We present an efficient labeling scheme for answering connectivity queries in graphs subject to a specified number of vertex failures. Our first result is a randomized construction of a labeling function that assigns vertices $O(f^3\log^5 n)$-bit labels, such that given the labels of $F\cup \{s,t\}$ where $|F|\leq f$, we can correctly report, with probability $1-1/\mathrm{poly}(n)$, whether $s$ and $t$ are connected in $G-F$. However, it is possible that over all $n^{O(f)}$ distinct queries, some are answered incorrectly. Our second result is a deterministic labeling function that produces $O(f^7 \log^{13} n)$-bit labels such that all connectivity queries are answered correctly. Both upper bounds are polynomially off from an $\Omega(f)$-bit lower bound. Our labeling schemes are based on a new low degree decomposition that improves the Duan-Pettie decomposition, and facilitates its distributed representation. We make heavy use of randomization to construct hitting sets, fault-tolerant graph sparsifiers, and in constructing linear sketches. Our derandomized labeling scheme combines a variety of techniques: the method of conditional expectations, hit-miss hash families, and $\epsilon$-nets for axis-aligned rectangles. The prior labeling scheme of Parter and Petruschka shows that $f=1$ and $f=2$ vertex faults can be handled with $O(\log n)$- and $O(\log^3 n)$-bit labels, respectively, and for $f>2$ vertex faults, $\tilde{O}(n^{1-1/2^{f-2}})$-bit labels suffice.

翻译：我们提出了一种高效标记方案，用于回答图中受指定数量顶点故障影响的连通性查询。首先，我们通过随机化方法构造一个标记函数，为每个顶点分配 $O(f^3\log^5 n)$ 比特的标记，使得给定 $F\cup \{s,t\}$ 的标记（其中 $|F|\leq f$），我们能够以概率 $1-1/\mathrm{poly}(n)$ 正确报告 $s$ 和 $t$ 在 $G-F$ 中是否连通。然而，在总共 $n^{O(f)}$ 个不同查询中，部分查询可能得到错误答案。其次，我们提出一个确定性标记函数，生成 $O(f^7 \log^{13} n)$ 比特的标记，使得所有连通性查询均能被正确回答。这两个上界与 $\Omega(f)$ 比特的下界之间仅相差多项式因子。我们的标记方案基于一种新的低度分解方法，改进了 Duan-Pettie 分解，并支持其分布式表示。我们大量使用随机化技术来构建命中集、容错图稀疏化器以及线性草图。去随机化标记方案结合了多种技术：条件期望法、命中-未命中哈希族以及轴对齐矩形的 $\epsilon$-网。此前 Parter 和 Petruschka 的标记方案表明，$f=1$ 和 $f=2$ 的顶点故障可分别用 $O(\log n)$ 和 $O(\log^3 n)$ 比特的标记处理，而对于 $f>2$ 的顶点故障，$\tilde{O}(n^{1-1/2^{f-2}})$ 比特的标记已足够。