QuickXsort: Efficient Sorting with n log n - 1.399n +o(n) Comparisons on Average

TL;DR

This paper introduces QuickXsort, a generalized sorting framework that achieves near-optimal average comparison counts, and demonstrates its efficiency with specific algorithms like QuickMergesort, which performs close to theoretical lower bounds.

Contribution

The paper generalizes QuickHeapsort to QuickXsort, providing new algorithms that nearly reach the theoretical minimum number of comparisons on average.

Findings

QuickXsort achieves approximately n log n - 1.399n comparisons on average.

QuickMergesort with MergeInsertion base case performs close to optimal.

Practical implementation of QuickMergesort is only 15% slower than STL-Introsort.

Abstract

In this paper we generalize the idea of QuickHeapsort leading to the notion of QuickXsort. Given some external sorting algorithm X, QuickXsort yields an internal sorting algorithm if X satisfies certain natural conditions. With QuickWeakHeapsort and QuickMergesort we present two examples for the QuickXsort-construction. Both are efficient algorithms that incur approximately n log n - 1.26n +o(n) comparisons on the average. A worst case of n log n + O(n) comparisons can be achieved without significantly affecting the average case. Furthermore, we describe an implementation of MergeInsertion for small n. Taking MergeInsertion as a base case for QuickMergesort, we establish a worst-case efficient sorting algorithm calling for n log n - 1.3999n + o(n) comparisons on average. QuickMergesort with constant size base cases shows the best performance on practical inputs: when sorting integers it is slower by only 15% to STL-Introsort.