An $f$-edge fault-tolerant distance sensitive oracle ($f$-DSO) with stretch $\sigma \geq 1$ is a data structure that preprocesses an input graph $G$. When queried with the triple $(s,t,F)$, where $s, t \in V$ and $F \subseteq E$ contains at most $f$ edges of $G$, the oracle returns an estimate $\widehat{d}_{G-F}(s,t)$ of the distance $d_{G-F}(s,t)$ between $s$ and $t$ in the graph $G-F$ such that $d_{G-F}(s,t) \leq \widehat{d}_{G-F}(s,t) \leq \sigma d_{G-F}(s,t)$. For any positive integer $k \ge 2$ and any $0 < \alpha < 1$, we present an $f$-DSO with sensitivity $f = o(\log n/\log\log n)$, stretch $2k-1$, space $O(n^{1+\frac{1}{k}+\alpha+o(1)})$, and an $\widetilde{O}(n^{1+\frac{1}{k} - \frac{\alpha}{k(f+1)}})$ query time. Prior to our work, there were only three known $f$-DSOs with subquadratic space. The first one by Chechik et al. [Algorithmica 2012] has a stretch of $(8k-2)(f+1)$, depending on $f$. Another approach is storing an $f$-edge fault-tolerant $(2k-1)$-spanner of $G$. The bottleneck is the large query time due to the size of any such spanner, which is $\Omega(n^{1+1/k})$ under the Erd\H{o}s girth conjecture. Bil\`o et al. [STOC 2023] gave a solution with stretch $3+\varepsilon$, query time $O(n^{\alpha})$ but space $O(n^{2-\frac{\alpha}{f+1}})$, approaching the quadratic barrier for large sensitivity. In the realm of subquadratic space, our $f$-DSOs are the first ones that guarantee, at the same time, large sensitivity, low stretch, and non-trivial query time. To obtain our results, we use the approximate distance oracles of Thorup and Zwick [JACM 2005], and the derandomization of the $f$-DSO of Weimann and Yuster [TALG 2013], that was recently given by Karthik and Parter [SODA 2021].

翻译：一个具有伸缩因子$\sigma \geq 1$的$f$边故障容忍距离敏感预言机（$f$-DSO）是一种预处理输入图$G$的数据结构。当使用三元组$(s,t,F)$查询时（其中$s, t \in V$，$F \subseteq E$包含至多$f$条$G$中的边），该预言机会返回图$G-F$中$s$和$t$之间距离$d_{G-F}(s,t)$的估计值$\widehat{d}_{G-F}(s,t)$，使得$d_{G-F}(s,t) \leq \widehat{d}_{G-F}(s,t) \leq \sigma d_{G-F}(s,t)$。对于任意正整数$k \ge 2$和任意$0 < \alpha < 1$，我们提出了一种$f$-DSO，其灵敏度$f = o(\log n/\log\log n)$，伸缩因子$2k-1$，空间复杂度$O(n^{1+\frac{1}{k}+\alpha+o(1)})$，查询时间$\widetilde{O}(n^{1+\frac{1}{k} - \frac{\alpha}{k(f+1)}})$。在我们工作之前，已知仅有三种具有次二次空间的$f$-DSO。第一种由Chechik等人[Algorithmica 2012]提出，其伸缩因子为$(8k-2)(f+1)$，依赖于$f$。另一种方法是存储$G$的一个$f$边故障容忍$(2k-1)$-拉伸子图。其瓶颈在于任何此类子图（在Erdős围长猜想下大小为$\Omega(n^{1+1/k})$）导致的较大查询时间。Bilò等人[STOC 2023]给出了一种解决方案，伸缩因子为$3+\varepsilon$，查询时间$O(n^{\alpha})$，但空间复杂度为$O(n^{2-\frac{\alpha}{f+1}})$，在大灵敏度情况下接近二次空间界限。在次二次空间领域，我们的$f$-DSO是首个同时保证大灵敏度、低拉伸和非平凡查询时间的方案。为获得我们的结果，我们使用了Thorup和Zwick[JACM 2005]的近似距离预言机，以及Karthik和Parter[SODA 2021]近期给出的Weimann和Yuster[TALG 2013]的$f$-DSO去随机化技术。