A nearly optimal upper bound for the self-stabilization time in Herman’s algorithm
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Self-stabilization algorithms are very important in designing fault-tolerant distributed systems. In this paper we consider Herman’s self-stabilization algorithm and study its expected termination time. McIver and Morgan have conjectured the optimal upper bound being \(0.148N^2\), where \(N\) denotes the number of processors. We present an elementary proof showing a bound of \(0.167N^2\), a sharp improvement compared with the best known bound \(0.521N^2\). Our proof is inspired by McIver and Morgan’s approach: we find a nearly optimal closed form of the expected stabilization time for any initial configuration, and apply the Lagrange multipliers method to give an upper bound.
KeywordsHerman’s algorithm Self-stabilization Lagrange multipliers method
- 4.Feller, W.: An Introduction to Probability Theory and Its Applications, vol. 1. Wiley, New York (1968)Google Scholar
- 5.Feng, Y., Zhang, L.: A tighter bound for the self-stabilization time in Herman’s algorithm. Inf. Process. Lett. 113(13), 486–488 (2013)Google Scholar
- 6.Feng, Y., Zhang, L.: A nearly optimal upper bound for the self-stabilization time in Herman’s algorithm. In: CONCUR, vol. 8704, pp. 342–356. Springer, Berlin (2014)Google Scholar