Distributed Distance Sensitivity Oracles

2024-11-20

arXiv.org

AI Summary

This paper studies single-edge-fault shortest-path distance queries on directed weighted graphs in the distributed CONGEST model: given a source $s$, a target $t$, and a failing edge $e$, quickly report the shortest $s$–$t$ distance avoiding $e$. It presents the first nontrivial distance sensitivity oracle in this model, establishes preprocessing–query round trade-offs, and proves unconditional information-theoretic lower bounds. Key contributions are: (1) an optimal $O(1)$-round query algorithm; (2) a low-complexity preprocessing scheme achieving sublinear total communication; and (3) a complete characterization of the distributed complexity of 2-APSiSP—the problem of computing the two edge-disjoint shortest paths between every pair of vertices—yielding tight $Theta(n)$-round upper and lower bounds, nearly matching the theoretical limit.

Technology Category

• Constraint Satisfaction and Optimization: Distributed CSP/Optimization
• Search and Optimization: Distributed Search
• 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
• Responsible Web: Human-perceived consequences of algorithmic deployment on the web

Abstract

We present results for the distance sensitivity oracle (DSO) problem, where one needs to preprocess a given directed weighted graph $G=(V,E)$ in order to answer queries about the shortest path distance from $s$ to $t$ in $G$ that avoids edge $e$, for any $s,t in V, e in E$. No non-trivial results are known for DSO in the distributed CONGEST model even though it is of importance to maintain efficient communication under an edge failure. We present DSO algorithms with different tradeoffs between preprocessing and query cost -- one that optimizes query response rounds, and another that prioritizes preprocessing rounds. We complement these algorithms with unconditional CONGEST lower bounds. Additionally, we present almost-optimal upper and lower bounds for the related all pairs second simple shortest path (2-APSiSP) problem.

Problem

• Develops algorithms for distance sensitivity oracles in distributed networks.
• Optimizes preprocessing and query costs for shortest path queries avoiding edge failures.
• Establishes upper and lower bounds for all pairs second simple shortest path problem.

Innovation

• Develops distributed DSO algorithms for edge failure scenarios
• Optimizes query response and preprocessing rounds tradeoffs
• Establishes bounds for 2-APSiSP problem in CONGEST model