On Smoothed Analysis of Quicksort and Hoare's Find

TL;DR

This paper conducts a smoothed analysis of Hoare's find and quicksort algorithms, examining their performance under perturbations and showing that median-of-three pivoting does not significantly improve efficiency.

Contribution

It provides the first smoothed analysis of Hoare's find and compares it with quicksort, establishing lower bounds for median-of-three pivot rules.

Findings

Smoothed analysis reveals the behavior of algorithms between worst and average cases.

Median-of-three pivoting does not significantly outperform classic pivot selection.

Lower bounds for the number of comparisons are established for median-of-three rules.

Abstract

We provide a smoothed analysis of Hoare's find algorithm and we revisit the smoothed analysis of quicksort. Hoare's find algorithm - often called quickselect - is an easy-to-implement algorithm for finding the k-th smallest element of a sequence. While the worst-case number of comparisons that Hoare's find needs is quadratic, the average-case number is linear. We analyze what happens between these two extremes by providing a smoothed analysis of the algorithm in terms of two different perturbation models: additive noise and partial permutations. Moreover, we provide lower bounds for the smoothed number of comparisons of quicksort and Hoare's find for the median-of-three pivot rule, which usually yields faster algorithms than always selecting the first element: The pivot is the median of the first, middle, and last element of the sequence. We show that median-of-three does not yield a significant improvement over the classic rule: the lower bounds for the classic rule carry over to median-of-three.