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Analysis of swaps in radix selection

Part of: Combinatorial probability Limit theorems Theory of data

Published online by Cambridge University Press:  01 July 2016

Amr Elmasry  and
Hosam Mahmoud
Amr Elmasry*
Affiliation:
University of Copenhagen
Hosam Mahmoud*
Affiliation:
The George Washington University
*
∗ Postal address: Department of Computer Science, University of Copenhagen, Universitetsparken 1, DK-2100 Copenhagen Ø, Denmark. Email address: [email protected]
∗∗ Postal address: Department of Statistics, The George Washington University, Washington, DC 20052, USA. Email address: [email protected]
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Abstract

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Radix Sort is a sorting algorithm based on analyzing digital data. We study the number of swaps made by Radix Select (a one-sided version of Radix Sort) to find an element with a randomly selected rank. This kind of grand average provides a smoothing over all individual distributions for specific fixed-order statistics. We give an exact analysis for the grand mean and an asymptotic analysis for the grand variance, obtained by poissonization, the Mellin transform, and depoissonization. The digital data model considered is the Bernoulli(p). The distributions involved in the swaps experience a phase change between the biased cases (p ≠ ½) and the unbiased case (p = ½). In the biased cases, the grand distribution for the number of swaps (when suitably scaled) converges to that of a perpetuity built from a two-point distribution. The tool for this proof is contraction in the Wasserstein metric space, and identifying the limit as the fixed-point solution of a distributional equation. In the unbiased case the same scaling for the number of swaps gives a limiting constant in probability.

Keywords

Random structurealgorithmorder statisticrecurrenceperpetuityphase changedigital datadigital sortingselectionMellin transformpoissonizationdepoissonization

MSC classification

Primary: 60C05: Combinatorial probability
Secondary: 60F05: Central limit and other weak theorems 68P10: Searching and sorting
Type
General Applied Probability
Information
Advances in Applied Probability , Volume 43 , Issue 2 , June 2011 , pp. 524 - 544
Copyright
Copyright © Applied Probability Trust 2011 

Footnotes

Part of this work was carried out while the author was at Max-Planck-Institut für Informatik. The author is currently on leave from Alexandria University of Egypt.

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