Dynamic Matchings in Convex Bipartite Graphs

Abstract

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 $O(\sqrt{|V|} \log^2{|V|})$-time amortized bound for this pair query.