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arXiv · 1504.01130

Proving the Herman-Protocol Conjecture

Maria Bruna·Radu Grigore·Stefan Kiefer·Joël Ouaknine·James Worrell
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Abstract

Herman's self-stabilisation algorithm, introduced 25 years ago, is a well-studied synchronous randomised protocol for enabling a ring of $N$ processes collectively holding any odd number of tokens to reach a stable state in which a single token remains. Determining the worst-case expected time to stabilisation is the central outstanding open problem about this protocol. It is known that there is a constant $h$ such that any initial configuration has expected stabilisation time at most $h N^2$. Ten years ago, McIver and Morgan established a lower bound of $4/27 \approx 0.148$ for $h$, achieved with three equally-spaced tokens, and conjectured this to be the optimal value of $h$. A series of papers over the last decade gradually reduced the upper bound on $h$, with the present record (achieved in 2014) standing at approximately $0.156$. In this paper, we prove McIver and Morgan's conjecture and establish that $h = 4/27$ is indeed optimal.

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Maria Bruna, Radu Grigore, Stefan Kiefer, Joël Ouaknine, James Worrell. 2015-04-05. Proving the Herman-Protocol Conjecture. https://doi.org/10.4230/lipics.icalp.2016.104

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