Asymptotic Analysis of a Leader Election Algorithm

TL;DR

This paper provides a precise asymptotic analysis of the average number of rounds in a leader election algorithm, including moments, distribution, and generalizations, confirming theoretical results with numerical data.

Contribution

It offers exact asymptotic expressions for the moments and distribution of the rounds in the leader election algorithm, extending analysis to generalized probability parameters.

Findings

Asymptotic limit of average rounds is approximately 2.4417.

Derived asymptotic expressions for second moments and distributions.

Identified a unique minimum of the expected rounds function for certain parameters.

Abstract

Itai and Rodeh showed that, on the average, the communication of a leader election algorithm takes no more than $LN$ bits, where $L \simeq 2.441716$ and $N$ denotes the size of the ring. We give a precise asymptotic analysis of the average number of rounds M(n) required by the algorithm, proving for example that $\dis M(\infty) := \lim\_{n\to \infty} M(n) = 2.441715879...$, where $n$ is the number of starting candidates in the election. Accurate asymptotic expressions of the second moment $M^{(2)}(n)$ of the discrete random variable at hand, its probability distribution, and the generalization to all moments are given. Corresponding asymptotic expansions $(n\to \infty)$ are provided for sufficiently large $j$, where $j$ counts the number of rounds. Our numerical results show that all computations perfectly fit the observed values. Finally, we investigate the generalization to probability $t/n$, where $t$ is a non negative real parameter. The real function $\dis M(\infty,t) := \lim\_{n\to \infty} M(n,t)$ is shown to admit \textit{one unique minimum} $M(\infty,t^{*})$ on the real segment $(0,2)$. Furthermore, the variations of $M(\infty,t)$ on thewhole real line are also studied in detail.