Simple and Fast Approximate Counting and Leader Election in Populations

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 provide simple protocols for approximate counting of the size of the population and for leader election. We show a protocol for leader election that terminates in $O(\frac{\log ^2{n}}{\log {m}})$ parallel time, where $1 \le m \le n$ 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. We also 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 constant $a>1$. This protocol assumes the existence of a unique leader in the population and stabilizes in $\varTheta {(\log {n})}$ parallel time, using constant number of states in every node, except from the unique leader which is required to use $\varTheta {(\log ^2{n})}$ states.

All authors were supported by the EEE/CS initiative NeST. The last author was also supported by the Leverhulme Research Centre for Functional Materials Design. This work was partially supported by the EPSRC Grant EP/P02002X/1 on Algorithmic Aspects of Temporal Graphs.