Space Complexity of Vertex Connectivity Oracles

TL;DR

This paper investigates the space complexity of vertex connectivity oracles, establishing lower bounds and presenting an optimal-space oracle that answers queries efficiently in undirected graphs.

Contribution

It provides a tight lower bound on space and constructs an optimal-space oracle with fast query time, improving understanding of efficiency limits.

Findings

Lower bound of (kn) bits on space complexity.

An oracle with optimal space and ( log n) query time.

Queries answered in ( log n) time, independent of k.

Abstract

A $k$-vertex connectivity oracle for undirected $G$ is a data structure that, given $u,v\in V(G)$, reports $\min\{k,\kappa(u,v)\}$, where $\kappa(u,v)$ 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 $\tilde{O}(kn)$ can even be encoded as a $\tilde{O}(k)$-bit labeling scheme so that vertex-connectivity queries can be answered in $\tilde{O}(k)$ 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 $\Omega(kn)$-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 $k$.