Efficient Sampling Methods for Discrete Distributions
Karl Bringmann, Konstantinos D. Panagiotou · Algorithmica · 2016
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 $$p_1, \dots , p_n$$ p 1 , ⋯ , p n . First, we consider the problem of sampling from the distribution where the i-th event has probability proportional to $$p_i$$ p i . Second, we study the problem of sampling a subset which includes the i-th event independently with probability $$p_i$$ p i . 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.