The coupon-collector problem revisited — a survey of engineering problems and computational methods

Abstract

A standard combinatorial problem is to estimate the number (T) of coupons, drawn at random, needed to complete a collection of all possible m types. Generalizations of this problem have found many engineering applications. The usefulness of the model is hampered by the difficulties in obtaining numerical results for moments or distributions. We show two computational paradigms that are well suited for this type of problems: one, following Flajolet et al. [21], is the calculus of generating functions over regular languages. We use it to provide relatively efficient answers to several questions about the sampling process – we show it is possible to compute arbitrarily accurate approximations for quantities such as E[T], in a time which is linear in m for any type distribution, while an exponential time is required for exact calculation. It also leads to a proof of a long-standing folk-theorem, concerning the extremality of uniform reference probabilities. The second method is a generalization of the Poisson transform [25], which we use to discuss statistical estimation procedures