On Stabilization in Herman's Algorithm

Herman’s algorithm is a synchronous randomized protocol for achieving self-stabilization in a token ring consisting of N processes. The interaction of tokens makes the dynamics of the protocol very difficult to analyze. In this paper we study the expected time to stabilization in terms of the initial configuration. It is straightforward that the algorithm achieves stabilization almost surely from any initial configuration, and it is known that the worst-case expected time to stabilization (with respect to the initial configuration) is Θ(N2 ). Our first contribution is to give an upper bound of 0.64N2 on the expected stabilization time, improving on previous upper bounds and reducing the gap with the best existing lower bound. We also introduce an asynchronous version of the protocol, showing a similar O(N2 ) convergence bound in this case. Assuming that errors arise from the corruption of some number k of bits, where k is fixed independently of the size of the ring, we show tha...

• Type: Journal
• Year: 2011
• Where: CORR
• Authors: Stefan Kiefer, Andrzej S. Murawski, Joël Ouaknine, James Worrell, Lijun Zhang
• Keywords: CORR 2011, Education, Expected Time, Initial Configuration, Upper Bounds