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.

翻译：分布式图染色是分布式计算中研究最为广泛的问题之一。存在一类经典的分布式图染色算法，称为局部迭代染色算法，最早由[Szegedy and Vishwanathan, STOC'93]的开创性工作正式定义。在此类算法中，每个顶点根据其局部邻域当前染色的预定函数迭代更新自身颜色。由于其框架的简洁性和自然性，局部迭代染色算法在理论和实践中均具有重要意义。本文提出一种运行时间为O(Δ^{3/4}logΔ)+log^*n的局部迭代(Δ+1)染色算法。这是首个运行时间关于Δ呈亚线性的局部迭代(Δ+1)染色算法，解答了近期突破性工作[Barenboim, Elkin, and Goldberg, JACM'21]中提出的主要公开问题。我们算法的关键是一个局部迭代过程，能在o(Δ)时间内将O(Δ^2)染色转化为(Δ+O(Δ^{3/4}logΔ))染色。在此过程中，我们处理编码（任意）缺陷染色的特殊正常染色，并以局部迭代方式平方级减少使用颜色数量。作为结果的主要应用，我们还给出一种自稳定分布式(Δ+1)染色算法，其稳定时间为O(Δ^{3/4}logΔ)+log^*n。据我们所知，这是首个稳定时间关于Δ呈亚线性的(Δ+1)染色自稳定算法。