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

Zeev Nutov

TL;DR

A very simple proof is shown that this is so even if G is-connected, answering the open question whether $\Omega(kn)$ bits are still necessary if $G$ is $k$-connected.

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 $Ω(kn)$ bits of space. They asked whether $Ω(kn)$ 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.

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

TL;DR

A very simple proof is shown that this is so even if G is-connected, answering the open question whether bits are still necessary if is -connected.

Abstract

A -connectivity oracle for a graph is a data structure that given determines whether there are at least internally disjoint -paths in . For undirected graphs, Pettie, Saranurak & Yin [STOC 2022, pp. 151-161] proved that any -connectivity oracle requires bits of space. They asked whether bits are still necessary if is -connected. We will show by a very simple proof that this is so even if is -connected, answering this open question.
Paper Structure (2 sections, 4 theorems)

This paper contains 2 sections, 4 theorems.

Table of Contents

  1. Introduction
  2. Proof of Theorem \ref{['t1']}

Key Result

Theorem 1

For any integers $k \ge 1$ and $1 \le p \le 2^{k^2/4}$ there exists a $k$-connected graph $G=G(k,p)$ with $n=k(p+1)$ nodes such that any $k$-connectivity oracle for $G$ requires at least $p \cdot k^2/4 = \left(1-\frac{1}{p+1}\right) \cdot kn/4$ bits of space. Consequently, a $k$-connectivity oracle

Theorems & Definitions (5)

  • Theorem 1
  • Definition 2
  • Lemma 3: LN
  • Corollary 4
  • Lemma 5