Asymptotic Analysis of a Leader Election Algorithm

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.