Given two vertex sets $S$ and $T$ in a graph, the $ST$-diameter is the maximum $s$-$t$-distance between vertices $s \in S$ and $t \in T$. We study the problem of estimating the $ST$-diameter of graphs that are subject to a small number of transient edge failures. An $f$-edge fault-tolerant $ST$-diameter oracle ($f$-FDO-$ST$) is a data structure that preprocesses a graph $G$, sets $S$, $T$, and a positive integer $f$. When queried with a set $F$ of at most $f$ failing edges, the oracle returns an estimate $\widehat{D}$ of the $ST$-diameter in $G-F$. The oracle is said to have stretch $σ\geq 1$ if $\operatorname{diam}(G{-}F,S,T) \leq \widehat{D} \leq σ\cdot \operatorname{diam}(G{-}F,S,T)$. We design new $f$-FDO-$ST$s by reducing their construction to that of all-pairs and single-source distance sensitivity oracles ($f$-DSOs). These are data structures that estimate the pairwise graph distances, or respectively the distances from a distinguished source, under up to $f$ failures. We obtain several new trade-offs between the size of the $ST$-diameter oracles, their stretch guarantees, query and preprocessing times by combining our black-box reductions with $f$-DSO results from the literature. We further provide a lower bound on the space requirement of approximate $ST$-diameter oracles. We prove that there exists a family of graphs for which any $f$-FDO-$ST$ with sensitivity $f \ge 2$ and stretch better than $5/3$ requires $Ω(n^{3/2})$ bits of space, regardless of the query time.

翻译：给定图$G$中两个顶点集$S$和$T$，$ST$直径定义为顶点$s\in S$与$t\in T$之间的最大$s$-$t$距离。本文研究在面临少量瞬态边故障情况下图$ST$直径的估计问题。一个$f$边故障容错$ST$直径预言机（$f$-FDO-$ST$）是一种数据结构，它对图$G$、集合$S$、$T$以及正整数$f$进行预处理。当查询包含至多$f$条故障边的集合$F$时，预言机返回$G-F$中$ST$直径的估计值$\widehat{D}$。若满足$\operatorname{diam}(G{-}F,S,T) \leq \widehat{D} \leq σ\cdot \operatorname{diam}(G{-}F,S,T)$，则称该预言机具有伸缩系数$σ\geq 1$。我们通过将$f$-FDO-$ST$的构造归约为全对和单源距离敏感预言机（$f$-DSOs），设计了新型$f$-FDO-$ST$。这些数据结构能在至多$f$个故障下分别估计图上成对距离或从指定源点出发的距离。通过将我们的黑盒归约与文献中的$f$-DSO结果相结合，我们在ST直径预言机的大小、伸缩性保证、查询时间和预处理时间之间获得了若干新的权衡关系。我们进一步给出了近似ST直径预言机空间需求的下界。我们证明存在一族图，使得任何敏感度$f \ge 2$且伸缩性优于$5/3$的$f$-FDO-$ST$需要$\Omega(n^{3/2})$比特空间，且与查询时间无关。