# Convergence of some leader election algorithms

Abstract : We start with a set of $n$ players. With some probability $P(n,k)$, 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 \emph{phases} (or rounds) before getting only one player? We present a probabilistic analysis of this algorithm under some conditions on the probability distributions $P(n,k)$, including stochastic monotonicity and the assumption that roughly a fixed proportion $\al$ of the players survive in each round. We prove a kind of convergence in distribution for $X_n - \log_{1/\!\alpha}(n)$; 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 $d\l(X_n, \lceil Z + \log_{1/\!\alpha} (n)\rceil\r) \to 0$, 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 algorithm further, including numerical results.
Keywords :
Document type :
Journal articles
Domain :

https://hal.archives-ouvertes.fr/hal-00461881
Contributor : Christian Lavault <>
Submitted on : Monday, March 8, 2010 - 6:53:51 PM
Last modification on : Thursday, June 4, 2020 - 10:34:02 AM
Long-term archiving on: : Friday, June 18, 2010 - 8:48:20 PM

### File

Franklin08.pdf
Files produced by the author(s)

### Identifiers

• HAL Id : hal-00461881, version 1

### Citation

Svante Janson, Christian Lavault, Guy Louchard. Convergence of some leader election algorithms. Discrete Mathematics and Theoretical Computer Science, DMTCS, 2008, 10 (3), p. 171-196. ⟨hal-00461881v1⟩

Record views