Limits and power of the simplest uniform and self-stabilizing phase clock algorithm

In this paper, the phase clock algorithm which stabilizes on general graphs is studied on anonymous rings. The authors showed that this K-clock algorithm works with K/spl ges/2D, where D is the diameter of the graph. We prove that this algorithm works on unidirectional (bidirectional) rings iff K satisfies K>K'K/K'+n(2K>2K'-K/K'+n, respectively) where K' is the greatest divisor of K (K'/spl ne/K) and n is the size of the ring. From this characterization, we show that any ring stabilizes with some K<2D if K is odd. We also prove that, if K is prime, unidirectional and bidirectional rings stabilize with K<2[n/2]/spl sime/D and K<2[n/3]/spl sime/4D/3, respectively. Finally, we generalize the algorithm to synchronize any ring with any clock value.