Herman协议猜想的证明
Proving the Herman-Protocol Conjecture
摘要：
赫尔曼的自稳定算法，介绍了25年前，是一个经过充分研究的同步随机协议，使环的$N$进程共同持有任何奇数令牌达到稳态，其中一个单一的令牌。确定最差情况下的预期稳定时间是本方案的核心未决问题。众所周知，存在一个常数$h$，使得任何初始配置的预期稳定时间最多为$h N^2$。十年前，McIver和Morgan为$h$建立了一个$4/27 \approx 0.148$的下限，用三个等间距的代币实现，并证明这是$h$的最佳值。过去十年的一系列论文逐渐降低了$h$的上限，目前的记录（2014年实现）约为$0.156$。本文证明了McIver和Morgan的猜想，并证明了$h = 4/27$确实是最优的.
英文摘要：
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.
机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。
