The new dual-pivot Quicksort by Vladimir Yaroslavskiy - used in Oracle's Java runtime library since version 7 - features intriguing asymmetries. They make a basic variant of this algorithm use less comparisons than classic single-pivot Quicksort. In this paper, we extend the analysis to the case where the two pivots are chosen as fixed order statistics of a random sample. Surprisingly, dual-pivot Quicksort then needs more comparisons than a corresponding version of classic Quicksort, so it is clear that counting comparisons is not sufficient to explain the running time advantages observed for Yaroslavskiy's algorithm in practice. Consequently, we take a more holistic approach and give also the precise leading term of the average number of swaps, the number of executed Java Bytecode instructions and the number of scanned elements, a new simple cost measure that approximates I/O costs in the memory hierarchy. We determine optimal order statistics for each of the cost measures. It turns out that the asymmetries in Yaroslavskiy's algorithm render pivots with a systematic skew more efficient than the symmetric choice. Moreover, we finally have a convincing explanation for the success of Yaroslavskiy's algorithm in practice: Compared with corresponding versions of classic single-pivot Quicksort, dual-pivot Quicksort needs significantly less I/Os, both with and without pivot sampling.
- Pub Date:
- November 2014
- Computer Science - Data Structures and Algorithms;
- Mathematics - Probability
- This article is identical (up to typograhical details) to the Algorithmica version available from Springerlink (see DOI). It is an extended and improved version of our corresponding article at the AofA 2014 conference [arXiv:1403.6602]