COIN-OR::LEMON - Graph Library

source: lemon-1.2/lemon/capacity_scaling.h @ 863:a93f1a27d831

Last change on this file since 863:a93f1a27d831 was 863:a93f1a27d831, checked in by Peter Kovacs <kpeter@…>, 10 years ago

Fix gcc 3.3 compilation error (#354)

gcc 3.3 requires that a class has a default constructor if it has
template named parameters. (That constructor can be protected.)

File size: 30.8 KB
Line 
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_CAPACITY_SCALING_H
20#define LEMON_CAPACITY_SCALING_H
21
22/// \ingroup min_cost_flow_algs
23///
24/// \file
25/// \brief Capacity Scaling algorithm for finding a minimum cost flow.
26
27#include <vector>
28#include <limits>
29#include <lemon/core.h>
30#include <lemon/bin_heap.h>
31
32namespace lemon {
33
34  /// \brief Default traits class of CapacityScaling algorithm.
35  ///
36  /// Default traits class of CapacityScaling algorithm.
37  /// \tparam GR Digraph type.
38  /// \tparam V The number type used for flow amounts, capacity bounds
39  /// and supply values. By default it is \c int.
40  /// \tparam C The number type used for costs and potentials.
41  /// By default it is the same as \c V.
42  template <typename GR, typename V = int, typename C = V>
43  struct CapacityScalingDefaultTraits
44  {
45    /// The type of the digraph
46    typedef GR Digraph;
47    /// The type of the flow amounts, capacity bounds and supply values
48    typedef V Value;
49    /// The type of the arc costs
50    typedef C Cost;
51
52    /// \brief The type of the heap used for internal Dijkstra computations.
53    ///
54    /// The type of the heap used for internal Dijkstra computations.
55    /// It must conform to the \ref lemon::concepts::Heap "Heap" concept,
56    /// its priority type must be \c Cost and its cross reference type
57    /// must be \ref RangeMap "RangeMap<int>".
58    typedef BinHeap<Cost, RangeMap<int> > Heap;
59  };
60
61  /// \addtogroup min_cost_flow_algs
62  /// @{
63
64  /// \brief Implementation of the Capacity Scaling algorithm for
65  /// finding a \ref min_cost_flow "minimum cost flow".
66  ///
67  /// \ref CapacityScaling implements the capacity scaling version
68  /// of the successive shortest path algorithm for finding a
69  /// \ref min_cost_flow "minimum cost flow" \ref amo93networkflows,
70  /// \ref edmondskarp72theoretical. It is an efficient dual
71  /// solution method.
72  ///
73  /// Most of the parameters of the problem (except for the digraph)
74  /// can be given using separate functions, and the algorithm can be
75  /// executed using the \ref run() function. If some parameters are not
76  /// specified, then default values will be used.
77  ///
78  /// \tparam GR The digraph type the algorithm runs on.
79  /// \tparam V The number type used for flow amounts, capacity bounds
80  /// and supply values in the algorithm. By default, it is \c int.
81  /// \tparam C The number type used for costs and potentials in the
82  /// algorithm. By default, it is the same as \c V.
83  /// \tparam TR The traits class that defines various types used by the
84  /// algorithm. By default, it is \ref CapacityScalingDefaultTraits
85  /// "CapacityScalingDefaultTraits<GR, V, C>".
86  /// In most cases, this parameter should not be set directly,
87  /// consider to use the named template parameters instead.
88  ///
89  /// \warning Both number types must be signed and all input data must
90  /// be integer.
91  /// \warning This algorithm does not support negative costs for such
92  /// arcs that have infinite upper bound.
93#ifdef DOXYGEN
94  template <typename GR, typename V, typename C, typename TR>
95#else
96  template < typename GR, typename V = int, typename C = V,
97             typename TR = CapacityScalingDefaultTraits<GR, V, C> >
98#endif
99  class CapacityScaling
100  {
101  public:
102
103    /// The type of the digraph
104    typedef typename TR::Digraph Digraph;
105    /// The type of the flow amounts, capacity bounds and supply values
106    typedef typename TR::Value Value;
107    /// The type of the arc costs
108    typedef typename TR::Cost Cost;
109
110    /// The type of the heap used for internal Dijkstra computations
111    typedef typename TR::Heap Heap;
112
113    /// The \ref CapacityScalingDefaultTraits "traits class" of the algorithm
114    typedef TR Traits;
115
116  public:
117
118    /// \brief Problem type constants for the \c run() function.
119    ///
120    /// Enum type containing the problem type constants that can be
121    /// returned by the \ref run() function of the algorithm.
122    enum ProblemType {
123      /// The problem has no feasible solution (flow).
124      INFEASIBLE,
125      /// The problem has optimal solution (i.e. it is feasible and
126      /// bounded), and the algorithm has found optimal flow and node
127      /// potentials (primal and dual solutions).
128      OPTIMAL,
129      /// The digraph contains an arc of negative cost and infinite
130      /// upper bound. It means that the objective function is unbounded
131      /// on that arc, however, note that it could actually be bounded
132      /// over the feasible flows, but this algroithm cannot handle
133      /// these cases.
134      UNBOUNDED
135    };
136 
137  private:
138
139    TEMPLATE_DIGRAPH_TYPEDEFS(GR);
140
141    typedef std::vector<int> IntVector;
142    typedef std::vector<Value> ValueVector;
143    typedef std::vector<Cost> CostVector;
144    typedef std::vector<char> BoolVector;
145    // Note: vector<char> is used instead of vector<bool> for efficiency reasons
146
147  private:
148
149    // Data related to the underlying digraph
150    const GR &_graph;
151    int _node_num;
152    int _arc_num;
153    int _res_arc_num;
154    int _root;
155
156    // Parameters of the problem
157    bool _have_lower;
158    Value _sum_supply;
159
160    // Data structures for storing the digraph
161    IntNodeMap _node_id;
162    IntArcMap _arc_idf;
163    IntArcMap _arc_idb;
164    IntVector _first_out;
165    BoolVector _forward;
166    IntVector _source;
167    IntVector _target;
168    IntVector _reverse;
169
170    // Node and arc data
171    ValueVector _lower;
172    ValueVector _upper;
173    CostVector _cost;
174    ValueVector _supply;
175
176    ValueVector _res_cap;
177    CostVector _pi;
178    ValueVector _excess;
179    IntVector _excess_nodes;
180    IntVector _deficit_nodes;
181
182    Value _delta;
183    int _factor;
184    IntVector _pred;
185
186  public:
187 
188    /// \brief Constant for infinite upper bounds (capacities).
189    ///
190    /// Constant for infinite upper bounds (capacities).
191    /// It is \c std::numeric_limits<Value>::infinity() if available,
192    /// \c std::numeric_limits<Value>::max() otherwise.
193    const Value INF;
194
195  private:
196
197    // Special implementation of the Dijkstra algorithm for finding
198    // shortest paths in the residual network of the digraph with
199    // respect to the reduced arc costs and modifying the node
200    // potentials according to the found distance labels.
201    class ResidualDijkstra
202    {
203    private:
204
205      int _node_num;
206      bool _geq;
207      const IntVector &_first_out;
208      const IntVector &_target;
209      const CostVector &_cost;
210      const ValueVector &_res_cap;
211      const ValueVector &_excess;
212      CostVector &_pi;
213      IntVector &_pred;
214     
215      IntVector _proc_nodes;
216      CostVector _dist;
217     
218    public:
219
220      ResidualDijkstra(CapacityScaling& cs) :
221        _node_num(cs._node_num), _geq(cs._sum_supply < 0),
222        _first_out(cs._first_out), _target(cs._target), _cost(cs._cost),
223        _res_cap(cs._res_cap), _excess(cs._excess), _pi(cs._pi),
224        _pred(cs._pred), _dist(cs._node_num)
225      {}
226
227      int run(int s, Value delta = 1) {
228        RangeMap<int> heap_cross_ref(_node_num, Heap::PRE_HEAP);
229        Heap heap(heap_cross_ref);
230        heap.push(s, 0);
231        _pred[s] = -1;
232        _proc_nodes.clear();
233
234        // Process nodes
235        while (!heap.empty() && _excess[heap.top()] > -delta) {
236          int u = heap.top(), v;
237          Cost d = heap.prio() + _pi[u], dn;
238          _dist[u] = heap.prio();
239          _proc_nodes.push_back(u);
240          heap.pop();
241
242          // Traverse outgoing residual arcs
243          int last_out = _geq ? _first_out[u+1] : _first_out[u+1] - 1;
244          for (int a = _first_out[u]; a != last_out; ++a) {
245            if (_res_cap[a] < delta) continue;
246            v = _target[a];
247            switch (heap.state(v)) {
248              case Heap::PRE_HEAP:
249                heap.push(v, d + _cost[a] - _pi[v]);
250                _pred[v] = a;
251                break;
252              case Heap::IN_HEAP:
253                dn = d + _cost[a] - _pi[v];
254                if (dn < heap[v]) {
255                  heap.decrease(v, dn);
256                  _pred[v] = a;
257                }
258                break;
259              case Heap::POST_HEAP:
260                break;
261            }
262          }
263        }
264        if (heap.empty()) return -1;
265
266        // Update potentials of processed nodes
267        int t = heap.top();
268        Cost dt = heap.prio();
269        for (int i = 0; i < int(_proc_nodes.size()); ++i) {
270          _pi[_proc_nodes[i]] += _dist[_proc_nodes[i]] - dt;
271        }
272
273        return t;
274      }
275
276    }; //class ResidualDijkstra
277
278  public:
279
280    /// \name Named Template Parameters
281    /// @{
282
283    template <typename T>
284    struct SetHeapTraits : public Traits {
285      typedef T Heap;
286    };
287
288    /// \brief \ref named-templ-param "Named parameter" for setting
289    /// \c Heap type.
290    ///
291    /// \ref named-templ-param "Named parameter" for setting \c Heap
292    /// type, which is used for internal Dijkstra computations.
293    /// It must conform to the \ref lemon::concepts::Heap "Heap" concept,
294    /// its priority type must be \c Cost and its cross reference type
295    /// must be \ref RangeMap "RangeMap<int>".
296    template <typename T>
297    struct SetHeap
298      : public CapacityScaling<GR, V, C, SetHeapTraits<T> > {
299      typedef  CapacityScaling<GR, V, C, SetHeapTraits<T> > Create;
300    };
301
302    /// @}
303
304  protected:
305
306    CapacityScaling() {}
307
308  public:
309
310    /// \brief Constructor.
311    ///
312    /// The constructor of the class.
313    ///
314    /// \param graph The digraph the algorithm runs on.
315    CapacityScaling(const GR& graph) :
316      _graph(graph), _node_id(graph), _arc_idf(graph), _arc_idb(graph),
317      INF(std::numeric_limits<Value>::has_infinity ?
318          std::numeric_limits<Value>::infinity() :
319          std::numeric_limits<Value>::max())
320    {
321      // Check the number types
322      LEMON_ASSERT(std::numeric_limits<Value>::is_signed,
323        "The flow type of CapacityScaling must be signed");
324      LEMON_ASSERT(std::numeric_limits<Cost>::is_signed,
325        "The cost type of CapacityScaling must be signed");
326
327      // Reset data structures
328      reset();
329    }
330
331    /// \name Parameters
332    /// The parameters of the algorithm can be specified using these
333    /// functions.
334
335    /// @{
336
337    /// \brief Set the lower bounds on the arcs.
338    ///
339    /// This function sets the lower bounds on the arcs.
340    /// If it is not used before calling \ref run(), the lower bounds
341    /// will be set to zero on all arcs.
342    ///
343    /// \param map An arc map storing the lower bounds.
344    /// Its \c Value type must be convertible to the \c Value type
345    /// of the algorithm.
346    ///
347    /// \return <tt>(*this)</tt>
348    template <typename LowerMap>
349    CapacityScaling& lowerMap(const LowerMap& map) {
350      _have_lower = true;
351      for (ArcIt a(_graph); a != INVALID; ++a) {
352        _lower[_arc_idf[a]] = map[a];
353        _lower[_arc_idb[a]] = map[a];
354      }
355      return *this;
356    }
357
358    /// \brief Set the upper bounds (capacities) on the arcs.
359    ///
360    /// This function sets the upper bounds (capacities) on the arcs.
361    /// If it is not used before calling \ref run(), the upper bounds
362    /// will be set to \ref INF on all arcs (i.e. the flow value will be
363    /// unbounded from above).
364    ///
365    /// \param map An arc map storing the upper bounds.
366    /// Its \c Value type must be convertible to the \c Value type
367    /// of the algorithm.
368    ///
369    /// \return <tt>(*this)</tt>
370    template<typename UpperMap>
371    CapacityScaling& upperMap(const UpperMap& map) {
372      for (ArcIt a(_graph); a != INVALID; ++a) {
373        _upper[_arc_idf[a]] = map[a];
374      }
375      return *this;
376    }
377
378    /// \brief Set the costs of the arcs.
379    ///
380    /// This function sets the costs of the arcs.
381    /// If it is not used before calling \ref run(), the costs
382    /// will be set to \c 1 on all arcs.
383    ///
384    /// \param map An arc map storing the costs.
385    /// Its \c Value type must be convertible to the \c Cost type
386    /// of the algorithm.
387    ///
388    /// \return <tt>(*this)</tt>
389    template<typename CostMap>
390    CapacityScaling& costMap(const CostMap& map) {
391      for (ArcIt a(_graph); a != INVALID; ++a) {
392        _cost[_arc_idf[a]] =  map[a];
393        _cost[_arc_idb[a]] = -map[a];
394      }
395      return *this;
396    }
397
398    /// \brief Set the supply values of the nodes.
399    ///
400    /// This function sets the supply values of the nodes.
401    /// If neither this function nor \ref stSupply() is used before
402    /// calling \ref run(), the supply of each node will be set to zero.
403    ///
404    /// \param map A node map storing the supply values.
405    /// Its \c Value type must be convertible to the \c Value type
406    /// of the algorithm.
407    ///
408    /// \return <tt>(*this)</tt>
409    template<typename SupplyMap>
410    CapacityScaling& supplyMap(const SupplyMap& map) {
411      for (NodeIt n(_graph); n != INVALID; ++n) {
412        _supply[_node_id[n]] = map[n];
413      }
414      return *this;
415    }
416
417    /// \brief Set single source and target nodes and a supply value.
418    ///
419    /// This function sets a single source node and a single target node
420    /// and the required flow value.
421    /// If neither this function nor \ref supplyMap() is used before
422    /// calling \ref run(), the supply of each node will be set to zero.
423    ///
424    /// Using this function has the same effect as using \ref supplyMap()
425    /// with such a map in which \c k is assigned to \c s, \c -k is
426    /// assigned to \c t and all other nodes have zero supply value.
427    ///
428    /// \param s The source node.
429    /// \param t The target node.
430    /// \param k The required amount of flow from node \c s to node \c t
431    /// (i.e. the supply of \c s and the demand of \c t).
432    ///
433    /// \return <tt>(*this)</tt>
434    CapacityScaling& stSupply(const Node& s, const Node& t, Value k) {
435      for (int i = 0; i != _node_num; ++i) {
436        _supply[i] = 0;
437      }
438      _supply[_node_id[s]] =  k;
439      _supply[_node_id[t]] = -k;
440      return *this;
441    }
442   
443    /// @}
444
445    /// \name Execution control
446    /// The algorithm can be executed using \ref run().
447
448    /// @{
449
450    /// \brief Run the algorithm.
451    ///
452    /// This function runs the algorithm.
453    /// The paramters can be specified using functions \ref lowerMap(),
454    /// \ref upperMap(), \ref costMap(), \ref supplyMap(), \ref stSupply().
455    /// For example,
456    /// \code
457    ///   CapacityScaling<ListDigraph> cs(graph);
458    ///   cs.lowerMap(lower).upperMap(upper).costMap(cost)
459    ///     .supplyMap(sup).run();
460    /// \endcode
461    ///
462    /// This function can be called more than once. All the given parameters
463    /// are kept for the next call, unless \ref resetParams() or \ref reset()
464    /// is used, thus only the modified parameters have to be set again.
465    /// If the underlying digraph was also modified after the construction
466    /// of the class (or the last \ref reset() call), then the \ref reset()
467    /// function must be called.
468    ///
469    /// \param factor The capacity scaling factor. It must be larger than
470    /// one to use scaling. If it is less or equal to one, then scaling
471    /// will be disabled.
472    ///
473    /// \return \c INFEASIBLE if no feasible flow exists,
474    /// \n \c OPTIMAL if the problem has optimal solution
475    /// (i.e. it is feasible and bounded), and the algorithm has found
476    /// optimal flow and node potentials (primal and dual solutions),
477    /// \n \c UNBOUNDED if the digraph contains an arc of negative cost
478    /// and infinite upper bound. It means that the objective function
479    /// is unbounded on that arc, however, note that it could actually be
480    /// bounded over the feasible flows, but this algroithm cannot handle
481    /// these cases.
482    ///
483    /// \see ProblemType
484    /// \see resetParams(), reset()
485    ProblemType run(int factor = 4) {
486      _factor = factor;
487      ProblemType pt = init();
488      if (pt != OPTIMAL) return pt;
489      return start();
490    }
491
492    /// \brief Reset all the parameters that have been given before.
493    ///
494    /// This function resets all the paramaters that have been given
495    /// before using functions \ref lowerMap(), \ref upperMap(),
496    /// \ref costMap(), \ref supplyMap(), \ref stSupply().
497    ///
498    /// It is useful for multiple \ref run() calls. Basically, all the given
499    /// parameters are kept for the next \ref run() call, unless
500    /// \ref resetParams() or \ref reset() is used.
501    /// If the underlying digraph was also modified after the construction
502    /// of the class or the last \ref reset() call, then the \ref reset()
503    /// function must be used, otherwise \ref resetParams() is sufficient.
504    ///
505    /// For example,
506    /// \code
507    ///   CapacityScaling<ListDigraph> cs(graph);
508    ///
509    ///   // First run
510    ///   cs.lowerMap(lower).upperMap(upper).costMap(cost)
511    ///     .supplyMap(sup).run();
512    ///
513    ///   // Run again with modified cost map (resetParams() is not called,
514    ///   // so only the cost map have to be set again)
515    ///   cost[e] += 100;
516    ///   cs.costMap(cost).run();
517    ///
518    ///   // Run again from scratch using resetParams()
519    ///   // (the lower bounds will be set to zero on all arcs)
520    ///   cs.resetParams();
521    ///   cs.upperMap(capacity).costMap(cost)
522    ///     .supplyMap(sup).run();
523    /// \endcode
524    ///
525    /// \return <tt>(*this)</tt>
526    ///
527    /// \see reset(), run()
528    CapacityScaling& resetParams() {
529      for (int i = 0; i != _node_num; ++i) {
530        _supply[i] = 0;
531      }
532      for (int j = 0; j != _res_arc_num; ++j) {
533        _lower[j] = 0;
534        _upper[j] = INF;
535        _cost[j] = _forward[j] ? 1 : -1;
536      }
537      _have_lower = false;
538      return *this;
539    }
540
541    /// \brief Reset the internal data structures and all the parameters
542    /// that have been given before.
543    ///
544    /// This function resets the internal data structures and all the
545    /// paramaters that have been given before using functions \ref lowerMap(),
546    /// \ref upperMap(), \ref costMap(), \ref supplyMap(), \ref stSupply().
547    ///
548    /// It is useful for multiple \ref run() calls. Basically, all the given
549    /// parameters are kept for the next \ref run() call, unless
550    /// \ref resetParams() or \ref reset() is used.
551    /// If the underlying digraph was also modified after the construction
552    /// of the class or the last \ref reset() call, then the \ref reset()
553    /// function must be used, otherwise \ref resetParams() is sufficient.
554    ///
555    /// See \ref resetParams() for examples.
556    ///
557    /// \return <tt>(*this)</tt>
558    ///
559    /// \see resetParams(), run()
560    CapacityScaling& reset() {
561      // Resize vectors
562      _node_num = countNodes(_graph);
563      _arc_num = countArcs(_graph);
564      _res_arc_num = 2 * (_arc_num + _node_num);
565      _root = _node_num;
566      ++_node_num;
567
568      _first_out.resize(_node_num + 1);
569      _forward.resize(_res_arc_num);
570      _source.resize(_res_arc_num);
571      _target.resize(_res_arc_num);
572      _reverse.resize(_res_arc_num);
573
574      _lower.resize(_res_arc_num);
575      _upper.resize(_res_arc_num);
576      _cost.resize(_res_arc_num);
577      _supply.resize(_node_num);
578     
579      _res_cap.resize(_res_arc_num);
580      _pi.resize(_node_num);
581      _excess.resize(_node_num);
582      _pred.resize(_node_num);
583
584      // Copy the graph
585      int i = 0, j = 0, k = 2 * _arc_num + _node_num - 1;
586      for (NodeIt n(_graph); n != INVALID; ++n, ++i) {
587        _node_id[n] = i;
588      }
589      i = 0;
590      for (NodeIt n(_graph); n != INVALID; ++n, ++i) {
591        _first_out[i] = j;
592        for (OutArcIt a(_graph, n); a != INVALID; ++a, ++j) {
593          _arc_idf[a] = j;
594          _forward[j] = true;
595          _source[j] = i;
596          _target[j] = _node_id[_graph.runningNode(a)];
597        }
598        for (InArcIt a(_graph, n); a != INVALID; ++a, ++j) {
599          _arc_idb[a] = j;
600          _forward[j] = false;
601          _source[j] = i;
602          _target[j] = _node_id[_graph.runningNode(a)];
603        }
604        _forward[j] = false;
605        _source[j] = i;
606        _target[j] = _root;
607        _reverse[j] = k;
608        _forward[k] = true;
609        _source[k] = _root;
610        _target[k] = i;
611        _reverse[k] = j;
612        ++j; ++k;
613      }
614      _first_out[i] = j;
615      _first_out[_node_num] = k;
616      for (ArcIt a(_graph); a != INVALID; ++a) {
617        int fi = _arc_idf[a];
618        int bi = _arc_idb[a];
619        _reverse[fi] = bi;
620        _reverse[bi] = fi;
621      }
622     
623      // Reset parameters
624      resetParams();
625      return *this;
626    }
627
628    /// @}
629
630    /// \name Query Functions
631    /// The results of the algorithm can be obtained using these
632    /// functions.\n
633    /// The \ref run() function must be called before using them.
634
635    /// @{
636
637    /// \brief Return the total cost of the found flow.
638    ///
639    /// This function returns the total cost of the found flow.
640    /// Its complexity is O(e).
641    ///
642    /// \note The return type of the function can be specified as a
643    /// template parameter. For example,
644    /// \code
645    ///   cs.totalCost<double>();
646    /// \endcode
647    /// It is useful if the total cost cannot be stored in the \c Cost
648    /// type of the algorithm, which is the default return type of the
649    /// function.
650    ///
651    /// \pre \ref run() must be called before using this function.
652    template <typename Number>
653    Number totalCost() const {
654      Number c = 0;
655      for (ArcIt a(_graph); a != INVALID; ++a) {
656        int i = _arc_idb[a];
657        c += static_cast<Number>(_res_cap[i]) *
658             (-static_cast<Number>(_cost[i]));
659      }
660      return c;
661    }
662
663#ifndef DOXYGEN
664    Cost totalCost() const {
665      return totalCost<Cost>();
666    }
667#endif
668
669    /// \brief Return the flow on the given arc.
670    ///
671    /// This function returns the flow on the given arc.
672    ///
673    /// \pre \ref run() must be called before using this function.
674    Value flow(const Arc& a) const {
675      return _res_cap[_arc_idb[a]];
676    }
677
678    /// \brief Return the flow map (the primal solution).
679    ///
680    /// This function copies the flow value on each arc into the given
681    /// map. The \c Value type of the algorithm must be convertible to
682    /// the \c Value type of the map.
683    ///
684    /// \pre \ref run() must be called before using this function.
685    template <typename FlowMap>
686    void flowMap(FlowMap &map) const {
687      for (ArcIt a(_graph); a != INVALID; ++a) {
688        map.set(a, _res_cap[_arc_idb[a]]);
689      }
690    }
691
692    /// \brief Return the potential (dual value) of the given node.
693    ///
694    /// This function returns the potential (dual value) of the
695    /// given node.
696    ///
697    /// \pre \ref run() must be called before using this function.
698    Cost potential(const Node& n) const {
699      return _pi[_node_id[n]];
700    }
701
702    /// \brief Return the potential map (the dual solution).
703    ///
704    /// This function copies the potential (dual value) of each node
705    /// into the given map.
706    /// The \c Cost type of the algorithm must be convertible to the
707    /// \c Value type of the map.
708    ///
709    /// \pre \ref run() must be called before using this function.
710    template <typename PotentialMap>
711    void potentialMap(PotentialMap &map) const {
712      for (NodeIt n(_graph); n != INVALID; ++n) {
713        map.set(n, _pi[_node_id[n]]);
714      }
715    }
716
717    /// @}
718
719  private:
720
721    // Initialize the algorithm
722    ProblemType init() {
723      if (_node_num <= 1) return INFEASIBLE;
724
725      // Check the sum of supply values
726      _sum_supply = 0;
727      for (int i = 0; i != _root; ++i) {
728        _sum_supply += _supply[i];
729      }
730      if (_sum_supply > 0) return INFEASIBLE;
731     
732      // Initialize vectors
733      for (int i = 0; i != _root; ++i) {
734        _pi[i] = 0;
735        _excess[i] = _supply[i];
736      }
737
738      // Remove non-zero lower bounds
739      const Value MAX = std::numeric_limits<Value>::max();
740      int last_out;
741      if (_have_lower) {
742        for (int i = 0; i != _root; ++i) {
743          last_out = _first_out[i+1];
744          for (int j = _first_out[i]; j != last_out; ++j) {
745            if (_forward[j]) {
746              Value c = _lower[j];
747              if (c >= 0) {
748                _res_cap[j] = _upper[j] < MAX ? _upper[j] - c : INF;
749              } else {
750                _res_cap[j] = _upper[j] < MAX + c ? _upper[j] - c : INF;
751              }
752              _excess[i] -= c;
753              _excess[_target[j]] += c;
754            } else {
755              _res_cap[j] = 0;
756            }
757          }
758        }
759      } else {
760        for (int j = 0; j != _res_arc_num; ++j) {
761          _res_cap[j] = _forward[j] ? _upper[j] : 0;
762        }
763      }
764
765      // Handle negative costs
766      for (int i = 0; i != _root; ++i) {
767        last_out = _first_out[i+1] - 1;
768        for (int j = _first_out[i]; j != last_out; ++j) {
769          Value rc = _res_cap[j];
770          if (_cost[j] < 0 && rc > 0) {
771            if (rc >= MAX) return UNBOUNDED;
772            _excess[i] -= rc;
773            _excess[_target[j]] += rc;
774            _res_cap[j] = 0;
775            _res_cap[_reverse[j]] += rc;
776          }
777        }
778      }
779     
780      // Handle GEQ supply type
781      if (_sum_supply < 0) {
782        _pi[_root] = 0;
783        _excess[_root] = -_sum_supply;
784        for (int a = _first_out[_root]; a != _res_arc_num; ++a) {
785          int ra = _reverse[a];
786          _res_cap[a] = -_sum_supply + 1;
787          _res_cap[ra] = 0;
788          _cost[a] = 0;
789          _cost[ra] = 0;
790        }
791      } else {
792        _pi[_root] = 0;
793        _excess[_root] = 0;
794        for (int a = _first_out[_root]; a != _res_arc_num; ++a) {
795          int ra = _reverse[a];
796          _res_cap[a] = 1;
797          _res_cap[ra] = 0;
798          _cost[a] = 0;
799          _cost[ra] = 0;
800        }
801      }
802
803      // Initialize delta value
804      if (_factor > 1) {
805        // With scaling
806        Value max_sup = 0, max_dem = 0, max_cap = 0;
807        for (int i = 0; i != _root; ++i) {
808          Value ex = _excess[i];
809          if ( ex > max_sup) max_sup =  ex;
810          if (-ex > max_dem) max_dem = -ex;
811          int last_out = _first_out[i+1] - 1;
812          for (int j = _first_out[i]; j != last_out; ++j) {
813            if (_res_cap[j] > max_cap) max_cap = _res_cap[j];
814          }
815        }
816        max_sup = std::min(std::min(max_sup, max_dem), max_cap);
817        for (_delta = 1; 2 * _delta <= max_sup; _delta *= 2) ;
818      } else {
819        // Without scaling
820        _delta = 1;
821      }
822
823      return OPTIMAL;
824    }
825
826    ProblemType start() {
827      // Execute the algorithm
828      ProblemType pt;
829      if (_delta > 1)
830        pt = startWithScaling();
831      else
832        pt = startWithoutScaling();
833
834      // Handle non-zero lower bounds
835      if (_have_lower) {
836        int limit = _first_out[_root];
837        for (int j = 0; j != limit; ++j) {
838          if (!_forward[j]) _res_cap[j] += _lower[j];
839        }
840      }
841
842      // Shift potentials if necessary
843      Cost pr = _pi[_root];
844      if (_sum_supply < 0 || pr > 0) {
845        for (int i = 0; i != _node_num; ++i) {
846          _pi[i] -= pr;
847        }       
848      }
849     
850      return pt;
851    }
852
853    // Execute the capacity scaling algorithm
854    ProblemType startWithScaling() {
855      // Perform capacity scaling phases
856      int s, t;
857      ResidualDijkstra _dijkstra(*this);
858      while (true) {
859        // Saturate all arcs not satisfying the optimality condition
860        int last_out;
861        for (int u = 0; u != _node_num; ++u) {
862          last_out = _sum_supply < 0 ?
863            _first_out[u+1] : _first_out[u+1] - 1;
864          for (int a = _first_out[u]; a != last_out; ++a) {
865            int v = _target[a];
866            Cost c = _cost[a] + _pi[u] - _pi[v];
867            Value rc = _res_cap[a];
868            if (c < 0 && rc >= _delta) {
869              _excess[u] -= rc;
870              _excess[v] += rc;
871              _res_cap[a] = 0;
872              _res_cap[_reverse[a]] += rc;
873            }
874          }
875        }
876
877        // Find excess nodes and deficit nodes
878        _excess_nodes.clear();
879        _deficit_nodes.clear();
880        for (int u = 0; u != _node_num; ++u) {
881          Value ex = _excess[u];
882          if (ex >=  _delta) _excess_nodes.push_back(u);
883          if (ex <= -_delta) _deficit_nodes.push_back(u);
884        }
885        int next_node = 0, next_def_node = 0;
886
887        // Find augmenting shortest paths
888        while (next_node < int(_excess_nodes.size())) {
889          // Check deficit nodes
890          if (_delta > 1) {
891            bool delta_deficit = false;
892            for ( ; next_def_node < int(_deficit_nodes.size());
893                    ++next_def_node ) {
894              if (_excess[_deficit_nodes[next_def_node]] <= -_delta) {
895                delta_deficit = true;
896                break;
897              }
898            }
899            if (!delta_deficit) break;
900          }
901
902          // Run Dijkstra in the residual network
903          s = _excess_nodes[next_node];
904          if ((t = _dijkstra.run(s, _delta)) == -1) {
905            if (_delta > 1) {
906              ++next_node;
907              continue;
908            }
909            return INFEASIBLE;
910          }
911
912          // Augment along a shortest path from s to t
913          Value d = std::min(_excess[s], -_excess[t]);
914          int u = t;
915          int a;
916          if (d > _delta) {
917            while ((a = _pred[u]) != -1) {
918              if (_res_cap[a] < d) d = _res_cap[a];
919              u = _source[a];
920            }
921          }
922          u = t;
923          while ((a = _pred[u]) != -1) {
924            _res_cap[a] -= d;
925            _res_cap[_reverse[a]] += d;
926            u = _source[a];
927          }
928          _excess[s] -= d;
929          _excess[t] += d;
930
931          if (_excess[s] < _delta) ++next_node;
932        }
933
934        if (_delta == 1) break;
935        _delta = _delta <= _factor ? 1 : _delta / _factor;
936      }
937
938      return OPTIMAL;
939    }
940
941    // Execute the successive shortest path algorithm
942    ProblemType startWithoutScaling() {
943      // Find excess nodes
944      _excess_nodes.clear();
945      for (int i = 0; i != _node_num; ++i) {
946        if (_excess[i] > 0) _excess_nodes.push_back(i);
947      }
948      if (_excess_nodes.size() == 0) return OPTIMAL;
949      int next_node = 0;
950
951      // Find shortest paths
952      int s, t;
953      ResidualDijkstra _dijkstra(*this);
954      while ( _excess[_excess_nodes[next_node]] > 0 ||
955              ++next_node < int(_excess_nodes.size()) )
956      {
957        // Run Dijkstra in the residual network
958        s = _excess_nodes[next_node];
959        if ((t = _dijkstra.run(s)) == -1) return INFEASIBLE;
960
961        // Augment along a shortest path from s to t
962        Value d = std::min(_excess[s], -_excess[t]);
963        int u = t;
964        int a;
965        if (d > 1) {
966          while ((a = _pred[u]) != -1) {
967            if (_res_cap[a] < d) d = _res_cap[a];
968            u = _source[a];
969          }
970        }
971        u = t;
972        while ((a = _pred[u]) != -1) {
973          _res_cap[a] -= d;
974          _res_cap[_reverse[a]] += d;
975          u = _source[a];
976        }
977        _excess[s] -= d;
978        _excess[t] += d;
979      }
980
981      return OPTIMAL;
982    }
983
984  }; //class CapacityScaling
985
986  ///@}
987
988} //namespace lemon
989
990#endif //LEMON_CAPACITY_SCALING_H
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