The problem of designing connectivity oracles supporting vertex failures is one of the basic data structures problems for undirected graphs. It is already well understood: previous works [Duan--Pettie STOC'10; Long--Saranurak FOCS'22] achieve query time linear in the number of failed vertices, and it is conditionally optimal as long as we require preprocessing time polynomial in the size of the graph and update time polynomial in the number of failed vertices. We revisit this problem in the paradigm of algorithms with predictions: we ask if the query time can be improved if the set of failed vertices can be predicted beforehand up to a small number of errors. More specifically, we design a data structure that, given a graph $G=(V,E)$ and a set of vertices predicted to fail $\widehat{D} \subseteq V$ of size $d=|\widehat{D}|$, preprocesses it in time $\tilde{O}(d|E|)$ and then can receive an update given as the symmetric difference between the predicted and the actual set of failed vertices $\widehat{D} \triangle D = (\widehat{D} \setminus D) \cup (D \setminus \widehat{D})$ of size $\eta = |\widehat{D} \triangle D|$, process it in time $\tilde{O}(\eta^4)$, and after that answer connectivity queries in $G \setminus D$ in time $O(\eta)$. Viewed from another perspective, our data structure provides an improvement over the state of the art for the \emph{fully dynamic subgraph connectivity problem} in the \emph{sensitivity setting} [Henzinger--Neumann ESA'16]. We argue that the preprocessing time and query time of our data structure are conditionally optimal under standard fine-grained complexity assumptions.

翻译：设计支持顶点故障的连通性预言机是无向图基本数据结构问题之一。该问题已得到充分研究：先前工作 [Duan--Pettie STOC'10；Long--Saranurak FOCS'22] 实现了与故障顶点数量线性相关的查询时间，且只要预处理时间为图规模的多项式且更新时间与故障顶点数多项式相关，该结果即为条件最优。我们在带预测算法范式下重新审视这一问题：若故障顶点集可提前预测且误差数较小，查询时间能否得到改善？具体而言，我们设计的数据结构给定图 $G=(V,E)$ 和预测故障顶点集 $\widehat{D} \subseteq V$（规模 $d=|\widehat{D}|$），在预处理时间 $\tilde{O}(d|E|)$ 后，可接收预测故障集与实际故障集的对称差 $\widehat{D} \triangle D = (\widehat{D} \setminus D) \cup (D \setminus \widehat{D})$（规模 $\eta = |\widehat{D} \triangle D|$）作为更新，在 $\tilde{O}(\eta^4)$ 时间内处理，随后在 $O(\eta)$ 时间内回答 $G \setminus D$ 中的连通性查询。从另一角度看，我们的数据结构改进了灵敏度设置下全动态子图连通性问题的现有技术 [Henzinger--Neumann ESA'16]。我们证明，在标准细粒度复杂度假设下，该数据结构的预处理时间和查询时间均为条件最优。