An Extended Note on the Comparison-optimal Dual Pivot Quickselect

TL;DR

This paper precisely determines the minimum average number of key comparisons needed by an optimal dual-pivot quickselect algorithm without sampling, revealing that it requires more comparisons than classical quickselect.

Contribution

It provides exact and asymptotic formulas for the comparison count of the optimal dual-pivot quickselect, extending understanding of its efficiency compared to classical methods.

Findings

Optimal dual-pivot quickselect requires more comparisons than classical quickselect.

Exact and asymptotic formulas for comparison counts are derived.

Main terms of asymptotic expansions are similar to classical quickselect.

Abstract

In this note the precise minimum number of key comparisons any dual-pivot quickselect algorithm (without sampling) needs on average is determined. The result is in the form of exact as well as asymptotic formul\ae{} of this number of a comparison-optimal algorithm. It turns out that the main terms of these asymptotic expansions coincide with the main terms of the corresponding analysis of the classical quickselect, but still---as this was shown for Yaroslavskiy quickselect---more comparisons are needed in the dual-pivot variant. The results are obtained by solving a second order differential equation for the generating function obtained from a recursive approach.