On $k$-connectivity oracles in $k$-connected graphs

TL;DR

This paper proves that any $k$-connectivity oracle for $k$-connected graphs requires at least $ ilde{ ext{Omega}}(kn)$ bits of space, confirming a conjecture about the space complexity in such graphs.

Contribution

The paper provides a simple proof that $k$-connectivity oracles need $ ilde{ ext{Omega}}(kn)$ bits even for $k$-connected graphs, answering an open question.

Findings

Any $k$-connectivity oracle requires $ ilde{ ext{Omega}}(kn)$ bits of space.

The proof simplifies previous complex arguments.

Confirms the space lower bound for $k$-connected graphs.

Abstract

A $k$-connectivity oracle for a graph $G=(V,E)$ is a data structure that given $s,t \in V$ determines whether there are at least $k+1$ internally disjoint $st$-paths in $G$. For undirected graphs, Pettie, Saranurak & Yin [STOC 2022, pp. 151-161] proved that any $k$-connectivity oracle requires $\Omega(kn)$ bits of space. They asked whether $\Omega(kn)$ bits are still necessary if $G$ is $k$-connected. We will show by a very simple proof that this is so even if $G$ is $k$-connected, answering this open question.