Analysis of Randomized Selection Algorithm Motivated by the LZ'77 Scheme

We consider a randomized selection algorithm that has n initial participants and a moderator. In each round of the process, each participant and the moderator throw a biased coin. Only the participants who throw the same result as the moderator stay in the game for subsequent rounds. With probability 1, all participants are eliminated in finitely many rounds. We let Mn denote the number of participants remaining in the game in the last nontrivial round. This simple algorithm has surprisingly many interesting applications. In particular, it models (asymptotically) the number of longest prefixes in the Lempel–Ziv ’77 data compression scheme. Such multiplicity was used recently in [13] to design an error-resilient LZ’77 scheme. We give precise asymptotic characteristics of the jth factorial moment of Mn for all j ∈ N. Also, we present a detailed asymptotic description of the exponential generating function for Mn. In particular, we exhibit periodic fluctuation in the distribution of Mn, and we prove that no limiting distribution exists (however, we observe that the asymptotic distribution follows the logarithmic series distribution plus some fluctuations). The results we develop are proved by probabilistic and analytical techniques of the analysis of algorithms. In particular, we utilize recurrence relations, analytical poissonization and depoissonization, the Mellin transform, and complex analysis.