State Complexity of Protocols With Leaders

Population protocols are a model of computation in which an arbitrary number of anonymous finite-memory agents are interacting in order to decide by stable consensus a predicate. In this paper, we focus on the counting predicates that asks, given an initial configuration, whether the number of agents in some initial state i is at least n. In 2018, Blondin, Esparza, and Jaax shown that with a fix number of leaders, there exists infinitely many n for which the counting predicate is stably computable by a protocol with at most O(log log(n)) states. We provide in this paper a matching lower-bound (up to a square root) that improves the inverse-Ackermannian lower-bound presented at PODC in 2021.