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：设计支持顶点失效的连通性预言机是无向图数据结构领域的基础问题之一。该问题目前已得到充分理解：先前的研究工作[Duan--Pettie STOC'10; Long--Saranurak FOCS'22]实现了与失效顶点数量成线性关系的查询时间，且只要预处理时间关于图规模为多项式级、更新时间关于失效顶点数量为多项式级，该结果在条件意义下是最优的。我们在算法预测范式下重新审视该问题：若失效顶点集合可被预先预测且仅存在少量误差，查询时间能否得到改进？具体而言，我们设计了一种数据结构，对于给定图$G=(V,E)$与预测失效顶点集合$\widehat{D} \subseteq V$（规模$d=|\widehat{D}|$），该结构能以$\tilde{O}(d|E|)$时间完成预处理，随后可接收以预测失效集与实际失效集对称差形式给出的更新$\widehat{D} \triangle D = (\widehat{D} \setminus D) \cup (D \setminus \widehat{D})$（规模$\eta = |\widehat{D} \triangle D|$），以$\tilde{O}(\eta^4)$时间处理更新，并在此后以$O(\eta)$时间应答关于$G \setminus D$的连通性查询。从另一视角看，我们的数据结构为\emph{灵敏度设置}下的\emph{全动态子图连通性问题}[Henzinger--Neumann ESA'16]提供了优于现有技术水平的解决方案。我们论证了在标准细粒度复杂性假设下，该数据结构的预处理时间与查询时间在条件意义下是最优的。