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的基本数据结构问题之一。该问题已得到充分研究：先前的工作[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，并且只要要求预处理时间与图规模呈多项式关系、更新时间与故障顶点数量呈多项式关系，该结果在条件意义下即是最优的。我们在带预测的算法范式下重新审视该问题：探究如果故障顶点集合能事先被预测（且预测误差较小），能否改进查询时间。具体而言，我们设计了一种数据结构：给定图$G=(V,E)$和预测将出故障的顶点集$\widehat{D} \subseteq V$（大小为$d=|\widehat{D}|$），该数据结构在$\tilde{O}(d|E|)$时间内完成预处理；随后接收一个更新，形式为预测故障集与实际故障集的对称差$\widehat{D} \triangle D = (\widehat{D} \setminus D) \cup (D \setminus \widehat{D})$（大小为$\eta = |\widehat{D} \triangle D|$），在$\tilde{O}(\eta^4)$时间内处理该更新；之后在$O(\eta)$时间内回答$G \setminus D$上的连通性查询。从另一角度而言，我们的数据结构改进了【敏感性设置下全动态子图连通性问题】[Henzinger--Neumann ESA'16]的现有技术水平。我们论证：在标准细粒度复杂度假设下，本数据结构的预处理时间和查询时间在条件意义下是最优的。