On the time and the bit complexity of distributed randomised anonymous ring colouring

We present and analyse a very simple randomised distributed vertex colouring algorithm for ring graphs. Its time complexity is log2n+o(logn) on average and 2log2n+o(logn) with probability 1−o(n−1). Since each message contains one bit, we deduce the same values for its bit complexity. Then we combine this algorithm with another and we obtain a 3-colouring algorithm for ring graphs. Thanks to an overlapping, we obtain once more the same values for the time complexities on average and with probability 1−o(n−1). The same results hold for the bit complexity. These results are obtained using the Mellin transform.We establish lower bounds (on average and with probability 1−o(n−1)) for the distributed randomised anonymous ring colouring problem. We prove that our algorithms match these lower bounds modulo a negligible additive function (negligible with respect to log2n).We assume that the ring is anonymous: unique identities are not available to distinguish the processes; we only assume that each vertex distinguishes between its neighbours. Furthermore, we do not assume that the size (or an upper bound on the size) of the ring is known.