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Hoare's Selection Algorithm: A Markov Chain Approach

Part of: Markov processes Theory of data

Published online by Cambridge University Press:  14 July 2016

Rudolf Grübel
Rudolf Grübel*
Affiliation:
Universität Hannover
*
∗Postal address: Institut für Mathematische Stochastik, Universität Hannover, Postfach 60 09, D-30060 Hannover, Germany. e-mail address: rgrubel@stochastik.uni-hannover.de
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Abstract

We obtain bounds for the distribution of the number of comparisons needed by Hoare's randomized selection algorithm FIND and give a new proof for Grübel and Rösler's (1996) result on the convergence of this distribution. Our approach is based on the construction and analysis of a suitable associated Markov chain. Some numerical results for the quantiles of the limit distributions are included, leading for example to the statement that, for a set S with n elements and n large, FIND will need with probability 0.9 about 4.72 x n comparisons to find the median of S.

Keywords

Randomized algorithmsstochastic orderingconvergence in distributionMarkov chains

MSC classification

Primary: 60J20: Applications of Markov chains and discrete-time Markov processes on general state spaces (social mobility, learning theory, industrial processes, etc.)
Secondary: 68P10: Searching and sorting
Type
Research Papers
Information
Journal of Applied Probability , Volume 35 , Issue 1 , March 1998 , pp. 36 - 45
Copyright
Copyright © Applied Probability Trust 1998 

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References

Devroye, L. (1984). Exponential bounds for the running time of a selection algorithm. J. Comput. System Sci. 29, 1–7.Google Scholar
Goldie, C. M. and Grübel, R. (1996). Perpetuities with thin tails. Adv. Appl. Prob. 28, 463–480.Google Scholar
Grübel, R. and Rösler, U. (1996). Asymptotic distribution theory for Hoare's selection algorithm. Adv. Appl. Prob. 28, 252–269.Google Scholar
Hoare, C. A. R. (1961). Algorithm 65, FIND. Commun. ACM 4, 321–322.Google Scholar
Paulsen, V. (1995). The moments of FIND. Preprint. Universität Kiel.Google Scholar