AI 中文总结
本文针对有向多源替换路径问题，提出一种随机组合算法，将其时间复杂度改进为$\tilde{O}(m\sqrt{\sigma n} + \sigma n^2)$，且算法简单、复杂度在组合算法中本质最优。
AI 中文摘要
在替换路径（RP）问题中，给定图$G=(V,E)$，其中$n=|V|$，$m=|E|$，以及两个顶点$s,t\in V$，要求对每条失效边$e\in E$，计算$G\setminus e$中从$s$到$t$的最短路径距离。多源替换路径（MSRP）问题是其自然推广：给定大小为$\sigma$的源点集合$S\subseteq V$，计算$S\times V$中所有顶点对的替换路径距离。本文提出一种随机组合算法，可在$\tilde{O}(m\sqrt{\sigma n} + \sigma n^2)$时间内解决无向有向图上的MSRP问题，且所有输出距离以高概率正确。该结果改进了有向图上已知最优的$\tilde{O}(m\min\{\sigma\sqrt{n}, n\} + \sigma n^2)$界，该界通过分别运行Chechik与Magen[ICALP'20]的单源RP算法、或构造并查询Bernstein与Karger[STOC'09]的全对距离敏感性预言机得到。由于Gupta、Jain与Modi[PODC'20]证明组合算法存在$m(\sigma n)^{1/2-o(1)}$的下界（即使在无向图上也成立），且加性项$\sigma n^2$与写下$\Theta(\sigma n^2)$个输出距离所需时间成正比，因此本文算法的运行时间在组合算法中本质上是紧的，且该算法极为简单。
英文摘要
In the replacement paths (RP) problem, we are given a graph $G = (V, E)$ with $n = |V|$ and $m = |E|$, together with two vertices $s, t \in V$, and are asked to compute the shortest-path distance from $s$ to $t$ in $G \setminus e$ for every failed edge $e \in E$. The multiple source replacement paths (MSRP) problem is its natural generalization: given a set $S \subseteq V$ of $σ$ sources, compute the replacement path distances for all pairs in $S \times V$. In this paper, we present a randomized combinatorial algorithm that solves MSRP on unweighted directed graphs in $\tilde{O}(m\sqrt{σn} + σn^2)$ time, with all the output distances correct with high probability. This improves the best known bound $\tilde{O}(m\min\{σ\sqrt{n}, n\} + σn^2)$ for directed graphs, which is obtained either by running the single source RP algorithm of Chechik and Magen [ICALP'20] from each source separately or by constructing and querying the all-pairs distance sensitivity oracle of Bernstein and Karger [STOC'09]. Our running time is essentially tight among combinatorial algorithms because Gupta, Jain, and Modi [PODC'20] proved a lower bound of $m{(σn)}^{1/2-o(1)}$ for such algorithms, which holds even on undirected graphs, and the additive term $σn^2$ is proportional to the time needed to write down the $Θ(σn^2)$ output distances. The algorithm is also remarkably simple.
Comments15 pages, 1 figure
