Robust Shattering Arguments

2026-06-26

AI Summary

Traditional graph shattering analyses fail beyond super-constant rounds due to their neglect of cumulative dependencies accumulated during the pre-shattering phase, thereby undermining the theoretical foundations of several distributed algorithms. This work proposes a novel shattering analysis framework that dispenses with the assumption of node independence, effectively handling long-range dependencies through a combination of probabilistic methods, dependency graph modeling, and decay-bound derivations. The framework provides, for the first time, rigorous shattering guarantees for the Fischer–Ghaffari Lovász Local Lemma (LLL) algorithm and constructs a counterexample to a classical lemma in the field. These robust analytical tools significantly strengthen the theoretical underpinnings of key distributed algorithms, including those for maximal independent set computation, (Δ+1)-coloring, and the distributed LLL.

Abstract

Graph shattering is a central technique underlying sublogarithmic-time distributed algorithms in the LOCAL model. Its analysis typically relies on bounding the probability that large sets of distant nodes remain unresolved, often via independence assumptions justified by locality. We show that these assumptions fail for pre-shattering procedures that run for super-constant rounds, where dependencies accumulate over time. As a result, several standard shattering arguments in the literature are incomplete, including those for maximal independent set, $(Δ+1)$-coloring, and the distributed Lovász Local Lemma (LLL). We provide a systematic repair of these analyses. Our main contribution is a corrected shattering analysis of the Fischer--Ghaffari LLL algorithm. In addition, we develop general tools that capture common patterns in modern algorithms and yield the required decay bounds without relying on independence. We also present explicit counterexamples to commonly used shattering lemmas. Overall, we establish a robust and reusable foundation for shattering arguments in the presence of long-range dependencies.

Problem

• graph shattering
• LOCAL model
• long-range dependencies
• Lovász Local Lemma
• distributed algorithms

Innovation

• shattering
• distributed algorithms
• Lovász Local Lemma
• long-range dependencies
• LOCAL model