QuickMergesort: Practically Efficient Constant-Factor Optimal Sorting

TL;DR

QuickMergesort combines Quicksort and Mergesort with median-of-sqrt(n) pivot selection to achieve near-optimal comparison counts and practical efficiency, outperforming existing sorting algorithms in experiments.

Contribution

The paper introduces a practical version of QuickMergesort with median-of-3 pivot selection that is both comparison-efficient and fast in practice, surpassing current library sorts.

Findings

QuickMergesort requires at most n log n - 0.75n comparisons on average.

The median-of-3 pivot selection maintains efficiency with minimal overhead.

Experimental results show it outperforms C++'s Introsort and Java's Dual-Pivot Quicksort.

Abstract

We consider the fundamental problem of internally sorting a sequence of $n$ elements. In its best theoretical setting QuickMergesort, a combination Quicksort with Mergesort with a Median-of-$\sqrt{n}$ pivot selection, requires at most $n \log n - 1.3999n + o(n)$ element comparisons on the average. The questions addressed in this paper is how to make this algorithm practical. As refined pivot selection usually adds much overhead, we show that the Median-of-3 pivot selection of QuickMergesort leads to at most $n \log n - 0{.}75n + o(n)$ element comparisons on average, while running fast on elementary data. The experiments show that QuickMergesort outperforms state-of-the-art library implementations, including C++'s Introsort and Java's Dual-Pivot Quicksort. Further trade-offs between a low running time and a low number of comparisons are studied. Moreover, we describe a practically efficient version with $n \log n + O(n)$ comparisons in the worst case.