\~{O}ptimal Dual Vertex Failure Connectivity Labels

TL;DR

This paper introduces the first sublinear vertex failure connectivity labeling schemes for general graphs, enabling efficient connectivity queries under multiple vertex failures with compact labels.

Contribution

It presents the first sublinear $f$-VFT labeling schemes for any $n$-vertex graph, including a novel 2-VFT scheme with $O( ext{log}^3 n)$ bits, advancing fault-tolerant connectivity labeling.

Findings

Developed 2-VFT connectivity labels with $O( ext{log}^3 n)$ bits.

Provided $f$-VFT labels with sub-linear length for $f=o( ext{log} ext{log} n)$.

Analyzed dual failure replacement paths using heavy-light tree decomposition.

Abstract

In this paper we present succinct labeling schemes for supporting connectivity queries under vertex faults. For a given $n$-vertex graph $G$, an $f$-VFT (resp., EFT) connectivity labeling scheme is a distributed data structure that assigns each of the graph edges and vertices a short label, such that given the labels of a vertex pair $u$ and $v$, and the labels of at most $f$ failing vertices (resp., edges) $F$, one can determine if $u$ and $v$ are connected in $G \setminus F$. The primary complexity measure is the length of the individual labels. Since their introduction by [Courcelle, Twigg, STACS '07], FT labeling schemes have been devised only for a limited collection of graph families. A recent work [Dory and Parter, PODC 2021] provided EFT labeling schemes for general graphs under edge failures, leaving the vertex failure case fairly open. We provide the first sublinear $f$-VFT labeling schemes for $f \geq 2$ for any $n$-vertex graph. Our key result is $2$-VFT connectivity labels with $O(\log^3 n)$ bits. Our constructions are based on analyzing the structure of dual failure replacement paths on top of the well-known heavy-light tree decomposition technique of [Sleator and Tarjan, STOC 1981]. We also provide $f$-VFT labels with sub-linear length (in $|V|$) for any $f=o(\log\log n)$, that are based on a reduction to the existing EFT labels.