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Algorithmica

, Volume 74, Issue 1, pp 485–506 | Cite as

Analysis of Quickselect Under Yaroslavskiy’s Dual-Pivoting Algorithm

  • Sebastian Wild
  • Markus E. Nebel
  • Hosam Mahmoud
Article

Abstract

There is excitement within the algorithms community about a new partitioning method introduced by Yaroslavskiy. This algorithm renders Quicksort slightly faster than the case when it runs under classic partitioning methods. We show that this improved performance in Quicksort is not sustained in Quickselect; a variant of Quicksort for finding order statistics. We investigate the number of comparisons made by Quickselect to find a key with a randomly selected rank under Yaroslavskiy’s algorithm. This grand averaging is a smoothing operator over all individual distributions for specific fixed order statistics. We give the exact grand average. The grand distribution of the number of comparison (when suitably scaled) is given as the fixed-point solution of a distributional equation of a contraction in the Zolotarev metric space. Our investigation shows that Quickselect under older partitioning methods slightly outperforms Quickselect under Yaroslavskiy’s algorithm, for an order statistic of a random rank. Similar results are obtained for extremal order statistics, where again we find the exact average, and the distribution for the number of comparisons (when suitably scaled). Both limiting distributions are of perpetuities (a sum of products of independent mixed continuous random variables).

Keywords

Quicksort Quickselect Average-case analysis Grand average Contraction 

Mathematics Subject Classification

Primary: 60C05 Secondary: 68P10 68P20 

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Copyright information

© Springer Science+Business Media New York 2014

Authors and Affiliations

  • Sebastian Wild
    • 1
  • Markus E. Nebel
    • 1
    • 2
  • Hosam Mahmoud
    • 3
  1. 1.Computer Science DepartmentUniversity of KaiserslauternKaiserslauternGermany
  2. 2.Department of Mathematics and Computer ScienceUniversity of Southern DenmarkOdenseDenmark
  3. 3.Department of StatisticsThe George Washington UniversityWashingtonUSA

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