Optimal Vertex Connectivity Oracles

01/02/2022

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by   Seth Pettie, et al.

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A k-vertex connectivity oracle for undirected G is a data structure that, given u,v∈ V(G), reports min{k,κ(u,v)}, where κ(u,v) is the pairwise vertex connectivity between u,v. There are three main measures of efficiency: construction time, query time, and space. Prior work of Izsak and Nutov shows that a data structure of total size Õ(kn) can even be encoded as a Õ(k)-bit labeling scheme so that vertex-connectivity queries can be answered in Õ(k) time. The construction time is polynomial, but unspecified. In this paper we address the top three complexity measures: Space, Query Time, and Construction Time. We give an Ω(kn)-bit lower bound on any vertex connectivity oracle. We construct an optimal-space connectivity oracle in max-flow time that answers queries in O(log n) time, independent of k.