Algorithms for Generating Small Random Samples

TL;DR

This paper addresses the problem of efficiently generating small random samples without replacement, focusing on the special cases $k=2$ and $k=3$. It introduces constant-time algorithms, RandomPair$(n)$ and RandomTriple$(n)$, that use two and three random numbers and resolve duplicates to ensure uniformity, with potential extensions to fixed larger $k$ via sampling networks. The authors benchmark these methods against standard general-purpose algorithms (reservoir, pool, and insertion sampling) using the $\rho\mu$ Java library and JMH, demonstrating substantial speedups for small $k$ and offering insights into hardware-friendly implementations. The work provides practical, open-source implementations and reproducible experiments, highlighting the impact on fast sampling tasks in areas like evolutionary algorithms and permutation-related computations.

Abstract

We present algorithms for generating small random samples without replacement. We consider two cases. We present an algorithm for sampling a pair of distinct integers, and an algorithm for sampling a triple of distinct integers. The worst-case runtime of both algorithms is constant, while the worst-case runtimes of common algorithms for the general case of sampling $k$ elements from a set of $n$ increase with $n$. Java implementations of both algorithms are included in the open source library $ρμ$.