Exact size counting in uniform population protocols in nearly logarithmic time

TL;DR

This paper introduces the first population protocol that accurately counts the exact size of a uniform, leaderless population in nearly logarithmic time, using a uniform transition algorithm without prior size knowledge.

Contribution

It presents the first sublinear-time, uniform population protocol for exact size counting and leader election, with significant improvements in time complexity and state efficiency.

Findings

Converges in O(log n log log n) time with high probability

Uses O(n^{60}) states for counting, O(n^{18}) for leader election

Reduces state complexity to O(n^{30}) and O(n^{9}) with increased time to O(log^2 n)

Abstract

We study population protocols: networks of anonymous agents that interact under a scheduler that picks pairs of agents uniformly at random. The _size counting problem_ is that of calculating the exact number $n$ of agents in the population, assuming no leader (each agent starts in the same state). We give the first protocol that solves this problem in sublinear time. The protocol converges in $O(\log n \log \log n)$ time and uses $O(n^{60})$ states ($O(1) + 60 \log n$ bits of memory per agent) with probability $1-O(\frac{\log \log n}{n})$. The time complexity is also $O(\log n \log \log n)$ in expectation. The time to converge is also $O(\log n \log \log n)$ in expectation. Crucially, unlike most published protocols with $\omega(1)$ states, our protocol is _uniform_: it uses the same transition algorithm for any population size, so does not need an estimate of the population size to be embedded into the algorithm. A sub-protocol is the first uniform sublinear-time leader election population protocol, taking $O(\log n \log \log n)$ time and $O(n^{18})$ states. The state complexity of both the counting and leader election protocols can be reduced to $O(n^{30})$ and $O(n^{9})$ respectively, while increasing the time to $O(\log^2 n)$.