Fast Approximate Counting and Leader Election in Populations

TL;DR

This paper introduces efficient protocols for leader election and population size estimation in population protocols, achieving trade-offs between time and space complexity with practical applications in distributed systems.

Contribution

It presents a leader election protocol with adjustable time and space complexity and a population size estimation protocol assuming a unique leader, both improving efficiency.

Findings

Leader election in $O( ext{log}_m(n) imes ext{log}_2 n)$ time with $O( ext{max}igrace{m, ext{log} nigrace})$ states.

Trade-off between time and space complexity by tuning parameter $m$.

Population size estimation within a polynomial factor in $n$, stabilizing in $O( ext{log} n)$ time.

Abstract

We study the problems of leader election and population size counting for population protocols: networks of finite-state anonymous agents that interact randomly under a uniform random scheduler. We show a protocol for leader election that terminates in $O(\log_m(n) \cdot \log_2 n)$ parallel time, where $m$ is a parameter, using $O(\max\{m,\log n\})$ states. By adjusting the parameter $m$ between a constant and $n$, we obtain a single leader election protocol whose time and space can be smoothly traded off between $O(\log^2 n)$ to $O(\log n)$ time and $O(\log n)$ to $O(n)$ states. Finally, we give a protocol which provides an upper bound $\hat{n}$ of the size $n$ of the population, where $\hat{n}$ is at most $n^a$ for some $a>1$. This protocol assumes the existence of a unique leader in the population and stabilizes in $\Theta{(\log{n})}$ parallel time, using constant number of states in every node, except the unique leader which is required to use $\Theta{(\log^2{n})}$ states.