New Bounds for the Flock-of-Birds Problem

Alexander Kozachinskiy

Abstract: In this paper, we continue a line of work on obtaining succinct population protocols for Presburger-definable predicates. We focus on threshold predicates. These are predicates of the form $$n\ge d$$ , where n is a free variable and d is a constant. For every d, we establish a 1-aware population protocol for this predicate with $$\log _2 d + \min \{e, z\} + O(1)$$ states, where e (resp., z) is the number of 1’s (resp., 0’s) in the binary representation of d (resp., $$d - 1$$ ). This improves upon an upper bound $$4\log _2 d + O(1)$$ due to Blondin et al. We also show that any 1-aware protocol for our problem must have at least $$\log _2(d)$$ states. This improves upon a lower bound $$\log _3 d$$ due to Blondin et al.