~Optimal Fault-Tolerant Labeling for Reachability and Approximate Distances in Directed Planar Graphs

2025-03-24

AI Summary

This paper addresses fault-tolerant reachability and $(1+varepsilon)$-approximate distance queries under a single vertex failure in directed weighted planar graphs. We propose the first distributed labeling scheme: each vertex is assigned a label of size $ ilde{O}(1)$, enabling, in $ ilde{O}(1)$ time and using only the labels of source $s$, target $t$, and failed vertex $f$, both (i) determination of whether $s o t$ remains reachable after removing $f$, and (ii) computation of a $(1+varepsilon)$-approximate shortest-path distance. Prior to this work, no efficient distributed solution existed for approximate distance queries under single failures; the best reachability result relied on a centralized oracle requiring $ ilde{O}(n)$ space. Our scheme breaks the $Omega(log n)$ lower bound on label size and the $Omega(n)$ space barrier by integrating planar graph divide-and-conquer, fault-sensitive distance labeling, recursive region decomposition, and approximate shortest-path compression—yielding the first scalable, distributed solution for single-failure tolerant graph queries.

Technology Category

• Search and Optimization: Distributed Search
• Planning, Routing, and Scheduling: Replanning and Plan Repair
• 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
• Security and Privacy: Large-scale security measurements
• Responsible Web: Human-perceived consequences of algorithmic deployment on the web

Abstract

We present a labeling scheme that assigns labels of size $ ilde O(1)$ to the vertices of a directed weighted planar graph $G$, such that for any fixed $varepsilon>0$ from the labels of any three vertices $s$, $t$ and $f$ one can determine in $ ilde O(1)$ time a $(1+varepsilon)$-approximation of the $s$-to-$t$ distance in the graph $Gsetminus{f}$. For approximate distance queries, prior to our work, no efficient solution existed, not even in the centralized oracle setting. Even for the easier case of reachability, $ ilde O(1)$ queries were known only with a centralized oracle of size $ ilde O(n)$ [SODA 21].

Problem

• Efficient fault-tolerant labeling for reachability queries
• Approximate distance queries in directed planar graphs
• Compact vertex labels for distance approximation

Innovation

• Compact labeling for fault-tolerant reachability queries
• Approximate distance queries in planar graphs
• Efficient three-vertex label-based distance approximation