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 Xn 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 α of the players survive in each round. We prove a kind of
more » ... We prove a kind of convergence in distribution for Xn − log 1/α 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(Xn, Z + log 1/α n) → 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 1982. We study the latter algorithm further, including numerical results. We consider a general leader election algorithm of the following type: We are given some random procedure that, given any set of n ≥ 2 individuals, eliminates some (but not all) individuals. If there is more that one survivor, we repeat the procedure with the set of survivors until only one (the winner) remains. We are interested in the (random) number X n of rounds required if we start with n individuals. (We set X 1 = 0, and have X n ≥ 1 for n ≥ 2.) We let N k be the number of individuals remaining after round k; thus X n := min{k : N k = 1}, where we start with N 0 = n. For convenience we may suppose that we continue with infinitely many rounds where nothing happens; thus N k is defined for all k ≥ 0 and N k = 1 for all k ≥ X n. 1365-8050