COIN-OR::LEMON - Graph Library

source: lemon-0.x/lemon/capacity_scaling.h @ 2553:bfced05fa852

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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
23///
24/// \file
25/// \brief The capacity scaling algorithm for finding a minimum cost
26/// flow.
27
28#include <lemon/graph_adaptor.h>
29#include <lemon/bin_heap.h>
30#include <vector>
31
32namespace lemon {
33
34  /// \addtogroup min_cost_flow
35  /// @{
36
37  /// \brief Implementation of the capacity scaling version of the
38  /// successive shortest path algorithm for finding a minimum cost
39  /// flow.
40  ///
41  /// \ref CapacityScaling implements the capacity scaling version
42  /// of the successive shortest path algorithm for finding a minimum
43  /// cost flow.
44  ///
45  /// \param Graph The directed graph type the algorithm runs on.
46  /// \param LowerMap The type of the lower bound map.
47  /// \param CapacityMap The type of the capacity (upper bound) map.
48  /// \param CostMap The type of the cost (length) map.
49  /// \param SupplyMap The type of the supply map.
50  ///
51  /// \warning
52  /// - Edge capacities and costs should be nonnegative integers.
53  ///   However \c CostMap::Value should be signed type.
54  /// - Supply values should be signed integers.
55  /// - \c LowerMap::Value must be convertible to
56  ///   \c CapacityMap::Value and \c CapacityMap::Value must be
57  ///   convertible to \c SupplyMap::Value.
58  ///
59  /// \author Peter Kovacs
60
61  template < typename Graph,
62             typename LowerMap = typename Graph::template EdgeMap<int>,
63             typename CapacityMap = LowerMap,
64             typename CostMap = typename Graph::template EdgeMap<int>,
65             typename SupplyMap = typename Graph::template NodeMap
66                                  <typename CapacityMap::Value> >
67  class CapacityScaling
68  {
69    typedef typename Graph::Node Node;
70    typedef typename Graph::NodeIt NodeIt;
71    typedef typename Graph::Edge Edge;
72    typedef typename Graph::EdgeIt EdgeIt;
73    typedef typename Graph::InEdgeIt InEdgeIt;
74    typedef typename Graph::OutEdgeIt OutEdgeIt;
75
76    typedef typename LowerMap::Value Lower;
77    typedef typename CapacityMap::Value Capacity;
78    typedef typename CostMap::Value Cost;
79    typedef typename SupplyMap::Value Supply;
80    typedef typename Graph::template EdgeMap<Capacity> CapacityRefMap;
81    typedef typename Graph::template NodeMap<Supply> SupplyRefMap;
82    typedef typename Graph::template NodeMap<Edge> PredMap;
83
84  public:
85
86    /// \brief Type to enable or disable capacity scaling.
87    enum ScalingEnum {
88      WITH_SCALING = 0,
89      WITHOUT_SCALING = -1
90    };
91
92    /// \brief The type of the flow map.
93    typedef CapacityRefMap FlowMap;
94    /// \brief The type of the potential map.
95    typedef typename Graph::template NodeMap<Cost> PotentialMap;
96
97  protected:
98
99    /// \brief Special implementation of the \ref Dijkstra algorithm
100    /// for finding shortest paths in the residual network of the graph
101    /// with respect to the reduced edge costs and modifying the
102    /// node potentials according to the distance of the nodes.
103    class ResidualDijkstra
104    {
105      typedef typename Graph::template NodeMap<Cost> CostNodeMap;
106      typedef typename Graph::template NodeMap<Edge> PredMap;
107
108      typedef typename Graph::template NodeMap<int> HeapCrossRef;
109      typedef BinHeap<Cost, HeapCrossRef> Heap;
110
111    protected:
112
113      /// \brief The directed graph the algorithm runs on.
114      const Graph &graph;
115
116      /// \brief The flow map.
117      const FlowMap &flow;
118      /// \brief The residual capacity map.
119      const CapacityRefMap &res_cap;
120      /// \brief The cost map.
121      const CostMap &cost;
122      /// \brief The excess map.
123      const SupplyRefMap &excess;
124
125      /// \brief The potential map.
126      PotentialMap &potential;
127
128      /// \brief The distance map.
129      CostNodeMap dist;
130      /// \brief The map of predecessors edges.
131      PredMap &pred;
132      /// \brief The processed (i.e. permanently labeled) nodes.
133      std::vector<Node> proc_nodes;
134
135    public:
136
137      /// \brief The constructor of the class.
138      ResidualDijkstra( const Graph &_graph,
139                        const FlowMap &_flow,
140                        const CapacityRefMap &_res_cap,
141                        const CostMap &_cost,
142                        const SupplyMap &_excess,
143                        PotentialMap &_potential,
144                        PredMap &_pred ) :
145        graph(_graph), flow(_flow), res_cap(_res_cap), cost(_cost),
146        excess(_excess), potential(_potential), dist(_graph),
147        pred(_pred)
148      {}
149
150      /// \brief Runs the algorithm from the given source node.
151      Node run(Node s, Capacity delta) {
152        HeapCrossRef heap_cross_ref(graph, Heap::PRE_HEAP);
153        Heap heap(heap_cross_ref);
154        heap.push(s, 0);
155        pred[s] = INVALID;
156        proc_nodes.clear();
157
158        // Processing nodes
159        while (!heap.empty() && excess[heap.top()] > -delta) {
160          Node u = heap.top(), v;
161          Cost d = heap.prio() - potential[u], nd;
162          dist[u] = heap.prio();
163          heap.pop();
164          proc_nodes.push_back(u);
165
166          // Traversing outgoing edges
167          for (OutEdgeIt e(graph, u); e != INVALID; ++e) {
168            if (res_cap[e] >= delta) {
169              v = graph.target(e);
170              switch(heap.state(v)) {
171              case Heap::PRE_HEAP:
172                heap.push(v, d + cost[e] + potential[v]);
173                pred[v] = e;
174                break;
175              case Heap::IN_HEAP:
176                nd = d + cost[e] + potential[v];
177                if (nd < heap[v]) {
178                  heap.decrease(v, nd);
179                  pred[v] = e;
180                }
181                break;
182              case Heap::POST_HEAP:
183                break;
184              }
185            }
186          }
187
188          // Traversing incoming edges
189          for (InEdgeIt e(graph, u); e != INVALID; ++e) {
190            if (flow[e] >= delta) {
191              v = graph.source(e);
192              switch(heap.state(v)) {
193              case Heap::PRE_HEAP:
194                heap.push(v, d - cost[e] + potential[v]);
195                pred[v] = e;
196                break;
197              case Heap::IN_HEAP:
198                nd = d - cost[e] + potential[v];
199                if (nd < heap[v]) {
200                  heap.decrease(v, nd);
201                  pred[v] = e;
202                }
203                break;
204              case Heap::POST_HEAP:
205                break;
206              }
207            }
208          }
209        }
210        if (heap.empty()) return INVALID;
211
212        // Updating potentials of processed nodes
213        Node t = heap.top();
214        Cost dt = heap.prio();
215        for (int i = 0; i < proc_nodes.size(); ++i)
216          potential[proc_nodes[i]] -= dist[proc_nodes[i]] - dt;
217
218        return t;
219      }
220
221    }; //class ResidualDijkstra
222
223  protected:
224
225    /// \brief The directed graph the algorithm runs on.
226    const Graph &graph;
227    /// \brief The original lower bound map.
228    const LowerMap *lower;
229    /// \brief The modified capacity map.
230    CapacityRefMap capacity;
231    /// \brief The cost map.
232    const CostMap &cost;
233    /// \brief The modified supply map.
234    SupplyRefMap supply;
235    /// \brief The sum of supply values equals zero.
236    bool valid_supply;
237
238    /// \brief The edge map of the current flow.
239    FlowMap flow;
240    /// \brief The potential node map.
241    PotentialMap potential;
242
243    /// \brief The residual capacity map.
244    CapacityRefMap res_cap;
245    /// \brief The excess map.
246    SupplyRefMap excess;
247    /// \brief The excess nodes (i.e. nodes with positive excess).
248    std::vector<Node> excess_nodes;
249    /// \brief The index of the next excess node.
250    int next_node;
251
252    /// \brief The scaling status (enabled or disabled).
253    ScalingEnum scaling;
254    /// \brief The delta parameter used for capacity scaling.
255    Capacity delta;
256    /// \brief The maximum number of phases.
257    Capacity phase_num;
258    /// \brief The deficit nodes.
259    std::vector<Node> deficit_nodes;
260
261    /// \brief Implementation of the \ref Dijkstra algorithm for
262    /// finding augmenting shortest paths in the residual network.
263    ResidualDijkstra dijkstra;
264    /// \brief The map of predecessors edges.
265    PredMap pred;
266
267  public :
268
269    /// \brief General constructor of the class (with lower bounds).
270    ///
271    /// General constructor of the class (with lower bounds).
272    ///
273    /// \param _graph The directed graph the algorithm runs on.
274    /// \param _lower The lower bounds of the edges.
275    /// \param _capacity The capacities (upper bounds) of the edges.
276    /// \param _cost The cost (length) values of the edges.
277    /// \param _supply The supply values of the nodes (signed).
278    CapacityScaling( const Graph &_graph,
279                     const LowerMap &_lower,
280                     const CapacityMap &_capacity,
281                     const CostMap &_cost,
282                     const SupplyMap &_supply ) :
283      graph(_graph), lower(&_lower), capacity(_graph), cost(_cost),
284      supply(_graph), flow(_graph, 0), potential(_graph, 0),
285      res_cap(_graph), excess(_graph), pred(_graph),
286      dijkstra(graph, flow, res_cap, cost, excess, potential, pred)
287    {
288      // Removing nonzero lower bounds
289      capacity = subMap(_capacity, _lower);
290      res_cap = capacity;
291      Supply sum = 0;
292      for (NodeIt n(graph); n != INVALID; ++n) {
293        Supply s = _supply[n];
294        for (InEdgeIt e(graph, n); e != INVALID; ++e)
295          s += _lower[e];
296        for (OutEdgeIt e(graph, n); e != INVALID; ++e)
297          s -= _lower[e];
298        supply[n] = s;
299        sum += s;
300      }
301      valid_supply = sum == 0;
302    }
303
304    /// \brief General constructor of the class (without lower bounds).
305    ///
306    /// General constructor of the class (without lower bounds).
307    ///
308    /// \param _graph The directed graph the algorithm runs on.
309    /// \param _capacity The capacities (upper bounds) of the edges.
310    /// \param _cost The cost (length) values of the edges.
311    /// \param _supply The supply values of the nodes (signed).
312    CapacityScaling( const Graph &_graph,
313                     const CapacityMap &_capacity,
314                     const CostMap &_cost,
315                     const SupplyMap &_supply ) :
316      graph(_graph), lower(NULL), capacity(_capacity), cost(_cost),
317      supply(_supply), flow(_graph, 0), potential(_graph, 0),
318      res_cap(_capacity), excess(_graph), pred(_graph),
319      dijkstra(graph, flow, res_cap, cost, excess, potential)
320    {
321      // Checking the sum of supply values
322      Supply sum = 0;
323      for (NodeIt n(graph); n != INVALID; ++n) sum += supply[n];
324      valid_supply = sum == 0;
325    }
326
327    /// \brief Simple constructor of the class (with lower bounds).
328    ///
329    /// Simple constructor of the class (with lower bounds).
330    ///
331    /// \param _graph The directed graph the algorithm runs on.
332    /// \param _lower The lower bounds of the edges.
333    /// \param _capacity The capacities (upper bounds) of the edges.
334    /// \param _cost The cost (length) values of the edges.
335    /// \param _s The source node.
336    /// \param _t The target node.
337    /// \param _flow_value The required amount of flow from node \c _s
338    /// to node \c _t (i.e. the supply of \c _s and the demand of
339    /// \c _t).
340    CapacityScaling( const Graph &_graph,
341                     const LowerMap &_lower,
342                     const CapacityMap &_capacity,
343                     const CostMap &_cost,
344                     Node _s, Node _t,
345                     Supply _flow_value ) :
346      graph(_graph), lower(&_lower), capacity(_graph), cost(_cost),
347      supply(_graph), flow(_graph, 0), potential(_graph, 0),
348      res_cap(_graph), excess(_graph), pred(_graph),
349      dijkstra(graph, flow, res_cap, cost, excess, potential)
350    {
351      // Removing nonzero lower bounds
352      capacity = subMap(_capacity, _lower);
353      res_cap = capacity;
354      for (NodeIt n(graph); n != INVALID; ++n) {
355        Supply s = 0;
356        if (n == _s) s =  _flow_value;
357        if (n == _t) s = -_flow_value;
358        for (InEdgeIt e(graph, n); e != INVALID; ++e)
359          s += _lower[e];
360        for (OutEdgeIt e(graph, n); e != INVALID; ++e)
361          s -= _lower[e];
362        supply[n] = s;
363      }
364      valid_supply = true;
365    }
366
367    /// \brief Simple constructor of the class (without lower bounds).
368    ///
369    /// Simple constructor of the class (without lower bounds).
370    ///
371    /// \param _graph The directed graph the algorithm runs on.
372    /// \param _capacity The capacities (upper bounds) of the edges.
373    /// \param _cost The cost (length) values of the edges.
374    /// \param _s The source node.
375    /// \param _t The target node.
376    /// \param _flow_value The required amount of flow from node \c _s
377    /// to node \c _t (i.e. the supply of \c _s and the demand of
378    /// \c _t).
379    CapacityScaling( const Graph &_graph,
380                     const CapacityMap &_capacity,
381                     const CostMap &_cost,
382                     Node _s, Node _t,
383                     Supply _flow_value ) :
384      graph(_graph), lower(NULL), capacity(_capacity), cost(_cost),
385      supply(_graph, 0), flow(_graph, 0), potential(_graph, 0),
386      res_cap(_capacity), excess(_graph), pred(_graph),
387      dijkstra(graph, flow, res_cap, cost, excess, potential)
388    {
389      supply[_s] =  _flow_value;
390      supply[_t] = -_flow_value;
391      valid_supply = true;
392    }
393
394    /// \brief Returns a const reference to the flow map.
395    ///
396    /// Returns a const reference to the flow map.
397    ///
398    /// \pre \ref run() must be called before using this function.
399    const FlowMap& flowMap() const {
400      return flow;
401    }
402
403    /// \brief Returns a const reference to the potential map (the dual
404    /// solution).
405    ///
406    /// Returns a const reference to the potential map (the dual
407    /// solution).
408    ///
409    /// \pre \ref run() must be called before using this function.
410    const PotentialMap& potentialMap() const {
411      return potential;
412    }
413
414    /// \brief Returns the total cost of the found flow.
415    ///
416    /// Returns the total cost of the found flow. The complexity of the
417    /// function is \f$ O(e) \f$.
418    ///
419    /// \pre \ref run() must be called before using this function.
420    Cost totalCost() const {
421      Cost c = 0;
422      for (EdgeIt e(graph); e != INVALID; ++e)
423        c += flow[e] * cost[e];
424      return c;
425    }
426
427    /// \brief Runs the algorithm.
428    ///
429    /// Runs the algorithm.
430    ///
431    /// \param scaling_mode The scaling mode. In case of WITH_SCALING
432    /// capacity scaling is enabled in the algorithm (this is the
433    /// default value) otherwise it is disabled.
434    /// If the maximum edge capacity and/or the amount of total supply
435    /// is small, the algorithm could be faster without scaling.
436    ///
437    /// \return \c true if a feasible flow can be found.
438    bool run(int scaling_mode = WITH_SCALING) {
439      return init(scaling_mode) && start();
440    }
441
442  protected:
443
444    /// \brief Initializes the algorithm.
445    bool init(int scaling_mode) {
446      if (!valid_supply) return false;
447      excess = supply;
448
449      // Initilaizing delta value
450      if (scaling_mode == WITH_SCALING) {
451        // With scaling
452        Supply max_sup = 0, max_dem = 0;
453        for (NodeIt n(graph); n != INVALID; ++n) {
454          if ( supply[n] > max_sup) max_sup =  supply[n];
455          if (-supply[n] > max_dem) max_dem = -supply[n];
456        }
457        if (max_dem < max_sup) max_sup = max_dem;
458        phase_num = 0;
459        for (delta = 1; 2 * delta <= max_sup; delta *= 2)
460          ++phase_num;
461      } else {
462        // Without scaling
463        delta = 1;
464      }
465      return true;
466    }
467
468    /// \brief Executes the algorithm.
469    bool start() {
470      if (delta > 1)
471        return startWithScaling();
472      else
473        return startWithoutScaling();
474    }
475
476    /// \brief Executes the capacity scaling version of the successive
477    /// shortest path algorithm.
478    bool startWithScaling() {
479      // Processing capacity scaling phases
480      Node s, t;
481      int phase_cnt = 0;
482      int factor = 4;
483      while (true) {
484        // Saturating all edges not satisfying the optimality condition
485        for (EdgeIt e(graph); e != INVALID; ++e) {
486          Node u = graph.source(e), v = graph.target(e);
487          Cost c = cost[e] - potential[u] + potential[v];
488          if (c < 0 && res_cap[e] >= delta) {
489            excess[u] -= res_cap[e];
490            excess[v] += res_cap[e];
491            flow[e] = capacity[e];
492            res_cap[e] = 0;
493          }
494          else if (c > 0 && flow[e] >= delta) {
495            excess[u] += flow[e];
496            excess[v] -= flow[e];
497            flow[e] = 0;
498            res_cap[e] = capacity[e];
499          }
500        }
501
502        // Finding excess nodes and deficit nodes
503        excess_nodes.clear();
504        deficit_nodes.clear();
505        for (NodeIt n(graph); n != INVALID; ++n) {
506          if (excess[n] >=  delta) excess_nodes.push_back(n);
507          if (excess[n] <= -delta) deficit_nodes.push_back(n);
508        }
509        next_node = 0;
510
511        // Finding augmenting shortest paths
512        while (next_node < excess_nodes.size()) {
513          // Checking deficit nodes
514          if (delta > 1) {
515            bool delta_deficit = false;
516            for (int i = 0; i < deficit_nodes.size(); ++i) {
517              if (excess[deficit_nodes[i]] <= -delta) {
518                delta_deficit = true;
519                break;
520              }
521            }
522            if (!delta_deficit) break;
523          }
524
525          // Running Dijkstra
526          s = excess_nodes[next_node];
527          if ((t = dijkstra.run(s, delta)) == INVALID) {
528            if (delta > 1) {
529              ++next_node;
530              continue;
531            }
532            return false;
533          }
534
535          // Augmenting along a shortest path from s to t.
536          Capacity d = excess[s] < -excess[t] ? excess[s] : -excess[t];
537          Node u = t;
538          Edge e;
539          if (d > delta) {
540            while ((e = pred[u]) != INVALID) {
541              Capacity rc;
542              if (u == graph.target(e)) {
543                rc = res_cap[e];
544                u = graph.source(e);
545              } else {
546                rc = flow[e];
547                u = graph.target(e);
548              }
549              if (rc < d) d = rc;
550            }
551          }
552          u = t;
553          while ((e = pred[u]) != INVALID) {
554            if (u == graph.target(e)) {
555              flow[e] += d;
556              res_cap[e] -= d;
557              u = graph.source(e);
558            } else {
559              flow[e] -= d;
560              res_cap[e] += d;
561              u = graph.target(e);
562            }
563          }
564          excess[s] -= d;
565          excess[t] += d;
566
567          if (excess[s] < delta) ++next_node;
568        }
569
570        if (delta == 1) break;
571        if (++phase_cnt > phase_num / 4) factor = 2;
572        delta = delta <= factor ? 1 : delta / factor;
573      }
574
575      // Handling nonzero lower bounds
576      if (lower) {
577        for (EdgeIt e(graph); e != INVALID; ++e)
578          flow[e] += (*lower)[e];
579      }
580      return true;
581    }
582
583    /// \brief Executes the successive shortest path algorithm without
584    /// capacity scaling.
585    bool startWithoutScaling() {
586      // Finding excess nodes
587      for (NodeIt n(graph); n != INVALID; ++n) {
588        if (excess[n] > 0) excess_nodes.push_back(n);
589      }
590      if (excess_nodes.size() == 0) return true;
591      next_node = 0;
592
593      // Finding shortest paths
594      Node s, t;
595      while ( excess[excess_nodes[next_node]] > 0 ||
596              ++next_node < excess_nodes.size() )
597      {
598        // Running Dijkstra
599        s = excess_nodes[next_node];
600        if ((t = dijkstra.run(s, 1)) == INVALID)
601          return false;
602
603        // Augmenting along a shortest path from s to t
604        Capacity d = excess[s] < -excess[t] ? excess[s] : -excess[t];
605        Node u = t;
606        Edge e;
607        while ((e = pred[u]) != INVALID) {
608          Capacity rc;
609          if (u == graph.target(e)) {
610            rc = res_cap[e];
611            u = graph.source(e);
612          } else {
613            rc = flow[e];
614            u = graph.target(e);
615          }
616          if (rc < d) d = rc;
617        }
618        u = t;
619        while ((e = pred[u]) != INVALID) {
620          if (u == graph.target(e)) {
621            flow[e] += d;
622            res_cap[e] -= d;
623            u = graph.source(e);
624          } else {
625            flow[e] -= d;
626            res_cap[e] += d;
627            u = graph.target(e);
628          }
629        }
630        excess[s] -= d;
631        excess[t] += d;
632      }
633
634      // Handling nonzero lower bounds
635      if (lower) {
636        for (EdgeIt e(graph); e != INVALID; ++e)
637          flow[e] += (*lower)[e];
638      }
639      return true;
640    }
641
642  }; //class CapacityScaling
643
644  ///@}
645
646} //namespace lemon
647
648#endif //LEMON_CAPACITY_SCALING_H
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