Leader election: A Markov chain approach

TL;DR

This paper analyzes a randomized leader election algorithm using Markov chains and discrete potential theory, providing insights into the asymptotic behavior of the process as the number of participants grows.

Contribution

It introduces two Markov chain models and applies potential theory and exponential order statistics to analyze leader election dynamics.

Findings

Asymptotic behavior of the number of rounds is characterized.

Distribution of remaining participants is described as the initial size grows.

Connections to exponential order statistics are established.

Abstract

A well-studied randomized election algorithm proceeds as follows: In each round the remaining candidates each toss a coin and leave the competition if they obtain heads. Of interest is the number of rounds required and the number of winners, both related to maxima of geometric random samples, as well as the number of remaining participants as a function of the number of rounds. We introduce two related Markov chains and use ideas and methods from discrete potential theory to analyse the respective asymptotic behaviour as the initial number of participants grows. One of the tools used is the approach via the R\'enyi-Sukhatme representation of exponential order statistics, which was first used in the leader election context by Bruss and Gr\"ubel in \cite{BrGr03}.