Deterministic Edge-Fault-Tolerant Connectivity Labeling Schemes with Nearly Optimal Label Size

2026-09-08

AI Summary

本文提出了一种使用O(log^2 n)位标签的确定性边故障容忍连通性标记方案，解决了在存在故障边集时判断图中两点间连通性的问题。

Abstract

For an undirected graph $G = (V,E)$ and a fault bound $f$, an edge-fault-tolerant connectivity labeling scheme assigns short labels to vertices and edges, so that for any vertex pair $(s,t)$ and failed edge set $F\subseteq E$ with $|F|\leq f$, the connectivity between $s$ and $t$ in $G-F$ can be answered by inspecting only the labels of $s$, $t$ and edges in $F$. In this paper, we present a labeling scheme that uses $O(\log^{2}n)$-bit labels that can be computed in deterministic polynomial time. This improves upon the previous $\tilde{O}(\sqrt{f})$ deterministic bound of [Long, Pettie, Saranurak'25], and even slightly improves the $O(\min\{f+\log n,\log^{2}n\log f\})$ randomized bound of [Dory, Parter'21] and [Long, Pettie, Saranurak'25] when $f = \Omega(\log^{2}n)$. Moreover, for a general $f$, this is the first labeling scheme that produces an $\tilde{O}(1)$-size labeling which is simultaneously correct across all queries. Our approach combines the cycle-space-based labeling scheme from Dory and Parter with a recent result by [Knauer'26] on sparse cycle bases.

Problem

• Edge-Fault-Tolerant
• Connectivity Labeling Schemes
• Optimal Label Size

Innovation

• Deterministic Polynomial Time
• Edge-Fault-Tolerant Connectivity
• Cycle-Space-Based Labeling Scheme
• Sparse Cycle Bases