Randomized selection revisited

TL;DR

This paper revisits the Floyd-Rivest selection algorithm, providing improved average-case analysis, extending it to nondistinct elements, and demonstrating its practical efficiency through computational results.

Contribution

It rectifies and extends Floyd-Rivest's analysis of the Select algorithm to nondistinct elements and confirms its practical efficiency.

Findings

Select requires at most n + min{k, n-k} + o(n) comparisons on average.

The analysis is corrected and extended to nondistinct elements.

Computational results show Select may be the best practical algorithm.

Abstract

We show that several versions of Floyd and Rivest's algorithm Select for finding the $k$th smallest of $n$ elements require at most $n+\min\{k,n-k\}+o(n)$ comparisons on average and with high probability. This rectifies the analysis of Floyd and Rivest, and extends it to the case of nondistinct elements. Our computational results confirm that Select may be the best algorithm in practice.