Better handling of inexact computation.
We do not use tolerance for excess, just for edges
3 * This file is a part of LEMON, a generic C++ optimization library
5 * Copyright (C) 2003-2006
6 * Egervary Jeno Kombinatorikus Optimalizalasi Kutatocsoport
7 * (Egervary Research Group on Combinatorial Optimization, EGRES).
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.
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
19 #ifndef LEMON_MAX_MATCHING_H
20 #define LEMON_MAX_MATCHING_H
23 #include <lemon/bits/invalid.h>
24 #include <lemon/unionfind.h>
25 #include <lemon/graph_utils.h>
29 ///\brief Maximum matching algorithm in undirected graph.
35 ///Edmonds' alternating forest maximum matching algorithm.
37 ///This class provides Edmonds' alternating forest matching
38 ///algorithm. The starting matching (if any) can be passed to the
39 ///algorithm using read-in functions \ref readNMapNode, \ref
40 ///readNMapEdge or \ref readEMapBool depending on the container. The
41 ///resulting maximum matching can be attained by write-out functions
42 ///\ref writeNMapNode, \ref writeNMapEdge or \ref writeEMapBool
43 ///depending on the preferred container.
45 ///The dual side of a matching is a map of the nodes to
46 ///MaxMatching::pos_enum, having values D, A and C showing the
47 ///Gallai-Edmonds decomposition of the graph. The nodes in D induce
48 ///a graph with factor-critical components, the nodes in A form the
49 ///barrier, and the nodes in C induce a graph having a perfect
50 ///matching. This decomposition can be attained by calling \ref
51 ///writePos after running the algorithm.
53 ///\param Graph The undirected graph type the algorithm runs on.
55 ///\author Jacint Szabo
56 template <typename Graph>
61 typedef typename Graph::Node Node;
62 typedef typename Graph::Edge Edge;
63 typedef typename Graph::UEdge UEdge;
64 typedef typename Graph::UEdgeIt UEdgeIt;
65 typedef typename Graph::NodeIt NodeIt;
66 typedef typename Graph::IncEdgeIt IncEdgeIt;
68 typedef typename Graph::template NodeMap<int> UFECrossRef;
69 typedef UnionFindEnum<UFECrossRef> UFE;
73 ///Indicates the Gallai-Edmonds decomposition of the graph.
75 ///Indicates the Gallai-Edmonds decomposition of the graph, which
76 ///shows an upper bound on the size of a maximum matching. The
77 ///nodes with pos_enum \c D induce a graph with factor-critical
78 ///components, the nodes in \c A form the canonical barrier, and the
79 ///nodes in \c C induce a graph having a perfect matching.
88 static const int HEUR_density=2;
90 typename Graph::template NodeMap<Node> _mate;
91 typename Graph::template NodeMap<pos_enum> position;
95 MaxMatching(const Graph& _g) : g(_g), _mate(_g,INVALID), position(_g) {}
97 ///Runs Edmonds' algorithm.
99 ///Runs Edmonds' algorithm for sparse graphs (number of edges <
100 ///2*number of nodes), and a heuristical Edmonds' algorithm with a
101 ///heuristic of postponing shrinks for dense graphs.
103 if ( countUEdges(g) < HEUR_density*countNodes(g) ) {
106 } else runEdmonds(1);
110 ///Runs Edmonds' algorithm.
112 ///If heur=0 it runs Edmonds' algorithm. If heur=1 it runs
113 ///Edmonds' algorithm with a heuristic of postponing shrinks,
114 ///giving a faster algorithm for dense graphs.
115 void runEdmonds( int heur = 1 ) {
117 //each vertex is put to C
118 for(NodeIt v(g); v!=INVALID; ++v)
121 typename Graph::template NodeMap<Node> ear(g,INVALID);
122 //undefined for the base nodes of the blossoms (i.e. for the
123 //representative elements of UFE blossom) and for the nodes in C
125 UFECrossRef blossom_base(g);
126 UFE blossom(blossom_base);
128 UFECrossRef tree_base(g);
131 //If these UFE's would be members of the class then also
132 //blossom_base and tree_base should be a member.
134 //We build only one tree and the other vertices uncovered by the
135 //matching belong to C. (They can be considered as singleton
136 //trees.) If this tree can be augmented or no more
137 //grow/augmentation/shrink is possible then we return to this
139 for(NodeIt v(g); v!=INVALID; ++v) {
140 if ( position[v]==C && _mate[v]==INVALID ) {
144 if ( heur == 1 ) lateShrink( v, ear, blossom, tree );
145 else normShrink( v, ear, blossom, tree );
151 ///Finds a greedy matching starting from the actual matching.
153 ///Starting form the actual matching stored, it finds a maximal
155 void greedyMatching() {
156 for(NodeIt v(g); v!=INVALID; ++v)
157 if ( _mate[v]==INVALID ) {
158 for( IncEdgeIt e(g,v); e!=INVALID ; ++e ) {
159 Node y=g.runningNode(e);
160 if ( _mate[y]==INVALID && y!=v ) {
169 ///Returns the size of the actual matching stored.
171 ///Returns the size of the actual matching stored. After \ref
172 ///run() it returns the size of a maximum matching in the graph.
175 for(NodeIt v(g); v!=INVALID; ++v) {
176 if ( _mate[v]!=INVALID ) {
184 ///Resets the actual matching to the empty matching.
186 ///Resets the actual matching to the empty matching.
188 void resetMatching() {
189 for(NodeIt v(g); v!=INVALID; ++v)
190 _mate.set(v,INVALID);
193 ///Returns the mate of a node in the actual matching.
195 ///Returns the mate of a \c node in the actual matching.
196 ///Returns INVALID if the \c node is not covered by the actual matching.
197 Node mate(Node& node) const {
201 ///Reads a matching from a \c Node valued \c Node map.
203 ///Reads a matching from a \c Node valued \c Node map. This map
204 ///must be \e symmetric, i.e. if \c map[u]==v then \c map[v]==u
205 ///must hold, and \c uv will be an edge of the matching.
206 template<typename NMapN>
207 void readNMapNode(NMapN& map) {
208 for(NodeIt v(g); v!=INVALID; ++v) {
213 ///Writes the stored matching to a \c Node valued \c Node map.
215 ///Writes the stored matching to a \c Node valued \c Node map. The
216 ///resulting map will be \e symmetric, i.e. if \c map[u]==v then \c
217 ///map[v]==u will hold, and now \c uv is an edge of the matching.
218 template<typename NMapN>
219 void writeNMapNode (NMapN& map) const {
220 for(NodeIt v(g); v!=INVALID; ++v) {
225 ///Reads a matching from an \c UEdge valued \c Node map.
227 ///Reads a matching from an \c UEdge valued \c Node map. \c
228 ///map[v] must be an \c UEdge incident to \c v. This map must
229 ///have the property that if \c g.oppositeNode(u,map[u])==v then
230 ///\c \c g.oppositeNode(v,map[v])==u holds, and now some edge
231 ///joining \c u to \c v will be an edge of the matching.
232 template<typename NMapE>
233 void readNMapEdge(NMapE& map) {
234 for(NodeIt v(g); v!=INVALID; ++v) {
237 _mate.set(v,g.oppositeNode(v,e));
241 ///Writes the matching stored to an \c UEdge valued \c Node map.
243 ///Writes the stored matching to an \c UEdge valued \c Node
244 ///map. \c map[v] will be an \c UEdge incident to \c v. This
245 ///map will have the property that if \c g.oppositeNode(u,map[u])
246 ///== v then \c map[u]==map[v] holds, and now this edge is an edge
248 template<typename NMapE>
249 void writeNMapEdge (NMapE& map) const {
250 typename Graph::template NodeMap<bool> todo(g,true);
251 for(NodeIt v(g); v!=INVALID; ++v) {
252 if ( todo[v] && _mate[v]!=INVALID ) {
254 for(IncEdgeIt e(g,v); e!=INVALID; ++e) {
255 if ( g.runningNode(e) == u ) {
268 ///Reads a matching from a \c bool valued \c Edge map.
270 ///Reads a matching from a \c bool valued \c Edge map. This map
271 ///must have the property that there are no two incident edges \c
272 ///e, \c f with \c map[e]==map[f]==true. The edges \c e with \c
273 ///map[e]==true form the matching.
274 template<typename EMapB>
275 void readEMapBool(EMapB& map) {
276 for(UEdgeIt e(g); e!=INVALID; ++e) {
287 ///Writes the matching stored to a \c bool valued \c Edge map.
289 ///Writes the matching stored to a \c bool valued \c Edge
290 ///map. This map will have the property that there are no two
291 ///incident edges \c e, \c f with \c map[e]==map[f]==true. The
292 ///edges \c e with \c map[e]==true form the matching.
293 template<typename EMapB>
294 void writeEMapBool (EMapB& map) const {
295 for(UEdgeIt e(g); e!=INVALID; ++e) map.set(e,false);
297 typename Graph::template NodeMap<bool> todo(g,true);
298 for(NodeIt v(g); v!=INVALID; ++v) {
299 if ( todo[v] && _mate[v]!=INVALID ) {
301 for(IncEdgeIt e(g,v); e!=INVALID; ++e) {
302 if ( g.runningNode(e) == u ) {
314 ///Writes the canonical decomposition of the graph after running
317 ///After calling any run methods of the class, it writes the
318 ///Gallai-Edmonds canonical decomposition of the graph. \c map
319 ///must be a node map of \ref pos_enum 's.
320 template<typename NMapEnum>
321 void writePos (NMapEnum& map) const {
322 for(NodeIt v(g); v!=INVALID; ++v) map.set(v,position[v]);
328 void lateShrink(Node v, typename Graph::template NodeMap<Node>& ear,
329 UFE& blossom, UFE& tree);
331 void normShrink(Node v, typename Graph::template NodeMap<Node>& ear,
332 UFE& blossom, UFE& tree);
334 void shrink(Node x,Node y, typename Graph::template NodeMap<Node>& ear,
335 UFE& blossom, UFE& tree,std::queue<Node>& Q);
337 void shrinkStep(Node& top, Node& middle, Node& bottom,
338 typename Graph::template NodeMap<Node>& ear,
339 UFE& blossom, UFE& tree, std::queue<Node>& Q);
341 bool growOrAugment(Node& y, Node& x, typename Graph::template
342 NodeMap<Node>& ear, UFE& blossom, UFE& tree,
343 std::queue<Node>& Q);
345 void augment(Node x, typename Graph::template NodeMap<Node>& ear,
346 UFE& blossom, UFE& tree);
351 // **********************************************************************
353 // **********************************************************************
356 template <typename Graph>
357 void MaxMatching<Graph>::lateShrink(Node v, typename Graph::template
358 NodeMap<Node>& ear, UFE& blossom, UFE& tree) {
359 //We have one tree which we grow, and also shrink but only if it cannot be
360 //postponed. If we augment then we return to the "for" cycle of
363 std::queue<Node> Q; //queue of the totally unscanned nodes
366 //queue of the nodes which must be scanned for a possible shrink
368 while ( !Q.empty() ) {
371 for( IncEdgeIt e(g,x); e!= INVALID; ++e ) {
372 Node y=g.runningNode(e);
373 //growOrAugment grows if y is covered by the matching and
374 //augments if not. In this latter case it returns 1.
375 if ( position[y]==C && growOrAugment(y, x, ear, blossom, tree, Q) ) return;
380 while ( !R.empty() ) {
384 for( IncEdgeIt e(g,x); e!=INVALID ; ++e ) {
385 Node y=g.runningNode(e);
387 if ( position[y] == D && blossom.find(x) != blossom.find(y) )
388 //Recall that we have only one tree.
389 shrink( x, y, ear, blossom, tree, Q);
391 while ( !Q.empty() ) {
394 for( IncEdgeIt e(g,x); e!= INVALID; ++e ) {
395 Node y=g.runningNode(e);
396 //growOrAugment grows if y is covered by the matching and
397 //augments if not. In this latter case it returns 1.
398 if ( position[y]==C && growOrAugment(y, x, ear, blossom, tree, Q) ) return;
403 } // while ( !R.empty() )
407 template <typename Graph>
408 void MaxMatching<Graph>::normShrink(Node v,
409 typename Graph::template
411 UFE& blossom, UFE& tree) {
412 //We have one tree, which we grow and shrink. If we augment then we
413 //return to the "for" cycle of runEdmonds().
415 std::queue<Node> Q; //queue of the unscanned nodes
417 while ( !Q.empty() ) {
422 for( IncEdgeIt e(g,x); e!=INVALID; ++e ) {
423 Node y=g.runningNode(e);
425 switch ( position[y] ) {
426 case D: //x and y must be in the same tree
427 if ( blossom.find(x) != blossom.find(y) )
428 //x and y are in the same tree
429 shrink( x, y, ear, blossom, tree, Q);
432 //growOrAugment grows if y is covered by the matching and
433 //augments if not. In this latter case it returns 1.
434 if ( growOrAugment(y, x, ear, blossom, tree, Q) ) return;
443 template <typename Graph>
444 void MaxMatching<Graph>::shrink(Node x,Node y, typename
445 Graph::template NodeMap<Node>& ear,
446 UFE& blossom, UFE& tree, std::queue<Node>& Q) {
447 //x and y are the two adjacent vertices in two blossoms.
449 typename Graph::template NodeMap<bool> path(g,false);
451 Node b=blossom.find(x);
454 while ( b!=INVALID ) {
455 b=blossom.find(ear[b]);
458 } //we go until the root through bases of blossoms and odd vertices
461 Node middle=blossom.find(top);
463 while ( !path[middle] )
464 shrinkStep(top, middle, bottom, ear, blossom, tree, Q);
465 //Until we arrive to a node on the path, we update blossom, tree
466 //and the positions of the odd nodes.
470 middle=blossom.find(top);
472 Node blossom_base=blossom.find(base);
473 while ( middle!=blossom_base )
474 shrinkStep(top, middle, bottom, ear, blossom, tree, Q);
475 //Until we arrive to a node on the path, we update blossom, tree
476 //and the positions of the odd nodes.
478 blossom.makeRep(base);
483 template <typename Graph>
484 void MaxMatching<Graph>::shrinkStep(Node& top, Node& middle, Node& bottom,
485 typename Graph::template
487 UFE& blossom, UFE& tree,
488 std::queue<Node>& Q) {
489 //We traverse a blossom and update everything.
493 while ( t!=middle ) {
498 bottom=_mate[middle];
499 position.set(bottom,D);
502 Node oldmiddle=middle;
503 middle=blossom.find(top);
505 tree.erase(oldmiddle);
506 blossom.insert(bottom);
507 blossom.join(bottom, oldmiddle);
508 blossom.join(top, oldmiddle);
512 template <typename Graph>
513 bool MaxMatching<Graph>::growOrAugment(Node& y, Node& x, typename Graph::template
514 NodeMap<Node>& ear, UFE& blossom, UFE& tree,
515 std::queue<Node>& Q) {
516 //x is in a blossom in the tree, y is outside. If y is covered by
517 //the matching we grow, otherwise we augment. In this case we
520 if ( _mate[y]!=INVALID ) { //grow
528 tree.join(y,blossom.find(x));
532 augment(x, ear, blossom, tree);
541 template <typename Graph>
542 void MaxMatching<Graph>::augment(Node x,
543 typename Graph::template NodeMap<Node>& ear,
544 UFE& blossom, UFE& tree) {
546 while ( v!=INVALID ) {
554 Node y=blossom.find(x);
555 for (typename UFE::ItemIt tit(tree, y); tit != INVALID; ++tit) {
556 if ( position[tit] == D ) {
557 for (typename UFE::ItemIt bit(blossom, tit); bit != INVALID; ++bit) {
558 position.set( bit ,C);
560 blossom.eraseClass(tit);
561 } else position.set( tit ,C);
568 } //END OF NAMESPACE LEMON
570 #endif //LEMON_MAX_MATCHING_H