在多个边缘故障下保持精确距离
Maintaining Exact Distances under Multiple Edge Failures
摘要：
我们提出了第一个紧凑的距离甲骨文，容忍多个故障，并保持精确的距离。给定一个无向赋权图$G = (V, E)$和一个任意大的常数$d$，我们构造一个预言机，给定顶点$u, v \in V$和一组$d$边失效$D$，输出$G - D$中$u$和$v$之间的精确距离（即去掉$D$中边的$G$）.我们的预言机具有空间复杂度$O(d n^4)$和查询时间$d^{O(d)}$。以前，在多个故障下有紧凑的近似距离预言机[Dahik，Cohen，Fiat和Kaplan，SODA'17; Duan，Gu和Ren，SODA'21]，但在$d$故障下的最佳精确距离预言机基本上需要$Ω(n^d)$空间[Duan和Pettie，SODA'09]。我们的远程oracle似乎需要$n^{Ω(d)}$时间来进行预处理;我们将其作为一个开放的问题来改进此预处理时间。
英文摘要：
We present the first compact distance oracle that tolerates multiple failures and maintains exact distances. Given an undirected weighted graph $G = (V, E)$ and an arbitrarily large constant $d$, we construct an oracle that given vertices $u, v \in V$ and a set of $d$ edge failures $D$, outputs the exact distance between $u$ and $v$ in $G - D$ (that is, $G$ with edges in $D$ removed). Our oracle has space complexity $O(d n^4)$ and query time $d^{O(d)}$. Previously, there were compact approximate distance oracles under multiple failures [Chechik, Cohen, Fiat, and Kaplan, SODA'17; Duan, Gu, and Ren, SODA'21], but the best exact distance oracles under $d$ failures require essentially $Ω(n^d)$ space [Duan and Pettie, SODA'09]. Our distance oracle seems to require $n^{Ω(d)}$ time to preprocess; we leave it as an open question to improve this preprocessing time.
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