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 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 $(Δ+1)$-coloring algorithm with $O(Δ^{3/4}\logΔ)+\log^*n$ running time. This is the first locally-iterative $(Δ+1)$-coloring algorithm with sublinear-in-$Δ$ 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 $O(Δ^2)$-coloring to a $(Δ+O(Δ^{3/4}\logΔ))$-coloring in $o(Δ)$ 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 $(Δ+1)$-coloring with $O(Δ^{3/4}\logΔ)+\log^*n$ stabilization time. To the best of our knowledge, this is the first self-stabilizing algorithm for $(Δ+1)$-coloring with sublinear-in-$Δ$ stabilization time.