Locally-iterative $(\Delta+1)$-Coloring in Sublinear (in $\Delta$) Rounds

TL;DR

This paper introduces a novel locally-iterative $( ext{Delta}+1)$-coloring algorithm that operates in sublinear rounds relative to $ ext{Delta}$, significantly advancing distributed graph coloring efficiency and stability.

Contribution

It presents the first locally-iterative $( ext{Delta}+1)$-coloring algorithm with sublinear-in-$ ext{Delta}$ runtime, answering a key open question and including a self-stabilizing variant.

Findings

Achieves $O( ext{Delta}^{3/4} ext{log} ext{Delta}) + ext{log}^* n$ running time

Transforms $O( ext{Delta}^2)$-coloring to near $( ext{Delta}+O( ext{Delta}^{3/4} ext{log} ext{Delta}))$-coloring efficiently

First sublinear-in-$ ext{Delta}$ stabilization time for $( ext{Delta}+1)$-coloring in self-stabilizing algorithms

Abstract

Distributed graph coloring is one of the most extensively studied problems in distributed computing. There is a canonical family of distributed graph coloring algorithms known as the locally-iterative coloring algorithms, first formalized in the seminal work of [Szegedy and Vishwanathan, STOC'93]. In such algorithms, every vertex iteratively updates its own color according to a predetermined function of the current coloring of its local neighborhood. Due to the simplicity and naturalness of its framework, locally-iterative coloring algorithms are of great significance both in theory and practice. In this paper, we give a locally-iterative $(\Delta+1)$-coloring algorithm with $O(\Delta^{3/4}\log\Delta)+\log^*n$ running time. This is the first locally-iterative $(\Delta+1)$-coloring algorithm with sublinear-in-$\Delta$ running time, and answers the main open question raised in a recent breakthrough [Barenboim, Elkin, and Goldberg, JACM'21]. A key component of our algorithm is a locally-iterative procedure that transforms an $O(\Delta^2)$-coloring to a $(\Delta+O(\Delta^{3/4}\log\Delta))$-coloring in $o(\Delta)$ time. Inside this procedure we work on special proper colorings that encode (arb)defective colorings, and reduce the number of used colors quadratically in a locally-iterative fashion. As a main application of our result, we also give a self-stabilizing distributed algorithm for $(\Delta+1)$-coloring with $O(\Delta^{3/4}\log\Delta)+\log^*n$ stabilization time. To the best of our knowledge, this is the first self-stabilizing algorithm for $(\Delta+1)$-coloring with sublinear-in-$\Delta$ stabilization time.