We study the \emph{sensitivity oracles problem for subgraph connectivity} in the \emph{decremental} and \emph{fully dynamic} settings. In the fully dynamic setting, we preprocess an $n$-vertices $m$-edges undirected graph $G$ with $n_{\rm off}$ deactivated vertices initially and the others are activated. Then we receive a single update $D\subseteq V(G)$ of size $|D| = d \leq d_{\star}$, representing vertices whose states will be switched. Finally, we get a sequence of queries, each of which asks the connectivity of two given vertices $u$ and $v$ in the activated subgraph. The decremental setting is a special case when there is no deactivated vertex initially, and it is also known as the \emph{vertex-failure connectivity oracles} problem. We present a better deterministic vertex-failure connectivity oracle with $\widehat{O}(d_{\star}m)$ preprocessing time, $\widetilde{O}(m)$ space, $\widetilde{O}(d^{2})$ update time and $O(d)$ query time, which improves the update time of the previous almost-optimal oracle [Long-Saranurak, FOCS 2022] from $\widehat{O}(d^{2})$ to $\widetilde{O}(d^{2})$. We also present a better deterministic fully dynamic sensitivity oracle for subgraph connectivity with $\widehat{O}(\min\{m(n_{\rm off} + d_{\star}),n^{\omega}\})$ preprocessing time, $\widetilde{O}(\min\{m(n_{\rm off} + d_{\star}),n^{2}\})$ space, $\widetilde{O}(d^{2})$ update time and $O(d)$ query time, which significantly improves the update time of the state of the art [Hu-Kosinas-Polak, 2023] from $\widetilde{O}(d^{4})$ to $\widetilde{O}(d^{2})$. Furthermore, our solution is even almost-optimal assuming popular fine-grained complexity conjectures.

翻译：我们研究递减和全动态设置下\emph{子图连通性的灵敏度预言机问题}。在全动态设置中，我们对一个具有$n$个顶点、$m$条边的无向图$G$进行预处理，初始时有$n_{\rm off}$个非激活顶点，其余顶点为激活状态。随后我们接收单个更新$D\subseteq V(G)$，其大小$|D| = d \leq d_{\star}$，表示状态将被切换的顶点集合。最后，我们得到一系列查询，每个查询询问激活子图中两个给定顶点$u$和$v$的连通性。当初始时没有非激活顶点时，递减设置是特殊情况，也称为\emph{顶点故障连通性预言机}问题。我们提出了一种更优的确定性顶点故障连通性预言机，具有$\widehat{O}(d_{\star}m)$预处理时间、$\widetilde{O}(m)$空间、$\widetilde{O}(d^{2})$更新时间和$O(d)$查询时间，将此前几乎最优预言机[Long-Saranurak, FOCS 2022]的更新时间从$\widehat{O}(d^{2})$改进为$\widetilde{O}(d^{2})$。我们还提出了一种更优的确定性全动态子图连通性灵敏度预言机，具有$\widehat{O}(\min\{m(n_{\rm off} + d_{\star}),n^{\omega}\})$预处理时间、$\widetilde{O}(\min\{m(n_{\rm off} + d_{\star}),n^{2}\})$空间、$\widetilde{O}(d^{2})$更新时间和$O(d)$查询时间，将现有最优方法[Hu-Kosinas-Polak, 2023]的更新时间从$\widetilde{O}(d^{4})$显著改进为$\widetilde{O}(d^{2})$。此外，在流行的细粒度复杂性假设下，我们的解决方案甚至是几乎最优的。