A Nearly Optimal Upper Bound for the Self-Stabilization Time in Herman’s Algorithm
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 self-stabilization 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 of it.
KeywordsGlobal Maximum Stabilization Time Lagrange Multiplier Method Strongly Connect Component State Markov Chain
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