Electronic Communications in Probability

A Characterization of the Set of Fixed Points of the Quicksort Transformation

James Fill and Svante Janson

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Abstract

The limiting distribution $\mu$ of the normalized number of key comparisons required by the Quicksort sorting algorithm is known to be the unique fixed point of a certain distributional transformation $T$ - unique, that is, subject to the constraints of zero mean and finite variance. We show that a distribution is a fixed point of $T$ if and only if it is the convolution of $\mu$ with a Cauchy distribution of arbitrary center and scale. In particular, therefore, $\mu$ is the unique fixed point of $T$ having zero mean.

Article information

Source
Electron. Commun. Probab., Volume 5 (2000), paper no. 9, 77-84.

Dates
Accepted: 26 May 2000
First available in Project Euclid: 2 March 2016

Permanent link to this document
https://projecteuclid.org/euclid.ecp/1456943502

Digital Object Identifier
doi:10.1214/ECP.v5-1021

Mathematical Reviews number (MathSciNet)
MR1781841

Zentralblatt MATH identifier
0943.68192

Keywords
Quicksort fixed point characteristic function smoothing transformation domain of attraction coupling integral equation

Rights
This work is licensed under aCreative Commons Attribution 3.0 License.

Citation

Fill, James; Janson, Svante. A Characterization of the Set of Fixed Points of the Quicksort Transformation. Electron. Commun. Probab. 5 (2000), paper no. 9, 77--84. doi:10.1214/ECP.v5-1021. https://projecteuclid.org/euclid.ecp/1456943502


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