Pivot Sampling in Dual-Pivot Quicksort

TL;DR

This paper analyzes the dual-pivot Quicksort algorithm, revealing how choosing skewed pivots based on sample order statistics can optimize performance depending on the cost measure, contrasting with classic median-based pivot selection.

Contribution

It extends the analysis of dual-pivot Quicksort to fixed sample order statistics, showing that skewed pivots outperform symmetric ones depending on the cost measure.

Findings

Skewed pivots can be more efficient than median pivots in dual-pivot Quicksort.

Optimal skew depends on the specific cost measure used.

Different cost measures lead to different optimal pivot choices.

Abstract

The new dual-pivot Quicksort by Vladimir Yaroslavskiy - used in Oracle's Java runtime library since version 7 - features intriguing asymmetries in its behavior. They were shown to cause a basic variant of this algorithm to use less comparisons than classic single-pivot Quicksort implementations. In this paper, we extend the analysis to the case where the two pivots are chosen as fixed order statistics of a random sample and give the precise leading term of the average number of comparisons, swaps and executed Java Bytecode instructions. It turns out that - unlike for classic Quicksort, where it is optimal to choose the pivot as median of the sample - the asymmetries in Yaroslavskiy's algorithm render pivots with a systematic skew more efficient than the symmetric choice. Moreover, the optimal skew heavily depends on the employed cost measure; most strikingly, abstract costs like the number of swaps and comparisons yield a very different result than counting Java Bytecode instructions, which can be assumed most closely related to actual running time.