source:lemon-0.x/src/work/jacint/max_matching.h@582:04cd483c2dbc

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