Analysis of Quickselect under Yaroslavskiy's Dual-Pivoting Algorithm

TL;DR

This paper analyzes the performance of Quickselect with Yaroslavskiy's dual-pivot partitioning, revealing it is less efficient than older methods for finding order statistics, with exact averages and distributional results derived.

Contribution

It provides the first detailed analysis of Quickselect under Yaroslavskiy's dual-pivot algorithm, including exact averages and distributional characterizations of comparisons.

Findings

Quickselect under Yaroslavskiy's algorithm performs worse than older methods for random rank statistics.

Exact grand averages of comparisons are derived for various order statistics.

Limiting distributions are characterized as perpetuities involving independent mixed random variables.

Abstract

There is excitement within the algorithms community about a new partitioning method introduced by Yaroslavskiy. This algorithm renders Quicksort slightly faster than the case when it runs under classic partitioning methods. We show that this improved performance in Quicksort is not sustained in Quickselect; a variant of Quicksort for finding order statistics. We investigate the number of comparisons made by Quickselect to find a key with a randomly selected rank under Yaroslavskiy's algorithm. This grand averaging is a smoothing operator over all individual distributions for specific fixed order statistics. We give the exact grand average. The grand distribution of the number of comparison (when suitably scaled) is given as the fixed-point solution of a distributional equation of a contraction in the Zolotarev metric space. Our investigation shows that Quickselect under older partitioning methods slightly outperforms Quickselect under Yaroslavskiy's algorithm, for an order statistic of a random rank. Similar results are obtained for extremal order statistics, where again we find the exact average, and the distribution for the number of comparisons (when suitably scaled). Both limiting distributions are of perpetuities (a sum of products of independent mixed continuous random variables).