COIN-OR::LEMON - Graph Library

source: lemon/lemon/capacity_scaling.h @ 1270:dceba191c00d

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1/* -*- mode: C++; indent-tabs-mode: nil; -*-
2 *
3 * This file is a part of LEMON, a generic C++ optimization library.
4 *
5 * Copyright (C) 2003-2013
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" \cite amo93networkflows,
70  /// \cite edmondskarp72theoretical. It is an efficient dual
71  /// solution method, which runs in polynomial time
72  /// \f$O(m\log U (n+m)\log n)\f$, where <i>U</i> denotes the maximum
73  /// of node supply and arc capacity values.
74  ///
75  /// This algorithm is typically slower than \ref CostScaling and
76  /// \ref NetworkSimplex, but in special cases, it can be more
77  /// efficient than them.
78  /// (For more information, see \ref min_cost_flow_algs "the module page".)
79  ///
80  /// Most of the parameters of the problem (except for the digraph)
81  /// can be given using separate functions, and the algorithm can be
82  /// executed using the \ref run() function. If some parameters are not
83  /// specified, then default values will be used.
84  ///
85  /// \tparam GR The digraph type the algorithm runs on.
86  /// \tparam V The number type used for flow amounts, capacity bounds
87  /// and supply values in the algorithm. By default, it is \c int.
88  /// \tparam C The number type used for costs and potentials in the
89  /// algorithm. By default, it is the same as \c V.
90  /// \tparam TR The traits class that defines various types used by the
91  /// algorithm. By default, it is \ref CapacityScalingDefaultTraits
92  /// "CapacityScalingDefaultTraits<GR, V, C>".
93  /// In most cases, this parameter should not be set directly,
94  /// consider to use the named template parameters instead.
95  ///
96  /// \warning Both \c V and \c C must be signed number types.
97  /// \warning Capacity bounds and supply values must be integer, but
98  /// arc costs can be arbitrary real numbers.
99  /// \warning This algorithm does not support negative costs for
100  /// arcs having infinite upper bound.
101#ifdef DOXYGEN
102  template <typename GR, typename V, typename C, typename TR>
103#else
104  template < typename GR, typename V = int, typename C = V,
105             typename TR = CapacityScalingDefaultTraits<GR, V, C> >
106#endif
107  class CapacityScaling
108  {
109  public:
110
111    /// The type of the digraph
112    typedef typename TR::Digraph Digraph;
113    /// The type of the flow amounts, capacity bounds and supply values
114    typedef typename TR::Value Value;
115    /// The type of the arc costs
116    typedef typename TR::Cost Cost;
117
118    /// The type of the heap used for internal Dijkstra computations
119    typedef typename TR::Heap Heap;
120
121    /// \brief The \ref lemon::CapacityScalingDefaultTraits "traits class"
122    /// of the algorithm
123    typedef TR Traits;
124
125  public:
126
127    /// \brief Problem type constants for the \c run() function.
128    ///
129    /// Enum type containing the problem type constants that can be
130    /// returned by the \ref run() function of the algorithm.
131    enum ProblemType {
132      /// The problem has no feasible solution (flow).
133      INFEASIBLE,
134      /// The problem has optimal solution (i.e. it is feasible and
135      /// bounded), and the algorithm has found optimal flow and node
136      /// potentials (primal and dual solutions).
137      OPTIMAL,
138      /// The digraph contains an arc of negative cost and infinite
139      /// upper bound. It means that the objective function is unbounded
140      /// on that arc, however, note that it could actually be bounded
141      /// over the feasible flows, but this algroithm cannot handle
142      /// these cases.
143      UNBOUNDED
144    };
145
146  private:
147
148    TEMPLATE_DIGRAPH_TYPEDEFS(GR);
149
150    typedef std::vector<int> IntVector;
151    typedef std::vector<Value> ValueVector;
152    typedef std::vector<Cost> CostVector;
153    typedef std::vector<char> BoolVector;
154    // Note: vector<char> is used instead of vector<bool> for efficiency reasons
155
156  private:
157
158    // Data related to the underlying digraph
159    const GR &_graph;
160    int _node_num;
161    int _arc_num;
162    int _res_arc_num;
163    int _root;
164
165    // Parameters of the problem
166    bool _have_lower;
167    Value _sum_supply;
168
169    // Data structures for storing the digraph
170    IntNodeMap _node_id;
171    IntArcMap _arc_idf;
172    IntArcMap _arc_idb;
173    IntVector _first_out;
174    BoolVector _forward;
175    IntVector _source;
176    IntVector _target;
177    IntVector _reverse;
178
179    // Node and arc data
180    ValueVector _lower;
181    ValueVector _upper;
182    CostVector _cost;
183    ValueVector _supply;
184
185    ValueVector _res_cap;
186    CostVector _pi;
187    ValueVector _excess;
188    IntVector _excess_nodes;
189    IntVector _deficit_nodes;
190
191    Value _delta;
192    int _factor;
193    IntVector _pred;
194
195  public:
196
197    /// \brief Constant for infinite upper bounds (capacities).
198    ///
199    /// Constant for infinite upper bounds (capacities).
200    /// It is \c std::numeric_limits<Value>::infinity() if available,
201    /// \c std::numeric_limits<Value>::max() otherwise.
202    const Value INF;
203
204  private:
205
206    // Special implementation of the Dijkstra algorithm for finding
207    // shortest paths in the residual network of the digraph with
208    // respect to the reduced arc costs and modifying the node
209    // potentials according to the found distance labels.
210    class ResidualDijkstra
211    {
212    private:
213
214      int _node_num;
215      bool _geq;
216      const IntVector &_first_out;
217      const IntVector &_target;
218      const CostVector &_cost;
219      const ValueVector &_res_cap;
220      const ValueVector &_excess;
221      CostVector &_pi;
222      IntVector &_pred;
223
224      IntVector _proc_nodes;
225      CostVector _dist;
226
227    public:
228
229      ResidualDijkstra(CapacityScaling& cs) :
230        _node_num(cs._node_num), _geq(cs._sum_supply < 0),
231        _first_out(cs._first_out), _target(cs._target), _cost(cs._cost),
232        _res_cap(cs._res_cap), _excess(cs._excess), _pi(cs._pi),
233        _pred(cs._pred), _dist(cs._node_num)
234      {}
235
236      int run(int s, Value delta = 1) {
237        RangeMap<int> heap_cross_ref(_node_num, Heap::PRE_HEAP);
238        Heap heap(heap_cross_ref);
239        heap.push(s, 0);
240        _pred[s] = -1;
241        _proc_nodes.clear();
242
243        // Process nodes
244        while (!heap.empty() && _excess[heap.top()] > -delta) {
245          int u = heap.top(), v;
246          Cost d = heap.prio() + _pi[u], dn;
247          _dist[u] = heap.prio();
248          _proc_nodes.push_back(u);
249          heap.pop();
250
251          // Traverse outgoing residual arcs
252          int last_out = _geq ? _first_out[u+1] : _first_out[u+1] - 1;
253          for (int a = _first_out[u]; a != last_out; ++a) {
254            if (_res_cap[a] < delta) continue;
255            v = _target[a];
256            switch (heap.state(v)) {
257              case Heap::PRE_HEAP:
258                heap.push(v, d + _cost[a] - _pi[v]);
259                _pred[v] = a;
260                break;
261              case Heap::IN_HEAP:
262                dn = d + _cost[a] - _pi[v];
263                if (dn < heap[v]) {
264                  heap.decrease(v, dn);
265                  _pred[v] = a;
266                }
267                break;
268              case Heap::POST_HEAP:
269                break;
270            }
271          }
272        }
273        if (heap.empty()) return -1;
274
275        // Update potentials of processed nodes
276        int t = heap.top();
277        Cost dt = heap.prio();
278        for (int i = 0; i < int(_proc_nodes.size()); ++i) {
279          _pi[_proc_nodes[i]] += _dist[_proc_nodes[i]] - dt;
280        }
281
282        return t;
283      }
284
285    }; //class ResidualDijkstra
286
287  public:
288
289    /// \name Named Template Parameters
290    /// @{
291
292    template <typename T>
293    struct SetHeapTraits : public Traits {
294      typedef T Heap;
295    };
296
297    /// \brief \ref named-templ-param "Named parameter" for setting
298    /// \c Heap type.
299    ///
300    /// \ref named-templ-param "Named parameter" for setting \c Heap
301    /// type, which is used for internal Dijkstra computations.
302    /// It must conform to the \ref lemon::concepts::Heap "Heap" concept,
303    /// its priority type must be \c Cost and its cross reference type
304    /// must be \ref RangeMap "RangeMap<int>".
305    template <typename T>
306    struct SetHeap
307      : public CapacityScaling<GR, V, C, SetHeapTraits<T> > {
308      typedef  CapacityScaling<GR, V, C, SetHeapTraits<T> > Create;
309    };
310
311    /// @}
312
313  protected:
314
315    CapacityScaling() {}
316
317  public:
318
319    /// \brief Constructor.
320    ///
321    /// The constructor of the class.
322    ///
323    /// \param graph The digraph the algorithm runs on.
324    CapacityScaling(const GR& graph) :
325      _graph(graph), _node_id(graph), _arc_idf(graph), _arc_idb(graph),
326      INF(std::numeric_limits<Value>::has_infinity ?
327          std::numeric_limits<Value>::infinity() :
328          std::numeric_limits<Value>::max())
329    {
330      // Check the number types
331      LEMON_ASSERT(std::numeric_limits<Value>::is_signed,
332        "The flow type of CapacityScaling must be signed");
333      LEMON_ASSERT(std::numeric_limits<Cost>::is_signed,
334        "The cost type of CapacityScaling must be signed");
335
336      // Reset data structures
337      reset();
338    }
339
340    /// \name Parameters
341    /// The parameters of the algorithm can be specified using these
342    /// functions.
343
344    /// @{
345
346    /// \brief Set the lower bounds on the arcs.
347    ///
348    /// This function sets the lower bounds on the arcs.
349    /// If it is not used before calling \ref run(), the lower bounds
350    /// will be set to zero on all arcs.
351    ///
352    /// \param map An arc map storing the lower bounds.
353    /// Its \c Value type must be convertible to the \c Value type
354    /// of the algorithm.
355    ///
356    /// \return <tt>(*this)</tt>
357    template <typename LowerMap>
358    CapacityScaling& lowerMap(const LowerMap& map) {
359      _have_lower = true;
360      for (ArcIt a(_graph); a != INVALID; ++a) {
361        _lower[_arc_idf[a]] = map[a];
362        _lower[_arc_idb[a]] = map[a];
363      }
364      return *this;
365    }
366
367    /// \brief Set the upper bounds (capacities) on the arcs.
368    ///
369    /// This function sets the upper bounds (capacities) on the arcs.
370    /// If it is not used before calling \ref run(), the upper bounds
371    /// will be set to \ref INF on all arcs (i.e. the flow value will be
372    /// unbounded from above).
373    ///
374    /// \param map An arc map storing the upper bounds.
375    /// Its \c Value type must be convertible to the \c Value type
376    /// of the algorithm.
377    ///
378    /// \return <tt>(*this)</tt>
379    template<typename UpperMap>
380    CapacityScaling& upperMap(const UpperMap& map) {
381      for (ArcIt a(_graph); a != INVALID; ++a) {
382        _upper[_arc_idf[a]] = map[a];
383      }
384      return *this;
385    }
386
387    /// \brief Set the costs of the arcs.
388    ///
389    /// This function sets the costs of the arcs.
390    /// If it is not used before calling \ref run(), the costs
391    /// will be set to \c 1 on all arcs.
392    ///
393    /// \param map An arc map storing the costs.
394    /// Its \c Value type must be convertible to the \c Cost type
395    /// of the algorithm.
396    ///
397    /// \return <tt>(*this)</tt>
398    template<typename CostMap>
399    CapacityScaling& costMap(const CostMap& map) {
400      for (ArcIt a(_graph); a != INVALID; ++a) {
401        _cost[_arc_idf[a]] =  map[a];
402        _cost[_arc_idb[a]] = -map[a];
403      }
404      return *this;
405    }
406
407    /// \brief Set the supply values of the nodes.
408    ///
409    /// This function sets the supply values of the nodes.
410    /// If neither this function nor \ref stSupply() is used before
411    /// calling \ref run(), the supply of each node will be set to zero.
412    ///
413    /// \param map A node map storing the supply values.
414    /// Its \c Value type must be convertible to the \c Value type
415    /// of the algorithm.
416    ///
417    /// \return <tt>(*this)</tt>
418    template<typename SupplyMap>
419    CapacityScaling& supplyMap(const SupplyMap& map) {
420      for (NodeIt n(_graph); n != INVALID; ++n) {
421        _supply[_node_id[n]] = map[n];
422      }
423      return *this;
424    }
425
426    /// \brief Set single source and target nodes and a supply value.
427    ///
428    /// This function sets a single source node and a single target node
429    /// and the required flow value.
430    /// If neither this function nor \ref supplyMap() is used before
431    /// calling \ref run(), the supply of each node will be set to zero.
432    ///
433    /// Using this function has the same effect as using \ref supplyMap()
434    /// with a map in which \c k is assigned to \c s, \c -k is
435    /// assigned to \c t and all other nodes have zero supply value.
436    ///
437    /// \param s The source node.
438    /// \param t The target node.
439    /// \param k The required amount of flow from node \c s to node \c t
440    /// (i.e. the supply of \c s and the demand of \c t).
441    ///
442    /// \return <tt>(*this)</tt>
443    CapacityScaling& stSupply(const Node& s, const Node& t, Value k) {
444      for (int i = 0; i != _node_num; ++i) {
445        _supply[i] = 0;
446      }
447      _supply[_node_id[s]] =  k;
448      _supply[_node_id[t]] = -k;
449      return *this;
450    }
451
452    /// @}
453
454    /// \name Execution control
455    /// The algorithm can be executed using \ref run().
456
457    /// @{
458
459    /// \brief Run the algorithm.
460    ///
461    /// This function runs the algorithm.
462    /// The paramters can be specified using functions \ref lowerMap(),
463    /// \ref upperMap(), \ref costMap(), \ref supplyMap(), \ref stSupply().
464    /// For example,
465    /// \code
466    ///   CapacityScaling<ListDigraph> cs(graph);
467    ///   cs.lowerMap(lower).upperMap(upper).costMap(cost)
468    ///     .supplyMap(sup).run();
469    /// \endcode
470    ///
471    /// This function can be called more than once. All the given parameters
472    /// are kept for the next call, unless \ref resetParams() or \ref reset()
473    /// is used, thus only the modified parameters have to be set again.
474    /// If the underlying digraph was also modified after the construction
475    /// of the class (or the last \ref reset() call), then the \ref reset()
476    /// function must be called.
477    ///
478    /// \param factor The capacity scaling factor. It must be larger than
479    /// one to use scaling. If it is less or equal to one, then scaling
480    /// will be disabled.
481    ///
482    /// \return \c INFEASIBLE if no feasible flow exists,
483    /// \n \c OPTIMAL if the problem has optimal solution
484    /// (i.e. it is feasible and bounded), and the algorithm has found
485    /// optimal flow and node potentials (primal and dual solutions),
486    /// \n \c UNBOUNDED if the digraph contains an arc of negative cost
487    /// and infinite upper bound. It means that the objective function
488    /// is unbounded on that arc, however, note that it could actually be
489    /// bounded over the feasible flows, but this algroithm cannot handle
490    /// these cases.
491    ///
492    /// \see ProblemType
493    /// \see resetParams(), reset()
494    ProblemType run(int factor = 4) {
495      _factor = factor;
496      ProblemType pt = init();
497      if (pt != OPTIMAL) return pt;
498      return start();
499    }
500
501    /// \brief Reset all the parameters that have been given before.
502    ///
503    /// This function resets all the paramaters that have been given
504    /// before using functions \ref lowerMap(), \ref upperMap(),
505    /// \ref costMap(), \ref supplyMap(), \ref stSupply().
506    ///
507    /// It is useful for multiple \ref run() calls. Basically, all the given
508    /// parameters are kept for the next \ref run() call, unless
509    /// \ref resetParams() or \ref reset() is used.
510    /// If the underlying digraph was also modified after the construction
511    /// of the class or the last \ref reset() call, then the \ref reset()
512    /// function must be used, otherwise \ref resetParams() is sufficient.
513    ///
514    /// For example,
515    /// \code
516    ///   CapacityScaling<ListDigraph> cs(graph);
517    ///
518    ///   // First run
519    ///   cs.lowerMap(lower).upperMap(upper).costMap(cost)
520    ///     .supplyMap(sup).run();
521    ///
522    ///   // Run again with modified cost map (resetParams() is not called,
523    ///   // so only the cost map have to be set again)
524    ///   cost[e] += 100;
525    ///   cs.costMap(cost).run();
526    ///
527    ///   // Run again from scratch using resetParams()
528    ///   // (the lower bounds will be set to zero on all arcs)
529    ///   cs.resetParams();
530    ///   cs.upperMap(capacity).costMap(cost)
531    ///     .supplyMap(sup).run();
532    /// \endcode
533    ///
534    /// \return <tt>(*this)</tt>
535    ///
536    /// \see reset(), run()
537    CapacityScaling& resetParams() {
538      for (int i = 0; i != _node_num; ++i) {
539        _supply[i] = 0;
540      }
541      for (int j = 0; j != _res_arc_num; ++j) {
542        _lower[j] = 0;
543        _upper[j] = INF;
544        _cost[j] = _forward[j] ? 1 : -1;
545      }
546      _have_lower = false;
547      return *this;
548    }
549
550    /// \brief Reset the internal data structures and all the parameters
551    /// that have been given before.
552    ///
553    /// This function resets the internal data structures and all the
554    /// paramaters that have been given before using functions \ref lowerMap(),
555    /// \ref upperMap(), \ref costMap(), \ref supplyMap(), \ref stSupply().
556    ///
557    /// It is useful for multiple \ref run() calls. Basically, all the given
558    /// parameters are kept for the next \ref run() call, unless
559    /// \ref resetParams() or \ref reset() is used.
560    /// If the underlying digraph was also modified after the construction
561    /// of the class or the last \ref reset() call, then the \ref reset()
562    /// function must be used, otherwise \ref resetParams() is sufficient.
563    ///
564    /// See \ref resetParams() for examples.
565    ///
566    /// \return <tt>(*this)</tt>
567    ///
568    /// \see resetParams(), run()
569    CapacityScaling& reset() {
570      // Resize vectors
571      _node_num = countNodes(_graph);
572      _arc_num = countArcs(_graph);
573      _res_arc_num = 2 * (_arc_num + _node_num);
574      _root = _node_num;
575      ++_node_num;
576
577      _first_out.resize(_node_num + 1);
578      _forward.resize(_res_arc_num);
579      _source.resize(_res_arc_num);
580      _target.resize(_res_arc_num);
581      _reverse.resize(_res_arc_num);
582
583      _lower.resize(_res_arc_num);
584      _upper.resize(_res_arc_num);
585      _cost.resize(_res_arc_num);
586      _supply.resize(_node_num);
587
588      _res_cap.resize(_res_arc_num);
589      _pi.resize(_node_num);
590      _excess.resize(_node_num);
591      _pred.resize(_node_num);
592
593      // Copy the graph
594      int i = 0, j = 0, k = 2 * _arc_num + _node_num - 1;
595      for (NodeIt n(_graph); n != INVALID; ++n, ++i) {
596        _node_id[n] = i;
597      }
598      i = 0;
599      for (NodeIt n(_graph); n != INVALID; ++n, ++i) {
600        _first_out[i] = j;
601        for (OutArcIt a(_graph, n); a != INVALID; ++a, ++j) {
602          _arc_idf[a] = j;
603          _forward[j] = true;
604          _source[j] = i;
605          _target[j] = _node_id[_graph.runningNode(a)];
606        }
607        for (InArcIt a(_graph, n); a != INVALID; ++a, ++j) {
608          _arc_idb[a] = j;
609          _forward[j] = false;
610          _source[j] = i;
611          _target[j] = _node_id[_graph.runningNode(a)];
612        }
613        _forward[j] = false;
614        _source[j] = i;
615        _target[j] = _root;
616        _reverse[j] = k;
617        _forward[k] = true;
618        _source[k] = _root;
619        _target[k] = i;
620        _reverse[k] = j;
621        ++j; ++k;
622      }
623      _first_out[i] = j;
624      _first_out[_node_num] = k;
625      for (ArcIt a(_graph); a != INVALID; ++a) {
626        int fi = _arc_idf[a];
627        int bi = _arc_idb[a];
628        _reverse[fi] = bi;
629        _reverse[bi] = fi;
630      }
631
632      // Reset parameters
633      resetParams();
634      return *this;
635    }
636
637    /// @}
638
639    /// \name Query Functions
640    /// The results of the algorithm can be obtained using these
641    /// functions.\n
642    /// The \ref run() function must be called before using them.
643
644    /// @{
645
646    /// \brief Return the total cost of the found flow.
647    ///
648    /// This function returns the total cost of the found flow.
649    /// Its complexity is O(m).
650    ///
651    /// \note The return type of the function can be specified as a
652    /// template parameter. For example,
653    /// \code
654    ///   cs.totalCost<double>();
655    /// \endcode
656    /// It is useful if the total cost cannot be stored in the \c Cost
657    /// type of the algorithm, which is the default return type of the
658    /// function.
659    ///
660    /// \pre \ref run() must be called before using this function.
661    template <typename Number>
662    Number totalCost() const {
663      Number c = 0;
664      for (ArcIt a(_graph); a != INVALID; ++a) {
665        int i = _arc_idb[a];
666        c += static_cast<Number>(_res_cap[i]) *
667             (-static_cast<Number>(_cost[i]));
668      }
669      return c;
670    }
671
672#ifndef DOXYGEN
673    Cost totalCost() const {
674      return totalCost<Cost>();
675    }
676#endif
677
678    /// \brief Return the flow on the given arc.
679    ///
680    /// This function returns the flow on the given arc.
681    ///
682    /// \pre \ref run() must be called before using this function.
683    Value flow(const Arc& a) const {
684      return _res_cap[_arc_idb[a]];
685    }
686
687    /// \brief Copy the flow values (the primal solution) into the
688    /// given map.
689    ///
690    /// This function copies the flow value on each arc into the given
691    /// map. The \c Value type of the algorithm must be convertible to
692    /// the \c Value type of the map.
693    ///
694    /// \pre \ref run() must be called before using this function.
695    template <typename FlowMap>
696    void flowMap(FlowMap &map) const {
697      for (ArcIt a(_graph); a != INVALID; ++a) {
698        map.set(a, _res_cap[_arc_idb[a]]);
699      }
700    }
701
702    /// \brief Return the potential (dual value) of the given node.
703    ///
704    /// This function returns the potential (dual value) of the
705    /// given node.
706    ///
707    /// \pre \ref run() must be called before using this function.
708    Cost potential(const Node& n) const {
709      return _pi[_node_id[n]];
710    }
711
712    /// \brief Copy the potential values (the dual solution) into the
713    /// given map.
714    ///
715    /// This function copies the potential (dual value) of each node
716    /// into the given map.
717    /// The \c Cost type of the algorithm must be convertible to the
718    /// \c Value type of the map.
719    ///
720    /// \pre \ref run() must be called before using this function.
721    template <typename PotentialMap>
722    void potentialMap(PotentialMap &map) const {
723      for (NodeIt n(_graph); n != INVALID; ++n) {
724        map.set(n, _pi[_node_id[n]]);
725      }
726    }
727
728    /// @}
729
730  private:
731
732    // Initialize the algorithm
733    ProblemType init() {
734      if (_node_num <= 1) return INFEASIBLE;
735
736      // Check the sum of supply values
737      _sum_supply = 0;
738      for (int i = 0; i != _root; ++i) {
739        _sum_supply += _supply[i];
740      }
741      if (_sum_supply > 0) return INFEASIBLE;
742
743      // Check lower and upper bounds
744      LEMON_DEBUG(checkBoundMaps(),
745          "Upper bounds must be greater or equal to the lower bounds");
746
747
748      // Initialize vectors
749      for (int i = 0; i != _root; ++i) {
750        _pi[i] = 0;
751        _excess[i] = _supply[i];
752      }
753
754      // Remove non-zero lower bounds
755      const Value MAX = std::numeric_limits<Value>::max();
756      int last_out;
757      if (_have_lower) {
758        for (int i = 0; i != _root; ++i) {
759          last_out = _first_out[i+1];
760          for (int j = _first_out[i]; j != last_out; ++j) {
761            if (_forward[j]) {
762              Value c = _lower[j];
763              if (c >= 0) {
764                _res_cap[j] = _upper[j] < MAX ? _upper[j] - c : INF;
765              } else {
766                _res_cap[j] = _upper[j] < MAX + c ? _upper[j] - c : INF;
767              }
768              _excess[i] -= c;
769              _excess[_target[j]] += c;
770            } else {
771              _res_cap[j] = 0;
772            }
773          }
774        }
775      } else {
776        for (int j = 0; j != _res_arc_num; ++j) {
777          _res_cap[j] = _forward[j] ? _upper[j] : 0;
778        }
779      }
780
781      // Handle negative costs
782      for (int i = 0; i != _root; ++i) {
783        last_out = _first_out[i+1] - 1;
784        for (int j = _first_out[i]; j != last_out; ++j) {
785          Value rc = _res_cap[j];
786          if (_cost[j] < 0 && rc > 0) {
787            if (rc >= MAX) return UNBOUNDED;
788            _excess[i] -= rc;
789            _excess[_target[j]] += rc;
790            _res_cap[j] = 0;
791            _res_cap[_reverse[j]] += rc;
792          }
793        }
794      }
795
796      // Handle GEQ supply type
797      if (_sum_supply < 0) {
798        _pi[_root] = 0;
799        _excess[_root] = -_sum_supply;
800        for (int a = _first_out[_root]; a != _res_arc_num; ++a) {
801          int ra = _reverse[a];
802          _res_cap[a] = -_sum_supply + 1;
803          _res_cap[ra] = 0;
804          _cost[a] = 0;
805          _cost[ra] = 0;
806        }
807      } else {
808        _pi[_root] = 0;
809        _excess[_root] = 0;
810        for (int a = _first_out[_root]; a != _res_arc_num; ++a) {
811          int ra = _reverse[a];
812          _res_cap[a] = 1;
813          _res_cap[ra] = 0;
814          _cost[a] = 0;
815          _cost[ra] = 0;
816        }
817      }
818
819      // Initialize delta value
820      if (_factor > 1) {
821        // With scaling
822        Value max_sup = 0, max_dem = 0, max_cap = 0;
823        for (int i = 0; i != _root; ++i) {
824          Value ex = _excess[i];
825          if ( ex > max_sup) max_sup =  ex;
826          if (-ex > max_dem) max_dem = -ex;
827          int last_out = _first_out[i+1] - 1;
828          for (int j = _first_out[i]; j != last_out; ++j) {
829            if (_res_cap[j] > max_cap) max_cap = _res_cap[j];
830          }
831        }
832        max_sup = std::min(std::min(max_sup, max_dem), max_cap);
833        for (_delta = 1; 2 * _delta <= max_sup; _delta *= 2) ;
834      } else {
835        // Without scaling
836        _delta = 1;
837      }
838
839      return OPTIMAL;
840    }
841
842    // Check if the upper bound is greater or equal to the lower bound
843    // on each arc.
844    bool checkBoundMaps() {
845      for (int j = 0; j != _res_arc_num; ++j) {
846        if (_upper[j] < _lower[j]) return false;
847      }
848      return true;
849    }
850
851    ProblemType start() {
852      // Execute the algorithm
853      ProblemType pt;
854      if (_delta > 1)
855        pt = startWithScaling();
856      else
857        pt = startWithoutScaling();
858
859      // Handle non-zero lower bounds
860      if (_have_lower) {
861        int limit = _first_out[_root];
862        for (int j = 0; j != limit; ++j) {
863          if (!_forward[j]) _res_cap[j] += _lower[j];
864        }
865      }
866
867      // Shift potentials if necessary
868      Cost pr = _pi[_root];
869      if (_sum_supply < 0 || pr > 0) {
870        for (int i = 0; i != _node_num; ++i) {
871          _pi[i] -= pr;
872        }
873      }
874
875      return pt;
876    }
877
878    // Execute the capacity scaling algorithm
879    ProblemType startWithScaling() {
880      // Perform capacity scaling phases
881      int s, t;
882      ResidualDijkstra _dijkstra(*this);
883      while (true) {
884        // Saturate all arcs not satisfying the optimality condition
885        int last_out;
886        for (int u = 0; u != _node_num; ++u) {
887          last_out = _sum_supply < 0 ?
888            _first_out[u+1] : _first_out[u+1] - 1;
889          for (int a = _first_out[u]; a != last_out; ++a) {
890            int v = _target[a];
891            Cost c = _cost[a] + _pi[u] - _pi[v];
892            Value rc = _res_cap[a];
893            if (c < 0 && rc >= _delta) {
894              _excess[u] -= rc;
895              _excess[v] += rc;
896              _res_cap[a] = 0;
897              _res_cap[_reverse[a]] += rc;
898            }
899          }
900        }
901
902        // Find excess nodes and deficit nodes
903        _excess_nodes.clear();
904        _deficit_nodes.clear();
905        for (int u = 0; u != _node_num; ++u) {
906          Value ex = _excess[u];
907          if (ex >=  _delta) _excess_nodes.push_back(u);
908          if (ex <= -_delta) _deficit_nodes.push_back(u);
909        }
910        int next_node = 0, next_def_node = 0;
911
912        // Find augmenting shortest paths
913        while (next_node < int(_excess_nodes.size())) {
914          // Check deficit nodes
915          if (_delta > 1) {
916            bool delta_deficit = false;
917            for ( ; next_def_node < int(_deficit_nodes.size());
918                    ++next_def_node ) {
919              if (_excess[_deficit_nodes[next_def_node]] <= -_delta) {
920                delta_deficit = true;
921                break;
922              }
923            }
924            if (!delta_deficit) break;
925          }
926
927          // Run Dijkstra in the residual network
928          s = _excess_nodes[next_node];
929          if ((t = _dijkstra.run(s, _delta)) == -1) {
930            if (_delta > 1) {
931              ++next_node;
932              continue;
933            }
934            return INFEASIBLE;
935          }
936
937          // Augment along a shortest path from s to t
938          Value d = std::min(_excess[s], -_excess[t]);
939          int u = t;
940          int a;
941          if (d > _delta) {
942            while ((a = _pred[u]) != -1) {
943              if (_res_cap[a] < d) d = _res_cap[a];
944              u = _source[a];
945            }
946          }
947          u = t;
948          while ((a = _pred[u]) != -1) {
949            _res_cap[a] -= d;
950            _res_cap[_reverse[a]] += d;
951            u = _source[a];
952          }
953          _excess[s] -= d;
954          _excess[t] += d;
955
956          if (_excess[s] < _delta) ++next_node;
957        }
958
959        if (_delta == 1) break;
960        _delta = _delta <= _factor ? 1 : _delta / _factor;
961      }
962
963      return OPTIMAL;
964    }
965
966    // Execute the successive shortest path algorithm
967    ProblemType startWithoutScaling() {
968      // Find excess nodes
969      _excess_nodes.clear();
970      for (int i = 0; i != _node_num; ++i) {
971        if (_excess[i] > 0) _excess_nodes.push_back(i);
972      }
973      if (_excess_nodes.size() == 0) return OPTIMAL;
974      int next_node = 0;
975
976      // Find shortest paths
977      int s, t;
978      ResidualDijkstra _dijkstra(*this);
979      while ( _excess[_excess_nodes[next_node]] > 0 ||
980              ++next_node < int(_excess_nodes.size()) )
981      {
982        // Run Dijkstra in the residual network
983        s = _excess_nodes[next_node];
984        if ((t = _dijkstra.run(s)) == -1) return INFEASIBLE;
985
986        // Augment along a shortest path from s to t
987        Value d = std::min(_excess[s], -_excess[t]);
988        int u = t;
989        int a;
990        if (d > 1) {
991          while ((a = _pred[u]) != -1) {
992            if (_res_cap[a] < d) d = _res_cap[a];
993            u = _source[a];
994          }
995        }
996        u = t;
997        while ((a = _pred[u]) != -1) {
998          _res_cap[a] -= d;
999          _res_cap[_reverse[a]] += d;
1000          u = _source[a];
1001        }
1002        _excess[s] -= d;
1003        _excess[t] += d;
1004      }
1005
1006      return OPTIMAL;
1007    }
1008
1009  }; //class CapacityScaling
1010
1011  ///@}
1012
1013} //namespace lemon
1014
1015#endif //LEMON_CAPACITY_SCALING_H
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