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.
翻译:在本文中,我们展示了支持顶端断层下连接查询的简明标签办法。对于一个给定的美元-顶端方正方块$G$,一个美元-VFT(resp.,EFT)连接标签办法是一个分布式数据结构,对每个图形边缘进行分配,对顶端边缘作出一个短标签,因此,鉴于顶端对面对美元和美元对面端断层的标签,最多为美元(resp.,边缘)的顶端螺旋(Rew.),人们可以确定美元和美元与美元之间是否连接在一起。对于美元/Setminus F$,主要复杂度是单个标签的长度。自从[Courcelle, Twigg,STACS '07]推出以来,FT标签办法只为有限的图形家庭设计。最近的一项工作[Dory和Parter,PODC 20211,为在边缘故障下的一般面板块提供了EFFT标签办法,使头部的底部值为美元-美元-美元。我们头部的底部的底部的内基结构故障也为2美元-FTFTFA结果。我们提供了第一位基的基的基的基的基底部的基件。