New Bounds for the Flock-of-Birds Problem
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In this paper, we continue a line of work on obtaining succinct population protocols for Presburger-definable predicates. More specifically, we focus on threshold predicates. These are predicates of the form n≥ 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 4log_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.
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