Generalized solution for the Herman Protocol Conjecture

Abstract

We have a cycle of $N$ nodes and there is a token on an odd number of nodes. At each step, each token independently moves to its clockwise neighbor or stays at its position with probability $\frac{1}{2}$. If two tokens arrive to the same node, then we remove both of them. The process ends when only one token remains. The question is that for a fixed $N$, which is the initial configuration that maximizes the expected number of steps $E(T)$. The Herman Protocol Conjecture says that the $3$-token configuration with distances $\lfloor\frac{N}{3}\rfloor$ and $\lceil\frac{N}{3}\rceil$ maximizes $E(T)$. We present a proof of this conjecture not only for $E(T)$ but also for $E\big(f(T)\big)$ for some function $f:\mathbb{N}\rightarrow\mathbb{R}^{+}$ which method applies for different generalizations of the problem