Optimal Partitioning for Dual-Pivot Quicksort

Abstract

Dual-pivot quicksort refers to variants of classical quicksort where in the partitioning step two pivots are used to split the input into three segments. This can be done in different ways, giving rise to different algorithms. Recently, a dual-pivot algorithm proposed by Yaroslavskiy received much attention, because a variant of it replaced the well-engineered quicksort algorithm in Sun's Java 7 runtime library. Nebel and Wild (ESA 2012) analyzed this algorithm and showed that on average it uses 1.9n ln n + O(n) comparisons to sort an input of size n, beating standard quicksort, which uses 2n ln n + O(n) comparisons. We introduce a model that captures all dual-pivot algorithms, give a unified analysis, and identify new dual-pivot algorithms that minimize the average number of key comparisons among all possible algorithms up to a linear term. This minimum is 1.8n ln n + O(n). For the case that the pivots are chosen from a small sample, we include a comparison of dual-pivot quicksort and classical quicksort. Specifically, we show that dual-pivot quicksort benefits from a skewed choice of pivots. We experimentally evaluate our algorithms and compare them to Yaroslavskiy's algorithm and the recently described three-pivot quicksort algorithm of Kushagra et al. (ALENEX 2014).