Reference:

Maria Bruna, Radu Grigore, Stefan Kiefer, Joël Ouaknine, and James Worrell. Proving the Herman-protocol conjecture. Technical report, arXiv.org, 2016. Available at https://arxiv.org/abs/1504.01130.

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 h N^2. Ten years ago, McIver and Morgan established a lower bound of 4/27 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.

Suggested BibTeX entry:

@techreport{16BGKOW-ICALP-TR,
    author = {Maria Bruna and Radu Grigore and Stefan Kiefer and Jo{\"e}l Ouaknine and James Worrell},
    institution = {arXiv.org},
    note = {Available at https://arxiv.org/abs/1504.01130},
    title = {Proving the {H}erman-Protocol Conjecture},
    year = {2016}
}