Better Decremental and Fully Dynamic Sensitivity Oracles for Subgraph Connectivity

We study the sensitivity oracles problem for subgraph connectivity in the decremental and fully dynamic settings. In the fully dynamic setting, we preprocess an n-vertices m-edges undirected graph G with n_ off deactivated vertices initially and the others are activated. Then we receive a single update D⊆ V(G) of size |D| = d ≤ d_⋆, 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 vertex-failure connectivity oracles problem. We present a better deterministic vertex-failure connectivity oracle with O(d_⋆m) preprocessing time, O(m) space, 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 O(d^2) to O(d^2). We also present a better deterministic fully dynamic sensitivity oracle for subgraph connectivity with O(min{m(n_ off + d_⋆),n^ω}) preprocessing time, O(min{m(n_ off + d_⋆),n^2}) space, 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 O(d^4) to O(d^2). Furthermore, our solution is even almost-optimal assuming popular fine-grained complexity conjectures.