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Asymptotic distribution theory for Hoare's selection algorithm

Part of: Limit theorems Theory of data Special processes

Published online by Cambridge University Press:  01 July 2016

Rudolf Grübel  and
Uwe Rösler
Rudolf Grübel*
Affiliation:
Universität Hannover and Christian-Albrecht-Universität Kiel
Uwe Rösler*
Affiliation:
Universität Hannover and Christian-Albrecht-Universität Kiel
*
* Postal address: Institut für Mathematische Stochastik, Universität Hannover, Postfach 60 09, D-30060 Hannover, Germany, and Mathematisches Seminar der Universität Kiel, Ludewig-Meyn-Str. 4, D-24098 Kiel, Germany.
* Postal address: Institut für Mathematische Stochastik, Universität Hannover, Postfach 60 09, D-30060 Hannover, Germany, and Mathematisches Seminar der Universität Kiel, Ludewig-Meyn-Str. 4, D-24098 Kiel, Germany.
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Abstract

We investigate the asymptotic behaviour of the distribution of the number of comparisons needed by a quicksort-style selection algorithm that finds the lth smallest in a set of n numbers. Letting n tend to infinity and considering the values l = 1, ···,n simultaneously we obtain a limiting stochastic process. This process admits various interpretations: it arises in connection with a representation of real numbers induced by nested random partitions and also in connection with expected path lengths of a random walk in a random environment on a binary tree.

Keywords

STOCHASTIC ALGORITHMSASYMPTOTIC DISTRIBUTIONORDER STATISTICSRANDOM BINARY NUMBERSRANDOM WALK IN A RANDOM ENVIRONMENT

MSC classification

Primary: 60F17: Functional limit theorems; invariance principles 68P10: Searching and sorting
Secondary: 60K99: None of the above, but in this section
Type
General Applied Probability
Information
Advances in Applied Probability , Volume 28 , Issue 1 , March 1996 , pp. 252 - 269
Copyright
Copyright © Applied Probability Trust 1996 

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