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arXiv 2402.09150cs.DS
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子图连通性的更好减量与全动态敏感性预言机

Better Decremental and Fully Dynamic Sensitivity Oracles for Subgraph Connectivity

Yaowei Long, Yunfan Wang

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AI总结：

本文研究了减量与全动态设置下的子图连通性敏感性预言机问题，提出了改进的确定性预言机，在保持近最优空间和查询时间的同时，显著降低了更新时间复杂度。

摘要：

我们研究减量和全动态设置下的子图连通性敏感性预言机问题。在全动态设置中，我们预处理一个具有 $n$ 个顶点、$m$ 条边的无向图 $G$，初始有 $n_{\rm off}$ 个停用顶点，其余为激活顶点。然后我们接收大小为 $|D| = d \leq d_{\star}$ 的单次更新 $D\subseteq V(G)$，表示状态将被切换的顶点。最后，我们得到一系列查询，每个查询询问给定顶点 $u$ 和 $v$ 在激活子图中的连通性。减量设置是初始没有停用顶点的特例，也称为顶点失效连通性预言机问题。我们提出了一个更好的确定性顶点失效连通性预言机，预处理时间为 $\widehat{O}(d_{\star}m)$，空间为 $\widetilde{O}(m)$，更新时间为 $\widetilde{O}(d^{2})$，查询时间为 $O(d)$，将先前近最优预言机 [Long-Saranurak, FOCS 2022] 的更新时间从 $\widehat{O}(d^{2})$ 改进为 $\widetilde{O}(d^{2})$。我们还提出了一个更好的确定性全动态子图连通性敏感性预言机，预处理时间为 $\widehat{O}(\min\{m(n_{\rm off} + d_{\star}),n^ω\})$，空间为 $\widetilde{O}(\min\{m(n_{\rm off} + d_{\star}),n^{2}\})$，更新时间为 $\widetilde{O}(d^{2})$，查询时间为 $O(d)$，将现有技术 [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，假设流行的细粒度复杂性猜想成立，我们的解决方案甚至几乎是最优的。

英文摘要：

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.

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