Probabilistic divide-and-conquer: a new exact simulation method, with integer partitions as an example

TL;DR

The paper introduces probabilistic divide-and-conquer, a novel method to enhance rejection sampling efficiency, demonstrated through integer partitions and coupon collector applications, significantly reducing rejection costs.

Contribution

It presents a new probabilistic divide-and-conquer approach that improves rejection sampling efficiency and introduces a variation for generating i.i.d. samples with sublinear cost.

Findings

Rejection cost reduced from order n^{3/4} to a constant for integer partitions.

Non-recursive application removes a large fraction of rejection sampling cost.

Cost for generating i.i.d. samples becomes sublinear in the number of samples.

Abstract

We propose a new method, probabilistic divide-and-conquer, for improving the success probability in rejection sampling. For the example of integer partitions, there is an ideal recursive scheme which improves the rejection cost from asymptotically order $n^{3/4}$ to a constant. We show other examples for which a non--recursive, one--time application of probabilistic divide-and-conquer removes a substantial fraction of the rejection sampling cost. We also present a variation of probabilistic divide-and-conquer for generating i.i.d. samples that exploits features of the coupon collector's problem, in order to obtain a cost that is sublinear in the number of samples.