Space Complexity of Vertex Connectivity Oracles

TL;DR

This work resolves a long-standing question on the space efficiency of vertex-connectivity oracles by proving a near-linear lower bound of $\Omega(kn)$ bits and delivering an optimal-space oracle that answers queries in $O(\log n)$ time, independent of $k$, within max-flow time. It refines the Izsak-Nutov labeling framework to achieve $O(\min\{kn\log n, m\log n\log k\})$ space, using structured label sets and efficient set-intersection queries. The paper further develops approximate Gomory-Hu trees for element connectivity, enabling $(1+\epsilon)$-approximate representations with running time $O(\epsilon^{-1}\log^6 n)\cdot T_{\mathrm{flow}}(m)$ and addressing obstacles posed by forbidden terminals in recursive constructions. Together, these results unify lower bounds with nearly optimal structural and algorithmic tools for both vertex and element connectivity, highlighting fundamental limits and near-optimal tradeoffs for connectivity oracles in graph algorithms.

Abstract

A $k$-vertex connectivity oracle for undirected $G$ is a data structure that, given $u,v\in V(G)$, 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 $\tilde{O}(kn)$ 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 $Ω(kn)$-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 $k$.

Figures (1)

• Figure 1: $P_0$ consists of $C(i,j)$ length-2 paths from $x_i$ to $z_j$. (a)$G_{1,x},G_{1,z}$ contain perfect matchings $M_{1,x},M_{1,z}$ with high probability. $P_1$ consists of $|Y_{1,x}|=n-1$ paths via $M_{1,x},M_{1,z}$. (b)$G_{2,x},G_{2,z}$ contain perfect matchings $M_{1,x},M_{1,z}$ with high probability. $P_2$ consists of $|Y_{2,x}|=\min\{n-1,4n-|P_0|-|P_1|\}$ paths via $M_{2,x},M_{2,z}$.

Theorems & Definitions (40)

• definition 2.1
• lemma 2.2
• lemma 2.3
• lemma 2.4
• proof
• definition 2.5
• lemma 2.6: Isolating cuts, vertex version with forbidden terminals
• proof
• theorem 3.1
• lemma 3.2