COIN-OR::LEMON - Graph Library

source: lemon-0.x/src/lemon/max_matching.h @ 1284:b941d044f87b

Last change on this file since 1284:b941d044f87b was 1234:49d018060749, checked in by Alpar Juttner, 15 years ago

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1/* -*- C++ -*-
2 * src/lemon/max_matching.h - Part of LEMON, a generic C++ optimization library
3 *
4 * Copyright (C) 2005 Egervary Jeno Kombinatorikus Optimalizalasi Kutatocsoport
5 * (Egervary Combinatorial Optimization Research Group, EGRES).
6 *
7 * Permission to use, modify and distribute this software is granted
8 * provided that this copyright notice appears in all copies. For
9 * precise terms see the accompanying LICENSE file.
10 *
11 * This software is provided "AS IS" with no warranty of any kind,
12 * express or implied, and with no claim as to its suitability for any
13 * purpose.
14 *
15 */
16
17#ifndef LEMON_MAX_MATCHING_H
18#define LEMON_MAX_MATCHING_H
19
20#include <queue>
21#include <lemon/invalid.h>
22#include <lemon/unionfind.h>
23#include <lemon/graph_utils.h>
24
25///\ingroup galgs
26///\file
27///\brief Maximum matching algorithm.
28
29namespace lemon {
30
31  /// \addtogroup galgs
32  /// @{
33
34  ///Edmonds' alternating forest maximum matching algorithm.
35
36  ///This class provides Edmonds' alternating forest matching
37  ///algorithm. The starting matching (if any) can be passed to the
38  ///algorithm using read-in functions \ref readNMapNode, \ref
39  ///readNMapEdge or \ref readEMapBool depending on the container. The
40  ///resulting maximum matching can be attained by write-out functions
41  ///\ref writeNMapNode, \ref writeNMapEdge or \ref writeEMapBool
42  ///depending on the preferred container.
43  ///
44  ///The dual side of a matching is a map of the nodes to
45  ///MaxMatching::pos_enum, having values D, A and C showing the
46  ///Gallai-Edmonds decomposition of the graph. The nodes in D induce
47  ///a graph with factor-critical components, the nodes in A form the
48  ///barrier, and the nodes in C induce a graph having a perfect
49  ///matching. This decomposition can be attained by calling \ref
50  ///writePos after running the algorithm.
51  ///
52  ///\param Graph The undirected graph type the algorithm runs on.
53  ///
54  ///\author Jacint Szabo 
55  template <typename Graph>
56  class MaxMatching {
57
58  protected:
59
60    typedef typename Graph::Node Node;
61    typedef typename Graph::Edge Edge;
62    typedef typename Graph::UndirEdge UndirEdge;
63    typedef typename Graph::UndirEdgeIt UndirEdgeIt;
64    typedef typename Graph::NodeIt NodeIt;
65    typedef typename Graph::IncEdgeIt IncEdgeIt;
66
67    typedef UnionFindEnum<Node, Graph::template NodeMap> UFE;
68
69  public:
70   
71    ///Indicates the Gallai-Edmonds decomposition of the graph.
72
73    ///Indicates the Gallai-Edmonds decomposition of the graph, which
74    ///shows an upper bound on the size of a maximum matching. The
75    ///nodes with pos_enum \c D induce a graph with factor-critical
76    ///components, the nodes in \c A form the canonical barrier, and the
77    ///nodes in \c C induce a graph having a perfect matching.
78    enum pos_enum {
79      D=0,
80      A=1,
81      C=2
82    };
83
84  protected:
85
86    static const int HEUR_density=2;
87    const Graph& g;
88    typename Graph::template NodeMap<Node> _mate;
89    typename Graph::template NodeMap<pos_enum> position;
90     
91  public:
92   
93    MaxMatching(const Graph& _g) : g(_g), _mate(_g,INVALID), position(_g) {}
94
95    ///Runs Edmonds' algorithm.
96
97    ///Runs Edmonds' algorithm for sparse graphs (number of edges <
98    ///2*number of nodes), and a heuristical Edmonds' algorithm with a
99    ///heuristic of postponing shrinks for dense graphs.
100    inline void run();
101
102    ///Runs Edmonds' algorithm.
103   
104    ///If heur=0 it runs Edmonds' algorithm. If heur=1 it runs
105    ///Edmonds' algorithm with a heuristic of postponing shrinks,
106    ///giving a faster algorithm for dense graphs. 
107    void runEdmonds( int heur );
108
109    ///Finds a greedy matching starting from the actual matching.
110   
111    ///Starting form the actual matching stored, it finds a maximal
112    ///greedy matching.
113    void greedyMatching();
114
115    ///Returns the size of the actual matching stored.
116
117    ///Returns the size of the actual matching stored. After \ref
118    ///run() it returns the size of a maximum matching in the graph.
119    int size() const;
120
121    ///Resets the actual matching to the empty matching.
122
123    ///Resets the actual matching to the empty matching. 
124    ///
125    void resetMatching();
126
127    ///Returns the mate of a node in the actual matching.
128
129    ///Returns the mate of a \c node in the actual matching.
130    ///Returns INVALID if the \c node is not covered by the actual matching.
131    Node mate(Node& node) const {
132      return _mate[node];
133    }
134
135    ///Reads a matching from a \c Node valued \c Node map.
136
137    ///Reads a matching from a \c Node valued \c Node map. This map
138    ///must be \e symmetric, i.e. if \c map[u]==v then \c map[v]==u
139    ///must hold, and \c uv will be an edge of the matching.
140    template<typename NMapN>
141    void readNMapNode(NMapN& map) {
142      for(NodeIt v(g); v!=INVALID; ++v) {
143        _mate.set(v,map[v]);   
144      }
145    }
146   
147    ///Writes the stored matching to a \c Node valued \c Node map.
148
149    ///Writes the stored matching to a \c Node valued \c Node map. The
150    ///resulting map will be \e symmetric, i.e. if \c map[u]==v then \c
151    ///map[v]==u will hold, and now \c uv is an edge of the matching.
152    template<typename NMapN>
153    void writeNMapNode (NMapN& map) const {
154      for(NodeIt v(g); v!=INVALID; ++v) {
155        map.set(v,_mate[v]);   
156      }
157    }
158
159    ///Reads a matching from an \c UndirEdge valued \c Node map.
160
161    ///Reads a matching from an \c UndirEdge valued \c Node map. \c
162    ///map[v] must be an \c UndirEdge incident to \c v. This map must
163    ///have the property that if \c g.oppositeNode(u,map[u])==v then
164    ///\c \c g.oppositeNode(v,map[v])==u holds, and now some edge
165    ///joining \c u to \c v will be an edge of the matching.
166    template<typename NMapE>
167    void readNMapEdge(NMapE& map) {
168     for(NodeIt v(g); v!=INVALID; ++v) {
169       UndirEdge e=map[v];
170        if ( e!=INVALID )
171          _mate.set(v,g.oppositeNode(v,e));
172      }
173    }
174   
175    ///Writes the matching stored to an \c UndirEdge valued \c Node map.
176
177    ///Writes the stored matching to an \c UndirEdge valued \c Node
178    ///map. \c map[v] will be an \c UndirEdge incident to \c v. This
179    ///map will have the property that if \c g.oppositeNode(u,map[u])
180    ///== v then \c map[u]==map[v] holds, and now this edge is an edge
181    ///of the matching.
182    template<typename NMapE>
183    void writeNMapEdge (NMapE& map)  const {
184      typename Graph::template NodeMap<bool> todo(g,true);
185      for(NodeIt v(g); v!=INVALID; ++v) {
186        if ( todo[v] && _mate[v]!=INVALID ) {
187          Node u=_mate[v];
188          for(IncEdgeIt e(g,v); e!=INVALID; ++e) {
189            if ( g.runningNode(e) == u ) {
190              map.set(u,e);
191              map.set(v,e);
192              todo.set(u,false);
193              todo.set(v,false);
194              break;
195            }
196          }
197        }
198      }
199    }
200
201
202    ///Reads a matching from a \c bool valued \c Edge map.
203   
204    ///Reads a matching from a \c bool valued \c Edge map. This map
205    ///must have the property that there are no two incident edges \c
206    ///e, \c f with \c map[e]==map[f]==true. The edges \c e with \c
207    ///map[e]==true form the matching.
208    template<typename EMapB>
209    void readEMapBool(EMapB& map) {
210      for(UndirEdgeIt e(g); e!=INVALID; ++e) {
211        if ( map[e] ) {
212          Node u=g.source(e);     
213          Node v=g.target(e);
214          _mate.set(u,v);
215          _mate.set(v,u);
216        }
217      }
218    }
219
220
221    ///Writes the matching stored to a \c bool valued \c Edge map.
222
223    ///Writes the matching stored to a \c bool valued \c Edge
224    ///map. This map will have the property that there are no two
225    ///incident edges \c e, \c f with \c map[e]==map[f]==true. The
226    ///edges \c e with \c map[e]==true form the matching.
227    template<typename EMapB>
228    void writeEMapBool (EMapB& map) const {
229      for(UndirEdgeIt e(g); e!=INVALID; ++e) map.set(e,false);
230
231      typename Graph::template NodeMap<bool> todo(g,true);
232      for(NodeIt v(g); v!=INVALID; ++v) {
233        if ( todo[v] && _mate[v]!=INVALID ) {
234          Node u=_mate[v];
235          for(IncEdgeIt e(g,v); e!=INVALID; ++e) {
236            if ( g.runningNode(e) == u ) {
237              map.set(e,true);
238              todo.set(u,false);
239              todo.set(v,false);
240              break;
241            }
242          }
243        }
244      }
245    }
246
247
248    ///Writes the canonical decomposition of the graph after running
249    ///the algorithm.
250
251    ///After calling any run methods of the class, it writes the
252    ///Gallai-Edmonds canonical decomposition of the graph. \c map
253    ///must be a node map of \ref pos_enum 's.
254    template<typename NMapEnum>
255    void writePos (NMapEnum& map) const {
256      for(NodeIt v(g); v!=INVALID; ++v)  map.set(v,position[v]);
257    }
258
259  private:
260
261 
262    void lateShrink(Node v, typename Graph::template NodeMap<Node>& ear, 
263                    UFE& blossom, UFE& tree);
264
265    void normShrink(Node v, typename Graph::template NodeMap<Node>& ear, 
266                    UFE& blossom, UFE& tree);
267
268    bool noShrinkStep(Node x, typename Graph::template NodeMap<Node>& ear, 
269                      UFE& blossom, UFE& tree, std::queue<Node>& Q);
270
271    void shrinkStep(Node& top, Node& middle, Node& bottom,
272                    typename Graph::template NodeMap<Node>& ear, 
273                    UFE& blossom, UFE& tree, std::queue<Node>& Q);
274
275    void augment(Node x, typename Graph::template NodeMap<Node>& ear, 
276                 UFE& blossom, UFE& tree);
277
278  };
279
280
281  // **********************************************************************
282  //  IMPLEMENTATIONS
283  // **********************************************************************
284
285
286  template <typename Graph>
287  void MaxMatching<Graph>::run() {
288    if ( countUndirEdges(g) < HEUR_density*countNodes(g) ) {
289      greedyMatching();
290      runEdmonds(0);
291    } else runEdmonds(1);
292  }
293
294
295  template <typename Graph>
296  void MaxMatching<Graph>::runEdmonds( int heur=1 ) {
297
298    for(NodeIt v(g); v!=INVALID; ++v)
299      position.set(v,C);     
300
301    typename Graph::template NodeMap<Node> ear(g,INVALID);
302    //undefined for the base nodes of the blossoms (i.e. for the
303    //representative elements of UFE blossom) and for the nodes in C
304
305    typename UFE::MapType blossom_base(g);
306    UFE blossom(blossom_base);
307    typename UFE::MapType tree_base(g);
308    UFE tree(tree_base);
309    //If these UFE's would be members of the class then also
310    //blossom_base and tree_base should be a member.
311
312    for(NodeIt v(g); v!=INVALID; ++v) {
313      if ( position[v]==C && _mate[v]==INVALID ) {
314        blossom.insert(v);
315        tree.insert(v);
316        position.set(v,D);
317        if ( heur == 1 ) lateShrink( v, ear, blossom, tree );
318        else normShrink( v, ear, blossom, tree );
319      }
320    }
321  }
322
323   
324  template <typename Graph>
325  void MaxMatching<Graph>::lateShrink(Node v, typename Graph::template NodeMap<Node>& ear, 
326                                      UFE& blossom, UFE& tree) {
327
328    std::queue<Node> Q;   //queue of the totally unscanned nodes
329    Q.push(v); 
330    std::queue<Node> R;   
331    //queue of the nodes which must be scanned for a possible shrink
332     
333    while ( !Q.empty() ) {
334      Node x=Q.front();
335      Q.pop();
336      if ( noShrinkStep( x, ear, blossom, tree, Q ) ) return;
337      else R.push(x);
338    }
339     
340    while ( !R.empty() ) {
341      Node x=R.front();
342      R.pop();
343       
344      for( IncEdgeIt e(g,x); e!=INVALID ; ++e ) {
345        Node y=g.runningNode(e);
346
347        if ( position[y] == D && blossom.find(x) != blossom.find(y) ) {
348          //x and y must be in the same tree
349       
350          typename Graph::template NodeMap<bool> path(g,false);
351
352          Node b=blossom.find(x);
353          path.set(b,true);
354          b=_mate[b];
355          while ( b!=INVALID ) {
356            b=blossom.find(ear[b]);
357            path.set(b,true);
358            b=_mate[b];
359          } //going till the root
360       
361          Node top=y;
362          Node middle=blossom.find(top);
363          Node bottom=x;
364          while ( !path[middle] )
365            shrinkStep(top, middle, bottom, ear, blossom, tree, Q);
366                 
367          Node base=middle;
368          top=x;
369          middle=blossom.find(top);
370          bottom=y;
371          Node blossom_base=blossom.find(base);
372          while ( middle!=blossom_base )
373            shrinkStep(top, middle, bottom, ear, blossom, tree, Q);
374                 
375          blossom.makeRep(base);
376        } // if shrink is needed
377
378        while ( !Q.empty() ) {
379          Node x=Q.front();
380          Q.pop();
381          if ( noShrinkStep(x, ear, blossom, tree, Q) ) return;
382          else R.push(x);
383        }
384      } //for e
385    } // while ( !R.empty() )
386  }
387
388
389  template <typename Graph>
390  void MaxMatching<Graph>::normShrink(Node v,
391                                      typename Graph::template
392                                      NodeMap<Node>& ear, 
393                                      UFE& blossom, UFE& tree) {
394    std::queue<Node> Q;   //queue of the unscanned nodes
395    Q.push(v); 
396    while ( !Q.empty() ) {
397
398      Node x=Q.front();
399      Q.pop();
400       
401      for( IncEdgeIt e(g,x); e!=INVALID; ++e ) {
402        Node y=g.runningNode(e);
403             
404        switch ( position[y] ) {
405        case D:          //x and y must be in the same tree
406
407          if ( blossom.find(x) != blossom.find(y) ) { //shrink
408            typename Graph::template NodeMap<bool> path(g,false);
409             
410            Node b=blossom.find(x);
411            path.set(b,true);
412            b=_mate[b];
413            while ( b!=INVALID ) {
414              b=blossom.find(ear[b]);
415              path.set(b,true);
416              b=_mate[b];
417            } //going till the root
418       
419            Node top=y;
420            Node middle=blossom.find(top);
421            Node bottom=x;
422            while ( !path[middle] )
423              shrinkStep(top, middle, bottom, ear, blossom, tree, Q);
424               
425            Node base=middle;
426            top=x;
427            middle=blossom.find(top);
428            bottom=y;
429            Node blossom_base=blossom.find(base);
430            while ( middle!=blossom_base )
431              shrinkStep(top, middle, bottom, ear, blossom, tree, Q);
432               
433            blossom.makeRep(base);
434          }
435          break;
436        case C:
437          if ( _mate[y]!=INVALID ) {   //grow
438
439            ear.set(y,x);
440            Node w=_mate[y];
441            blossom.insert(w);
442            position.set(y,A);
443            position.set(w,D);
444            tree.insert(y);
445            tree.insert(w);
446            tree.join(y,blossom.find(x)); 
447            tree.join(w,y); 
448            Q.push(w);
449          } else {                 //augment 
450            augment(x, ear, blossom, tree);
451            _mate.set(x,y);
452            _mate.set(y,x);
453            return;
454          } //if
455          break;
456        default: break;
457        }
458      }
459    }
460  }
461
462  template <typename Graph>
463  void MaxMatching<Graph>::greedyMatching() {
464    for(NodeIt v(g); v!=INVALID; ++v)
465      if ( _mate[v]==INVALID ) {
466        for( IncEdgeIt e(g,v); e!=INVALID ; ++e ) {
467          Node y=g.runningNode(e);
468          if ( _mate[y]==INVALID && y!=v ) {
469            _mate.set(v,y);
470            _mate.set(y,v);
471            break;
472          }
473        }
474      }
475  }
476   
477  template <typename Graph>
478  int MaxMatching<Graph>::size() const {
479    int s=0;
480    for(NodeIt v(g); v!=INVALID; ++v) {
481      if ( _mate[v]!=INVALID ) {
482        ++s;
483      }
484    }
485    return s/2;
486  }
487
488  template <typename Graph>
489  void MaxMatching<Graph>::resetMatching() {
490    for(NodeIt v(g); v!=INVALID; ++v)
491      _mate.set(v,INVALID);     
492  }
493
494  template <typename Graph>
495  bool MaxMatching<Graph>::noShrinkStep(Node x,
496                                        typename Graph::template
497                                        NodeMap<Node>& ear, 
498                                        UFE& blossom, UFE& tree,
499                                        std::queue<Node>& Q) {
500    for( IncEdgeIt e(g,x); e!= INVALID; ++e ) {
501      Node y=g.runningNode(e);
502       
503      if ( position[y]==C ) {
504        if ( _mate[y]!=INVALID ) {       //grow
505          ear.set(y,x);
506          Node w=_mate[y];
507          blossom.insert(w);
508          position.set(y,A);
509          position.set(w,D);
510          tree.insert(y);
511          tree.insert(w);
512          tree.join(y,blossom.find(x)); 
513          tree.join(w,y); 
514          Q.push(w);
515        } else {                      //augment
516          augment(x, ear, blossom, tree);
517          _mate.set(x,y);
518          _mate.set(y,x);
519          return true;
520        }
521      }
522    }
523    return false;
524  }
525
526  template <typename Graph>
527  void MaxMatching<Graph>::shrinkStep(Node& top, Node& middle, Node& bottom,
528                                      typename Graph::template
529                                      NodeMap<Node>& ear, 
530                                      UFE& blossom, UFE& tree,
531                                      std::queue<Node>& Q) {
532    ear.set(top,bottom);
533    Node t=top;
534    while ( t!=middle ) {
535      Node u=_mate[t];
536      t=ear[u];
537      ear.set(t,u);
538    }
539    bottom=_mate[middle];
540    position.set(bottom,D);
541    Q.push(bottom);
542    top=ear[bottom];           
543    Node oldmiddle=middle;
544    middle=blossom.find(top);
545    tree.erase(bottom);
546    tree.erase(oldmiddle);
547    blossom.insert(bottom);
548    blossom.join(bottom, oldmiddle);
549    blossom.join(top, oldmiddle);
550  }
551
552  template <typename Graph>
553  void MaxMatching<Graph>::augment(Node x,
554                                   typename Graph::template NodeMap<Node>& ear, 
555                                   UFE& blossom, UFE& tree) {
556    Node v=_mate[x];
557    while ( v!=INVALID ) {
558       
559      Node u=ear[v];
560      _mate.set(v,u);
561      Node tmp=v;
562      v=_mate[u];
563      _mate.set(u,tmp);
564    }
565    typename UFE::ItemIt it;
566    for (tree.first(it,blossom.find(x)); tree.valid(it); tree.next(it)) {   
567      if ( position[it] == D ) {
568        typename UFE::ItemIt b_it;
569        for (blossom.first(b_it,it); blossom.valid(b_it); blossom.next(b_it)) { 
570          position.set( b_it ,C);
571        }
572        blossom.eraseClass(it);
573      } else position.set( it ,C);
574    }
575    tree.eraseClass(x);
576
577  }
578
579  /// @}
580 
581} //END OF NAMESPACE LEMON
582
583#endif //LEMON_MAX_MATCHING_H
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