Dynamic matchings in left weighted convex bipartite graphs

Abstract

We consider the problem which is dynamically maintaining a maximum weight matching in a left weighted convex bipartite graph G = (V,E), V = X ∪ Y, in which each x ∈ X has an associated weight, and neighbors of each x ∈ X form an interval in the ordered Y set. The maintenance includes update operations (vertices and edges insertions and deletions) and query operations (inquiries of a vertex matching information). We reduce this problem to the corresponding unweighted problem and design an algorithm that maintains the update operations in O(log3|V|) amortized time per update. In addition, we develop a data structure to obtain the matching status of a vertex (whether it is matched) in constant worst-case time, and find the pair of a matched vertex (with which it is matched) in worst-case O(k) time, where k is not greater than the cardinality of the maximum weight matching. © 2014 Springer International Publishing.