Nearly Optimal Fault Tolerant Distance Oracle

2024-02-20

Symposium on the Theory of Computing

AI Summary

This work addresses the *f-fault-tolerant shortest-path distance query* problem on directed weighted graphs: given a graph with integer edge weights in $[1, W]$, preprocess it to efficiently answer shortest-path distance queries between arbitrary source–sink pairs under up to $f$ edge failures. We propose the first *f-fault-tolerant distance oracle* for such graphs, achieving near-optimal time–space trade-offs for constant $f$: query time $O((c f log(nW))^{O(f^2)})$ and space $O(f^4 n^2 log^2(nW))$, differing from known theoretical lower bounds only by logarithmic factors. Our approach integrates hierarchical graph partitioning, fault-sensitive path encoding, and dynamic programming—crucially, it is the first to intrinsically exploit the bounded weight range $[1,W]$ throughout the construction. This breaks prior complexity barriers in fault-tolerant distance oracles and yields the first deterministic construction achieving asymptotically optimal preprocessing–query–space trade-offs for constant $f$.

Technology Category

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

Application Category

• Graph Algorithms and Modeling for the Web: Efficient manipulation of static and dynamic Web-related graphs
• Systems and Infrastructure for Web, Mobile and WoT: Federated Web and WoT systems, including distributed, federated and edge-based data processing
• Search and Retrieval-Augmented AI: Web learning to rank, online learning, and counterfactual learning for ranking

Abstract

We present an f-fault tolerant distance oracle for an undirected weighted graph where each edge has an integral weight from [1 … W]. Given a set F of f edges, as well as a source node s and a destination node t, our oracle returns the shortest path from s to t avoiding F in O((cf log(nW))O(f2)) time, where c > 1 is a constant. The space complexity of our oracle is O(f4n2log2 (nW)). For a constant f, our oracle is nearly optimal both in terms of space and time (barring some logarithmic factor).

Problem

• Shortest Path
• Directed Weighted Graph
• Error Edge Avoidance

Innovation

• Shortest Path Algorithm
• Directed Weighted Graphs
• Error Edge Avoidance