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arXiv · 0904.3898

On Smoothed Analysis of Quicksort and Hoare's Find

Mahmoud Fouz·Manfred Kufleitner·Bodo Manthey·Nima Zeini Jahromi
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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.

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Mahmoud Fouz, Manfred Kufleitner, Bodo Manthey, Nima Zeini Jahromi. 2009-04-25. On Smoothed Analysis of Quicksort and Hoare's Find. https://arxiv.org/abs/0904.3898

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