COIN-OR::LEMON - Graph Library

source: lemon-1.2/lemon/cost_scaling.h @ 812:4b1b378823dc

Last change on this file since 812:4b1b378823dc was 812:4b1b378823dc, checked in by Peter Kovacs <kpeter@…>, 10 years ago

Small doc improvements + unifications in MCF classes (#180)

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