COIN-OR::LEMON - Graph Library

source: lemon/lemon/capacity_scaling.h @ 899:cc9e0c15d747

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1/* -*- C++ -*-
2 *
3 * This file is a part of LEMON, a generic C++ optimization library
4 *
5 * Copyright (C) 2003-2008
6 * Egervary Jeno Kombinatorikus Optimalizalasi Kutatocsoport
7 * (Egervary Research Group on Combinatorial Optimization, EGRES).
8 *
9 * Permission to use, modify and distribute this software is granted
10 * provided that this copyright notice appears in all copies. For
11 * precise terms see the accompanying LICENSE file.
12 *
13 * This software is provided "AS IS" with no warranty of any kind,
14 * express or implied, and with no claim as to its suitability for any
15 * purpose.
16 *
17 */
18
19#ifndef LEMON_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<char> BoolVector;
143    typedef std::vector<Value> ValueVector;
144    typedef std::vector<Cost> CostVector;
145
146  private:
147
148    // Data related to the underlying digraph
149    const GR &_graph;
150    int _node_num;
151    int _arc_num;
152    int _res_arc_num;
153    int _root;
154
155    // Parameters of the problem
156    bool _have_lower;
157    Value _sum_supply;
158
159    // Data structures for storing the digraph
160    IntNodeMap _node_id;
161    IntArcMap _arc_idf;
162    IntArcMap _arc_idb;
163    IntVector _first_out;
164    BoolVector _forward;
165    IntVector _source;
166    IntVector _target;
167    IntVector _reverse;
168
169    // Node and arc data
170    ValueVector _lower;
171    ValueVector _upper;
172    CostVector _cost;
173    ValueVector _supply;
174
175    ValueVector _res_cap;
176    CostVector _pi;
177    ValueVector _excess;
178    IntVector _excess_nodes;
179    IntVector _deficit_nodes;
180
181    Value _delta;
182    int _factor;
183    IntVector _pred;
184
185  public:
186 
187    /// \brief Constant for infinite upper bounds (capacities).
188    ///
189    /// Constant for infinite upper bounds (capacities).
190    /// It is \c std::numeric_limits<Value>::infinity() if available,
191    /// \c std::numeric_limits<Value>::max() otherwise.
192    const Value INF;
193
194  private:
195
196    // Special implementation of the Dijkstra algorithm for finding
197    // shortest paths in the residual network of the digraph with
198    // respect to the reduced arc costs and modifying the node
199    // potentials according to the found distance labels.
200    class ResidualDijkstra
201    {
202    private:
203
204      int _node_num;
205      bool _geq;
206      const IntVector &_first_out;
207      const IntVector &_target;
208      const CostVector &_cost;
209      const ValueVector &_res_cap;
210      const ValueVector &_excess;
211      CostVector &_pi;
212      IntVector &_pred;
213     
214      IntVector _proc_nodes;
215      CostVector _dist;
216     
217    public:
218
219      ResidualDijkstra(CapacityScaling& cs) :
220        _node_num(cs._node_num), _geq(cs._sum_supply < 0),
221        _first_out(cs._first_out), _target(cs._target), _cost(cs._cost),
222        _res_cap(cs._res_cap), _excess(cs._excess), _pi(cs._pi),
223        _pred(cs._pred), _dist(cs._node_num)
224      {}
225
226      int run(int s, Value delta = 1) {
227        RangeMap<int> heap_cross_ref(_node_num, Heap::PRE_HEAP);
228        Heap heap(heap_cross_ref);
229        heap.push(s, 0);
230        _pred[s] = -1;
231        _proc_nodes.clear();
232
233        // Process nodes
234        while (!heap.empty() && _excess[heap.top()] > -delta) {
235          int u = heap.top(), v;
236          Cost d = heap.prio() + _pi[u], dn;
237          _dist[u] = heap.prio();
238          _proc_nodes.push_back(u);
239          heap.pop();
240
241          // Traverse outgoing residual arcs
242          int last_out = _geq ? _first_out[u+1] : _first_out[u+1] - 1;
243          for (int a = _first_out[u]; a != last_out; ++a) {
244            if (_res_cap[a] < delta) continue;
245            v = _target[a];
246            switch (heap.state(v)) {
247              case Heap::PRE_HEAP:
248                heap.push(v, d + _cost[a] - _pi[v]);
249                _pred[v] = a;
250                break;
251              case Heap::IN_HEAP:
252                dn = d + _cost[a] - _pi[v];
253                if (dn < heap[v]) {
254                  heap.decrease(v, dn);
255                  _pred[v] = a;
256                }
257                break;
258              case Heap::POST_HEAP:
259                break;
260            }
261          }
262        }
263        if (heap.empty()) return -1;
264
265        // Update potentials of processed nodes
266        int t = heap.top();
267        Cost dt = heap.prio();
268        for (int i = 0; i < int(_proc_nodes.size()); ++i) {
269          _pi[_proc_nodes[i]] += _dist[_proc_nodes[i]] - dt;
270        }
271
272        return t;
273      }
274
275    }; //class ResidualDijkstra
276
277  public:
278
279    /// \name Named Template Parameters
280    /// @{
281
282    template <typename T>
283    struct SetHeapTraits : public Traits {
284      typedef T Heap;
285    };
286
287    /// \brief \ref named-templ-param "Named parameter" for setting
288    /// \c Heap type.
289    ///
290    /// \ref named-templ-param "Named parameter" for setting \c Heap
291    /// type, which is used for internal Dijkstra computations.
292    /// It must conform to the \ref lemon::concepts::Heap "Heap" concept,
293    /// its priority type must be \c Cost and its cross reference type
294    /// must be \ref RangeMap "RangeMap<int>".
295    template <typename T>
296    struct SetHeap
297      : public CapacityScaling<GR, V, C, SetHeapTraits<T> > {
298      typedef  CapacityScaling<GR, V, C, SetHeapTraits<T> > Create;
299    };
300
301    /// @}
302
303  public:
304
305    /// \brief Constructor.
306    ///
307    /// The constructor of the class.
308    ///
309    /// \param graph The digraph the algorithm runs on.
310    CapacityScaling(const GR& graph) :
311      _graph(graph), _node_id(graph), _arc_idf(graph), _arc_idb(graph),
312      INF(std::numeric_limits<Value>::has_infinity ?
313          std::numeric_limits<Value>::infinity() :
314          std::numeric_limits<Value>::max())
315    {
316      // Check the number types
317      LEMON_ASSERT(std::numeric_limits<Value>::is_signed,
318        "The flow type of CapacityScaling must be signed");
319      LEMON_ASSERT(std::numeric_limits<Cost>::is_signed,
320        "The cost type of CapacityScaling must be signed");
321
322      // Reset data structures
323      reset();
324    }
325
326    /// \name Parameters
327    /// The parameters of the algorithm can be specified using these
328    /// functions.
329
330    /// @{
331
332    /// \brief Set the lower bounds on the arcs.
333    ///
334    /// This function sets the lower bounds on the arcs.
335    /// If it is not used before calling \ref run(), the lower bounds
336    /// will be set to zero on all arcs.
337    ///
338    /// \param map An arc map storing the lower bounds.
339    /// Its \c Value type must be convertible to the \c Value type
340    /// of the algorithm.
341    ///
342    /// \return <tt>(*this)</tt>
343    template <typename LowerMap>
344    CapacityScaling& lowerMap(const LowerMap& map) {
345      _have_lower = true;
346      for (ArcIt a(_graph); a != INVALID; ++a) {
347        _lower[_arc_idf[a]] = map[a];
348        _lower[_arc_idb[a]] = map[a];
349      }
350      return *this;
351    }
352
353    /// \brief Set the upper bounds (capacities) on the arcs.
354    ///
355    /// This function sets the upper bounds (capacities) on the arcs.
356    /// If it is not used before calling \ref run(), the upper bounds
357    /// will be set to \ref INF on all arcs (i.e. the flow value will be
358    /// unbounded from above).
359    ///
360    /// \param map An arc map storing the upper bounds.
361    /// Its \c Value type must be convertible to the \c Value type
362    /// of the algorithm.
363    ///
364    /// \return <tt>(*this)</tt>
365    template<typename UpperMap>
366    CapacityScaling& upperMap(const UpperMap& map) {
367      for (ArcIt a(_graph); a != INVALID; ++a) {
368        _upper[_arc_idf[a]] = map[a];
369      }
370      return *this;
371    }
372
373    /// \brief Set the costs of the arcs.
374    ///
375    /// This function sets the costs of the arcs.
376    /// If it is not used before calling \ref run(), the costs
377    /// will be set to \c 1 on all arcs.
378    ///
379    /// \param map An arc map storing the costs.
380    /// Its \c Value type must be convertible to the \c Cost type
381    /// of the algorithm.
382    ///
383    /// \return <tt>(*this)</tt>
384    template<typename CostMap>
385    CapacityScaling& costMap(const CostMap& map) {
386      for (ArcIt a(_graph); a != INVALID; ++a) {
387        _cost[_arc_idf[a]] =  map[a];
388        _cost[_arc_idb[a]] = -map[a];
389      }
390      return *this;
391    }
392
393    /// \brief Set the supply values of the nodes.
394    ///
395    /// This function sets the supply values of the nodes.
396    /// If neither this function nor \ref stSupply() is used before
397    /// calling \ref run(), the supply of each node will be set to zero.
398    ///
399    /// \param map A node map storing the supply values.
400    /// Its \c Value type must be convertible to the \c Value type
401    /// of the algorithm.
402    ///
403    /// \return <tt>(*this)</tt>
404    template<typename SupplyMap>
405    CapacityScaling& supplyMap(const SupplyMap& map) {
406      for (NodeIt n(_graph); n != INVALID; ++n) {
407        _supply[_node_id[n]] = map[n];
408      }
409      return *this;
410    }
411
412    /// \brief Set single source and target nodes and a supply value.
413    ///
414    /// This function sets a single source node and a single target node
415    /// and the required flow value.
416    /// If neither this function nor \ref supplyMap() is used before
417    /// calling \ref run(), the supply of each node will be set to zero.
418    ///
419    /// Using this function has the same effect as using \ref supplyMap()
420    /// with such a map in which \c k is assigned to \c s, \c -k is
421    /// assigned to \c t and all other nodes have zero supply value.
422    ///
423    /// \param s The source node.
424    /// \param t The target node.
425    /// \param k The required amount of flow from node \c s to node \c t
426    /// (i.e. the supply of \c s and the demand of \c t).
427    ///
428    /// \return <tt>(*this)</tt>
429    CapacityScaling& stSupply(const Node& s, const Node& t, Value k) {
430      for (int i = 0; i != _node_num; ++i) {
431        _supply[i] = 0;
432      }
433      _supply[_node_id[s]] =  k;
434      _supply[_node_id[t]] = -k;
435      return *this;
436    }
437   
438    /// @}
439
440    /// \name Execution control
441    /// The algorithm can be executed using \ref run().
442
443    /// @{
444
445    /// \brief Run the algorithm.
446    ///
447    /// This function runs the algorithm.
448    /// The paramters can be specified using functions \ref lowerMap(),
449    /// \ref upperMap(), \ref costMap(), \ref supplyMap(), \ref stSupply().
450    /// For example,
451    /// \code
452    ///   CapacityScaling<ListDigraph> cs(graph);
453    ///   cs.lowerMap(lower).upperMap(upper).costMap(cost)
454    ///     .supplyMap(sup).run();
455    /// \endcode
456    ///
457    /// This function can be called more than once. All the given parameters
458    /// are kept for the next call, unless \ref resetParams() or \ref reset()
459    /// is used, thus only the modified parameters have to be set again.
460    /// If the underlying digraph was also modified after the construction
461    /// of the class (or the last \ref reset() call), then the \ref reset()
462    /// function must be called.
463    ///
464    /// \param factor The capacity scaling factor. It must be larger than
465    /// one to use scaling. If it is less or equal to one, then scaling
466    /// will be disabled.
467    ///
468    /// \return \c INFEASIBLE if no feasible flow exists,
469    /// \n \c OPTIMAL if the problem has optimal solution
470    /// (i.e. it is feasible and bounded), and the algorithm has found
471    /// optimal flow and node potentials (primal and dual solutions),
472    /// \n \c UNBOUNDED if the digraph contains an arc of negative cost
473    /// and infinite upper bound. It means that the objective function
474    /// is unbounded on that arc, however, note that it could actually be
475    /// bounded over the feasible flows, but this algroithm cannot handle
476    /// these cases.
477    ///
478    /// \see ProblemType
479    /// \see resetParams(), reset()
480    ProblemType run(int factor = 4) {
481      _factor = factor;
482      ProblemType pt = init();
483      if (pt != OPTIMAL) return pt;
484      return start();
485    }
486
487    /// \brief Reset all the parameters that have been given before.
488    ///
489    /// This function resets all the paramaters that have been given
490    /// before using functions \ref lowerMap(), \ref upperMap(),
491    /// \ref costMap(), \ref supplyMap(), \ref stSupply().
492    ///
493    /// It is useful for multiple \ref run() calls. Basically, all the given
494    /// parameters are kept for the next \ref run() call, unless
495    /// \ref resetParams() or \ref reset() is used.
496    /// If the underlying digraph was also modified after the construction
497    /// of the class or the last \ref reset() call, then the \ref reset()
498    /// function must be used, otherwise \ref resetParams() is sufficient.
499    ///
500    /// For example,
501    /// \code
502    ///   CapacityScaling<ListDigraph> cs(graph);
503    ///
504    ///   // First run
505    ///   cs.lowerMap(lower).upperMap(upper).costMap(cost)
506    ///     .supplyMap(sup).run();
507    ///
508    ///   // Run again with modified cost map (resetParams() is not called,
509    ///   // so only the cost map have to be set again)
510    ///   cost[e] += 100;
511    ///   cs.costMap(cost).run();
512    ///
513    ///   // Run again from scratch using resetParams()
514    ///   // (the lower bounds will be set to zero on all arcs)
515    ///   cs.resetParams();
516    ///   cs.upperMap(capacity).costMap(cost)
517    ///     .supplyMap(sup).run();
518    /// \endcode
519    ///
520    /// \return <tt>(*this)</tt>
521    ///
522    /// \see reset(), run()
523    CapacityScaling& resetParams() {
524      for (int i = 0; i != _node_num; ++i) {
525        _supply[i] = 0;
526      }
527      for (int j = 0; j != _res_arc_num; ++j) {
528        _lower[j] = 0;
529        _upper[j] = INF;
530        _cost[j] = _forward[j] ? 1 : -1;
531      }
532      _have_lower = false;
533      return *this;
534    }
535
536    /// \brief Reset the internal data structures and all the parameters
537    /// that have been given before.
538    ///
539    /// This function resets the internal data structures and all the
540    /// paramaters that have been given before using functions \ref lowerMap(),
541    /// \ref upperMap(), \ref costMap(), \ref supplyMap(), \ref stSupply().
542    ///
543    /// It is useful for multiple \ref run() calls. Basically, all the given
544    /// parameters are kept for the next \ref run() call, unless
545    /// \ref resetParams() or \ref reset() is used.
546    /// If the underlying digraph was also modified after the construction
547    /// of the class or the last \ref reset() call, then the \ref reset()
548    /// function must be used, otherwise \ref resetParams() is sufficient.
549    ///
550    /// See \ref resetParams() for examples.
551    ///
552    /// \return <tt>(*this)</tt>
553    ///
554    /// \see resetParams(), run()
555    CapacityScaling& reset() {
556      // Resize vectors
557      _node_num = countNodes(_graph);
558      _arc_num = countArcs(_graph);
559      _res_arc_num = 2 * (_arc_num + _node_num);
560      _root = _node_num;
561      ++_node_num;
562
563      _first_out.resize(_node_num + 1);
564      _forward.resize(_res_arc_num);
565      _source.resize(_res_arc_num);
566      _target.resize(_res_arc_num);
567      _reverse.resize(_res_arc_num);
568
569      _lower.resize(_res_arc_num);
570      _upper.resize(_res_arc_num);
571      _cost.resize(_res_arc_num);
572      _supply.resize(_node_num);
573     
574      _res_cap.resize(_res_arc_num);
575      _pi.resize(_node_num);
576      _excess.resize(_node_num);
577      _pred.resize(_node_num);
578
579      // Copy the graph
580      int i = 0, j = 0, k = 2 * _arc_num + _node_num - 1;
581      for (NodeIt n(_graph); n != INVALID; ++n, ++i) {
582        _node_id[n] = i;
583      }
584      i = 0;
585      for (NodeIt n(_graph); n != INVALID; ++n, ++i) {
586        _first_out[i] = j;
587        for (OutArcIt a(_graph, n); a != INVALID; ++a, ++j) {
588          _arc_idf[a] = j;
589          _forward[j] = true;
590          _source[j] = i;
591          _target[j] = _node_id[_graph.runningNode(a)];
592        }
593        for (InArcIt a(_graph, n); a != INVALID; ++a, ++j) {
594          _arc_idb[a] = j;
595          _forward[j] = false;
596          _source[j] = i;
597          _target[j] = _node_id[_graph.runningNode(a)];
598        }
599        _forward[j] = false;
600        _source[j] = i;
601        _target[j] = _root;
602        _reverse[j] = k;
603        _forward[k] = true;
604        _source[k] = _root;
605        _target[k] = i;
606        _reverse[k] = j;
607        ++j; ++k;
608      }
609      _first_out[i] = j;
610      _first_out[_node_num] = k;
611      for (ArcIt a(_graph); a != INVALID; ++a) {
612        int fi = _arc_idf[a];
613        int bi = _arc_idb[a];
614        _reverse[fi] = bi;
615        _reverse[bi] = fi;
616      }
617     
618      // Reset parameters
619      resetParams();
620      return *this;
621    }
622
623    /// @}
624
625    /// \name Query Functions
626    /// The results of the algorithm can be obtained using these
627    /// functions.\n
628    /// The \ref run() function must be called before using them.
629
630    /// @{
631
632    /// \brief Return the total cost of the found flow.
633    ///
634    /// This function returns the total cost of the found flow.
635    /// Its complexity is O(e).
636    ///
637    /// \note The return type of the function can be specified as a
638    /// template parameter. For example,
639    /// \code
640    ///   cs.totalCost<double>();
641    /// \endcode
642    /// It is useful if the total cost cannot be stored in the \c Cost
643    /// type of the algorithm, which is the default return type of the
644    /// function.
645    ///
646    /// \pre \ref run() must be called before using this function.
647    template <typename Number>
648    Number totalCost() const {
649      Number c = 0;
650      for (ArcIt a(_graph); a != INVALID; ++a) {
651        int i = _arc_idb[a];
652        c += static_cast<Number>(_res_cap[i]) *
653             (-static_cast<Number>(_cost[i]));
654      }
655      return c;
656    }
657
658#ifndef DOXYGEN
659    Cost totalCost() const {
660      return totalCost<Cost>();
661    }
662#endif
663
664    /// \brief Return the flow on the given arc.
665    ///
666    /// This function returns the flow on the given arc.
667    ///
668    /// \pre \ref run() must be called before using this function.
669    Value flow(const Arc& a) const {
670      return _res_cap[_arc_idb[a]];
671    }
672
673    /// \brief Return the flow map (the primal solution).
674    ///
675    /// This function copies the flow value on each arc into the given
676    /// map. The \c Value type of the algorithm must be convertible to
677    /// the \c Value type of the map.
678    ///
679    /// \pre \ref run() must be called before using this function.
680    template <typename FlowMap>
681    void flowMap(FlowMap &map) const {
682      for (ArcIt a(_graph); a != INVALID; ++a) {
683        map.set(a, _res_cap[_arc_idb[a]]);
684      }
685    }
686
687    /// \brief Return the potential (dual value) of the given node.
688    ///
689    /// This function returns the potential (dual value) of the
690    /// given node.
691    ///
692    /// \pre \ref run() must be called before using this function.
693    Cost potential(const Node& n) const {
694      return _pi[_node_id[n]];
695    }
696
697    /// \brief Return the potential map (the dual solution).
698    ///
699    /// This function copies the potential (dual value) of each node
700    /// into the given map.
701    /// The \c Cost type of the algorithm must be convertible to the
702    /// \c Value type of the map.
703    ///
704    /// \pre \ref run() must be called before using this function.
705    template <typename PotentialMap>
706    void potentialMap(PotentialMap &map) const {
707      for (NodeIt n(_graph); n != INVALID; ++n) {
708        map.set(n, _pi[_node_id[n]]);
709      }
710    }
711
712    /// @}
713
714  private:
715
716    // Initialize the algorithm
717    ProblemType init() {
718      if (_node_num <= 1) return INFEASIBLE;
719
720      // Check the sum of supply values
721      _sum_supply = 0;
722      for (int i = 0; i != _root; ++i) {
723        _sum_supply += _supply[i];
724      }
725      if (_sum_supply > 0) return INFEASIBLE;
726     
727      // Initialize vectors
728      for (int i = 0; i != _root; ++i) {
729        _pi[i] = 0;
730        _excess[i] = _supply[i];
731      }
732
733      // Remove non-zero lower bounds
734      const Value MAX = std::numeric_limits<Value>::max();
735      int last_out;
736      if (_have_lower) {
737        for (int i = 0; i != _root; ++i) {
738          last_out = _first_out[i+1];
739          for (int j = _first_out[i]; j != last_out; ++j) {
740            if (_forward[j]) {
741              Value c = _lower[j];
742              if (c >= 0) {
743                _res_cap[j] = _upper[j] < MAX ? _upper[j] - c : INF;
744              } else {
745                _res_cap[j] = _upper[j] < MAX + c ? _upper[j] - c : INF;
746              }
747              _excess[i] -= c;
748              _excess[_target[j]] += c;
749            } else {
750              _res_cap[j] = 0;
751            }
752          }
753        }
754      } else {
755        for (int j = 0; j != _res_arc_num; ++j) {
756          _res_cap[j] = _forward[j] ? _upper[j] : 0;
757        }
758      }
759
760      // Handle negative costs
761      for (int i = 0; i != _root; ++i) {
762        last_out = _first_out[i+1] - 1;
763        for (int j = _first_out[i]; j != last_out; ++j) {
764          Value rc = _res_cap[j];
765          if (_cost[j] < 0 && rc > 0) {
766            if (rc >= MAX) return UNBOUNDED;
767            _excess[i] -= rc;
768            _excess[_target[j]] += rc;
769            _res_cap[j] = 0;
770            _res_cap[_reverse[j]] += rc;
771          }
772        }
773      }
774     
775      // Handle GEQ supply type
776      if (_sum_supply < 0) {
777        _pi[_root] = 0;
778        _excess[_root] = -_sum_supply;
779        for (int a = _first_out[_root]; a != _res_arc_num; ++a) {
780          int ra = _reverse[a];
781          _res_cap[a] = -_sum_supply + 1;
782          _res_cap[ra] = 0;
783          _cost[a] = 0;
784          _cost[ra] = 0;
785        }
786      } else {
787        _pi[_root] = 0;
788        _excess[_root] = 0;
789        for (int a = _first_out[_root]; a != _res_arc_num; ++a) {
790          int ra = _reverse[a];
791          _res_cap[a] = 1;
792          _res_cap[ra] = 0;
793          _cost[a] = 0;
794          _cost[ra] = 0;
795        }
796      }
797
798      // Initialize delta value
799      if (_factor > 1) {
800        // With scaling
801        Value max_sup = 0, max_dem = 0;
802        for (int i = 0; i != _node_num; ++i) {
803          Value ex = _excess[i];
804          if ( ex > max_sup) max_sup =  ex;
805          if (-ex > max_dem) max_dem = -ex;
806        }
807        Value max_cap = 0;
808        for (int j = 0; j != _res_arc_num; ++j) {
809          if (_res_cap[j] > max_cap) max_cap = _res_cap[j];
810        }
811        max_sup = std::min(std::min(max_sup, max_dem), max_cap);
812        for (_delta = 1; 2 * _delta <= max_sup; _delta *= 2) ;
813      } else {
814        // Without scaling
815        _delta = 1;
816      }
817
818      return OPTIMAL;
819    }
820
821    ProblemType start() {
822      // Execute the algorithm
823      ProblemType pt;
824      if (_delta > 1)
825        pt = startWithScaling();
826      else
827        pt = startWithoutScaling();
828
829      // Handle non-zero lower bounds
830      if (_have_lower) {
831        int limit = _first_out[_root];
832        for (int j = 0; j != limit; ++j) {
833          if (!_forward[j]) _res_cap[j] += _lower[j];
834        }
835      }
836
837      // Shift potentials if necessary
838      Cost pr = _pi[_root];
839      if (_sum_supply < 0 || pr > 0) {
840        for (int i = 0; i != _node_num; ++i) {
841          _pi[i] -= pr;
842        }       
843      }
844     
845      return pt;
846    }
847
848    // Execute the capacity scaling algorithm
849    ProblemType startWithScaling() {
850      // Perform capacity scaling phases
851      int s, t;
852      ResidualDijkstra _dijkstra(*this);
853      while (true) {
854        // Saturate all arcs not satisfying the optimality condition
855        int last_out;
856        for (int u = 0; u != _node_num; ++u) {
857          last_out = _sum_supply < 0 ?
858            _first_out[u+1] : _first_out[u+1] - 1;
859          for (int a = _first_out[u]; a != last_out; ++a) {
860            int v = _target[a];
861            Cost c = _cost[a] + _pi[u] - _pi[v];
862            Value rc = _res_cap[a];
863            if (c < 0 && rc >= _delta) {
864              _excess[u] -= rc;
865              _excess[v] += rc;
866              _res_cap[a] = 0;
867              _res_cap[_reverse[a]] += rc;
868            }
869          }
870        }
871
872        // Find excess nodes and deficit nodes
873        _excess_nodes.clear();
874        _deficit_nodes.clear();
875        for (int u = 0; u != _node_num; ++u) {
876          Value ex = _excess[u];
877          if (ex >=  _delta) _excess_nodes.push_back(u);
878          if (ex <= -_delta) _deficit_nodes.push_back(u);
879        }
880        int next_node = 0, next_def_node = 0;
881
882        // Find augmenting shortest paths
883        while (next_node < int(_excess_nodes.size())) {
884          // Check deficit nodes
885          if (_delta > 1) {
886            bool delta_deficit = false;
887            for ( ; next_def_node < int(_deficit_nodes.size());
888                    ++next_def_node ) {
889              if (_excess[_deficit_nodes[next_def_node]] <= -_delta) {
890                delta_deficit = true;
891                break;
892              }
893            }
894            if (!delta_deficit) break;
895          }
896
897          // Run Dijkstra in the residual network
898          s = _excess_nodes[next_node];
899          if ((t = _dijkstra.run(s, _delta)) == -1) {
900            if (_delta > 1) {
901              ++next_node;
902              continue;
903            }
904            return INFEASIBLE;
905          }
906
907          // Augment along a shortest path from s to t
908          Value d = std::min(_excess[s], -_excess[t]);
909          int u = t;
910          int a;
911          if (d > _delta) {
912            while ((a = _pred[u]) != -1) {
913              if (_res_cap[a] < d) d = _res_cap[a];
914              u = _source[a];
915            }
916          }
917          u = t;
918          while ((a = _pred[u]) != -1) {
919            _res_cap[a] -= d;
920            _res_cap[_reverse[a]] += d;
921            u = _source[a];
922          }
923          _excess[s] -= d;
924          _excess[t] += d;
925
926          if (_excess[s] < _delta) ++next_node;
927        }
928
929        if (_delta == 1) break;
930        _delta = _delta <= _factor ? 1 : _delta / _factor;
931      }
932
933      return OPTIMAL;
934    }
935
936    // Execute the successive shortest path algorithm
937    ProblemType startWithoutScaling() {
938      // Find excess nodes
939      _excess_nodes.clear();
940      for (int i = 0; i != _node_num; ++i) {
941        if (_excess[i] > 0) _excess_nodes.push_back(i);
942      }
943      if (_excess_nodes.size() == 0) return OPTIMAL;
944      int next_node = 0;
945
946      // Find shortest paths
947      int s, t;
948      ResidualDijkstra _dijkstra(*this);
949      while ( _excess[_excess_nodes[next_node]] > 0 ||
950              ++next_node < int(_excess_nodes.size()) )
951      {
952        // Run Dijkstra in the residual network
953        s = _excess_nodes[next_node];
954        if ((t = _dijkstra.run(s)) == -1) return INFEASIBLE;
955
956        // Augment along a shortest path from s to t
957        Value d = std::min(_excess[s], -_excess[t]);
958        int u = t;
959        int a;
960        if (d > 1) {
961          while ((a = _pred[u]) != -1) {
962            if (_res_cap[a] < d) d = _res_cap[a];
963            u = _source[a];
964          }
965        }
966        u = t;
967        while ((a = _pred[u]) != -1) {
968          _res_cap[a] -= d;
969          _res_cap[_reverse[a]] += d;
970          u = _source[a];
971        }
972        _excess[s] -= d;
973        _excess[t] += d;
974      }
975
976      return OPTIMAL;
977    }
978
979  }; //class CapacityScaling
980
981  ///@}
982
983} //namespace lemon
984
985#endif //LEMON_CAPACITY_SCALING_H
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