We give an improved connectivity oracle under vertex failures. After a set of $k$ vertices fails, our oracle performs an $O(k^{6})$-time update independent of the graph size $n$, and then answers pairwise connectivity queries in optimal $O(k)$ time. For constant $k$, it uses near-linear space and can be built in near-linear preprocessing time. In contrast, all prior oracles with $n$-independent update time[PSS+22, vdBS19] either require $Ω(n^{2})$ space or incur $2^{2^{O(k)}}$ update and query time. Moreover, their preprocessing time is polynomially large in $n$, far from near-linear. Our oracle builds on the unbreakable decomposition framework of[PSS+22], but introduces three new ingredients: (i) shortcutting over the tree decomposition to reduce space from quadratic to near-linear, (ii) bootstrapping that leverages $n$-dependent oracles internally to obtain near-linear preprocessing, and (iii) a new patch set mechanism that yields conditionally optimal $O(k)$ query time.

翻译：我们提出了一种改进的顶点故障下的连接性预言。在发生 $k$ 个顶点故障后，该预言机执行一次独立于图规模 $n$ 的 $O(k^{6})$ 时间更新，随后在最优的 $O(k)$ 时间内回答成对连接性查询。对于常数 $k$，该结构使用近线性空间，并可在近线性预处理时间内构建。相比之下，所有先前具有与 $n$ 无关更新时间的预言机[PSS+22, vdBS19]要么需要 $\Omega(n^{2})$ 空间，要么导致 $2^{2^{O(k)}}$ 的更新与查询时间。此外，其预处理时间在 $n$ 上多项式级大，远非近线性。我们的预言机基于[PSS+22]的不可破分解框架，但引入了三项新要素：(i) 对树分解进行捷径化，将空间从二次降低至近线性；(ii) 自举方法，内部利用依赖 $n$ 的预言机以获得近线性预处理；(iii) 新的补丁集机制，实现条件最优的 $O(k)$ 查询时间。