Do not append -Wall -W to CXXFLAGS when it is set on the commandline.
2 * lemon/max_matching.h - Part of LEMON, a generic C++ optimization library
4 * Copyright (C) 2006 Egervary Jeno Kombinatorikus Optimalizalasi Kutatocsoport
5 * (Egervary Research Group on Combinatorial Optimization, EGRES).
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.
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
17 #ifndef LEMON_MAX_MATCHING_H
18 #define LEMON_MAX_MATCHING_H
21 #include <lemon/invalid.h>
22 #include <lemon/unionfind.h>
23 #include <lemon/graph_utils.h>
27 ///\brief Maximum matching algorithm.
34 ///Edmonds' alternating forest maximum matching algorithm.
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.
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.
52 ///\param Graph The undirected graph type the algorithm runs on.
54 ///\author Jacint Szabo
55 template <typename Graph>
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;
67 typedef UnionFindEnum<Node, Graph::template NodeMap> UFE;
71 ///Indicates the Gallai-Edmonds decomposition of the graph.
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.
86 static const int HEUR_density=2;
88 typename Graph::template NodeMap<Node> _mate;
89 typename Graph::template NodeMap<pos_enum> position;
93 MaxMatching(const Graph& _g) : g(_g), _mate(_g,INVALID), position(_g) {}
95 ///Runs Edmonds' algorithm.
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.
101 if ( countUndirEdges(g) < HEUR_density*countNodes(g) ) {
104 } else runEdmonds(1);
108 ///Runs Edmonds' algorithm.
110 ///If heur=0 it runs Edmonds' algorithm. If heur=1 it runs
111 ///Edmonds' algorithm with a heuristic of postponing shrinks,
112 ///giving a faster algorithm for dense graphs.
113 void runEdmonds( int heur = 1 ) {
115 for(NodeIt v(g); v!=INVALID; ++v)
118 typename Graph::template NodeMap<Node> ear(g,INVALID);
119 //undefined for the base nodes of the blossoms (i.e. for the
120 //representative elements of UFE blossom) and for the nodes in C
122 typename UFE::MapType blossom_base(g);
123 UFE blossom(blossom_base);
124 typename UFE::MapType tree_base(g);
126 //If these UFE's would be members of the class then also
127 //blossom_base and tree_base should be a member.
129 for(NodeIt v(g); v!=INVALID; ++v) {
130 if ( position[v]==C && _mate[v]==INVALID ) {
134 if ( heur == 1 ) lateShrink( v, ear, blossom, tree );
135 else normShrink( v, ear, blossom, tree );
141 ///Finds a greedy matching starting from the actual matching.
143 ///Starting form the actual matching stored, it finds a maximal
145 void greedyMatching() {
146 for(NodeIt v(g); v!=INVALID; ++v)
147 if ( _mate[v]==INVALID ) {
148 for( IncEdgeIt e(g,v); e!=INVALID ; ++e ) {
149 Node y=g.runningNode(e);
150 if ( _mate[y]==INVALID && y!=v ) {
159 ///Returns the size of the actual matching stored.
161 ///Returns the size of the actual matching stored. After \ref
162 ///run() it returns the size of a maximum matching in the graph.
165 for(NodeIt v(g); v!=INVALID; ++v) {
166 if ( _mate[v]!=INVALID ) {
174 ///Resets the actual matching to the empty matching.
176 ///Resets the actual matching to the empty matching.
178 void resetMatching() {
179 for(NodeIt v(g); v!=INVALID; ++v)
180 _mate.set(v,INVALID);
183 ///Returns the mate of a node in the actual matching.
185 ///Returns the mate of a \c node in the actual matching.
186 ///Returns INVALID if the \c node is not covered by the actual matching.
187 Node mate(Node& node) const {
191 ///Reads a matching from a \c Node valued \c Node map.
193 ///Reads a matching from a \c Node valued \c Node map. This map
194 ///must be \e symmetric, i.e. if \c map[u]==v then \c map[v]==u
195 ///must hold, and \c uv will be an edge of the matching.
196 template<typename NMapN>
197 void readNMapNode(NMapN& map) {
198 for(NodeIt v(g); v!=INVALID; ++v) {
203 ///Writes the stored matching to a \c Node valued \c Node map.
205 ///Writes the stored matching to a \c Node valued \c Node map. The
206 ///resulting map will be \e symmetric, i.e. if \c map[u]==v then \c
207 ///map[v]==u will hold, and now \c uv is an edge of the matching.
208 template<typename NMapN>
209 void writeNMapNode (NMapN& map) const {
210 for(NodeIt v(g); v!=INVALID; ++v) {
215 ///Reads a matching from an \c UndirEdge valued \c Node map.
217 ///Reads a matching from an \c UndirEdge valued \c Node map. \c
218 ///map[v] must be an \c UndirEdge incident to \c v. This map must
219 ///have the property that if \c g.oppositeNode(u,map[u])==v then
220 ///\c \c g.oppositeNode(v,map[v])==u holds, and now some edge
221 ///joining \c u to \c v will be an edge of the matching.
222 template<typename NMapE>
223 void readNMapEdge(NMapE& map) {
224 for(NodeIt v(g); v!=INVALID; ++v) {
227 _mate.set(v,g.oppositeNode(v,e));
231 ///Writes the matching stored to an \c UndirEdge valued \c Node map.
233 ///Writes the stored matching to an \c UndirEdge valued \c Node
234 ///map. \c map[v] will be an \c UndirEdge incident to \c v. This
235 ///map will have the property that if \c g.oppositeNode(u,map[u])
236 ///== v then \c map[u]==map[v] holds, and now this edge is an edge
238 template<typename NMapE>
239 void writeNMapEdge (NMapE& map) const {
240 typename Graph::template NodeMap<bool> todo(g,true);
241 for(NodeIt v(g); v!=INVALID; ++v) {
242 if ( todo[v] && _mate[v]!=INVALID ) {
244 for(IncEdgeIt e(g,v); e!=INVALID; ++e) {
245 if ( g.runningNode(e) == u ) {
258 ///Reads a matching from a \c bool valued \c Edge map.
260 ///Reads a matching from a \c bool valued \c Edge map. This map
261 ///must have the property that there are no two incident edges \c
262 ///e, \c f with \c map[e]==map[f]==true. The edges \c e with \c
263 ///map[e]==true form the matching.
264 template<typename EMapB>
265 void readEMapBool(EMapB& map) {
266 for(UndirEdgeIt e(g); e!=INVALID; ++e) {
277 ///Writes the matching stored to a \c bool valued \c Edge map.
279 ///Writes the matching stored to a \c bool valued \c Edge
280 ///map. This map will have the property that there are no two
281 ///incident edges \c e, \c f with \c map[e]==map[f]==true. The
282 ///edges \c e with \c map[e]==true form the matching.
283 template<typename EMapB>
284 void writeEMapBool (EMapB& map) const {
285 for(UndirEdgeIt e(g); e!=INVALID; ++e) map.set(e,false);
287 typename Graph::template NodeMap<bool> todo(g,true);
288 for(NodeIt v(g); v!=INVALID; ++v) {
289 if ( todo[v] && _mate[v]!=INVALID ) {
291 for(IncEdgeIt e(g,v); e!=INVALID; ++e) {
292 if ( g.runningNode(e) == u ) {
304 ///Writes the canonical decomposition of the graph after running
307 ///After calling any run methods of the class, it writes the
308 ///Gallai-Edmonds canonical decomposition of the graph. \c map
309 ///must be a node map of \ref pos_enum 's.
310 template<typename NMapEnum>
311 void writePos (NMapEnum& map) const {
312 for(NodeIt v(g); v!=INVALID; ++v) map.set(v,position[v]);
318 void lateShrink(Node v, typename Graph::template NodeMap<Node>& ear,
319 UFE& blossom, UFE& tree);
321 void normShrink(Node v, typename Graph::template NodeMap<Node>& ear,
322 UFE& blossom, UFE& tree);
324 bool noShrinkStep(Node x, typename Graph::template NodeMap<Node>& ear,
325 UFE& blossom, UFE& tree, std::queue<Node>& Q);
327 void shrinkStep(Node& top, Node& middle, Node& bottom,
328 typename Graph::template NodeMap<Node>& ear,
329 UFE& blossom, UFE& tree, std::queue<Node>& Q);
331 void augment(Node x, typename Graph::template NodeMap<Node>& ear,
332 UFE& blossom, UFE& tree);
337 // **********************************************************************
339 // **********************************************************************
342 template <typename Graph>
343 void MaxMatching<Graph>::lateShrink(Node v, typename Graph::template NodeMap<Node>& ear,
344 UFE& blossom, UFE& tree) {
346 std::queue<Node> Q; //queue of the totally unscanned nodes
349 //queue of the nodes which must be scanned for a possible shrink
351 while ( !Q.empty() ) {
354 if ( noShrinkStep( x, ear, blossom, tree, Q ) ) return;
358 while ( !R.empty() ) {
362 for( IncEdgeIt e(g,x); e!=INVALID ; ++e ) {
363 Node y=g.runningNode(e);
365 if ( position[y] == D && blossom.find(x) != blossom.find(y) ) {
366 //x and y must be in the same tree
368 typename Graph::template NodeMap<bool> path(g,false);
370 Node b=blossom.find(x);
373 while ( b!=INVALID ) {
374 b=blossom.find(ear[b]);
377 } //going till the root
380 Node middle=blossom.find(top);
382 while ( !path[middle] )
383 shrinkStep(top, middle, bottom, ear, blossom, tree, Q);
387 middle=blossom.find(top);
389 Node blossom_base=blossom.find(base);
390 while ( middle!=blossom_base )
391 shrinkStep(top, middle, bottom, ear, blossom, tree, Q);
393 blossom.makeRep(base);
394 } // if shrink is needed
396 while ( !Q.empty() ) {
399 if ( noShrinkStep(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 std::queue<Node> Q; //queue of the unscanned nodes
414 while ( !Q.empty() ) {
419 for( IncEdgeIt e(g,x); e!=INVALID; ++e ) {
420 Node y=g.runningNode(e);
422 switch ( position[y] ) {
423 case D: //x and y must be in the same tree
425 if ( blossom.find(x) != blossom.find(y) ) { //shrink
426 typename Graph::template NodeMap<bool> path(g,false);
428 Node b=blossom.find(x);
431 while ( b!=INVALID ) {
432 b=blossom.find(ear[b]);
435 } //going till the root
438 Node middle=blossom.find(top);
440 while ( !path[middle] )
441 shrinkStep(top, middle, bottom, ear, blossom, tree, Q);
445 middle=blossom.find(top);
447 Node blossom_base=blossom.find(base);
448 while ( middle!=blossom_base )
449 shrinkStep(top, middle, bottom, ear, blossom, tree, Q);
451 blossom.makeRep(base);
455 if ( _mate[y]!=INVALID ) { //grow
464 tree.join(y,blossom.find(x));
468 augment(x, ear, blossom, tree);
480 template <typename Graph>
481 bool MaxMatching<Graph>::noShrinkStep(Node x,
482 typename Graph::template
484 UFE& blossom, UFE& tree,
485 std::queue<Node>& Q) {
486 for( IncEdgeIt e(g,x); e!= INVALID; ++e ) {
487 Node y=g.runningNode(e);
489 if ( position[y]==C ) {
490 if ( _mate[y]!=INVALID ) { //grow
498 tree.join(y,blossom.find(x));
502 augment(x, ear, blossom, tree);
512 template <typename Graph>
513 void MaxMatching<Graph>::shrinkStep(Node& top, Node& middle, Node& bottom,
514 typename Graph::template
516 UFE& blossom, UFE& tree,
517 std::queue<Node>& Q) {
520 while ( t!=middle ) {
525 bottom=_mate[middle];
526 position.set(bottom,D);
529 Node oldmiddle=middle;
530 middle=blossom.find(top);
532 tree.erase(oldmiddle);
533 blossom.insert(bottom);
534 blossom.join(bottom, oldmiddle);
535 blossom.join(top, oldmiddle);
538 template <typename Graph>
539 void MaxMatching<Graph>::augment(Node x,
540 typename Graph::template NodeMap<Node>& ear,
541 UFE& blossom, UFE& tree) {
543 while ( v!=INVALID ) {
551 typename UFE::ItemIt it;
552 for (tree.first(it,blossom.find(x)); tree.valid(it); tree.next(it)) {
553 if ( position[it] == D ) {
554 typename UFE::ItemIt b_it;
555 for (blossom.first(b_it,it); blossom.valid(b_it); blossom.next(b_it)) {
556 position.set( b_it ,C);
558 blossom.eraseClass(it);
559 } else position.set( it ,C);
567 } //END OF NAMESPACE LEMON
569 #endif //LEMON_MAX_MATCHING_H