# Proving the Herman-Protocol Conjecture

Bruna, Maria and Grigore, Radu and Kiefer, Stefan and Ouaknine, Joël and Worrell, James (2016) Proving the Herman-Protocol Conjecture. In: Chatzigiannakis, Ioannis and Mitzenmacher, Michael and Rabani, Yuval and Sangiorgi, Davide, eds. 43rd International Colloquium on Automata, Languages, and Programming. ICALP 2016, Rome, Italy, July 12-15, 2016. Leibniz International Proceedings in Informatics . Schloss Dagstuhl--Leibniz-Zentrum fuer Informatik, Saarbrücken/Wadern, Germany, 104:1-104:12. ISBN 978-3-95977-013-2. (doi:10.4230/LIPIcs.ICALP.2016.104)

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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 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.

Item Type: Book section 10.4230/LIPIcs.ICALP.2016.104 Q Science > QA Mathematics (inc Computing science) > QA 75 Electronic computers. Computer science Faculties > Sciences > School of Computing > Programming Languages and Systems Group Radu Grigore 03 May 2016 11:30 UTC 20 Feb 2020 14:19 UTC https://kar.kent.ac.uk/id/eprint/55083 (The current URI for this page, for reference purposes) https://orcid.org/0000-0003-1128-0311