A Lower Bound on Probabilistic Algorithms for Distributive Ring Coloring

Suppose that n processors are arranged in a ring and can communicate only with their immediate neighbors. It is shown that any probabilistic algorithm for 3 coloring the ring must take at least $\frac{1}{2}\log^* n - 2$ rounds, otherwise the probability that all processors are colored legally is less than $\frac{1}{2}$. A similar time bound holds for selecting a maximal independent set. The bound is tight (up to a constant factor) in light of the deterministic algorithms of Cole and Vishkin [Inform, and Control, 70 (1986), pp. 32–53] and extends the lower bound for deterministic algorithms of Linial [Proc. 28th IEEE Foundations of Computer Science Symposium, 1987, pp. 331–335].