On Stabilization in Herman's Algorithm

TL;DR

This paper analyzes the expected stabilization time of Herman's algorithm, providing new bounds, extending to asynchronous versions, and demonstrating rapid recovery from bounded errors, with faster stabilization in typical random initial states.

Contribution

It offers improved upper bounds on stabilization time, introduces an asynchronous variant, and shows quick recovery from bounded errors, advancing understanding of Herman's algorithm's efficiency.

Findings

Expected stabilization time is at most 0.64 N^2.

Asynchronous version also stabilizes in O(N^2) time.

Quick recovery from fixed, bounded errors with O(N) expected time.

Abstract

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 Theta(N^2). Our first contribution is to give an upper bound of 0.64 N^2 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(N^2) 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 that the expected time to stabilization is O(N). This reveals a hitherto unknown and highly desirable property of Herman's algorithm: it recovers quickly from bounded errors. We also show that if the initial configuration arises by resetting each bit independently and uniformly at random, then stabilization is significantly faster than in the worst case.