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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 \c D induce a graph with factor-critical
58    ///components, the nodes in \c A form the canonical barrier, and the
59    ///nodes in \c 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
243    ///of \ref pos_enum 's.
244    template<typename NMapEnum>
245    void writePos (NMapEnum& map) const {
246      NodeIt v;
247      for( G.first(v); G.valid(v); G.next(v)) map.set(v,position[v]);
248    }
249
250  private:
251
252    void lateShrink(Node v, typename Graph::template NodeMap<Node>& ear,
253                    UFE& blossom, UFE& tree);
254
255    void normShrink(Node v, typename Graph::NodeMap<Node>& ear,
256                    UFE& blossom, UFE& tree);
257
258    bool noShrinkStep(Node x, typename Graph::NodeMap<Node>& ear,
259                      UFE& blossom, UFE& tree, std::queue<Node>& Q);
260
261    void shrinkStep(Node& top, Node& middle, Node& bottom, typename Graph::NodeMap<Node>& ear,
262                    UFE& blossom, UFE& tree, std::queue<Node>& Q);
263
264    void augment(Node x, typename Graph::NodeMap<Node>& ear,
265                 UFE& blossom, UFE& tree);
266
267  };
268
269
270  // **********************************************************************
271  //  IMPLEMENTATIONS
272  // **********************************************************************
273
274
275  template <typename Graph>
276  void MaxMatching<Graph>::run() {
277    if ( G.edgeNum() > 2*G.nodeNum() ) {
278      greedyMatching();
279      runEdmonds(1);
280    } else runEdmonds(0);
281  }
282
283  template <typename Graph>
284  void MaxMatching<Graph>::runEdmonds( int heur=1 ) {
285
286    typename Graph::template NodeMap<Node> ear(G,INVALID);
287    //undefined for the base nodes of the blossoms (i.e. for the
288    //representative elements of UFE blossom) and for the nodes in C
289
290    typename UFE::MapType blossom_base(G);
291    UFE blossom(blossom_base);
292    typename UFE::MapType tree_base(G);
293    UFE tree(tree_base);
294
295    NodeIt v;
296    for( G.first(v); G.valid(v); G.next(v) ) {
297      if ( position[v]==C && mate[v]==INVALID ) {
298        blossom.insert(v);
299        tree.insert(v);
300        position.set(v,D);
301        if ( heur == 1 ) lateShrink( v, ear, blossom, tree );
302        else normShrink( v, ear, blossom, tree );
303      }
304    }
305  }
306
307  template <typename Graph>
308  void MaxMatching<Graph>::lateShrink(Node v, typename Graph::template NodeMap<Node>& ear,
309                                      UFE& blossom, UFE& tree) {
310
311    std::queue<Node> Q;   //queue of the totally unscanned nodes
312    Q.push(v);
313    std::queue<Node> R;
314    //queue of the nodes which must be scanned for a possible shrink
315
316    while ( !Q.empty() ) {
317      Node x=Q.front();
318      Q.pop();
319      if ( noShrinkStep( x, ear, blossom, tree, Q ) ) return;
320      else R.push(x);
321    }
322
323    while ( !R.empty() ) {
324      Node x=R.front();
325      R.pop();
326
327      OutEdgeIt e;
328      for( G.first(e,x); G.valid(e); G.next(e) ) {
329        Node y=G.bNode(e);
330
331        if ( position[y] == D && blossom.find(x) != blossom.find(y) ) {
332          //x and y must be in the same tree
333
334          typename Graph::template NodeMap<bool> path(G,false);
335
336          Node b=blossom.find(x);
337          path.set(b,true);
338          b=mate[b];
339          while ( b!=INVALID ) {
340            b=blossom.find(ear[b]);
341            path.set(b,true);
342            b=mate[b];
343          } //going till the root
344
345          Node top=y;
346          Node middle=blossom.find(top);
347          Node bottom=x;
348          while ( !path[middle] )
349            shrinkStep(top, middle, bottom, ear, blossom, tree, Q);
350
351          Node base=middle;
352          top=x;
353          middle=blossom.find(top);
354          bottom=y;
355          Node blossom_base=blossom.find(base);
356          while ( middle!=blossom_base )
357            shrinkStep(top, middle, bottom, ear, blossom, tree, Q);
358
359          blossom.makeRep(base);
360        } // if shrink is needed
361
362        while ( !Q.empty() ) {
363          Node x=Q.front();
364          Q.pop();
365          if ( noShrinkStep(x, ear, blossom, tree, Q) ) return;
366          else R.push(x);
367        }
368      } //for e
369    } // while ( !R.empty() )
370  }
371
372  template <typename Graph>
373  void MaxMatching<Graph>::normShrink(Node v, typename Graph::NodeMap<Node>& ear,
374                                      UFE& blossom, UFE& tree) {
375
376    std::queue<Node> Q;   //queue of the unscanned nodes
377    Q.push(v);
378    while ( !Q.empty() ) {
379      Node x=Q.front();
380      Q.pop();
381
382      OutEdgeIt e;
383      for( G.first(e,x); G.valid(e); G.next(e) ) {
384        Node y=G.bNode(e);
385
386        switch ( position[y] ) {
387        case D:          //x and y must be in the same tree
388          if ( blossom.find(x) != blossom.find(y) ) { //shrink
389            typename Graph::template NodeMap<bool> path(G,false);
390
391            Node b=blossom.find(x);
392            path.set(b,true);
393            b=mate[b];
394            while ( b!=INVALID ) {
395              b=blossom.find(ear[b]);
396              path.set(b,true);
397              b=mate[b];
398            } //going till the root
399
400            Node top=y;
401            Node middle=blossom.find(top);
402            Node bottom=x;
403            while ( !path[middle] )
404              shrinkStep(top, middle, bottom, ear, blossom, tree, Q);
405
406            Node base=middle;
407            top=x;
408            middle=blossom.find(top);
409            bottom=y;
410            Node blossom_base=blossom.find(base);
411            while ( middle!=blossom_base )
412              shrinkStep(top, middle, bottom, ear, blossom, tree, Q);
413
414            blossom.makeRep(base);
415          }
416          break;
417        case C:
418          if ( mate[y]!=INVALID ) {   //grow
419            ear.set(y,x);
420            Node w=mate[y];
421            blossom.insert(w);
422            position.set(y,A);
423            position.set(w,D);
424            tree.insert(y);
425            tree.insert(w);
426            tree.join(y,blossom.find(x));
427            tree.join(w,y);
428            Q.push(w);
429          } else {                 //augment
430            augment(x, ear, blossom, tree);
431            mate.set(x,y);
432            mate.set(y,x);
433            return;
434          } //if
435          break;
436        default: break;
437        }
438      }
439    }
440  }
441
442  template <typename Graph>
443  void MaxMatching<Graph>::greedyMatching() {
444    NodeIt v;
445    for( G.first(v); G.valid(v); G.next(v) )
446      if ( mate[v]==INVALID ) {
447        OutEdgeIt e;
448        for( G.first(e,v); G.valid(e); G.next(e) ) {
449          Node y=G.bNode(e);
450          if ( mate[y]==INVALID && y!=v ) {
451            mate.set(v,y);
452            mate.set(y,v);
453            break;
454          }
455        }
456      }
457  }
458
459  template <typename Graph>
460  int MaxMatching<Graph>::size() const {
461    int s=0;
462    NodeIt v;
463    for(G.first(v); G.valid(v); G.next(v) ) {
464      if ( G.valid(mate[v]) ) {
465        ++s;
466      }
467    }
468    return (int)s/2;
469  }
470
471  template <typename Graph>
472  void MaxMatching<Graph>::resetPos() {
473    NodeIt v;
474    for( G.first(v); G.valid(v); G.next(v))
475      position.set(v,C);
476  }
477
478  template <typename Graph>
479  void MaxMatching<Graph>::resetMatching() {
480    NodeIt v;
481    for( G.first(v); G.valid(v); G.next(v))
482      mate.set(v,INVALID);
483  }
484
485  template <typename Graph>
486  bool MaxMatching<Graph>::noShrinkStep(Node x, typename Graph::NodeMap<Node>& ear,
487                                        UFE& blossom, UFE& tree, std::queue<Node>& Q) {
488    OutEdgeIt e;
489    for( G.first(e,x); G.valid(e); G.next(e) ) {
490      Node y=G.bNode(e);
491
492      if ( position[y]==C ) {
493        if ( mate[y]!=INVALID ) {       //grow
494          ear.set(y,x);
495          Node w=mate[y];
496          blossom.insert(w);
497          position.set(y,A);
498          position.set(w,D);
499          tree.insert(y);
500          tree.insert(w);
501          tree.join(y,blossom.find(x));
502          tree.join(w,y);
503          Q.push(w);
504        } else {                      //augment
505          augment(x, ear, blossom, tree);
506          mate.set(x,y);
507          mate.set(y,x);
508          return true;
509        }
510      }
511    }
512    return false;
513  }
514
515  template <typename Graph>
516  void MaxMatching<Graph>::shrinkStep(Node& top, Node& middle, Node& bottom, typename Graph::NodeMap<Node>& ear,
517                                      UFE& blossom, UFE& tree, std::queue<Node>& Q) {
518    ear.set(top,bottom);
519    Node t=top;
520    while ( t!=middle ) {
521      Node u=mate[t];
522      t=ear[u];
523      ear.set(t,u);
524    }
525    bottom=mate[middle];
526    position.set(bottom,D);
527    Q.push(bottom);
528    top=ear[bottom];
529    Node oldmiddle=middle;
530    middle=blossom.find(top);
531    tree.erase(bottom);
532    tree.erase(oldmiddle);
533    blossom.insert(bottom);
534    blossom.join(bottom, oldmiddle);
535    blossom.join(top, oldmiddle);
536  }
537
538  template <typename Graph>
539  void MaxMatching<Graph>::augment(Node x, typename Graph::NodeMap<Node>& ear,
540                                   UFE& blossom, UFE& tree) {
541    Node v=mate[x];
542    while ( G.valid(v) ) {
543
544      Node u=ear[v];
545      mate.set(v,u);
546      Node tmp=v;
547      v=mate[u];
548      mate.set(u,tmp);
549    }
550    typename UFE::ItemIt it;
551    for (tree.first(it,blossom.find(x)); tree.valid(it); tree.next(it)) {
552      if ( position[it] == D ) {
553        typename UFE::ItemIt b_it;
554        for (blossom.first(b_it,it); blossom.valid(b_it); blossom.next(b_it)) {
555          position.set( b_it ,C);
556        }
557        blossom.eraseClass(it);
558      } else position.set( it ,C);
559    }
560    tree.eraseClass(x);
561  }
562
563
564
565  /// @}
566
567} //END OF NAMESPACE HUGO
568
569#endif //EDMONDS_H
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