We present succinct labeling schemes for answering connectivity queries in graphs subject to a specified number of vertex failures. An $f$-vertex/edge fault tolerant ($f$-V/EFT) connectivity labeling is a scheme that produces succinct labels for the vertices (and possibly to the edges) of an $n$-vertex graph $G$, such that given only the labels of two vertices $s,t$ and of at most $f$ faulty vertices/edges $F$, one can infer if $s$ and $t$ are connected in $G-F$. The primary complexity measure is the maximum label length (in bits). The $f$-EFT setting is relatively well understood: [Dory and Parter, PODC 2021] gave a randomized scheme with succinct labels of $O(\log^3 n)$ bits, which was subsequently derandomized by [Izumi et al., PODC 2023] with $\tilde{O}(f^2)$-bit labels. As both noted, handling vertex faults is more challenging. The known bounds for the $f$-VFT setting are far away: [Parter and Petruschka, DISC 2022] gave $\tilde{O}(n^{1-1/2^{\Theta(f)}})$-bit labels, which is linear in $n$ already for $f =\Omega(\log\log n)$. In this work we present an efficient $f$-VFT connectivity labeling scheme using $poly(f, \log n)$ bits. Specifically, we present a randomized scheme with $O(f^3 \log^5 n)$-bit labels, and a derandomized version with $O(f^7 \log^{13} n)$-bit labels, compared to an $\Omega(f)$-bit lower bound on the required label length. Our schemes are based on a new low-degree graph decomposition that improves on [Duan and Pettie, SODA 2017], and facilitates its distributed representation into labels. Finally, we show that our labels naturally yield routing schemes avoiding a given set of at most $f$ vertex failures with table and header sizes of only $poly(f,\log n)$ bits. This improves significantly over the linear size bounds implied by the EFT routing scheme of Dory and Parter.

翻译：我们提出简洁标记方案，用于回答图中在指定数量顶点故障条件下的连通性查询。$f$顶点/边故障容忍（$f$-V/EFT）连通性标记是一种方案，可为$n$顶点图$G$的顶点（以及可能边）生成简洁标记，使得仅通过两个顶点$s,t$的标记及至多$f$个故障顶点/边$F$的标记，即可推断$s$和$t$在$G-F$中是否连通。主要复杂度度量是最大标记长度（以比特计）。$f$-EFT场景已得到较好理解：[Dory and Parter, PODC 2021]提出了$O(\log^3 n)$比特的随机化方案，随后[Izumi et al., PODC 2023]通过$\tilde{O}(f^2)$比特标记实现了去随机化。正如两者所言，处理顶点故障更具挑战性。已知$f$-VFT场景的边界差距显著：[Parter and Petruschka, DISC 2022]给出了$\tilde{O}(n^{1-1/2^{\Theta(f)}})$比特标记，当$f =\Omega(\log\log n)$时已为$n$的线性量级。本文提出一种高效的$f$-VFT连通性标记方案，使用$poly(f, \log n)$比特。具体而言，我们提出$O(f^3 \log^5 n)$比特的随机化方案，以及$O(f^7 \log^{13} n)$比特的去随机化版本，而所需标记长度的下界为$\Omega(f)$比特。我们的方案基于一种新的低度图分解，该分解改进了[Duan and Pettie, SODA 2017]的工作，并促使其分布式表示融入标记中。最后，我们证明这些标记可自然地导出路由方案，避免给定至多$f$个顶点故障集，其表和头部大小仅为$poly(f,\log n)$比特。这显著优于Dory和Parter的EFT路由方案所隐含的线性尺寸边界。