New Bounds for the Flock-of-Birds Problem

TL;DR

This paper develops more efficient population protocols for threshold predicates, reducing the number of states needed and establishing tighter bounds, advancing the understanding of succinctness in population protocols.

Contribution

It introduces a new population protocol with fewer states for threshold predicates and tightens the bounds on the minimal states required, improving previous results.

Findings

New protocol with log2(d) + min{e,z} + O(1) states

Improved upper bound over previous work

Tighter lower bound of log2(d) states

Abstract

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 $n\ge d$, 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.