eng
episciences.org
Discrete Mathematics & Theoretical Computer Science
1365-8050
2008-01-01
Vol. 10 no. 3
Analysis of Algorithms
10.46298/dmtcs.437
437
journal article
Convergence of some leader election algorithms
Svante Janson
Christian Lavault
Guy Louchard
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.
https://dmtcs.episciences.org/437/pdf
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]