Deterministic Sensitivity Oracles for Diameter, Eccentricities and All Pairs Distances

TL;DR

This paper develops space-efficient deterministic and randomized data structures, called oracles, for quickly approximating extremal and pairwise distances in directed graphs with edge failures, advancing fault-tolerant graph analysis.

Contribution

It introduces the first fault-tolerant eccentricity oracle for dual failures with subcubic space and presents a hierarchical derandomization framework for sensitivity oracles.

Findings

First fault-tolerant eccentricity oracle for dual failures in subcubic space

Lower bounds on approximation vs. space and diameter vs. space trade-offs

First deterministic distance sensitivity oracle with subcubic preprocessing time

Abstract

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.