# Analysis of leader election algorithms

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1 LIPN - Laboratoire d-Informatique de Paris-Nord

Abstract : We start with a set of n players. With some probability $Pn, k$, we kill $n−k$ players; the other ones stay alive, and we repeat with them. What is the distribution of the number $X n$ of phases or rounds before getting only one player? We present a probabilistic analysis of this algorithm under some conditions on the probability distributions $Pn, k$, including stochastic monotonicity and the assumption that roughly a fixed proportion $\alpha$ of the players survive in each round. We prove a kind of convergence in distribution for $\lceilX n−\log {1-\alpha}n ceil$; as in many other similar problems there are oscillations and no true limit distribution, but suitable subsequences converge, and there is an absolutely continuous random variable $Z$ such that $dXn, \lceilZ+\log {1-\alpha}n ceil \to 0$, where d is either the total variation distance or the Wasserstein distance. Applications of the general result include the leader election algorithm where players are eliminated by independent coin tosses and a variation of the leader election algorithm proposed by W.R. Franklin. We study the latter algorithmfurther, including numerical results.

Keywords : Leader election analysis of algorithms probability

Autor: Christian Lavault -

Fuente: https://hal.archives-ouvertes.fr/

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