SIAM Journal on Computing, Volume 49, Issue 6, Page 1363-1396, January 2020.
We introduce new data structures for answering connectivity queries in graphs subject to batched vertex failures. A deterministic structure processes a batch of $d\leq d_\star$ failed vertices in $\tilde{O}(d^3)$ time and thereafter answers connectivity queries in $O(d)$ time. It occupies space $O(d_\star m\log n)$. We develop a randomized Monte Carlo version of our data structure with update time $\tilde{O}(d^2)$, query time $O(d)$, and space $\tilde{O}(m)$ for any failure bound $d\le n$. This is the first connectivity oracle for general graphs that can efficiently deal with an unbounded number of vertex failures. We also develop a more efficient Monte Carlo edge failure connectivity oracle. Using space $O(n\log^2 n)$, $d$ edge failures are processed in $O(d\log d\log\log n)$ time, and thereafter, connectivity queries are answered in $O(\log\log n)$ time, which are correct with high probability. Our data structures are based on a new decomposition theorem for an undirected graph $G=(V,E)$, which is of independent interest. It states that for any terminal set $U\subseteq V$ we can remove a set $B$ of $|U|/(s-2)$ vertices such that the remaining graph contains a Steiner forest for $U-B$ with maximum degree $s$.

中文翻译：

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易受顶点故障影响的图形连接Oracle

SIAM计算学报，第49卷，第6期，第1363-1396页，2020年1月。
我们引入了新的数据结构，用于在受批顶点故障影响的图中回答连接查询。确定性结构在$ \ tilde {O}（d ^ 3）$时间内处理一批$ d \ leq d_ \ star $失败的顶点，然后在$ O（d）$时间内回答连接查询。它占用空间$ O（d_ \ star m \ log n）$。我们开发了数据结构的随机蒙特卡洛版本，其中更新时间$ \ tilde {O}（d ^ 2）$，查询时间$ O（d）$和任意空间$ \ tilde {O}（m）$失败限制$ d \ le n $。这是通用图的第一个连接预言，可以有效处理无数个顶点失败。我们还开发了一个更有效的蒙特卡洛边缘故障连接oracle。使用空间$ O（n \ log ^ 2 n）$，在$ O（d \ log d \ log \ log n）$时间内处理$ d $边缘故障，此后，连接查询以$ O（\ log \ log n）$的时间回答，这很可能是正确的。我们的数据结构基于一个新的分解定理，该分解定理是无向图$ G =（V，E）$的一个独立关注点。它指出，对于任何终端集$ U \ subseteq V $，我们都可以删除$ | U | /（s-2）$顶点的集合$ B $，以便剩余图包含$ UB $的Steiner林，且最大程度$ s $。