论文标题

使用亚皮的预处理时间改善距离灵敏度甲骨文

Improved Distance Sensitivity Oracles with Subcubic Preprocessing Time

论文作者

Ren, Hanlin

论文摘要

我们考虑构建距离灵敏度甲环(DSO)的问题。 Given a directed graph $G=(V, E)$ with edge weights in $\{1, 2, \dots, M\}$, we need to preprocess it into a data structure, and answer the following queries: given vertices $u,v\in V$ and a failed vertex or edge $f\in (V\cup E)$, output the length of the shortest path from $u$ to $v$ that does not go through $ f $。我们的主要结果是一个简单的DSO,带有$ \ tilde {o}(n^{2.7233} m)$预处理时间和$ o(1)$查询时间。此外,如果输入图是无方向性的,则可以将预处理时间提高到$ \ tilde {o}(n^{2.6865} m)$。预处理算法是随机分配的,正确概率$ \ ge 1-1/n^c $,对于可以任意大大的常数$ c $。以前,有一个dso,带有$ \ tilde {o}(n^{2.8729} m)$预处理时间和$ \ operatatorName {polylog}(n)$查询时间[Chechik and Cohen,stoc'20]。 我们的DSO的核心是[Bernstein和Karger,Stoc'09]的以下观察结果:如果有一个DSO和预处理时间$ p $和查询时间$ Q $,那么我们可以使用预处理时间$ p+p+p+\ tilde {o} {o}(n^2)\ cdot q $和Query Time $ o(1)$(1)。 (在这里$ \ tilde {o}(\ cdot)$ hides $ \ permatatorName {polylog}(n)$因子。

We consider the problem of building Distance Sensitivity Oracles (DSOs). Given a directed graph $G=(V, E)$ with edge weights in $\{1, 2, \dots, M\}$, we need to preprocess it into a data structure, and answer the following queries: given vertices $u,v\in V$ and a failed vertex or edge $f\in (V\cup E)$, output the length of the shortest path from $u$ to $v$ that does not go through $f$. Our main result is a simple DSO with $\tilde{O}(n^{2.7233}M)$ preprocessing time and $O(1)$ query time. Moreover, if the input graph is undirected, the preprocessing time can be improved to $\tilde{O}(n^{2.6865}M)$. The preprocessing algorithm is randomized with correct probability $\ge 1-1/n^C$, for a constant $C$ that can be made arbitrarily large. Previously, there is a DSO with $\tilde{O}(n^{2.8729}M)$ preprocessing time and $\operatorname{polylog}(n)$ query time [Chechik and Cohen, STOC'20]. At the core of our DSO is the following observation from [Bernstein and Karger, STOC'09]: if there is a DSO with preprocessing time $P$ and query time $Q$, then we can construct a DSO with preprocessing time $P+\tilde{O}(n^2)\cdot Q$ and query time $O(1)$. (Here $\tilde{O}(\cdot)$ hides $\operatorname{polylog}(n)$ factors.)

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