A statistical view on exchanges in Quickselect

TL;DR

This paper analyzes the distribution of key exchanges in Quickselect, providing a limit theorem, explicit convergence rates, and tools for statistical analysis, applicable to various algorithm variants and cost measures.

Contribution

It introduces a limit law for key exchanges in Quickselect with explicit convergence rates and simulation methods, extending to different pivot choices and cost measures.

Findings

Limit law characterized by a recursive distributional equation.

Explicit convergence rate in the Kolmogorov--Smirnov metric.

Algorithm for exact simulation from the limit distribution.

Abstract

In this paper we study the number of key exchanges required by Hoare's FIND algorithm (also called Quickselect) when operating on a uniformly distributed random permutation and selecting an independent uniformly distributed rank. After normalization we give a limit theorem where the limit law is a perpetuity characterized by a recursive distributional equation. To make the limit theorem usable for statistical methods and statistical experiments we provide an explicit rate of convergence in the Kolmogorov--Smirnov metric, a numerical table of the limit law's distribution function and an algorithm for exact simulation from the limit distribution. We also investigate the limit law's density. This case study provides a program applicable to other cost measures, alternative models for the rank selected and more balanced choices of the pivot element such as median-of-$2t+1$ versions of Quickselect as well as further variations of the algorithm.