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Dual-Pivot Quicksort: Optimality, Analysis and Zeros of Associated Lattice Paths

Part of: Theory of data Combinatorics Theory of computing Discrete mathematics in relation to computer science Algorithms - Computer Science

Published online by Cambridge University Press:  14 August 2018

MARTIN AUMÜLLER ,
MARTIN DIETZFELBINGER ,
CLEMENS HEUBERGER ,
DANIEL KRENN  and
HELMUT PRODINGER
MARTIN AUMÜLLER
Affiliation:
IT University of Copenhagen, Rued Langgaards Vej 7, 2300 Copenhagen, Denmark (e-mail: maau@itu.dk)
MARTIN DIETZFELBINGER
Affiliation:
Fakultät für Informatik und Automatisierung, Technische Universität Ilmenau, Helmholtzplatz 5, 98693 Ilmenau, Germany (e-mail: martin.dietzfelbinger@tu-ilmenau.de)
CLEMENS HEUBERGER
Affiliation:
Institut für Mathematik, Alpen-Adria-Universität Klagenfurt, Universitätsstraße 65–67, 9020 Klagenfurt am Wörthersee, Austria (e-mail: clemens.heuberger@aau.at, math@danielkrenn.at, daniel.krenn@aau.at)
DANIEL KRENN
Affiliation:
Institut für Mathematik, Alpen-Adria-Universität Klagenfurt, Universitätsstraße 65–67, 9020 Klagenfurt am Wörthersee, Austria (e-mail: clemens.heuberger@aau.at, math@danielkrenn.at, daniel.krenn@aau.at)
HELMUT PRODINGER
Affiliation:
Department of Mathematical Sciences, Stellenbosch University, 7602 Stellenbosch, South Africa (e-mail: hproding@sun.ac.za)
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Abstract

We present an average-case analysis of a variant of dual-pivot quicksort. We show that the algorithmic partitioning strategy used is optimal, that is, it minimizes the expected number of key comparisons. For the analysis, we calculate the expected number of comparisons exactly as well as asymptotically; in particular, we provide exact expressions for the linear, logarithmic and constant terms.

An essential step is the analysis of zeros of lattice paths in a certain probability model. Along the way a combinatorial identity is proved.

MSC classification

Primary: 05A16: Asymptotic enumeration
Secondary: 68R05: Combinatorics 68P10: Searching and sorting 68Q25: Analysis of algorithms and problem complexity 68W40: Analysis of algorithms
Type
Paper
Information
Combinatorics, Probability and Computing , Volume 28 , Special Issue 4: Analysis of Algorithms , July 2019 , pp. 485 - 518
Copyright
Copyright © Cambridge University Press 2018 

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Footnotes

†

Supported by the Austrian Science Fund (FWF): P 24644-N26 and by the Karl Popper Kolleg ‘Modeling–Simulation–Optimization' funded by the Alpen-Adria-Universität Klagenfurt and by the Carinthian Economic Promotion Fund (KWF).

‡

Supported by an incentive grant of the National Research Foundation of South Africa.

§

An extended abstract containing the ideas of the asymptotic analysis of the dual-pivot quicksort strategies ‘Count’ and ‘Clairvoyant’, as well as the lattice path analysis of this article appeared as [3], and an appendix containing proofs is available as arXiv:1602.04031v1. This article contains additionally a proof that ‘Count’ is indeed the optimal strategy. This led to a major restructuring; the analysis now focuses on this strategy. Moreover, some proofs have been simplified.

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