On a Decentralized $(\Delta{+}1)$-Graph Coloring Algorithm

TL;DR

This paper analyzes a decentralized graph coloring algorithm, introducing a variant that improves recoloring efficiency and confirming the tightness of existing bounds in worst-case scenarios.

Contribution

It introduces a new variant of a decentralized coloring algorithm with improved expected recoloring bounds and proves the tightness of existing bounds in adversarial cases.

Findings

New variant requires $O(n ext{log}\Delta)$ expected recolorings.

The $O(n ext{log}\Delta)$ bound generalizes the coupon collector problem.

The $O(n ext{Delta})$ bound is tight in adversarial scenarios.

Abstract

We consider a decentralized graph coloring model where each vertex only knows its own color and whether some neighbor has the same color as it. The networking community has studied this model extensively due to its applications to channel selection, rate adaptation, etc. Here, we analyze variants of a simple algorithm of Bhartia et al. [Proc., ACM MOBIHOC, 2016]. In particular, we introduce a variant which requires only $O(n\log\Delta)$ expected recolorings that generalizes the coupon collector problem. Finally, we show that the $O(n\Delta)$ bound Bhartia et al. achieve for their algorithm still holds and is tight in adversarial scenarios.