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(log2 |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 log2 |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(√|V| log2 |V|)-time amortized bound for this pair query. © Springer-Verlag Berlin Heidelberg 2007.
CITATION STYLE
Brodal, G. S., Georgiadis, L., Hansen, K. A., & Katriel, I. (2007). Dynamic matchings in convex bipartite graphs. In Lecture Notes in Computer Science (including subseries Lecture Notes in Artificial Intelligence and Lecture Notes in Bioinformatics) (Vol. 4708 LNCS, pp. 406–417). Springer Verlag. https://doi.org/10.1007/978-3-540-74456-6_37
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