 8) {
499      _alpha = factor;
500      ProblemType pt = init();
501      if (pt != OPTIMAL) return pt;
502      start(method);
503      return OPTIMAL;
504    }
505
506    /// \brief Reset all the parameters that have been given before.
507    ///
508    /// This function resets all the paramaters that have been given
509    /// before using functions \ref lowerMap(), \ref upperMap(),
510    /// \ref costMap(), \ref supplyMap(), \ref stSupply().
511    ///
512    /// It is useful for multiple \ref run() calls. Basically, all the given
513    /// parameters are kept for the next \ref run() call, unless
514    /// \ref resetParams() or \ref reset() is used.
515    /// If the underlying digraph was also modified after the construction
516    /// of the class or the last \ref reset() call, then the \ref reset()
517    /// function must be used, otherwise \ref resetParams() is sufficient.
518    ///
519    /// For example,
520    /// \code
521    ///   CostScaling<ListDigraph> cs(graph);
522    ///
523    ///   // First run
524    ///   cs.lowerMap(lower).upperMap(upper).costMap(cost)
525    ///     .supplyMap(sup).run();
526    ///
527    ///   // Run again with modified cost map (resetParams() is not called,
528    ///   // so only the cost map have to be set again)
529    ///   cost[e] += 100;
530    ///   cs.costMap(cost).run();
531    ///
532    ///   // Run again from scratch using resetParams()
533    ///   // (the lower bounds will be set to zero on all arcs)
534    ///   cs.resetParams();
535    ///   cs.upperMap(capacity).costMap(cost)
536    ///     .supplyMap(sup).run();
537    /// \endcode
538    ///
539    /// \return <tt>(*this)</tt>
540    ///
541    /// \see reset(), run()
542    CostScaling& resetParams() {
543      for (int i = 0; i != _res_node_num; ++i) {
544        _supply[i] = 0;
545      }
546      int limit = _first_out[_root];
547      for (int j = 0; j != limit; ++j) {
548        _lower[j] = 0;
549        _upper[j] = INF;
550        _scost[j] = _forward[j] ? 1 : -1;
551      }
552      for (int j = limit; j != _res_arc_num; ++j) {
553        _lower[j] = 0;
554        _upper[j] = INF;
555        _scost[j] = 0;
556        _scost[_reverse[j]] = 0;
557      }     
558      _have_lower = false;
559      return *this;
560    }
561
562    /// \brief Reset all the parameters that have been given before.
563    ///
564    /// This function resets all the paramaters that have been given
565    /// before using functions \ref lowerMap(), \ref upperMap(),
566    /// \ref costMap(), \ref supplyMap(), \ref stSupply().
567    ///
568    /// It is useful for multiple run() calls. If this function is not
569    /// used, all the parameters given before are kept for the next
570    /// \ref run() call.
571    /// However, the underlying digraph must not be modified after this
572    /// class have been constructed, since it copies and extends the graph.
573    /// \return <tt>(*this)</tt>
574    CostScaling& reset() {
575      // Resize vectors
576      _node_num = countNodes(_graph);
577      _arc_num = countArcs(_graph);
578      _res_node_num = _node_num + 1;
579      _res_arc_num = 2 * (_arc_num + _node_num);
580      _root = _node_num;
581
582      _first_out.resize(_res_node_num + 1);
583      _forward.resize(_res_arc_num);
584      _source.resize(_res_arc_num);
585      _target.resize(_res_arc_num);
586      _reverse.resize(_res_arc_num);
587
588      _lower.resize(_res_arc_num);
589      _upper.resize(_res_arc_num);
590      _scost.resize(_res_arc_num);
591      _supply.resize(_res_node_num);
592     
593      _res_cap.resize(_res_arc_num);
594      _cost.resize(_res_arc_num);
595      _pi.resize(_res_node_num);
596      _excess.resize(_res_node_num);
597      _next_out.resize(_res_node_num);
598
599      _arc_vec.reserve(_res_arc_num);
600      _cost_vec.reserve(_res_arc_num);
601
602      // Copy the graph
603      int i = 0, j = 0, k = 2 * _arc_num + _node_num;
604      for (NodeIt n(_graph); n != INVALID; ++n, ++i) {
605        _node_id[n] = i;
606      }
607      i = 0;
608      for (NodeIt n(_graph); n != INVALID; ++n, ++i) {
609        _first_out[i] = j;
610        for (OutArcIt a(_graph, n); a != INVALID; ++a, ++j) {
611          _arc_idf[a] = j;
612          _forward[j] = true;
613          _source[j] = i;
614          _target[j] = _node_id[_graph.runningNode(a)];
615        }
616        for (InArcIt a(_graph, n); a != INVALID; ++a, ++j) {
617          _arc_idb[a] = j;
618          _forward[j] = false;
619          _source[j] = i;
620          _target[j] = _node_id[_graph.runningNode(a)];
621        }
622        _forward[j] = false;
623        _source[j] = i;
624        _target[j] = _root;
625        _reverse[j] = k;
626        _forward[k] = true;
627        _source[k] = _root;
628        _target[k] = i;
629        _reverse[k] = j;
630        ++j; ++k;
631      }
632      _first_out[i] = j;
633      _first_out[_res_node_num] = k;
634      for (ArcIt a(_graph); a != INVALID; ++a) {
635        int fi = _arc_idf[a];
636        int bi = _arc_idb[a];
637        _reverse[fi] = bi;
638        _reverse[bi] = fi;
639      }
640     
641      // Reset parameters
642      resetParams();
643      return *this;
644    }
645
646    /// @}
647
648    /// \name Query Functions
649    /// The results of the algorithm can be obtained using these
650    /// functions.\n
651    /// The \ref run() function must be called before using them.
652
653    /// @{
654
655    /// \brief Return the total cost of the found flow.
656    ///
657    /// This function returns the total cost of the found flow.
658    /// Its complexity is O(e).
659    ///
660    /// \note The return type of the function can be specified as a
661    /// template parameter. For example,
662    /// \code
663    ///   cs.totalCost<double>();
664    /// \endcode
665    /// It is useful if the total cost cannot be stored in the \c Cost
666    /// type of the algorithm, which is the default return type of the
667    /// function.
668    ///
669    /// \pre \ref run() must be called before using this function.
670    template <typename Number>
671    Number totalCost() const {
672      Number c = 0;
673      for (ArcIt a(_graph); a != INVALID; ++a) {
674        int i = _arc_idb[a];
675        c += static_cast<Number>(_res_cap[i]) *
676             (-static_cast<Number>(_scost[i]));
677      }
678      return c;
679    }
680
681#ifndef DOXYGEN
682    Cost totalCost() const {
683      return totalCost<Cost>();
684    }
685#endif
686
687    /// \brief Return the flow on the given arc.
688    ///
689    /// This function returns the flow on the given arc.
690    ///
691    /// \pre \ref run() must be called before using this function.
692    Value flow(const Arc& a) const {
693      return _res_cap[_arc_idb[a]];
694    }
695
696    /// \brief Return the flow map (the primal solution).
697    ///
698    /// This function copies the flow value on each arc into the given
699    /// map. The \c Value type of the algorithm must be convertible to
700    /// the \c Value type of the map.
701    ///
702    /// \pre \ref run() must be called before using this function.
703    template <typename FlowMap>
704    void flowMap(FlowMap &map) const {
705      for (ArcIt a(_graph); a != INVALID; ++a) {
706        map.set(a, _res_cap[_arc_idb[a]]);
707      }
708    }
709
710    /// \brief Return the potential (dual value) of the given node.
711    ///
712    /// This function returns the potential (dual value) of the
713    /// given node.
714    ///
715    /// \pre \ref run() must be called before using this function.
716    Cost potential(const Node& n) const {
717      return static_cast<Cost>(_pi[_node_id[n]]);
718    }
719
720    /// \brief Return the potential map (the dual solution).
721    ///
722    /// This function copies the potential (dual value) of each node
723    /// into the given map.
724    /// The \c Cost type of the algorithm must be convertible to the
725    /// \c Value type of the map.
726    ///
727    /// \pre \ref run() must be called before using this function.
728    template <typename PotentialMap>
729    void potentialMap(PotentialMap &map) const {
730      for (NodeIt n(_graph); n != INVALID; ++n) {
731        map.set(n, static_cast<Cost>(_pi[_node_id[n]]));
732      }
733    }
734
735    /// @}
736
737  private:
738
739    // Initialize the algorithm
740    ProblemType init() {
741      if (_res_node_num <= 1) return INFEASIBLE;
742
743      // Check the sum of supply values
744      _sum_supply = 0;
745      for (int i = 0; i != _root; ++i) {
746        _sum_supply += _supply[i];
747      }
748      if (_sum_supply > 0) return INFEASIBLE;
749     
750
751      // Initialize vectors
752      for (int i = 0; i != _res_node_num; ++i) {
753        _pi[i] = 0;
754        _excess[i] = _supply[i];
755      }
756     
757      // Remove infinite upper bounds and check negative arcs
758      const Value MAX = std::numeric_limits<Value>::max();
759      int last_out;
760      if (_have_lower) {
761        for (int i = 0; i != _root; ++i) {
762          last_out = _first_out[i+1];
763          for (int j = _first_out[i]; j != last_out; ++j) {
764            if (_forward[j]) {
765              Value c = _scost[j] < 0 ? _upper[j] : _lower[j];
766              if (c >= MAX) return UNBOUNDED;
767              _excess[i] -= c;
768              _excess[_target[j]] += c;
769            }
770          }
771        }
772      } else {
773        for (int i = 0; i != _root; ++i) {
774          last_out = _first_out[i+1];
775          for (int j = _first_out[i]; j != last_out; ++j) {
776            if (_forward[j] && _scost[j] < 0) {
777              Value c = _upper[j];
778              if (c >= MAX) return UNBOUNDED;
779              _excess[i] -= c;
780              _excess[_target[j]] += c;
781            }
782          }
783        }
784      }
785      Value ex, max_cap = 0;
786      for (int i = 0; i != _res_node_num; ++i) {
787        ex = _excess[i];
788        _excess[i] = 0;
789        if (ex < 0) max_cap -= ex;
790      }
791      for (int j = 0; j != _res_arc_num; ++j) {
792        if (_upper[j] >= MAX) _upper[j] = max_cap;
793      }
794
795      // Initialize the large cost vector and the epsilon parameter
796      _epsilon = 0;
797      LargeCost lc;
798      for (int i = 0; i != _root; ++i) {
799        last_out = _first_out[i+1];
800        for (int j = _first_out[i]; j != last_out; ++j) {
801          lc = static_cast<LargeCost>(_scost[j]) * _res_node_num * _alpha;
802          _cost[j] = lc;
803          if (lc > _epsilon) _epsilon = lc;
804        }
805      }
806      _epsilon /= _alpha;
807
808      // Initialize maps for Circulation and remove non-zero lower bounds
809      ConstMap<Arc, Value> low(0);
810      typedef typename Digraph::template ArcMap<Value> ValueArcMap;
811      typedef typename Digraph::template NodeMap<Value> ValueNodeMap;
812      ValueArcMap cap(_graph), flow(_graph);
813      ValueNodeMap sup(_graph);
814      for (NodeIt n(_graph); n != INVALID; ++n) {
815        sup[n] = _supply[_node_id[n]];
816      }
817      if (_have_lower) {
818        for (ArcIt a(_graph); a != INVALID; ++a) {
819          int j = _arc_idf[a];
820          Value c = _lower[j];
821          cap[a] = _upper[j] - c;
822          sup[_graph.source(a)] -= c;
823          sup[_graph.target(a)] += c;
824        }
825      } else {
826        for (ArcIt a(_graph); a != INVALID; ++a) {
827          cap[a] = _upper[_arc_idf[a]];
828        }
829      }
830
831      // Find a feasible flow using Circulation
832      Circulation<Digraph, ConstMap<Arc, Value>, ValueArcMap, ValueNodeMap>
833        circ(_graph, low, cap, sup);
834      if (!circ.flowMap(flow).run()) return INFEASIBLE;
835
836      // Set residual capacities and handle GEQ supply type
837      if (_sum_supply < 0) {
838        for (ArcIt a(_graph); a != INVALID; ++a) {
839          Value fa = flow[a];
840          _res_cap[_arc_idf[a]] = cap[a] - fa;
841          _res_cap[_arc_idb[a]] = fa;
842          sup[_graph.source(a)] -= fa;
843          sup[_graph.target(a)] += fa;
844        }
845        for (NodeIt n(_graph); n != INVALID; ++n) {
846          _excess[_node_id[n]] = sup[n];
847        }
848        for (int a = _first_out[_root]; a != _res_arc_num; ++a) {
849          int u = _target[a];
850          int ra = _reverse[a];
851          _res_cap[a] = -_sum_supply + 1;
852          _res_cap[ra] = -_excess[u];
853          _cost[a] = 0;
854          _cost[ra] = 0;
855          _excess[u] = 0;
856        }
857      } else {
858        for (ArcIt a(_graph); a != INVALID; ++a) {
859          Value fa = flow[a];
860          _res_cap[_arc_idf[a]] = cap[a] - fa;
861          _res_cap[_arc_idb[a]] = fa;
862        }
863        for (int a = _first_out[_root]; a != _res_arc_num; ++a) {
864          int ra = _reverse[a];
865          _res_cap[a] = 1;
866          _res_cap[ra] = 0;
867          _cost[a] = 0;
868          _cost[ra] = 0;
869        }
870      }
871     
872      return OPTIMAL;
873    }
874
875    // Execute the algorithm and transform the results
876    void start(Method method) {
877      // Maximum path length for partial augment
878      const int MAX_PATH_LENGTH = 4;
879     
880      // Execute the algorithm
881      switch (method) {
882        case PUSH:
883          startPush();
884          break;
885        case AUGMENT:
886          startAugment();
887          break;
888        case PARTIAL_AUGMENT:
889          startAugment(MAX_PATH_LENGTH);
890          break;
891      }
892
893      // Compute node potentials for the original costs
894      _arc_vec.clear();
895      _cost_vec.clear();
896      for (int j = 0; j != _res_arc_num; ++j) {
897        if (_res_cap[j] > 0) {
898          _arc_vec.push_back(IntPair(_source[j], _target[j]));
899          _cost_vec.push_back(_scost[j]);
900        }
901      }
902      _sgr.build(_res_node_num, _arc_vec.begin(), _arc_vec.end());
903
904      typename BellmanFord<StaticDigraph, LargeCostArcMap>
905        ::template SetDistMap<LargeCostNodeMap>::Create bf(_sgr, _cost_map);
906      bf.distMap(_pi_map);
907      bf.init(0);
908      bf.start();
909
910      // Handle non-zero lower bounds
911      if (_have_lower) {
912        int limit = _first_out[_root];
913        for (int j = 0; j != limit; ++j) {
914          if (!_forward[j]) _res_cap[j] += _lower[j];
915        }
916      }
917    }
918
919    /// Execute the algorithm performing augment and relabel operations
920    void startAugment(int max_length = std::numeric_limits<int>::max()) {
921      // Paramters for heuristics
922      const int BF_HEURISTIC_EPSILON_BOUND = 1000;
923      const int BF_HEURISTIC_BOUND_FACTOR  = 3;
924
925      // Perform cost scaling phases
926      IntVector pred_arc(_res_node_num);
927      std::vector<int> path_nodes;
928      for ( ; _epsilon >= 1; _epsilon = _epsilon < _alpha && _epsilon > 1 ?
929                                        1 : _epsilon / _alpha )
930      {
931        // "Early Termination" heuristic: use Bellman-Ford algorithm
932        // to check if the current flow is optimal
933        if (_epsilon <= BF_HEURISTIC_EPSILON_BOUND) {
934          _arc_vec.clear();
935          _cost_vec.clear();
936          for (int j = 0; j != _res_arc_num; ++j) {
937            if (_res_cap[j] > 0) {
938              _arc_vec.push_back(IntPair(_source[j], _target[j]));
939              _cost_vec.push_back(_cost[j] + 1);
940            }
941          }
942          _sgr.build(_res_node_num, _arc_vec.begin(), _arc_vec.end());
943
944          BellmanFord<StaticDigraph, LargeCostArcMap> bf(_sgr, _cost_map);
945          bf.init(0);
946          bool done = false;
947          int K = int(BF_HEURISTIC_BOUND_FACTOR * sqrt(_res_node_num));
948          for (int i = 0; i < K && !done; ++i)
949            done = bf.processNextWeakRound();
950          if (done) break;
951        }
952
953        // Saturate arcs not satisfying the optimality condition
954        for (int a = 0; a != _res_arc_num; ++a) {
955          if (_res_cap[a] > 0 &&
956              _cost[a] + _pi[_source[a]] - _pi[_target[a]] < 0) {
957            Value delta = _res_cap[a];
958            _excess[_source[a]] -= delta;
959            _excess[_target[a]] += delta;
960            _res_cap[a] = 0;
961            _res_cap[_reverse[a]] += delta;
962          }
963        }
964       
965        // Find active nodes (i.e. nodes with positive excess)
966        for (int u = 0; u != _res_node_num; ++u) {
967          if (_excess[u] > 0) _active_nodes.push_back(u);
968        }
969
970        // Initialize the next arcs
971        for (int u = 0; u != _res_node_num; ++u) {
972          _next_out[u] = _first_out[u];
973        }
974
975        // Perform partial augment and relabel operations
976        while (true) {
977          // Select an active node (FIFO selection)
978          while (_active_nodes.size() > 0 &&
979                 _excess[_active_nodes.front()] <= 0) {
980            _active_nodes.pop_front();
981          }
982          if (_active_nodes.size() == 0) break;
983          int start = _active_nodes.front();
984          path_nodes.clear();
985          path_nodes.push_back(start);
986
987          // Find an augmenting path from the start node
988          int tip = start;
989          while (_excess[tip] >= 0 &&
990                 int(path_nodes.size()) <= max_length) {
991            int u;
992            LargeCost min_red_cost, rc;
993            int last_out = _sum_supply < 0 ?
994              _first_out[tip+1] : _first_out[tip+1] - 1;
995            for (int a = _next_out[tip]; a != last_out; ++a) {
996              if (_res_cap[a] > 0 &&
997                  _cost[a] + _pi[_source[a]] - _pi[_target[a]] < 0) {
998                u = _target[a];
999                pred_arc[u] = a;
1000                _next_out[tip] = a;
1001                tip = u;
1002                path_nodes.push_back(tip);
1003                goto next_step;
1004              }
1005            }
1006
1007            // Relabel tip node
1008            min_red_cost = std::numeric_limits<LargeCost>::max() / 2;
1009            for (int a = _first_out[tip]; a != last_out; ++a) {
1010              rc = _cost[a] + _pi[_source[a]] - _pi[_target[a]];
1011              if (_res_cap[a] > 0 && rc < min_red_cost) {
1012                min_red_cost = rc;
1013              }
1014            }
1015            _pi[tip] -= min_red_cost + _epsilon;
1016
1017            // Reset the next arc of tip
1018            _next_out[tip] = _first_out[tip];
1019
1020            // Step back
1021            if (tip != start) {
1022              path_nodes.pop_back();
1023              tip = path_nodes.back();
1024            }
1025
1026          next_step: ;
1027          }
1028
1029          // Augment along the found path (as much flow as possible)
1030          Value delta;
1031          int u, v = path_nodes.front(), pa;
1032          for (int i = 1; i < int(path_nodes.size()); ++i) {
1033            u = v;
1034            v = path_nodes[i];
1035            pa = pred_arc[v];
1036            delta = std::min(_res_cap[pa], _excess[u]);
1037            _res_cap[pa] -= delta;
1038            _res_cap[_reverse[pa]] += delta;
1039            _excess[u] -= delta;
1040            _excess[v] += delta;
1041            if (_excess[v] > 0 && _excess[v] <= delta)
1042              _active_nodes.push_back(v);
1043          }
1044        }
1045      }
1046    }
1047
1048    /// Execute the algorithm performing push and relabel operations
1049    void startPush() {
1050      // Paramters for heuristics
1051      const int BF_HEURISTIC_EPSILON_BOUND = 1000;
1052      const int BF_HEURISTIC_BOUND_FACTOR  = 3;
1053
1054      // Perform cost scaling phases
1055      BoolVector hyper(_res_node_num, false);
1056      for ( ; _epsilon >= 1; _epsilon = _epsilon < _alpha && _epsilon > 1 ?
1057                                        1 : _epsilon / _alpha )
1058      {
1059        // "Early Termination" heuristic: use Bellman-Ford algorithm
1060        // to check if the current flow is optimal
1061        if (_epsilon <= BF_HEURISTIC_EPSILON_BOUND) {
1062          _arc_vec.clear();
1063          _cost_vec.clear();
1064          for (int j = 0; j != _res_arc_num; ++j) {
1065            if (_res_cap[j] > 0) {
1066              _arc_vec.push_back(IntPair(_source[j], _target[j]));
1067              _cost_vec.push_back(_cost[j] + 1);
1068            }
1069          }
1070          _sgr.build(_res_node_num, _arc_vec.begin(), _arc_vec.end());
1071
1072          BellmanFord<StaticDigraph, LargeCostArcMap> bf(_sgr, _cost_map);
1073          bf.init(0);
1074          bool done = false;
1075          int K = int(BF_HEURISTIC_BOUND_FACTOR * sqrt(_res_node_num));
1076          for (int i = 0; i < K && !done; ++i)
1077            done = bf.processNextWeakRound();
1078          if (done) break;
1079        }
1080
1081        // Saturate arcs not satisfying the optimality condition
1082        for (int a = 0; a != _res_arc_num; ++a) {
1083          if (_res_cap[a] > 0 &&
1084              _cost[a] + _pi[_source[a]] - _pi[_target[a]] < 0) {
1085            Value delta = _res_cap[a];
1086            _excess[_source[a]] -= delta;
1087            _excess[_target[a]] += delta;
1088            _res_cap[a] = 0;
1089            _res_cap[_reverse[a]] += delta;
1090          }
1091        }
1092
1093        // Find active nodes (i.e. nodes with positive excess)
1094        for (int u = 0; u != _res_node_num; ++u) {
1095          if (_excess[u] > 0) _active_nodes.push_back(u);
1096        }
1097
1098        // Initialize the next arcs
1099        for (int u = 0; u != _res_node_num; ++u) {
1100          _next_out[u] = _first_out[u];
1101        }
1102
1103        // Perform push and relabel operations
1104        while (_active_nodes.size() > 0) {
1105          LargeCost min_red_cost, rc;
1106          Value delta;
1107          int n, t, a, last_out = _res_arc_num;
1108
1109          // Select an active node (FIFO selection)
1110        next_node:
1111          n = _active_nodes.front();
1112          last_out = _sum_supply < 0 ?
1113            _first_out[n+1] : _first_out[n+1] - 1;
1114
1115          // Perform push operations if there are admissible arcs
1116          if (_excess[n] > 0) {
1117            for (a = _next_out[n]; a != last_out; ++a) {
1118              if (_res_cap[a] > 0 &&
1119                  _cost[a] + _pi[_source[a]] - _pi[_target[a]] < 0) {
1120                delta = std::min(_res_cap[a], _excess[n]);
1121                t = _target[a];
1122
1123                // Push-look-ahead heuristic
1124                Value ahead = -_excess[t];
1125                int last_out_t = _sum_supply < 0 ?
1126                  _first_out[t+1] : _first_out[t+1] - 1;
1127                for (int ta = _next_out[t]; ta != last_out_t; ++ta) {
1128                  if (_res_cap[ta] > 0 &&
1129                      _cost[ta] + _pi[_source[ta]] - _pi[_target[ta]] < 0)
1130                    ahead += _res_cap[ta];
1131                  if (ahead >= delta) break;
1132                }
1133                if (ahead < 0) ahead = 0;
1134
1135                // Push flow along the arc
1136                if (ahead < delta) {
1137                  _res_cap[a] -= ahead;
1138                  _res_cap[_reverse[a]] += ahead;
1139                  _excess[n] -= ahead;
1140                  _excess[t] += ahead;
1141                  _active_nodes.push_front(t);
1142                  hyper[t] = true;
1143                  _next_out[n] = a;
1144                  goto next_node;
1145                } else {
1146                  _res_cap[a] -= delta;
1147                  _res_cap[_reverse[a]] += delta;
1148                  _excess[n] -= delta;
1149                  _excess[t] += delta;
1150                  if (_excess[t] > 0 && _excess[t] <= delta)
1151                    _active_nodes.push_back(t);
1152                }
1153
1154                if (_excess[n] == 0) {
1155                  _next_out[n] = a;
1156                  goto remove_nodes;
1157                }
1158              }
1159            }
1160            _next_out[n] = a;
1161          }
1162
1163          // Relabel the node if it is still active (or hyper)
1164          if (_excess[n] > 0 || hyper[n]) {
1165            min_red_cost = std::numeric_limits<LargeCost>::max() / 2;
1166            for (int a = _first_out[n]; a != last_out; ++a) {
1167              rc = _cost[a] + _pi[_source[a]] - _pi[_target[a]];
1168              if (_res_cap[a] > 0 && rc < min_red_cost) {
1169                min_red_cost = rc;
1170              }
1171            }
1172            _pi[n] -= min_red_cost + _epsilon;
1173            hyper[n] = false;
1174
1175            // Reset the next arc
1176            _next_out[n] = _first_out[n];
1177          }
1178       
1179          // Remove nodes that are not active nor hyper
1180        remove_nodes:
1181          while ( _active_nodes.size() > 0 &&
1182                  _excess[_active_nodes.front()] <= 0 &&
1183                  !hyper[_active_nodes.front()] ) {
1184            _active_nodes.pop_front();
1185          }
1186        }
1187      }
1188    }
1189
1190  }; //class CostScaling
1191
1192  ///@}
1193
1194} //namespace lemon
1195
1196#endif //LEMON_COST_SCALING_H
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