Loosely-Stabilizing Phase Clocks and the Adaptive Majority Problem

We present a loosely-stabilizing phase clock for population protocols. In the population model we are given a system of $n$ identical agents which interact in a sequence of randomly chosen pairs. Our phase clock is leaderless and it requires $O(\log n)$ states. It runs forever and is, at any point of time, in a synchronous state w.h.p. When started in an arbitrary configuration, it recovers rapidly and enters a synchronous configuration within $O(n\log n)$ interactions w.h.p. Once the clock is synchronized, it stays in a synchronous configuration for at least poly $n$ parallel time w.h.p. We use our clock to design a loosely-stabilizing protocol that solves the comparison problem introduced by Alistarh et al., 2021. In this problem, a subset of agents has at any time either $A$ or $B$ as input. The goal is to keep track which of the two opinions is (momentarily) the majority. We show that if the majority has a support of at least $\Omega(\log n)$ agents and a sufficiently large bias is present, then the protocol converges to a correct output within $O(n\log n)$ interactions and stays in a correct configuration for poly $n$ interactions, w.h.p.