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Proving the Herman-Protocol Conjecture

Authors Maria Bruna, Radu Grigore, Stefan Kiefer, Joël Ouaknine, James Worrell

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Maria Bruna
Radu Grigore
Stefan Kiefer
Joël Ouaknine
James Worrell

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Maria Bruna, Radu Grigore, Stefan Kiefer, Joël Ouaknine, and James Worrell. Proving the Herman-Protocol Conjecture. In 43rd International Colloquium on Automata, Languages, and Programming (ICALP 2016). Leibniz International Proceedings in Informatics (LIPIcs), Volume 55, pp. 104:1-104:12, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2016) https://doi.org/10.4230/LIPIcs.ICALP.2016.104

Abstract

Herman's self-stabilization algorithm, introduced 25 years ago, is a well-studied synchronous randomized 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 stabilization 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 stabilization time at most hN2. Ten years ago, McIver and Morgan established a lower bound of 4/27 ~ 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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Keywords
  • randomized protocols
  • self-stabilization
  • Lyapunov function
  • expected time

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