QuickXsort: A Fast Sorting Scheme in Theory and Practice

Abstract

QuickXsortis a highly efficient in-place sequential sorting scheme that mixesHoare’sQuicksortalgorithm with X, where X can be chosen from a wider rangeof other known sorting algorithms, likeHeapsort,InsertionsortandMergesort.Its major advantage is thatQuickXsortcan be in-place even if X is not. In thiswork we provide general transfer theorems expressing the number of comparisonsofQuickXsortin terms of the number of comparisons of X. More specifically,if pivots are chosen as medians of (not too fast) growing size samples, the aver-age number of comparisons ofQuickXsortand X differ only byo(n)-terms. Formedian-of-kpivot selection for some constantk, the difference is a linear term whosecoefficient we compute precisely. For instance, median-of-threeQuickMergesortuses at mostnlgn−0.8358n+O(logn)comparisons. Furthermore, we examine thepossibility of sorting base cases with some other algorithm using even less compar-isons. By doing so the average-case number of comparisons can be reduced down tonlgn−1.4112n+o(n)for a remaining gap of only 0.0315ncomparisons to the knownlower bound (while using onlyO(logn)additional space andO(nlogn)time over-all). Implementations of these sorting strategies show that the algorithms challengewell-established library implementations like Musser’sIntrosort