Better Decremental and Fully Dynamic Sensitivity Oracles for Subgraph Connectivity

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 $\widehat{O}(\min\{m(n_{\rm off} + d_{\star}),n^ω\})$ 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.