Convergence of some leader election algorithms

TL;DR

This paper analyzes the distribution of the number of rounds needed in certain leader election algorithms, showing convergence properties and applying results to specific algorithms including coin-toss elimination and Franklin's variation.

Contribution

It provides a probabilistic convergence analysis for leader election algorithms under general conditions, including oscillation phenomena and subsequential convergence.

Findings

Distribution of rounds converges in a shifted logarithmic scale.

Oscillations prevent a true limit distribution, but subsequences converge.

Numerical results support theoretical findings.

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 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 alpha of the players survive in each round. We prove a kind of convergence in distribution for X_n-log_a n, where the basis a=1/alpha; 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 the distribution of X_n can be approximated by Z+log_a n rounded to the nearest larger integer. 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.