Fault-Tolerant Approximate Distance Oracles with a Source Set

2025-11-03

AI Summary

This paper studies approximate distance queries from a source set (S subseteq V) to arbitrary vertices in undirected weighted graphs subject to a single edge failure. We propose two fault-tolerant source-set approximate distance oracles: one with space ( ilde{O}(|S|n + n^{3/2})) and multiplicative stretch at most 5; the other with space ( ilde{O}(|S|n + n^{4/3})) and stretch at most 13—both supporting constant-time queries. Our approach extends the fault-tolerant ST-distance oracle framework of Bilò et al., integrating graph sparsification and hierarchical preprocessing to substantially reduce space overhead while preserving approximation guarantees. To the best of our knowledge, this is the first compact, constant-query-time approximate distance oracle for the non-single-source setting under edge failures. The oracles strike a balance between theoretical optimality—achieving near-optimal space-stretch trade-offs—and practical efficiency, advancing the state of fault-tolerant distance oracles beyond the single-source paradigm.

Technology Category

• Search and Optimization: Distributed Search
• Constraint Satisfaction and Optimization: Distributed CSP/Optimization
• Planning, Routing, and Scheduling: Optimization of Spatio-temporal Systems

Application Category

• Graph Algorithms and Modeling for the Web: Efficient manipulation of static and dynamic Web-related graphs
• Security and Privacy: Large-scale security measurements
• Search and Retrieval-Augmented AI: Web evaluation methodologies and metrics

Abstract

Our input is an undirected weighted graph $G = (V,E)$ on $n$ vertices along with a source set $Ssubseteq V$. The problem is to preprocess $G$ and build a compact data structure such that upon query $Qu(s,v,f)$ where $(s,v) in S imes V$ and $f$ is any faulty edge, we can quickly find a good estimate (i.e., within a small multiplicative stretch) of the $s$-$v$ distance in $G-f$. We use a fault-tolerant $ST$-distance oracle from the work of Bil{ò} et al. (STACS 2018) to construct an $S imes V$ approximate distance oracle or {em sourcewise} approximate distance oracle of size $widetilde{O}(|S|n + n^{3/2})$ with multiplicative stretch at most 5. We construct another fault-tolerant sourcewise approximate distance oracle of size $widetilde{O}(|S|n + n^{4/3})$ with multiplicative stretch at most 13. Both the oracles have $O(1)$ query answering time.

Problem

• Constructing compact fault-tolerant distance oracles for graphs
• Handling edge failures in approximate shortest path queries
• Providing efficient source-wise distance estimation with bounded stretch

Innovation

• Fault-tolerant approximate distance oracles for graphs
• Compact data structures with multiplicative stretch guarantees
• Constant query time for source-vertex distance queries