We construct data structures for extremal and pairwise distances in directed graphs in the presence of transient edge failures. Henzinger et al. [ITCS 2017] initiated the study of fault-tolerant (sensitivity) oracles for the diameter and vertex eccentricities. We extend this with a special focus on space efficiency. We present several new data structures, among them the first fault-tolerant eccentricity oracle for dual failures in subcubic space. We further prove lower bounds that show limits to approximation vs. space and diameter vs. space trade-offs for fault-tolerant oracles. They highlight key differences between data structures for undirected and directed graphs. Initially, our oracles are randomized leaning on a sampling technique frequently used in sensitivity analysis. Building on the work of Alon, Chechik, and Cohen [ICALP 2019] as well as Karthik and Parter [SODA 2021], we develop a hierarchical framework to derandomize fault-tolerant data structures. We first apply it to our own diameter and eccentricity oracles and then show its versatility by derandomizing algorithms from the literature: the distance sensitivity oracle of Ren [JCSS 2022] and the Single-Source Replacement Path algorithm of Chechik and Magen [ICALP 2020]. This way, we obtain the first deterministic distance sensitivity oracle with subcubic preprocessing time.
翻译:我们以瞬时边缘失灵时,在定向图表中为极端偏差和双向距离建造数据结构。[ITRS 2017] 开始对直径和顶心偏心的断层(灵敏度)触角进行研究。我们特别以空间效率为重点扩大这一结构。我们展示了几个新的数据结构,其中包括在亚紫外空间双重失灵时首个不易偏差偏心或触角;我们进一步证明,低界限显示接近空间和直径与空间对偏差或触角之间空间交易的极限。它们突出非定向图和定向图的数据结构之间的关键差异。最初,我们的触角随机依赖敏感分析中经常使用的取样技术。我们以Alon、Chechik和Chen[ICR2019]以及Karthik和Parter[SODO20211]的工作为基础,我们开发了一个分级框架,以解错容数据结构。我们首先将它应用到我们自己的直径和偏心或偏心或偏切度图上的数据结构之间,然后用Slorvicle的分辨率分析方法显示其偏差性。Schechinialal-Sqlicalalalalxalalalalalalalmaxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxx。