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Algorithmica

, Volume 75, Issue 4, pp 632–683 | Cite as

Analysis of Pivot Sampling in Dual-Pivot Quicksort: A Holistic Analysis of Yaroslavskiy’s Partitioning Scheme

  • Markus E. Nebel
  • Sebastian Wild
  • Conrado Martínez
Article

Abstract

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.

Keywords

Quicksort Dual-pivot Yaroslavskiy’s partitioning method  Median of three Average-case analysis I/O operations External-memory model 

Notes

Acknowledgments

We thank two anonymous reviewers for their careful reading and helpful comments.

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Copyright information

© Springer Science+Business Media New York 2015

Authors and Affiliations

  • Markus E. Nebel
    • 1
    • 2
  • Sebastian Wild
    • 1
  • Conrado Martínez
    • 3
  1. 1.Computer Science DepartmentUniversity of KaiserslauternKaiserslauternGermany
  2. 2.Department of Mathematics and Computer ScienceUniversity of Southern DenmarkOdenseDenmark
  3. 3.Department of Computer ScienceUniv. Politècnica de CatalunyaBarcelonaSpain

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