COIN-OR::LEMON - Graph Library

source: lemon-1.2/lemon/cost_scaling.h @ 808:9c428bb2b105

Last change on this file since 808:9c428bb2b105 was 808:9c428bb2b105, checked in by Peter Kovacs <kpeter@…>, 10 years ago

Port CostScaling? from SVN -r3524 (#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/adaptors.h>
34#include <lemon/circulation.h>
35#include <lemon/bellman_ford.h>
36
37namespace lemon {
38
39  /// \addtogroup min_cost_flow_algs
40  /// @{
41
42  /// \brief Implementation of the cost scaling algorithm for finding a
43  /// minimum cost flow.
44  ///
45  /// \ref CostScaling implements the cost scaling algorithm performing
46  /// augment/push and relabel operations for finding a minimum cost
47  /// flow.
48  ///
49  /// \tparam Digraph The digraph type the algorithm runs on.
50  /// \tparam LowerMap The type of the lower bound map.
51  /// \tparam CapacityMap The type of the capacity (upper bound) map.
52  /// \tparam CostMap The type of the cost (length) map.
53  /// \tparam SupplyMap The type of the supply map.
54  ///
55  /// \warning
56  /// - Arc capacities and costs should be \e non-negative \e integers.
57  /// - Supply values should be \e signed \e integers.
58  /// - The value types of the maps should be convertible to each other.
59  /// - \c CostMap::Value must be signed type.
60  ///
61  /// \note Arc costs are multiplied with the number of nodes during
62  /// the algorithm so overflow problems may arise more easily than with
63  /// other minimum cost flow algorithms.
64  /// If it is available, <tt>long long int</tt> type is used instead of
65  /// <tt>long int</tt> in the inside computations.
66  ///
67  /// \author Peter Kovacs
68  template < typename Digraph,
69             typename LowerMap = typename Digraph::template ArcMap<int>,
70             typename CapacityMap = typename Digraph::template ArcMap<int>,
71             typename CostMap = typename Digraph::template ArcMap<int>,
72             typename SupplyMap = typename Digraph::template NodeMap<int> >
73  class CostScaling
74  {
75    TEMPLATE_DIGRAPH_TYPEDEFS(Digraph);
76
77    typedef typename CapacityMap::Value Capacity;
78    typedef typename CostMap::Value Cost;
79    typedef typename SupplyMap::Value Supply;
80    typedef typename Digraph::template ArcMap<Capacity> CapacityArcMap;
81    typedef typename Digraph::template NodeMap<Supply> SupplyNodeMap;
82
83    typedef ResidualDigraph< const Digraph,
84                             CapacityArcMap, CapacityArcMap > ResDigraph;
85    typedef typename ResDigraph::Arc ResArc;
86
87#if defined __GNUC__ && !defined __STRICT_ANSI__
88    typedef long long int LCost;
89#else
90    typedef long int LCost;
91#endif
92    typedef typename Digraph::template ArcMap<LCost> LargeCostMap;
93
94  public:
95
96    /// The type of the flow map.
97    typedef typename Digraph::template ArcMap<Capacity> FlowMap;
98    /// The type of the potential map.
99    typedef typename Digraph::template NodeMap<LCost> PotentialMap;
100
101  private:
102
103    /// \brief Map adaptor class for handling residual arc costs.
104    ///
105    /// Map adaptor class for handling residual arc costs.
106    template <typename Map>
107    class ResidualCostMap : public MapBase<ResArc, typename Map::Value>
108    {
109    private:
110
111      const Map &_cost_map;
112
113    public:
114
115      ///\e
116      ResidualCostMap(const Map &cost_map) :
117        _cost_map(cost_map) {}
118
119      ///\e
120      inline typename Map::Value operator[](const ResArc &e) const {
121        return ResDigraph::forward(e) ? _cost_map[e] : -_cost_map[e];
122      }
123
124    }; //class ResidualCostMap
125
126    /// \brief Map adaptor class for handling reduced arc costs.
127    ///
128    /// Map adaptor class for handling reduced arc costs.
129    class ReducedCostMap : public MapBase<Arc, LCost>
130    {
131    private:
132
133      const Digraph &_gr;
134      const LargeCostMap &_cost_map;
135      const PotentialMap &_pot_map;
136
137    public:
138
139      ///\e
140      ReducedCostMap( const Digraph &gr,
141                      const LargeCostMap &cost_map,
142                      const PotentialMap &pot_map ) :
143        _gr(gr), _cost_map(cost_map), _pot_map(pot_map) {}
144
145      ///\e
146      inline LCost operator[](const Arc &e) const {
147        return _cost_map[e] + _pot_map[_gr.source(e)]
148                            - _pot_map[_gr.target(e)];
149      }
150
151    }; //class ReducedCostMap
152
153  private:
154
155    // The digraph the algorithm runs on
156    const Digraph &_graph;
157    // The original lower bound map
158    const LowerMap *_lower;
159    // The modified capacity map
160    CapacityArcMap _capacity;
161    // The original cost map
162    const CostMap &_orig_cost;
163    // The scaled cost map
164    LargeCostMap _cost;
165    // The modified supply map
166    SupplyNodeMap _supply;
167    bool _valid_supply;
168
169    // Arc map of the current flow
170    FlowMap *_flow;
171    bool _local_flow;
172    // Node map of the current potentials
173    PotentialMap *_potential;
174    bool _local_potential;
175
176    // The residual cost map
177    ResidualCostMap<LargeCostMap> _res_cost;
178    // The residual digraph
179    ResDigraph *_res_graph;
180    // The reduced cost map
181    ReducedCostMap *_red_cost;
182    // The excess map
183    SupplyNodeMap _excess;
184    // The epsilon parameter used for cost scaling
185    LCost _epsilon;
186    // The scaling factor
187    int _alpha;
188
189  public:
190
191    /// \brief General constructor (with lower bounds).
192    ///
193    /// General constructor (with lower bounds).
194    ///
195    /// \param digraph The digraph the algorithm runs on.
196    /// \param lower The lower bounds of the arcs.
197    /// \param capacity The capacities (upper bounds) of the arcs.
198    /// \param cost The cost (length) values of the arcs.
199    /// \param supply The supply values of the nodes (signed).
200    CostScaling( const Digraph &digraph,
201                 const LowerMap &lower,
202                 const CapacityMap &capacity,
203                 const CostMap &cost,
204                 const SupplyMap &supply ) :
205      _graph(digraph), _lower(&lower), _capacity(digraph), _orig_cost(cost),
206      _cost(digraph), _supply(digraph), _flow(NULL), _local_flow(false),
207      _potential(NULL), _local_potential(false), _res_cost(_cost),
208      _res_graph(NULL), _red_cost(NULL), _excess(digraph, 0)
209    {
210      // Check the sum of supply values
211      Supply sum = 0;
212      for (NodeIt n(_graph); n != INVALID; ++n) sum += _supply[n];
213      _valid_supply = sum == 0;
214     
215      for (ArcIt e(_graph); e != INVALID; ++e) _capacity[e] = capacity[e];
216      for (NodeIt n(_graph); n != INVALID; ++n) _supply[n] = supply[n];
217
218      // Remove non-zero lower bounds
219      for (ArcIt e(_graph); e != INVALID; ++e) {
220        if (lower[e] != 0) {
221          _capacity[e] -= lower[e];
222          _supply[_graph.source(e)] -= lower[e];
223          _supply[_graph.target(e)] += lower[e];
224        }
225      }
226    }
227/*
228    /// \brief General constructor (without lower bounds).
229    ///
230    /// General constructor (without lower bounds).
231    ///
232    /// \param digraph The digraph the algorithm runs on.
233    /// \param capacity The capacities (upper bounds) of the arcs.
234    /// \param cost The cost (length) values of the arcs.
235    /// \param supply The supply values of the nodes (signed).
236    CostScaling( const Digraph &digraph,
237                 const CapacityMap &capacity,
238                 const CostMap &cost,
239                 const SupplyMap &supply ) :
240      _graph(digraph), _lower(NULL), _capacity(capacity), _orig_cost(cost),
241      _cost(digraph), _supply(supply), _flow(NULL), _local_flow(false),
242      _potential(NULL), _local_potential(false), _res_cost(_cost),
243      _res_graph(NULL), _red_cost(NULL), _excess(digraph, 0)
244    {
245      // Check the sum of supply values
246      Supply sum = 0;
247      for (NodeIt n(_graph); n != INVALID; ++n) sum += _supply[n];
248      _valid_supply = sum == 0;
249    }
250
251    /// \brief Simple constructor (with lower bounds).
252    ///
253    /// Simple constructor (with lower bounds).
254    ///
255    /// \param digraph The digraph the algorithm runs on.
256    /// \param lower The lower bounds of the arcs.
257    /// \param capacity The capacities (upper bounds) of the arcs.
258    /// \param cost The cost (length) values of the arcs.
259    /// \param s The source node.
260    /// \param t The target node.
261    /// \param flow_value The required amount of flow from node \c s
262    /// to node \c t (i.e. the supply of \c s and the demand of \c t).
263    CostScaling( const Digraph &digraph,
264                 const LowerMap &lower,
265                 const CapacityMap &capacity,
266                 const CostMap &cost,
267                 Node s, Node t,
268                 Supply flow_value ) :
269      _graph(digraph), _lower(&lower), _capacity(capacity), _orig_cost(cost),
270      _cost(digraph), _supply(digraph, 0), _flow(NULL), _local_flow(false),
271      _potential(NULL), _local_potential(false), _res_cost(_cost),
272      _res_graph(NULL), _red_cost(NULL), _excess(digraph, 0)
273    {
274      // Remove non-zero lower bounds
275      _supply[s] =  flow_value;
276      _supply[t] = -flow_value;
277      for (ArcIt e(_graph); e != INVALID; ++e) {
278        if (lower[e] != 0) {
279          _capacity[e] -= lower[e];
280          _supply[_graph.source(e)] -= lower[e];
281          _supply[_graph.target(e)] += lower[e];
282        }
283      }
284      _valid_supply = true;
285    }
286
287    /// \brief Simple constructor (without lower bounds).
288    ///
289    /// Simple constructor (without lower bounds).
290    ///
291    /// \param digraph The digraph the algorithm runs on.
292    /// \param capacity The capacities (upper bounds) of the arcs.
293    /// \param cost The cost (length) values of the arcs.
294    /// \param s The source node.
295    /// \param t The target node.
296    /// \param flow_value The required amount of flow from node \c s
297    /// to node \c t (i.e. the supply of \c s and the demand of \c t).
298    CostScaling( const Digraph &digraph,
299                 const CapacityMap &capacity,
300                 const CostMap &cost,
301                 Node s, Node t,
302                 Supply flow_value ) :
303      _graph(digraph), _lower(NULL), _capacity(capacity), _orig_cost(cost),
304      _cost(digraph), _supply(digraph, 0), _flow(NULL), _local_flow(false),
305      _potential(NULL), _local_potential(false), _res_cost(_cost),
306      _res_graph(NULL), _red_cost(NULL), _excess(digraph, 0)
307    {
308      _supply[s] =  flow_value;
309      _supply[t] = -flow_value;
310      _valid_supply = true;
311    }
312*/
313    /// Destructor.
314    ~CostScaling() {
315      if (_local_flow) delete _flow;
316      if (_local_potential) delete _potential;
317      delete _res_graph;
318      delete _red_cost;
319    }
320
321    /// \brief Set the flow map.
322    ///
323    /// Set the flow map.
324    ///
325    /// \return \c (*this)
326    CostScaling& flowMap(FlowMap &map) {
327      if (_local_flow) {
328        delete _flow;
329        _local_flow = false;
330      }
331      _flow = &map;
332      return *this;
333    }
334
335    /// \brief Set the potential map.
336    ///
337    /// Set the potential map.
338    ///
339    /// \return \c (*this)
340    CostScaling& potentialMap(PotentialMap &map) {
341      if (_local_potential) {
342        delete _potential;
343        _local_potential = false;
344      }
345      _potential = &map;
346      return *this;
347    }
348
349    /// \name Execution control
350
351    /// @{
352
353    /// \brief Run the algorithm.
354    ///
355    /// Run the algorithm.
356    ///
357    /// \param partial_augment By default the algorithm performs
358    /// partial augment and relabel operations in the cost scaling
359    /// phases. Set this parameter to \c false for using local push and
360    /// relabel operations instead.
361    ///
362    /// \return \c true if a feasible flow can be found.
363    bool run(bool partial_augment = true) {
364      if (partial_augment) {
365        return init() && startPartialAugment();
366      } else {
367        return init() && startPushRelabel();
368      }
369    }
370
371    /// @}
372
373    /// \name Query Functions
374    /// The result of the algorithm can be obtained using these
375    /// functions.\n
376    /// \ref lemon::CostScaling::run() "run()" must be called before
377    /// using them.
378
379    /// @{
380
381    /// \brief Return a const reference to the arc map storing the
382    /// found flow.
383    ///
384    /// Return a const reference to the arc map storing the found flow.
385    ///
386    /// \pre \ref run() must be called before using this function.
387    const FlowMap& flowMap() const {
388      return *_flow;
389    }
390
391    /// \brief Return a const reference to the node map storing the
392    /// found potentials (the dual solution).
393    ///
394    /// Return a const reference to the node map storing the found
395    /// potentials (the dual solution).
396    ///
397    /// \pre \ref run() must be called before using this function.
398    const PotentialMap& potentialMap() const {
399      return *_potential;
400    }
401
402    /// \brief Return the flow on the given arc.
403    ///
404    /// Return the flow on the given arc.
405    ///
406    /// \pre \ref run() must be called before using this function.
407    Capacity flow(const Arc& arc) const {
408      return (*_flow)[arc];
409    }
410
411    /// \brief Return the potential of the given node.
412    ///
413    /// Return the potential of the given node.
414    ///
415    /// \pre \ref run() must be called before using this function.
416    Cost potential(const Node& node) const {
417      return (*_potential)[node];
418    }
419
420    /// \brief Return the total cost of the found flow.
421    ///
422    /// Return the total cost of the found flow. The complexity of the
423    /// function is \f$ O(e) \f$.
424    ///
425    /// \pre \ref run() must be called before using this function.
426    Cost totalCost() const {
427      Cost c = 0;
428      for (ArcIt e(_graph); e != INVALID; ++e)
429        c += (*_flow)[e] * _orig_cost[e];
430      return c;
431    }
432
433    /// @}
434
435  private:
436
437    /// Initialize the algorithm.
438    bool init() {
439      if (!_valid_supply) return false;
440      // The scaling factor
441      _alpha = 8;
442
443      // Initialize flow and potential maps
444      if (!_flow) {
445        _flow = new FlowMap(_graph);
446        _local_flow = true;
447      }
448      if (!_potential) {
449        _potential = new PotentialMap(_graph);
450        _local_potential = true;
451      }
452
453      _red_cost = new ReducedCostMap(_graph, _cost, *_potential);
454      _res_graph = new ResDigraph(_graph, _capacity, *_flow);
455
456      // Initialize the scaled cost map and the epsilon parameter
457      Cost max_cost = 0;
458      int node_num = countNodes(_graph);
459      for (ArcIt e(_graph); e != INVALID; ++e) {
460        _cost[e] = LCost(_orig_cost[e]) * node_num * _alpha;
461        if (_orig_cost[e] > max_cost) max_cost = _orig_cost[e];
462      }
463      _epsilon = max_cost * node_num;
464
465      // Find a feasible flow using Circulation
466      Circulation< Digraph, ConstMap<Arc, Capacity>, CapacityArcMap,
467                   SupplyMap >
468        circulation( _graph, constMap<Arc>(Capacity(0)), _capacity,
469                     _supply );
470      return circulation.flowMap(*_flow).run();
471    }
472
473    /// Execute the algorithm performing partial augmentation and
474    /// relabel operations.
475    bool startPartialAugment() {
476      // Paramters for heuristics
477//      const int BF_HEURISTIC_EPSILON_BOUND = 1000;
478//      const int BF_HEURISTIC_BOUND_FACTOR  = 3;
479      // Maximum augment path length
480      const int MAX_PATH_LENGTH = 4;
481
482      // Variables
483      typename Digraph::template NodeMap<Arc> pred_arc(_graph);
484      typename Digraph::template NodeMap<bool> forward(_graph);
485      typename Digraph::template NodeMap<OutArcIt> next_out(_graph);
486      typename Digraph::template NodeMap<InArcIt> next_in(_graph);
487      typename Digraph::template NodeMap<bool> next_dir(_graph);
488      std::deque<Node> active_nodes;
489      std::vector<Node> path_nodes;
490
491//      int node_num = countNodes(_graph);
492      for ( ; _epsilon >= 1; _epsilon = _epsilon < _alpha && _epsilon > 1 ?
493                                        1 : _epsilon / _alpha )
494      {
495/*
496        // "Early Termination" heuristic: use Bellman-Ford algorithm
497        // to check if the current flow is optimal
498        if (_epsilon <= BF_HEURISTIC_EPSILON_BOUND) {
499          typedef ShiftMap< ResidualCostMap<LargeCostMap> > ShiftCostMap;
500          ShiftCostMap shift_cost(_res_cost, 1);
501          BellmanFord<ResDigraph, ShiftCostMap> bf(*_res_graph, shift_cost);
502          bf.init(0);
503          bool done = false;
504          int K = int(BF_HEURISTIC_BOUND_FACTOR * sqrt(node_num));
505          for (int i = 0; i < K && !done; ++i)
506            done = bf.processNextWeakRound();
507          if (done) break;
508        }
509*/
510        // Saturate arcs not satisfying the optimality condition
511        Capacity delta;
512        for (ArcIt e(_graph); e != INVALID; ++e) {
513          if (_capacity[e] - (*_flow)[e] > 0 && (*_red_cost)[e] < 0) {
514            delta = _capacity[e] - (*_flow)[e];
515            _excess[_graph.source(e)] -= delta;
516            _excess[_graph.target(e)] += delta;
517            (*_flow)[e] = _capacity[e];
518          }
519          if ((*_flow)[e] > 0 && -(*_red_cost)[e] < 0) {
520            _excess[_graph.target(e)] -= (*_flow)[e];
521            _excess[_graph.source(e)] += (*_flow)[e];
522            (*_flow)[e] = 0;
523          }
524        }
525
526        // Find active nodes (i.e. nodes with positive excess)
527        for (NodeIt n(_graph); n != INVALID; ++n) {
528          if (_excess[n] > 0) active_nodes.push_back(n);
529        }
530
531        // Initialize the next arc maps
532        for (NodeIt n(_graph); n != INVALID; ++n) {
533          next_out[n] = OutArcIt(_graph, n);
534          next_in[n] = InArcIt(_graph, n);
535          next_dir[n] = true;
536        }
537
538        // Perform partial augment and relabel operations
539        while (active_nodes.size() > 0) {
540          // Select an active node (FIFO selection)
541          if (_excess[active_nodes[0]] <= 0) {
542            active_nodes.pop_front();
543            continue;
544          }
545          Node start = active_nodes[0];
546          path_nodes.clear();
547          path_nodes.push_back(start);
548
549          // Find an augmenting path from the start node
550          Node u, tip = start;
551          LCost min_red_cost;
552          while ( _excess[tip] >= 0 &&
553                  int(path_nodes.size()) <= MAX_PATH_LENGTH )
554          {
555            if (next_dir[tip]) {
556              for (OutArcIt e = next_out[tip]; e != INVALID; ++e) {
557                if (_capacity[e] - (*_flow)[e] > 0 && (*_red_cost)[e] < 0) {
558                  u = _graph.target(e);
559                  pred_arc[u] = e;
560                  forward[u] = true;
561                  next_out[tip] = e;
562                  tip = u;
563                  path_nodes.push_back(tip);
564                  goto next_step;
565                }
566              }
567              next_dir[tip] = false;
568            }
569            for (InArcIt e = next_in[tip]; e != INVALID; ++e) {
570              if ((*_flow)[e] > 0 && -(*_red_cost)[e] < 0) {
571                u = _graph.source(e);
572                pred_arc[u] = e;
573                forward[u] = false;
574                next_in[tip] = e;
575                tip = u;
576                path_nodes.push_back(tip);
577                goto next_step;
578              }
579            }
580
581            // Relabel tip node
582            min_red_cost = std::numeric_limits<LCost>::max() / 2;
583            for (OutArcIt oe(_graph, tip); oe != INVALID; ++oe) {
584              if ( _capacity[oe] - (*_flow)[oe] > 0 &&
585                   (*_red_cost)[oe] < min_red_cost )
586                min_red_cost = (*_red_cost)[oe];
587            }
588            for (InArcIt ie(_graph, tip); ie != INVALID; ++ie) {
589              if ((*_flow)[ie] > 0 && -(*_red_cost)[ie] < min_red_cost)
590                min_red_cost = -(*_red_cost)[ie];
591            }
592            (*_potential)[tip] -= min_red_cost + _epsilon;
593
594            // Reset the next arc maps
595            next_out[tip] = OutArcIt(_graph, tip);
596            next_in[tip] = InArcIt(_graph, tip);
597            next_dir[tip] = true;
598
599            // Step back
600            if (tip != start) {
601              path_nodes.pop_back();
602              tip = path_nodes[path_nodes.size()-1];
603            }
604
605          next_step:
606            continue;
607          }
608
609          // Augment along the found path (as much flow as possible)
610          Capacity delta;
611          for (int i = 1; i < int(path_nodes.size()); ++i) {
612            u = path_nodes[i];
613            delta = forward[u] ?
614              _capacity[pred_arc[u]] - (*_flow)[pred_arc[u]] :
615              (*_flow)[pred_arc[u]];
616            delta = std::min(delta, _excess[path_nodes[i-1]]);
617            (*_flow)[pred_arc[u]] += forward[u] ? delta : -delta;
618            _excess[path_nodes[i-1]] -= delta;
619            _excess[u] += delta;
620            if (_excess[u] > 0 && _excess[u] <= delta) active_nodes.push_back(u);
621          }
622        }
623      }
624
625      // Compute node potentials for the original costs
626      ResidualCostMap<CostMap> res_cost(_orig_cost);
627      BellmanFord< ResDigraph, ResidualCostMap<CostMap> >
628        bf(*_res_graph, res_cost);
629      bf.init(0); bf.start();
630      for (NodeIt n(_graph); n != INVALID; ++n)
631        (*_potential)[n] = bf.dist(n);
632
633      // Handle non-zero lower bounds
634      if (_lower) {
635        for (ArcIt e(_graph); e != INVALID; ++e)
636          (*_flow)[e] += (*_lower)[e];
637      }
638      return true;
639    }
640
641    /// Execute the algorithm performing push and relabel operations.
642    bool startPushRelabel() {
643      // Paramters for heuristics
644//      const int BF_HEURISTIC_EPSILON_BOUND = 1000;
645//      const int BF_HEURISTIC_BOUND_FACTOR  = 3;
646
647      typename Digraph::template NodeMap<bool> hyper(_graph, false);
648      typename Digraph::template NodeMap<Arc> pred_arc(_graph);
649      typename Digraph::template NodeMap<bool> forward(_graph);
650      typename Digraph::template NodeMap<OutArcIt> next_out(_graph);
651      typename Digraph::template NodeMap<InArcIt> next_in(_graph);
652      typename Digraph::template NodeMap<bool> next_dir(_graph);
653      std::deque<Node> active_nodes;
654
655//      int node_num = countNodes(_graph);
656      for ( ; _epsilon >= 1; _epsilon = _epsilon < _alpha && _epsilon > 1 ?
657                                        1 : _epsilon / _alpha )
658      {
659/*
660        // "Early Termination" heuristic: use Bellman-Ford algorithm
661        // to check if the current flow is optimal
662        if (_epsilon <= BF_HEURISTIC_EPSILON_BOUND) {
663          typedef ShiftMap< ResidualCostMap<LargeCostMap> > ShiftCostMap;
664          ShiftCostMap shift_cost(_res_cost, 1);
665          BellmanFord<ResDigraph, ShiftCostMap> bf(*_res_graph, shift_cost);
666          bf.init(0);
667          bool done = false;
668          int K = int(BF_HEURISTIC_BOUND_FACTOR * sqrt(node_num));
669          for (int i = 0; i < K && !done; ++i)
670            done = bf.processNextWeakRound();
671          if (done) break;
672        }
673*/
674
675        // Saturate arcs not satisfying the optimality condition
676        Capacity delta;
677        for (ArcIt e(_graph); e != INVALID; ++e) {
678          if (_capacity[e] - (*_flow)[e] > 0 && (*_red_cost)[e] < 0) {
679            delta = _capacity[e] - (*_flow)[e];
680            _excess[_graph.source(e)] -= delta;
681            _excess[_graph.target(e)] += delta;
682            (*_flow)[e] = _capacity[e];
683          }
684          if ((*_flow)[e] > 0 && -(*_red_cost)[e] < 0) {
685            _excess[_graph.target(e)] -= (*_flow)[e];
686            _excess[_graph.source(e)] += (*_flow)[e];
687            (*_flow)[e] = 0;
688          }
689        }
690
691        // Find active nodes (i.e. nodes with positive excess)
692        for (NodeIt n(_graph); n != INVALID; ++n) {
693          if (_excess[n] > 0) active_nodes.push_back(n);
694        }
695
696        // Initialize the next arc maps
697        for (NodeIt n(_graph); n != INVALID; ++n) {
698          next_out[n] = OutArcIt(_graph, n);
699          next_in[n] = InArcIt(_graph, n);
700          next_dir[n] = true;
701        }
702
703        // Perform push and relabel operations
704        while (active_nodes.size() > 0) {
705          // Select an active node (FIFO selection)
706          Node n = active_nodes[0], t;
707          bool relabel_enabled = true;
708
709          // Perform push operations if there are admissible arcs
710          if (_excess[n] > 0 && next_dir[n]) {
711            OutArcIt e = next_out[n];
712            for ( ; e != INVALID; ++e) {
713              if (_capacity[e] - (*_flow)[e] > 0 && (*_red_cost)[e] < 0) {
714                delta = std::min(_capacity[e] - (*_flow)[e], _excess[n]);
715                t = _graph.target(e);
716
717                // Push-look-ahead heuristic
718                Capacity ahead = -_excess[t];
719                for (OutArcIt oe(_graph, t); oe != INVALID; ++oe) {
720                  if (_capacity[oe] - (*_flow)[oe] > 0 && (*_red_cost)[oe] < 0)
721                    ahead += _capacity[oe] - (*_flow)[oe];
722                }
723                for (InArcIt ie(_graph, t); ie != INVALID; ++ie) {
724                  if ((*_flow)[ie] > 0 && -(*_red_cost)[ie] < 0)
725                    ahead += (*_flow)[ie];
726                }
727                if (ahead < 0) ahead = 0;
728
729                // Push flow along the arc
730                if (ahead < delta) {
731                  (*_flow)[e] += ahead;
732                  _excess[n] -= ahead;
733                  _excess[t] += ahead;
734                  active_nodes.push_front(t);
735                  hyper[t] = true;
736                  relabel_enabled = false;
737                  break;
738                } else {
739                  (*_flow)[e] += delta;
740                  _excess[n] -= delta;
741                  _excess[t] += delta;
742                  if (_excess[t] > 0 && _excess[t] <= delta)
743                    active_nodes.push_back(t);
744                }
745
746                if (_excess[n] == 0) break;
747              }
748            }
749            if (e != INVALID) {
750              next_out[n] = e;
751            } else {
752              next_dir[n] = false;
753            }
754          }
755
756          if (_excess[n] > 0 && !next_dir[n]) {
757            InArcIt e = next_in[n];
758            for ( ; e != INVALID; ++e) {
759              if ((*_flow)[e] > 0 && -(*_red_cost)[e] < 0) {
760                delta = std::min((*_flow)[e], _excess[n]);
761                t = _graph.source(e);
762
763                // Push-look-ahead heuristic
764                Capacity ahead = -_excess[t];
765                for (OutArcIt oe(_graph, t); oe != INVALID; ++oe) {
766                  if (_capacity[oe] - (*_flow)[oe] > 0 && (*_red_cost)[oe] < 0)
767                    ahead += _capacity[oe] - (*_flow)[oe];
768                }
769                for (InArcIt ie(_graph, t); ie != INVALID; ++ie) {
770                  if ((*_flow)[ie] > 0 && -(*_red_cost)[ie] < 0)
771                    ahead += (*_flow)[ie];
772                }
773                if (ahead < 0) ahead = 0;
774
775                // Push flow along the arc
776                if (ahead < delta) {
777                  (*_flow)[e] -= ahead;
778                  _excess[n] -= ahead;
779                  _excess[t] += ahead;
780                  active_nodes.push_front(t);
781                  hyper[t] = true;
782                  relabel_enabled = false;
783                  break;
784                } else {
785                  (*_flow)[e] -= delta;
786                  _excess[n] -= delta;
787                  _excess[t] += delta;
788                  if (_excess[t] > 0 && _excess[t] <= delta)
789                    active_nodes.push_back(t);
790                }
791
792                if (_excess[n] == 0) break;
793              }
794            }
795            next_in[n] = e;
796          }
797
798          // Relabel the node if it is still active (or hyper)
799          if (relabel_enabled && (_excess[n] > 0 || hyper[n])) {
800            LCost min_red_cost = std::numeric_limits<LCost>::max() / 2;
801            for (OutArcIt oe(_graph, n); oe != INVALID; ++oe) {
802              if ( _capacity[oe] - (*_flow)[oe] > 0 &&
803                   (*_red_cost)[oe] < min_red_cost )
804                min_red_cost = (*_red_cost)[oe];
805            }
806            for (InArcIt ie(_graph, n); ie != INVALID; ++ie) {
807              if ((*_flow)[ie] > 0 && -(*_red_cost)[ie] < min_red_cost)
808                min_red_cost = -(*_red_cost)[ie];
809            }
810            (*_potential)[n] -= min_red_cost + _epsilon;
811            hyper[n] = false;
812
813            // Reset the next arc maps
814            next_out[n] = OutArcIt(_graph, n);
815            next_in[n] = InArcIt(_graph, n);
816            next_dir[n] = true;
817          }
818
819          // Remove nodes that are not active nor hyper
820          while ( active_nodes.size() > 0 &&
821                  _excess[active_nodes[0]] <= 0 &&
822                  !hyper[active_nodes[0]] ) {
823            active_nodes.pop_front();
824          }
825        }
826      }
827
828      // Compute node potentials for the original costs
829      ResidualCostMap<CostMap> res_cost(_orig_cost);
830      BellmanFord< ResDigraph, ResidualCostMap<CostMap> >
831        bf(*_res_graph, res_cost);
832      bf.init(0); bf.start();
833      for (NodeIt n(_graph); n != INVALID; ++n)
834        (*_potential)[n] = bf.dist(n);
835
836      // Handle non-zero lower bounds
837      if (_lower) {
838        for (ArcIt e(_graph); e != INVALID; ++e)
839          (*_flow)[e] += (*_lower)[e];
840      }
841      return true;
842    }
843
844  }; //class CostScaling
845
846  ///@}
847
848} //namespace lemon
849
850#endif //LEMON_COST_SCALING_H
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