Efficient Sampling Methods for Discrete Distributions

Abstract

We study the fundamental problem of the exact and efficient generation of random values from a finite and discrete probability distribution. Suppose that we are given n distinct events with associated probabilities p1, ⋯ , pn. First, we consider the problem of sampling from the distribution where the i-th event has probability proportional to pi. Second, we study the problem of sampling a subset which includes the i-th event independently with probability pi. For both problems we present on two different classes of inputs—sorted and general probabilities—efficient data structures consisting of a preprocessing and a query algorithm. Varying the allotted preprocessing time yields a trade-off between preprocessing and query time, which we prove to be asymptotically optimal everywhere.