## Electronic Journal of Probability

### A Gaussian limit process for optimal FIND algorithms

#### Abstract

We consider versions of the FIND algorithm where the pivot element used is the median of a subset chosen uniformly at random from the data. For the median selection we assume that subsamples of size asymptotic to $c \cdot n^\alpha$ are chosen, where $0 < \alpha \leq \frac{1}{2}$, $c > 0$ and $n$ is the size of the data set to be split. We consider the complexity of FIND as a process in the rank to be selected and measured by the number of key comparisons required. After normalization we show weak convergence of the complexity to a centered Gaussian process as $n \to \infty$, which depends on $\alpha$. The proof relies on a contraction argument for probability distributions on càdlàg functions. We also identify the covariance function of the Gaussian limit process and discuss path and tail properties.

#### Article information

Source
Electron. J. Probab., Volume 19 (2014), paper no. 3, 28 pp.

Dates
Accepted: 6 January 2014
First available in Project Euclid: 4 June 2016

https://projecteuclid.org/euclid.ejp/1465065645

Digital Object Identifier
doi:10.1214/EJP.v19-2933

Mathematical Reviews number (MathSciNet)
MR3164756

Zentralblatt MATH identifier
1358.68085

Rights

#### Citation

Sulzbach, Henning; Neininger, Ralph; Drmota, Michael. A Gaussian limit process for optimal FIND algorithms. Electron. J. Probab. 19 (2014), paper no. 3, 28 pp. doi:10.1214/EJP.v19-2933. https://projecteuclid.org/euclid.ejp/1465065645

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