COIN-OR::LEMON - Graph Library

source: lemon-1.2/lemon/capacity_scaling.h @ 813:25804ef35064

Last change on this file since 813:25804ef35064 was 813:25804ef35064, checked in by Peter Kovacs <kpeter@…>, 10 years ago

Add citations to the scaling MCF algorithms (#180, #184)
and improve the doc of their group.

File size: 29.2 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  ///
84  /// \warning Both number types must be signed and all input data must
85  /// be integer.
86  /// \warning This algorithm does not support negative costs for such
87  /// arcs that have infinite upper bound.
88#ifdef DOXYGEN
89  template <typename GR, typename V, typename C, typename TR>
90#else
91  template < typename GR, typename V = int, typename C = V,
92             typename TR = CapacityScalingDefaultTraits<GR, V, C> >
93#endif
94  class CapacityScaling
95  {
96  public:
97
98    /// The type of the digraph
99    typedef typename TR::Digraph Digraph;
100    /// The type of the flow amounts, capacity bounds and supply values
101    typedef typename TR::Value Value;
102    /// The type of the arc costs
103    typedef typename TR::Cost Cost;
104
105    /// The type of the heap used for internal Dijkstra computations
106    typedef typename TR::Heap Heap;
107
108    /// The \ref CapacityScalingDefaultTraits "traits class" of the algorithm
109    typedef TR Traits;
110
111  public:
112
113    /// \brief Problem type constants for the \c run() function.
114    ///
115    /// Enum type containing the problem type constants that can be
116    /// returned by the \ref run() function of the algorithm.
117    enum ProblemType {
118      /// The problem has no feasible solution (flow).
119      INFEASIBLE,
120      /// The problem has optimal solution (i.e. it is feasible and
121      /// bounded), and the algorithm has found optimal flow and node
122      /// potentials (primal and dual solutions).
123      OPTIMAL,
124      /// The digraph contains an arc of negative cost and infinite
125      /// upper bound. It means that the objective function is unbounded
126      /// on that arc, however, note that it could actually be bounded
127      /// over the feasible flows, but this algroithm cannot handle
128      /// these cases.
129      UNBOUNDED
130    };
131 
132  private:
133
134    TEMPLATE_DIGRAPH_TYPEDEFS(GR);
135
136    typedef std::vector<int> IntVector;
137    typedef std::vector<char> BoolVector;
138    typedef std::vector<Value> ValueVector;
139    typedef std::vector<Cost> CostVector;
140
141  private:
142
143    // Data related to the underlying digraph
144    const GR &_graph;
145    int _node_num;
146    int _arc_num;
147    int _res_arc_num;
148    int _root;
149
150    // Parameters of the problem
151    bool _have_lower;
152    Value _sum_supply;
153
154    // Data structures for storing the digraph
155    IntNodeMap _node_id;
156    IntArcMap _arc_idf;
157    IntArcMap _arc_idb;
158    IntVector _first_out;
159    BoolVector _forward;
160    IntVector _source;
161    IntVector _target;
162    IntVector _reverse;
163
164    // Node and arc data
165    ValueVector _lower;
166    ValueVector _upper;
167    CostVector _cost;
168    ValueVector _supply;
169
170    ValueVector _res_cap;
171    CostVector _pi;
172    ValueVector _excess;
173    IntVector _excess_nodes;
174    IntVector _deficit_nodes;
175
176    Value _delta;
177    int _factor;
178    IntVector _pred;
179
180  public:
181 
182    /// \brief Constant for infinite upper bounds (capacities).
183    ///
184    /// Constant for infinite upper bounds (capacities).
185    /// It is \c std::numeric_limits<Value>::infinity() if available,
186    /// \c std::numeric_limits<Value>::max() otherwise.
187    const Value INF;
188
189  private:
190
191    // Special implementation of the Dijkstra algorithm for finding
192    // shortest paths in the residual network of the digraph with
193    // respect to the reduced arc costs and modifying the node
194    // potentials according to the found distance labels.
195    class ResidualDijkstra
196    {
197    private:
198
199      int _node_num;
200      bool _geq;
201      const IntVector &_first_out;
202      const IntVector &_target;
203      const CostVector &_cost;
204      const ValueVector &_res_cap;
205      const ValueVector &_excess;
206      CostVector &_pi;
207      IntVector &_pred;
208     
209      IntVector _proc_nodes;
210      CostVector _dist;
211     
212    public:
213
214      ResidualDijkstra(CapacityScaling& cs) :
215        _node_num(cs._node_num), _geq(cs._sum_supply < 0),
216        _first_out(cs._first_out), _target(cs._target), _cost(cs._cost),
217        _res_cap(cs._res_cap), _excess(cs._excess), _pi(cs._pi),
218        _pred(cs._pred), _dist(cs._node_num)
219      {}
220
221      int run(int s, Value delta = 1) {
222        RangeMap<int> heap_cross_ref(_node_num, Heap::PRE_HEAP);
223        Heap heap(heap_cross_ref);
224        heap.push(s, 0);
225        _pred[s] = -1;
226        _proc_nodes.clear();
227
228        // Process nodes
229        while (!heap.empty() && _excess[heap.top()] > -delta) {
230          int u = heap.top(), v;
231          Cost d = heap.prio() + _pi[u], dn;
232          _dist[u] = heap.prio();
233          _proc_nodes.push_back(u);
234          heap.pop();
235
236          // Traverse outgoing residual arcs
237          int last_out = _geq ? _first_out[u+1] : _first_out[u+1] - 1;
238          for (int a = _first_out[u]; a != last_out; ++a) {
239            if (_res_cap[a] < delta) continue;
240            v = _target[a];
241            switch (heap.state(v)) {
242              case Heap::PRE_HEAP:
243                heap.push(v, d + _cost[a] - _pi[v]);
244                _pred[v] = a;
245                break;
246              case Heap::IN_HEAP:
247                dn = d + _cost[a] - _pi[v];
248                if (dn < heap[v]) {
249                  heap.decrease(v, dn);
250                  _pred[v] = a;
251                }
252                break;
253              case Heap::POST_HEAP:
254                break;
255            }
256          }
257        }
258        if (heap.empty()) return -1;
259
260        // Update potentials of processed nodes
261        int t = heap.top();
262        Cost dt = heap.prio();
263        for (int i = 0; i < int(_proc_nodes.size()); ++i) {
264          _pi[_proc_nodes[i]] += _dist[_proc_nodes[i]] - dt;
265        }
266
267        return t;
268      }
269
270    }; //class ResidualDijkstra
271
272  public:
273
274    /// \name Named Template Parameters
275    /// @{
276
277    template <typename T>
278    struct SetHeapTraits : public Traits {
279      typedef T Heap;
280    };
281
282    /// \brief \ref named-templ-param "Named parameter" for setting
283    /// \c Heap type.
284    ///
285    /// \ref named-templ-param "Named parameter" for setting \c Heap
286    /// type, which is used for internal Dijkstra computations.
287    /// It must conform to the \ref lemon::concepts::Heap "Heap" concept,
288    /// its priority type must be \c Cost and its cross reference type
289    /// must be \ref RangeMap "RangeMap<int>".
290    template <typename T>
291    struct SetHeap
292      : public CapacityScaling<GR, V, C, SetHeapTraits<T> > {
293      typedef  CapacityScaling<GR, V, C, SetHeapTraits<T> > Create;
294    };
295
296    /// @}
297
298  public:
299
300    /// \brief Constructor.
301    ///
302    /// The constructor of the class.
303    ///
304    /// \param graph The digraph the algorithm runs on.
305    CapacityScaling(const GR& graph) :
306      _graph(graph), _node_id(graph), _arc_idf(graph), _arc_idb(graph),
307      INF(std::numeric_limits<Value>::has_infinity ?
308          std::numeric_limits<Value>::infinity() :
309          std::numeric_limits<Value>::max())
310    {
311      // Check the number types
312      LEMON_ASSERT(std::numeric_limits<Value>::is_signed,
313        "The flow type of CapacityScaling must be signed");
314      LEMON_ASSERT(std::numeric_limits<Cost>::is_signed,
315        "The cost type of CapacityScaling must be signed");
316
317      // Resize vectors
318      _node_num = countNodes(_graph);
319      _arc_num = countArcs(_graph);
320      _res_arc_num = 2 * (_arc_num + _node_num);
321      _root = _node_num;
322      ++_node_num;
323
324      _first_out.resize(_node_num + 1);
325      _forward.resize(_res_arc_num);
326      _source.resize(_res_arc_num);
327      _target.resize(_res_arc_num);
328      _reverse.resize(_res_arc_num);
329
330      _lower.resize(_res_arc_num);
331      _upper.resize(_res_arc_num);
332      _cost.resize(_res_arc_num);
333      _supply.resize(_node_num);
334     
335      _res_cap.resize(_res_arc_num);
336      _pi.resize(_node_num);
337      _excess.resize(_node_num);
338      _pred.resize(_node_num);
339
340      // Copy the graph
341      int i = 0, j = 0, k = 2 * _arc_num + _node_num - 1;
342      for (NodeIt n(_graph); n != INVALID; ++n, ++i) {
343        _node_id[n] = i;
344      }
345      i = 0;
346      for (NodeIt n(_graph); n != INVALID; ++n, ++i) {
347        _first_out[i] = j;
348        for (OutArcIt a(_graph, n); a != INVALID; ++a, ++j) {
349          _arc_idf[a] = j;
350          _forward[j] = true;
351          _source[j] = i;
352          _target[j] = _node_id[_graph.runningNode(a)];
353        }
354        for (InArcIt a(_graph, n); a != INVALID; ++a, ++j) {
355          _arc_idb[a] = j;
356          _forward[j] = false;
357          _source[j] = i;
358          _target[j] = _node_id[_graph.runningNode(a)];
359        }
360        _forward[j] = false;
361        _source[j] = i;
362        _target[j] = _root;
363        _reverse[j] = k;
364        _forward[k] = true;
365        _source[k] = _root;
366        _target[k] = i;
367        _reverse[k] = j;
368        ++j; ++k;
369      }
370      _first_out[i] = j;
371      _first_out[_node_num] = k;
372      for (ArcIt a(_graph); a != INVALID; ++a) {
373        int fi = _arc_idf[a];
374        int bi = _arc_idb[a];
375        _reverse[fi] = bi;
376        _reverse[bi] = fi;
377      }
378     
379      // Reset parameters
380      reset();
381    }
382
383    /// \name Parameters
384    /// The parameters of the algorithm can be specified using these
385    /// functions.
386
387    /// @{
388
389    /// \brief Set the lower bounds on the arcs.
390    ///
391    /// This function sets the lower bounds on the arcs.
392    /// If it is not used before calling \ref run(), the lower bounds
393    /// will be set to zero on all arcs.
394    ///
395    /// \param map An arc map storing the lower bounds.
396    /// Its \c Value type must be convertible to the \c Value type
397    /// of the algorithm.
398    ///
399    /// \return <tt>(*this)</tt>
400    template <typename LowerMap>
401    CapacityScaling& lowerMap(const LowerMap& map) {
402      _have_lower = true;
403      for (ArcIt a(_graph); a != INVALID; ++a) {
404        _lower[_arc_idf[a]] = map[a];
405        _lower[_arc_idb[a]] = map[a];
406      }
407      return *this;
408    }
409
410    /// \brief Set the upper bounds (capacities) on the arcs.
411    ///
412    /// This function sets the upper bounds (capacities) on the arcs.
413    /// If it is not used before calling \ref run(), the upper bounds
414    /// will be set to \ref INF on all arcs (i.e. the flow value will be
415    /// unbounded from above).
416    ///
417    /// \param map An arc map storing the upper bounds.
418    /// Its \c Value type must be convertible to the \c Value type
419    /// of the algorithm.
420    ///
421    /// \return <tt>(*this)</tt>
422    template<typename UpperMap>
423    CapacityScaling& upperMap(const UpperMap& map) {
424      for (ArcIt a(_graph); a != INVALID; ++a) {
425        _upper[_arc_idf[a]] = map[a];
426      }
427      return *this;
428    }
429
430    /// \brief Set the costs of the arcs.
431    ///
432    /// This function sets the costs of the arcs.
433    /// If it is not used before calling \ref run(), the costs
434    /// will be set to \c 1 on all arcs.
435    ///
436    /// \param map An arc map storing the costs.
437    /// Its \c Value type must be convertible to the \c Cost type
438    /// of the algorithm.
439    ///
440    /// \return <tt>(*this)</tt>
441    template<typename CostMap>
442    CapacityScaling& costMap(const CostMap& map) {
443      for (ArcIt a(_graph); a != INVALID; ++a) {
444        _cost[_arc_idf[a]] =  map[a];
445        _cost[_arc_idb[a]] = -map[a];
446      }
447      return *this;
448    }
449
450    /// \brief Set the supply values of the nodes.
451    ///
452    /// This function sets the supply values of the nodes.
453    /// If neither this function nor \ref stSupply() is used before
454    /// calling \ref run(), the supply of each node will be set to zero.
455    ///
456    /// \param map A node map storing the supply values.
457    /// Its \c Value type must be convertible to the \c Value type
458    /// of the algorithm.
459    ///
460    /// \return <tt>(*this)</tt>
461    template<typename SupplyMap>
462    CapacityScaling& supplyMap(const SupplyMap& map) {
463      for (NodeIt n(_graph); n != INVALID; ++n) {
464        _supply[_node_id[n]] = map[n];
465      }
466      return *this;
467    }
468
469    /// \brief Set single source and target nodes and a supply value.
470    ///
471    /// This function sets a single source node and a single target node
472    /// and the required flow value.
473    /// If neither this function nor \ref supplyMap() is used before
474    /// calling \ref run(), the supply of each node will be set to zero.
475    ///
476    /// Using this function has the same effect as using \ref supplyMap()
477    /// with such a map in which \c k is assigned to \c s, \c -k is
478    /// assigned to \c t and all other nodes have zero supply value.
479    ///
480    /// \param s The source node.
481    /// \param t The target node.
482    /// \param k The required amount of flow from node \c s to node \c t
483    /// (i.e. the supply of \c s and the demand of \c t).
484    ///
485    /// \return <tt>(*this)</tt>
486    CapacityScaling& stSupply(const Node& s, const Node& t, Value k) {
487      for (int i = 0; i != _node_num; ++i) {
488        _supply[i] = 0;
489      }
490      _supply[_node_id[s]] =  k;
491      _supply[_node_id[t]] = -k;
492      return *this;
493    }
494   
495    /// @}
496
497    /// \name Execution control
498    /// The algorithm can be executed using \ref run().
499
500    /// @{
501
502    /// \brief Run the algorithm.
503    ///
504    /// This function runs the algorithm.
505    /// The paramters can be specified using functions \ref lowerMap(),
506    /// \ref upperMap(), \ref costMap(), \ref supplyMap(), \ref stSupply().
507    /// For example,
508    /// \code
509    ///   CapacityScaling<ListDigraph> cs(graph);
510    ///   cs.lowerMap(lower).upperMap(upper).costMap(cost)
511    ///     .supplyMap(sup).run();
512    /// \endcode
513    ///
514    /// This function can be called more than once. All the parameters
515    /// that have been given are kept for the next call, unless
516    /// \ref reset() is called, thus only the modified parameters
517    /// have to be set again. See \ref reset() for examples.
518    /// However, the underlying digraph must not be modified after this
519    /// class have been constructed, since it copies and extends the graph.
520    ///
521    /// \param factor The capacity scaling factor. It must be larger than
522    /// one to use scaling. If it is less or equal to one, then scaling
523    /// will be disabled.
524    ///
525    /// \return \c INFEASIBLE if no feasible flow exists,
526    /// \n \c OPTIMAL if the problem has optimal solution
527    /// (i.e. it is feasible and bounded), and the algorithm has found
528    /// optimal flow and node potentials (primal and dual solutions),
529    /// \n \c UNBOUNDED if the digraph contains an arc of negative cost
530    /// and infinite upper bound. It means that the objective function
531    /// is unbounded on that arc, however, note that it could actually be
532    /// bounded over the feasible flows, but this algroithm cannot handle
533    /// these cases.
534    ///
535    /// \see ProblemType
536    ProblemType run(int factor = 4) {
537      _factor = factor;
538      ProblemType pt = init();
539      if (pt != OPTIMAL) return pt;
540      return start();
541    }
542
543    /// \brief Reset all the parameters that have been given before.
544    ///
545    /// This function resets all the paramaters that have been given
546    /// before using functions \ref lowerMap(), \ref upperMap(),
547    /// \ref costMap(), \ref supplyMap(), \ref stSupply().
548    ///
549    /// It is useful for multiple run() calls. If this function is not
550    /// used, all the parameters given before are kept for the next
551    /// \ref run() call.
552    /// However, the underlying digraph must not be modified after this
553    /// class have been constructed, since it copies and extends the graph.
554    ///
555    /// For example,
556    /// \code
557    ///   CapacityScaling<ListDigraph> cs(graph);
558    ///
559    ///   // First run
560    ///   cs.lowerMap(lower).upperMap(upper).costMap(cost)
561    ///     .supplyMap(sup).run();
562    ///
563    ///   // Run again with modified cost map (reset() is not called,
564    ///   // so only the cost map have to be set again)
565    ///   cost[e] += 100;
566    ///   cs.costMap(cost).run();
567    ///
568    ///   // Run again from scratch using reset()
569    ///   // (the lower bounds will be set to zero on all arcs)
570    ///   cs.reset();
571    ///   cs.upperMap(capacity).costMap(cost)
572    ///     .supplyMap(sup).run();
573    /// \endcode
574    ///
575    /// \return <tt>(*this)</tt>
576    CapacityScaling& reset() {
577      for (int i = 0; i != _node_num; ++i) {
578        _supply[i] = 0;
579      }
580      for (int j = 0; j != _res_arc_num; ++j) {
581        _lower[j] = 0;
582        _upper[j] = INF;
583        _cost[j] = _forward[j] ? 1 : -1;
584      }
585      _have_lower = false;
586      return *this;
587    }
588
589    /// @}
590
591    /// \name Query Functions
592    /// The results of the algorithm can be obtained using these
593    /// functions.\n
594    /// The \ref run() function must be called before using them.
595
596    /// @{
597
598    /// \brief Return the total cost of the found flow.
599    ///
600    /// This function returns the total cost of the found flow.
601    /// Its complexity is O(e).
602    ///
603    /// \note The return type of the function can be specified as a
604    /// template parameter. For example,
605    /// \code
606    ///   cs.totalCost<double>();
607    /// \endcode
608    /// It is useful if the total cost cannot be stored in the \c Cost
609    /// type of the algorithm, which is the default return type of the
610    /// function.
611    ///
612    /// \pre \ref run() must be called before using this function.
613    template <typename Number>
614    Number totalCost() const {
615      Number c = 0;
616      for (ArcIt a(_graph); a != INVALID; ++a) {
617        int i = _arc_idb[a];
618        c += static_cast<Number>(_res_cap[i]) *
619             (-static_cast<Number>(_cost[i]));
620      }
621      return c;
622    }
623
624#ifndef DOXYGEN
625    Cost totalCost() const {
626      return totalCost<Cost>();
627    }
628#endif
629
630    /// \brief Return the flow on the given arc.
631    ///
632    /// This function returns the flow on the given arc.
633    ///
634    /// \pre \ref run() must be called before using this function.
635    Value flow(const Arc& a) const {
636      return _res_cap[_arc_idb[a]];
637    }
638
639    /// \brief Return the flow map (the primal solution).
640    ///
641    /// This function copies the flow value on each arc into the given
642    /// map. The \c Value type of the algorithm must be convertible to
643    /// the \c Value type of the map.
644    ///
645    /// \pre \ref run() must be called before using this function.
646    template <typename FlowMap>
647    void flowMap(FlowMap &map) const {
648      for (ArcIt a(_graph); a != INVALID; ++a) {
649        map.set(a, _res_cap[_arc_idb[a]]);
650      }
651    }
652
653    /// \brief Return the potential (dual value) of the given node.
654    ///
655    /// This function returns the potential (dual value) of the
656    /// given node.
657    ///
658    /// \pre \ref run() must be called before using this function.
659    Cost potential(const Node& n) const {
660      return _pi[_node_id[n]];
661    }
662
663    /// \brief Return the potential map (the dual solution).
664    ///
665    /// This function copies the potential (dual value) of each node
666    /// into the given map.
667    /// The \c Cost type of the algorithm must be convertible to the
668    /// \c Value type of the map.
669    ///
670    /// \pre \ref run() must be called before using this function.
671    template <typename PotentialMap>
672    void potentialMap(PotentialMap &map) const {
673      for (NodeIt n(_graph); n != INVALID; ++n) {
674        map.set(n, _pi[_node_id[n]]);
675      }
676    }
677
678    /// @}
679
680  private:
681
682    // Initialize the algorithm
683    ProblemType init() {
684      if (_node_num == 0) return INFEASIBLE;
685
686      // Check the sum of supply values
687      _sum_supply = 0;
688      for (int i = 0; i != _root; ++i) {
689        _sum_supply += _supply[i];
690      }
691      if (_sum_supply > 0) return INFEASIBLE;
692     
693      // Initialize vectors
694      for (int i = 0; i != _root; ++i) {
695        _pi[i] = 0;
696        _excess[i] = _supply[i];
697      }
698
699      // Remove non-zero lower bounds
700      const Value MAX = std::numeric_limits<Value>::max();
701      int last_out;
702      if (_have_lower) {
703        for (int i = 0; i != _root; ++i) {
704          last_out = _first_out[i+1];
705          for (int j = _first_out[i]; j != last_out; ++j) {
706            if (_forward[j]) {
707              Value c = _lower[j];
708              if (c >= 0) {
709                _res_cap[j] = _upper[j] < MAX ? _upper[j] - c : INF;
710              } else {
711                _res_cap[j] = _upper[j] < MAX + c ? _upper[j] - c : INF;
712              }
713              _excess[i] -= c;
714              _excess[_target[j]] += c;
715            } else {
716              _res_cap[j] = 0;
717            }
718          }
719        }
720      } else {
721        for (int j = 0; j != _res_arc_num; ++j) {
722          _res_cap[j] = _forward[j] ? _upper[j] : 0;
723        }
724      }
725
726      // Handle negative costs
727      for (int i = 0; i != _root; ++i) {
728        last_out = _first_out[i+1] - 1;
729        for (int j = _first_out[i]; j != last_out; ++j) {
730          Value rc = _res_cap[j];
731          if (_cost[j] < 0 && rc > 0) {
732            if (rc >= MAX) return UNBOUNDED;
733            _excess[i] -= rc;
734            _excess[_target[j]] += rc;
735            _res_cap[j] = 0;
736            _res_cap[_reverse[j]] += rc;
737          }
738        }
739      }
740     
741      // Handle GEQ supply type
742      if (_sum_supply < 0) {
743        _pi[_root] = 0;
744        _excess[_root] = -_sum_supply;
745        for (int a = _first_out[_root]; a != _res_arc_num; ++a) {
746          int ra = _reverse[a];
747          _res_cap[a] = -_sum_supply + 1;
748          _res_cap[ra] = 0;
749          _cost[a] = 0;
750          _cost[ra] = 0;
751        }
752      } else {
753        _pi[_root] = 0;
754        _excess[_root] = 0;
755        for (int a = _first_out[_root]; a != _res_arc_num; ++a) {
756          int ra = _reverse[a];
757          _res_cap[a] = 1;
758          _res_cap[ra] = 0;
759          _cost[a] = 0;
760          _cost[ra] = 0;
761        }
762      }
763
764      // Initialize delta value
765      if (_factor > 1) {
766        // With scaling
767        Value max_sup = 0, max_dem = 0;
768        for (int i = 0; i != _node_num; ++i) {
769          Value ex = _excess[i];
770          if ( ex > max_sup) max_sup =  ex;
771          if (-ex > max_dem) max_dem = -ex;
772        }
773        Value max_cap = 0;
774        for (int j = 0; j != _res_arc_num; ++j) {
775          if (_res_cap[j] > max_cap) max_cap = _res_cap[j];
776        }
777        max_sup = std::min(std::min(max_sup, max_dem), max_cap);
778        for (_delta = 1; 2 * _delta <= max_sup; _delta *= 2) ;
779      } else {
780        // Without scaling
781        _delta = 1;
782      }
783
784      return OPTIMAL;
785    }
786
787    ProblemType start() {
788      // Execute the algorithm
789      ProblemType pt;
790      if (_delta > 1)
791        pt = startWithScaling();
792      else
793        pt = startWithoutScaling();
794
795      // Handle non-zero lower bounds
796      if (_have_lower) {
797        int limit = _first_out[_root];
798        for (int j = 0; j != limit; ++j) {
799          if (!_forward[j]) _res_cap[j] += _lower[j];
800        }
801      }
802
803      // Shift potentials if necessary
804      Cost pr = _pi[_root];
805      if (_sum_supply < 0 || pr > 0) {
806        for (int i = 0; i != _node_num; ++i) {
807          _pi[i] -= pr;
808        }       
809      }
810     
811      return pt;
812    }
813
814    // Execute the capacity scaling algorithm
815    ProblemType startWithScaling() {
816      // Perform capacity scaling phases
817      int s, t;
818      ResidualDijkstra _dijkstra(*this);
819      while (true) {
820        // Saturate all arcs not satisfying the optimality condition
821        int last_out;
822        for (int u = 0; u != _node_num; ++u) {
823          last_out = _sum_supply < 0 ?
824            _first_out[u+1] : _first_out[u+1] - 1;
825          for (int a = _first_out[u]; a != last_out; ++a) {
826            int v = _target[a];
827            Cost c = _cost[a] + _pi[u] - _pi[v];
828            Value rc = _res_cap[a];
829            if (c < 0 && rc >= _delta) {
830              _excess[u] -= rc;
831              _excess[v] += rc;
832              _res_cap[a] = 0;
833              _res_cap[_reverse[a]] += rc;
834            }
835          }
836        }
837
838        // Find excess nodes and deficit nodes
839        _excess_nodes.clear();
840        _deficit_nodes.clear();
841        for (int u = 0; u != _node_num; ++u) {
842          Value ex = _excess[u];
843          if (ex >=  _delta) _excess_nodes.push_back(u);
844          if (ex <= -_delta) _deficit_nodes.push_back(u);
845        }
846        int next_node = 0, next_def_node = 0;
847
848        // Find augmenting shortest paths
849        while (next_node < int(_excess_nodes.size())) {
850          // Check deficit nodes
851          if (_delta > 1) {
852            bool delta_deficit = false;
853            for ( ; next_def_node < int(_deficit_nodes.size());
854                    ++next_def_node ) {
855              if (_excess[_deficit_nodes[next_def_node]] <= -_delta) {
856                delta_deficit = true;
857                break;
858              }
859            }
860            if (!delta_deficit) break;
861          }
862
863          // Run Dijkstra in the residual network
864          s = _excess_nodes[next_node];
865          if ((t = _dijkstra.run(s, _delta)) == -1) {
866            if (_delta > 1) {
867              ++next_node;
868              continue;
869            }
870            return INFEASIBLE;
871          }
872
873          // Augment along a shortest path from s to t
874          Value d = std::min(_excess[s], -_excess[t]);
875          int u = t;
876          int a;
877          if (d > _delta) {
878            while ((a = _pred[u]) != -1) {
879              if (_res_cap[a] < d) d = _res_cap[a];
880              u = _source[a];
881            }
882          }
883          u = t;
884          while ((a = _pred[u]) != -1) {
885            _res_cap[a] -= d;
886            _res_cap[_reverse[a]] += d;
887            u = _source[a];
888          }
889          _excess[s] -= d;
890          _excess[t] += d;
891
892          if (_excess[s] < _delta) ++next_node;
893        }
894
895        if (_delta == 1) break;
896        _delta = _delta <= _factor ? 1 : _delta / _factor;
897      }
898
899      return OPTIMAL;
900    }
901
902    // Execute the successive shortest path algorithm
903    ProblemType startWithoutScaling() {
904      // Find excess nodes
905      _excess_nodes.clear();
906      for (int i = 0; i != _node_num; ++i) {
907        if (_excess[i] > 0) _excess_nodes.push_back(i);
908      }
909      if (_excess_nodes.size() == 0) return OPTIMAL;
910      int next_node = 0;
911
912      // Find shortest paths
913      int s, t;
914      ResidualDijkstra _dijkstra(*this);
915      while ( _excess[_excess_nodes[next_node]] > 0 ||
916              ++next_node < int(_excess_nodes.size()) )
917      {
918        // Run Dijkstra in the residual network
919        s = _excess_nodes[next_node];
920        if ((t = _dijkstra.run(s)) == -1) return INFEASIBLE;
921
922        // Augment along a shortest path from s to t
923        Value d = std::min(_excess[s], -_excess[t]);
924        int u = t;
925        int a;
926        if (d > 1) {
927          while ((a = _pred[u]) != -1) {
928            if (_res_cap[a] < d) d = _res_cap[a];
929            u = _source[a];
930          }
931        }
932        u = t;
933        while ((a = _pred[u]) != -1) {
934          _res_cap[a] -= d;
935          _res_cap[_reverse[a]] += d;
936          u = _source[a];
937        }
938        _excess[s] -= d;
939        _excess[t] += d;
940      }
941
942      return OPTIMAL;
943    }
944
945  }; //class CapacityScaling
946
947  ///@}
948
949} //namespace lemon
950
951#endif //LEMON_CAPACITY_SCALING_H
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