Proving the Herman-Protocol Conjecture

TL;DR

This paper proves the long-standing conjecture that the worst-case expected stabilization time for Herman's self-stabilisation algorithm is exactly 4/27 times the square of the number of processes, confirming the optimality of previous bounds.

Contribution

It rigorously proves McIver and Morgan's conjecture that the constant h in the expected stabilization time bound is 4/27, resolving a 25-year open problem.

Findings

Confirmed the optimal constant h as 4/27 for the expected stabilization time.

Validated the conjecture that the worst-case initial configuration is three equally-spaced tokens.

Closed the gap between known upper and lower bounds on the stabilization time constant.

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.