Self-Stabilizing Phase Clocks and the Adaptive Majority Problem
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We present a self-stabilising 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(log n) parallel time 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 self-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 initial majority has a support of at least Ω(log n) agents and a sufficiently large bias is present, then the protocol converges to a correct output within O(log n) time and stays in a correct configuration for poly n time, w.h.p.
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