Near-Optimal Deterministic Vertex-Failure Connectivity Oracles

TL;DR

This paper introduces a deterministic data structure for vertex-failure connectivity queries that is nearly optimal in space, preprocessing, update, and query times, significantly improving previous algorithms.

Contribution

It presents a new deterministic data structure that achieves near-optimal complexity bounds for vertex-failure connectivity, improving upon prior work in space and update efficiency.

Findings

Preprocessing time is (md_{\u221a})

Update time is (d^{2})

Query time is (d)

Abstract

We revisit the vertex-failure connectivity oracle problem. This is one of the most basic graph data structure problems under vertex updates, yet its complexity is still not well-understood. We essentially settle the complexity of this problem by showing a new data structure whose space, preprocessing time, update time, and query time are simultaneously optimal up to sub-polynomial factors assuming popular conjectures. Moreover, the data structure is deterministic. More precisely, for any integer $d_{\star}$, the data structure preprocesses a graph $G$ with $n$ vertices and $m$ edges in $\hat{O}(md_{\star})$ time and uses $\tilde{O}(\min\{m,nd_{\star}\})$ space. Then, given the vertex set $D$ to be deleted where $|D|=d\le d_{\star}$, it takes $\hat{O}(d^{2})$ updates time. Finally, given any vertex pair $(u,v)$, it checks if $u$ and $v$ are connected in $G\setminus D$ in $O(d)$ time. This improves the previously best deterministic algorithm by Duan and Pettie (SODA 2017) in both space and update time by a factor of $d$. It also significantly speeds up the $\Omega(\min\{mn,n^{\omega}\})$ preprocessing time of all known (even randomized) algorithms with update time at most $\tilde{O}(d^{5})$.