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Erschienen in:
2007 | OriginalPaper | Buchkapitel
Dynamic Matchings in Convex Bipartite Graphs
verfasst von : Gerth Stølting Brodal, Loukas Georgiadis, Kristoffer Arnsfelt Hansen, Irit Katriel
Erschienen in: Mathematical Foundations of Computer Science 2007
Verlag: Springer Berlin Heidelberg
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We consider the problem of maintaining a maximum matching in a convex bipartite graph
G
= (
V
,
E
) under a set of update operations which includes insertions and deletions of vertices and edges. It is not hard to show that it is impossible to maintain an explicit representation of a maximum matching in sub-linear time per operation, even in the amortized sense. Despite this difficulty, we develop a data structure which maintains the set of vertices that participate in a maximum matching in
O
(log
2
|
V
|) amortized time per update and reports the status of a vertex (matched or unmatched) in constant worst-case time. Our structure can report the mate of a matched vertex in the maximum matching in worst-case
O
( min {
k
log
2
|
V
| + log|
V
|, |
V
| log|
V
|}) time, where
k
is the number of update operations since the last query for the same pair of vertices was made. In addition, we give an
$O(\sqrt{|V|} \log^2{|V|})$
-time amortized bound for this pair query.