Locally-iterative (Δ + 1)-coloring in sublinear (in Δ) rounds

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 Szegedy and Vishwanathan (1993) [6]. 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 runtime $O({\Delta }^{3/4}\log \Delta )+{\log }^{⁎}n$, using messages of size $O(\log n)$ bits. This is the first locally-iterative $(\Delta +1)$-coloring algorithm with sublinear-in-Δ runtime, and answers the main open question raised by previous best result (Barenboim et al. (2021) [16]). The key component of our algorithm is a new locally-iterative procedure that transforms an $O\left({\Delta }^2\right)$-coloring to a $(\Delta +O({\Delta }^{3/4}\log \Delta \left)\right)$-coloring in $o\left(\Delta \right)$ time. As an application of our result, we also devise a self-stabilizing algorithm for $(\Delta +1)$-coloring with $O({\Delta }^{3/4}\log \Delta )+{\log }^{⁎}n$ stabilization time, using $O(\log n)$-bit messages. To the best of our knowledge, this is the first self-stabilizing algorithm for $(\Delta +1)$-coloring in the CONGEST model with sublinear-in-Δ stabilization time.