Self-stabilizing $(\varDelta +1)$-Coloring in Sublinear (in $\varDelta$) Rounds via Locally-Iterative Algorithms

Abstract

Fault-tolerance is a central theme in distributed computing. Self-stabilization is a key property that guarantees a distributed system starting from an arbitrary state eventually converges to a desired behavior. Such strong level of fault-tolerance is often desirable due to the error-prone nature of distributed systems. Developing fast and robust coloring algorithms has been a central topic in the study of distributed graph algorithms. In this paper, we give a $(\varDelta +1)$-coloring algorithm with $O(\varDelta ^{3/4}\log \varDelta )+\log ^*{n}$ stabilization time, only using messages of size $O(\log {n})$, on input graphs of n vertices and maximum degree $\varDelta$. This is the first self-stabilizing $(\varDelta +1)$-coloring algorithm with sublinear-in-$\varDelta$ stabilization time. The key building block of our algorithm is a new locally-iterative $(\varDelta +1)$-coloring algorithm with $O(\varDelta ^{3/4}\log \varDelta )+\log ^*{n}$ runtime. To the best of our knowledge, this is the first locally-iterative $(\varDelta +1)$-coloring algorithm with sublinear-in-$\varDelta$ runtime. This answers an open question raised in [Barenboim, Elkin, and Goldberg, JACM ’21].