COIN-OR::LEMON - Graph Library

source: lemon/lemon/capacity_scaling.h @ 891:75e6020b19b1

Last change on this file since 891:75e6020b19b1 was 891:75e6020b19b1, checked in by Peter Kovacs <kpeter@…>, 10 years ago

Add doc for the traits class parameters (#315)

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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      // Resize vectors
323      _node_num = countNodes(_graph);
324      _arc_num = countArcs(_graph);
325      _res_arc_num = 2 * (_arc_num + _node_num);
326      _root = _node_num;
327      ++_node_num;
328
329      _first_out.resize(_node_num + 1);
330      _forward.resize(_res_arc_num);
331      _source.resize(_res_arc_num);
332      _target.resize(_res_arc_num);
333      _reverse.resize(_res_arc_num);
334
335      _lower.resize(_res_arc_num);
336      _upper.resize(_res_arc_num);
337      _cost.resize(_res_arc_num);
338      _supply.resize(_node_num);
339     
340      _res_cap.resize(_res_arc_num);
341      _pi.resize(_node_num);
342      _excess.resize(_node_num);
343      _pred.resize(_node_num);
344
345      // Copy the graph
346      int i = 0, j = 0, k = 2 * _arc_num + _node_num - 1;
347      for (NodeIt n(_graph); n != INVALID; ++n, ++i) {
348        _node_id[n] = i;
349      }
350      i = 0;
351      for (NodeIt n(_graph); n != INVALID; ++n, ++i) {
352        _first_out[i] = j;
353        for (OutArcIt a(_graph, n); a != INVALID; ++a, ++j) {
354          _arc_idf[a] = j;
355          _forward[j] = true;
356          _source[j] = i;
357          _target[j] = _node_id[_graph.runningNode(a)];
358        }
359        for (InArcIt a(_graph, n); a != INVALID; ++a, ++j) {
360          _arc_idb[a] = j;
361          _forward[j] = false;
362          _source[j] = i;
363          _target[j] = _node_id[_graph.runningNode(a)];
364        }
365        _forward[j] = false;
366        _source[j] = i;
367        _target[j] = _root;
368        _reverse[j] = k;
369        _forward[k] = true;
370        _source[k] = _root;
371        _target[k] = i;
372        _reverse[k] = j;
373        ++j; ++k;
374      }
375      _first_out[i] = j;
376      _first_out[_node_num] = k;
377      for (ArcIt a(_graph); a != INVALID; ++a) {
378        int fi = _arc_idf[a];
379        int bi = _arc_idb[a];
380        _reverse[fi] = bi;
381        _reverse[bi] = fi;
382      }
383     
384      // Reset parameters
385      reset();
386    }
387
388    /// \name Parameters
389    /// The parameters of the algorithm can be specified using these
390    /// functions.
391
392    /// @{
393
394    /// \brief Set the lower bounds on the arcs.
395    ///
396    /// This function sets the lower bounds on the arcs.
397    /// If it is not used before calling \ref run(), the lower bounds
398    /// will be set to zero on all arcs.
399    ///
400    /// \param map An arc map storing the lower bounds.
401    /// Its \c Value type must be convertible to the \c Value type
402    /// of the algorithm.
403    ///
404    /// \return <tt>(*this)</tt>
405    template <typename LowerMap>
406    CapacityScaling& lowerMap(const LowerMap& map) {
407      _have_lower = true;
408      for (ArcIt a(_graph); a != INVALID; ++a) {
409        _lower[_arc_idf[a]] = map[a];
410        _lower[_arc_idb[a]] = map[a];
411      }
412      return *this;
413    }
414
415    /// \brief Set the upper bounds (capacities) on the arcs.
416    ///
417    /// This function sets the upper bounds (capacities) on the arcs.
418    /// If it is not used before calling \ref run(), the upper bounds
419    /// will be set to \ref INF on all arcs (i.e. the flow value will be
420    /// unbounded from above).
421    ///
422    /// \param map An arc map storing the upper bounds.
423    /// Its \c Value type must be convertible to the \c Value type
424    /// of the algorithm.
425    ///
426    /// \return <tt>(*this)</tt>
427    template<typename UpperMap>
428    CapacityScaling& upperMap(const UpperMap& map) {
429      for (ArcIt a(_graph); a != INVALID; ++a) {
430        _upper[_arc_idf[a]] = map[a];
431      }
432      return *this;
433    }
434
435    /// \brief Set the costs of the arcs.
436    ///
437    /// This function sets the costs of the arcs.
438    /// If it is not used before calling \ref run(), the costs
439    /// will be set to \c 1 on all arcs.
440    ///
441    /// \param map An arc map storing the costs.
442    /// Its \c Value type must be convertible to the \c Cost type
443    /// of the algorithm.
444    ///
445    /// \return <tt>(*this)</tt>
446    template<typename CostMap>
447    CapacityScaling& costMap(const CostMap& map) {
448      for (ArcIt a(_graph); a != INVALID; ++a) {
449        _cost[_arc_idf[a]] =  map[a];
450        _cost[_arc_idb[a]] = -map[a];
451      }
452      return *this;
453    }
454
455    /// \brief Set the supply values of the nodes.
456    ///
457    /// This function sets the supply values of the nodes.
458    /// If neither this function nor \ref stSupply() is used before
459    /// calling \ref run(), the supply of each node will be set to zero.
460    ///
461    /// \param map A node map storing the supply values.
462    /// Its \c Value type must be convertible to the \c Value type
463    /// of the algorithm.
464    ///
465    /// \return <tt>(*this)</tt>
466    template<typename SupplyMap>
467    CapacityScaling& supplyMap(const SupplyMap& map) {
468      for (NodeIt n(_graph); n != INVALID; ++n) {
469        _supply[_node_id[n]] = map[n];
470      }
471      return *this;
472    }
473
474    /// \brief Set single source and target nodes and a supply value.
475    ///
476    /// This function sets a single source node and a single target node
477    /// and the required flow value.
478    /// If neither this function nor \ref supplyMap() is used before
479    /// calling \ref run(), the supply of each node will be set to zero.
480    ///
481    /// Using this function has the same effect as using \ref supplyMap()
482    /// with such a map in which \c k is assigned to \c s, \c -k is
483    /// assigned to \c t and all other nodes have zero supply value.
484    ///
485    /// \param s The source node.
486    /// \param t The target node.
487    /// \param k The required amount of flow from node \c s to node \c t
488    /// (i.e. the supply of \c s and the demand of \c t).
489    ///
490    /// \return <tt>(*this)</tt>
491    CapacityScaling& stSupply(const Node& s, const Node& t, Value k) {
492      for (int i = 0; i != _node_num; ++i) {
493        _supply[i] = 0;
494      }
495      _supply[_node_id[s]] =  k;
496      _supply[_node_id[t]] = -k;
497      return *this;
498    }
499   
500    /// @}
501
502    /// \name Execution control
503    /// The algorithm can be executed using \ref run().
504
505    /// @{
506
507    /// \brief Run the algorithm.
508    ///
509    /// This function runs the algorithm.
510    /// The paramters can be specified using functions \ref lowerMap(),
511    /// \ref upperMap(), \ref costMap(), \ref supplyMap(), \ref stSupply().
512    /// For example,
513    /// \code
514    ///   CapacityScaling<ListDigraph> cs(graph);
515    ///   cs.lowerMap(lower).upperMap(upper).costMap(cost)
516    ///     .supplyMap(sup).run();
517    /// \endcode
518    ///
519    /// This function can be called more than once. All the parameters
520    /// that have been given are kept for the next call, unless
521    /// \ref reset() is called, thus only the modified parameters
522    /// have to be set again. See \ref reset() for examples.
523    /// However, the underlying digraph must not be modified after this
524    /// class have been constructed, since it copies and extends the graph.
525    ///
526    /// \param factor The capacity scaling factor. It must be larger than
527    /// one to use scaling. If it is less or equal to one, then scaling
528    /// will be disabled.
529    ///
530    /// \return \c INFEASIBLE if no feasible flow exists,
531    /// \n \c OPTIMAL if the problem has optimal solution
532    /// (i.e. it is feasible and bounded), and the algorithm has found
533    /// optimal flow and node potentials (primal and dual solutions),
534    /// \n \c UNBOUNDED if the digraph contains an arc of negative cost
535    /// and infinite upper bound. It means that the objective function
536    /// is unbounded on that arc, however, note that it could actually be
537    /// bounded over the feasible flows, but this algroithm cannot handle
538    /// these cases.
539    ///
540    /// \see ProblemType
541    ProblemType run(int factor = 4) {
542      _factor = factor;
543      ProblemType pt = init();
544      if (pt != OPTIMAL) return pt;
545      return start();
546    }
547
548    /// \brief Reset all the parameters that have been given before.
549    ///
550    /// This function resets all the paramaters that have been given
551    /// before using functions \ref lowerMap(), \ref upperMap(),
552    /// \ref costMap(), \ref supplyMap(), \ref stSupply().
553    ///
554    /// It is useful for multiple run() calls. If this function is not
555    /// used, all the parameters given before are kept for the next
556    /// \ref run() call.
557    /// However, the underlying digraph must not be modified after this
558    /// class have been constructed, since it copies and extends the graph.
559    ///
560    /// For example,
561    /// \code
562    ///   CapacityScaling<ListDigraph> cs(graph);
563    ///
564    ///   // First run
565    ///   cs.lowerMap(lower).upperMap(upper).costMap(cost)
566    ///     .supplyMap(sup).run();
567    ///
568    ///   // Run again with modified cost map (reset() is not called,
569    ///   // so only the cost map have to be set again)
570    ///   cost[e] += 100;
571    ///   cs.costMap(cost).run();
572    ///
573    ///   // Run again from scratch using reset()
574    ///   // (the lower bounds will be set to zero on all arcs)
575    ///   cs.reset();
576    ///   cs.upperMap(capacity).costMap(cost)
577    ///     .supplyMap(sup).run();
578    /// \endcode
579    ///
580    /// \return <tt>(*this)</tt>
581    CapacityScaling& reset() {
582      for (int i = 0; i != _node_num; ++i) {
583        _supply[i] = 0;
584      }
585      for (int j = 0; j != _res_arc_num; ++j) {
586        _lower[j] = 0;
587        _upper[j] = INF;
588        _cost[j] = _forward[j] ? 1 : -1;
589      }
590      _have_lower = false;
591      return *this;
592    }
593
594    /// @}
595
596    /// \name Query Functions
597    /// The results of the algorithm can be obtained using these
598    /// functions.\n
599    /// The \ref run() function must be called before using them.
600
601    /// @{
602
603    /// \brief Return the total cost of the found flow.
604    ///
605    /// This function returns the total cost of the found flow.
606    /// Its complexity is O(e).
607    ///
608    /// \note The return type of the function can be specified as a
609    /// template parameter. For example,
610    /// \code
611    ///   cs.totalCost<double>();
612    /// \endcode
613    /// It is useful if the total cost cannot be stored in the \c Cost
614    /// type of the algorithm, which is the default return type of the
615    /// function.
616    ///
617    /// \pre \ref run() must be called before using this function.
618    template <typename Number>
619    Number totalCost() const {
620      Number c = 0;
621      for (ArcIt a(_graph); a != INVALID; ++a) {
622        int i = _arc_idb[a];
623        c += static_cast<Number>(_res_cap[i]) *
624             (-static_cast<Number>(_cost[i]));
625      }
626      return c;
627    }
628
629#ifndef DOXYGEN
630    Cost totalCost() const {
631      return totalCost<Cost>();
632    }
633#endif
634
635    /// \brief Return the flow on the given arc.
636    ///
637    /// This function returns the flow on the given arc.
638    ///
639    /// \pre \ref run() must be called before using this function.
640    Value flow(const Arc& a) const {
641      return _res_cap[_arc_idb[a]];
642    }
643
644    /// \brief Return the flow map (the primal solution).
645    ///
646    /// This function copies the flow value on each arc into the given
647    /// map. The \c Value type of the algorithm must be convertible to
648    /// the \c Value type of the map.
649    ///
650    /// \pre \ref run() must be called before using this function.
651    template <typename FlowMap>
652    void flowMap(FlowMap &map) const {
653      for (ArcIt a(_graph); a != INVALID; ++a) {
654        map.set(a, _res_cap[_arc_idb[a]]);
655      }
656    }
657
658    /// \brief Return the potential (dual value) of the given node.
659    ///
660    /// This function returns the potential (dual value) of the
661    /// given node.
662    ///
663    /// \pre \ref run() must be called before using this function.
664    Cost potential(const Node& n) const {
665      return _pi[_node_id[n]];
666    }
667
668    /// \brief Return the potential map (the dual solution).
669    ///
670    /// This function copies the potential (dual value) of each node
671    /// into the given map.
672    /// The \c Cost type of the algorithm must be convertible to the
673    /// \c Value type of the map.
674    ///
675    /// \pre \ref run() must be called before using this function.
676    template <typename PotentialMap>
677    void potentialMap(PotentialMap &map) const {
678      for (NodeIt n(_graph); n != INVALID; ++n) {
679        map.set(n, _pi[_node_id[n]]);
680      }
681    }
682
683    /// @}
684
685  private:
686
687    // Initialize the algorithm
688    ProblemType init() {
689      if (_node_num <= 1) return INFEASIBLE;
690
691      // Check the sum of supply values
692      _sum_supply = 0;
693      for (int i = 0; i != _root; ++i) {
694        _sum_supply += _supply[i];
695      }
696      if (_sum_supply > 0) return INFEASIBLE;
697     
698      // Initialize vectors
699      for (int i = 0; i != _root; ++i) {
700        _pi[i] = 0;
701        _excess[i] = _supply[i];
702      }
703
704      // Remove non-zero lower bounds
705      const Value MAX = std::numeric_limits<Value>::max();
706      int last_out;
707      if (_have_lower) {
708        for (int i = 0; i != _root; ++i) {
709          last_out = _first_out[i+1];
710          for (int j = _first_out[i]; j != last_out; ++j) {
711            if (_forward[j]) {
712              Value c = _lower[j];
713              if (c >= 0) {
714                _res_cap[j] = _upper[j] < MAX ? _upper[j] - c : INF;
715              } else {
716                _res_cap[j] = _upper[j] < MAX + c ? _upper[j] - c : INF;
717              }
718              _excess[i] -= c;
719              _excess[_target[j]] += c;
720            } else {
721              _res_cap[j] = 0;
722            }
723          }
724        }
725      } else {
726        for (int j = 0; j != _res_arc_num; ++j) {
727          _res_cap[j] = _forward[j] ? _upper[j] : 0;
728        }
729      }
730
731      // Handle negative costs
732      for (int i = 0; i != _root; ++i) {
733        last_out = _first_out[i+1] - 1;
734        for (int j = _first_out[i]; j != last_out; ++j) {
735          Value rc = _res_cap[j];
736          if (_cost[j] < 0 && rc > 0) {
737            if (rc >= MAX) return UNBOUNDED;
738            _excess[i] -= rc;
739            _excess[_target[j]] += rc;
740            _res_cap[j] = 0;
741            _res_cap[_reverse[j]] += rc;
742          }
743        }
744      }
745     
746      // Handle GEQ supply type
747      if (_sum_supply < 0) {
748        _pi[_root] = 0;
749        _excess[_root] = -_sum_supply;
750        for (int a = _first_out[_root]; a != _res_arc_num; ++a) {
751          int ra = _reverse[a];
752          _res_cap[a] = -_sum_supply + 1;
753          _res_cap[ra] = 0;
754          _cost[a] = 0;
755          _cost[ra] = 0;
756        }
757      } else {
758        _pi[_root] = 0;
759        _excess[_root] = 0;
760        for (int a = _first_out[_root]; a != _res_arc_num; ++a) {
761          int ra = _reverse[a];
762          _res_cap[a] = 1;
763          _res_cap[ra] = 0;
764          _cost[a] = 0;
765          _cost[ra] = 0;
766        }
767      }
768
769      // Initialize delta value
770      if (_factor > 1) {
771        // With scaling
772        Value max_sup = 0, max_dem = 0;
773        for (int i = 0; i != _node_num; ++i) {
774          Value ex = _excess[i];
775          if ( ex > max_sup) max_sup =  ex;
776          if (-ex > max_dem) max_dem = -ex;
777        }
778        Value max_cap = 0;
779        for (int j = 0; j != _res_arc_num; ++j) {
780          if (_res_cap[j] > max_cap) max_cap = _res_cap[j];
781        }
782        max_sup = std::min(std::min(max_sup, max_dem), max_cap);
783        for (_delta = 1; 2 * _delta <= max_sup; _delta *= 2) ;
784      } else {
785        // Without scaling
786        _delta = 1;
787      }
788
789      return OPTIMAL;
790    }
791
792    ProblemType start() {
793      // Execute the algorithm
794      ProblemType pt;
795      if (_delta > 1)
796        pt = startWithScaling();
797      else
798        pt = startWithoutScaling();
799
800      // Handle non-zero lower bounds
801      if (_have_lower) {
802        int limit = _first_out[_root];
803        for (int j = 0; j != limit; ++j) {
804          if (!_forward[j]) _res_cap[j] += _lower[j];
805        }
806      }
807
808      // Shift potentials if necessary
809      Cost pr = _pi[_root];
810      if (_sum_supply < 0 || pr > 0) {
811        for (int i = 0; i != _node_num; ++i) {
812          _pi[i] -= pr;
813        }       
814      }
815     
816      return pt;
817    }
818
819    // Execute the capacity scaling algorithm
820    ProblemType startWithScaling() {
821      // Perform capacity scaling phases
822      int s, t;
823      ResidualDijkstra _dijkstra(*this);
824      while (true) {
825        // Saturate all arcs not satisfying the optimality condition
826        int last_out;
827        for (int u = 0; u != _node_num; ++u) {
828          last_out = _sum_supply < 0 ?
829            _first_out[u+1] : _first_out[u+1] - 1;
830          for (int a = _first_out[u]; a != last_out; ++a) {
831            int v = _target[a];
832            Cost c = _cost[a] + _pi[u] - _pi[v];
833            Value rc = _res_cap[a];
834            if (c < 0 && rc >= _delta) {
835              _excess[u] -= rc;
836              _excess[v] += rc;
837              _res_cap[a] = 0;
838              _res_cap[_reverse[a]] += rc;
839            }
840          }
841        }
842
843        // Find excess nodes and deficit nodes
844        _excess_nodes.clear();
845        _deficit_nodes.clear();
846        for (int u = 0; u != _node_num; ++u) {
847          Value ex = _excess[u];
848          if (ex >=  _delta) _excess_nodes.push_back(u);
849          if (ex <= -_delta) _deficit_nodes.push_back(u);
850        }
851        int next_node = 0, next_def_node = 0;
852
853        // Find augmenting shortest paths
854        while (next_node < int(_excess_nodes.size())) {
855          // Check deficit nodes
856          if (_delta > 1) {
857            bool delta_deficit = false;
858            for ( ; next_def_node < int(_deficit_nodes.size());
859                    ++next_def_node ) {
860              if (_excess[_deficit_nodes[next_def_node]] <= -_delta) {
861                delta_deficit = true;
862                break;
863              }
864            }
865            if (!delta_deficit) break;
866          }
867
868          // Run Dijkstra in the residual network
869          s = _excess_nodes[next_node];
870          if ((t = _dijkstra.run(s, _delta)) == -1) {
871            if (_delta > 1) {
872              ++next_node;
873              continue;
874            }
875            return INFEASIBLE;
876          }
877
878          // Augment along a shortest path from s to t
879          Value d = std::min(_excess[s], -_excess[t]);
880          int u = t;
881          int a;
882          if (d > _delta) {
883            while ((a = _pred[u]) != -1) {
884              if (_res_cap[a] < d) d = _res_cap[a];
885              u = _source[a];
886            }
887          }
888          u = t;
889          while ((a = _pred[u]) != -1) {
890            _res_cap[a] -= d;
891            _res_cap[_reverse[a]] += d;
892            u = _source[a];
893          }
894          _excess[s] -= d;
895          _excess[t] += d;
896
897          if (_excess[s] < _delta) ++next_node;
898        }
899
900        if (_delta == 1) break;
901        _delta = _delta <= _factor ? 1 : _delta / _factor;
902      }
903
904      return OPTIMAL;
905    }
906
907    // Execute the successive shortest path algorithm
908    ProblemType startWithoutScaling() {
909      // Find excess nodes
910      _excess_nodes.clear();
911      for (int i = 0; i != _node_num; ++i) {
912        if (_excess[i] > 0) _excess_nodes.push_back(i);
913      }
914      if (_excess_nodes.size() == 0) return OPTIMAL;
915      int next_node = 0;
916
917      // Find shortest paths
918      int s, t;
919      ResidualDijkstra _dijkstra(*this);
920      while ( _excess[_excess_nodes[next_node]] > 0 ||
921              ++next_node < int(_excess_nodes.size()) )
922      {
923        // Run Dijkstra in the residual network
924        s = _excess_nodes[next_node];
925        if ((t = _dijkstra.run(s)) == -1) return INFEASIBLE;
926
927        // Augment along a shortest path from s to t
928        Value d = std::min(_excess[s], -_excess[t]);
929        int u = t;
930        int a;
931        if (d > 1) {
932          while ((a = _pred[u]) != -1) {
933            if (_res_cap[a] < d) d = _res_cap[a];
934            u = _source[a];
935          }
936        }
937        u = t;
938        while ((a = _pred[u]) != -1) {
939          _res_cap[a] -= d;
940          _res_cap[_reverse[a]] += d;
941          u = _source[a];
942        }
943        _excess[s] -= d;
944        _excess[t] += d;
945      }
946
947      return OPTIMAL;
948    }
949
950  }; //class CapacityScaling
951
952  ///@}
953
954} //namespace lemon
955
956#endif //LEMON_CAPACITY_SCALING_H
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