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.

Source : oai:HAL:hal-00461881v1

Volume: Vol. 10 no. 3

Section: Analysis of Algorithms

Published on: January 1, 2008

Submitted on: March 26, 2015

Keywords: Analysis of algorithms,distributed election algorithms,probability,[MATH.MATH-PR] Mathematics [math]/Probability [math.PR],[INFO.INFO-DC] Computer Science [cs]/Distributed, Parallel, and Cluster Computing [cs.DC],[MATH.MATH-CO] Mathematics [math]/Combinatorics [math.CO]

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