A Gaussian limit process for optimal FIND algorithms

  • 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⋅nα are chosen, where 0<α≤12, 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→∞, which depends on α. 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.

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Metadaten
Author:Henning Sulzbach, Ralph NeiningerGND, Michael Drmota
URN:urn:nbn:de:hebis:30:3-327510
DOI:https://doi.org/10.1214/EJP.v19-2933
ISSN:1083-6489
Parent Title (English):Electronic journal of probability
Document Type:Article
Language:English
Date of Publication (online):2014/01/06
Date of first Publication:2014/01/06
Publishing Institution:Universitätsbibliothek Johann Christian Senckenberg
Release Date:2014/01/27
Tag:FIND algorithm; Gaussian process; Quickselect; complexity; contraction method; functional limit theorem; key comparisons
Volume:19
Issue:3
Page Number:28
Note:
This work is licensed under a Creative Commons Attribution 3.0 License. http://creativecommons.org/licenses/by/3.0/
HeBIS-PPN:363411305
Institutes:Informatik und Mathematik / Mathematik
Dewey Decimal Classification:5 Naturwissenschaften und Mathematik / 51 Mathematik / 510 Mathematik
MSC-Classification:60-XX PROBABILITY THEORY AND STOCHASTIC PROCESSES (For additional applications, see 11Kxx, 62-XX, 90-XX, 91-XX, 92-XX, 93-XX, 94-XX) / 60Cxx Combinatorial probability / 60C05 Combinatorial probability
60-XX PROBABILITY THEORY AND STOCHASTIC PROCESSES (For additional applications, see 11Kxx, 62-XX, 90-XX, 91-XX, 92-XX, 93-XX, 94-XX) / 60Fxx Limit theorems [See also 28Dxx, 60B12] / 60F17 Functional limit theorems; invariance principles
60-XX PROBABILITY THEORY AND STOCHASTIC PROCESSES (For additional applications, see 11Kxx, 62-XX, 90-XX, 91-XX, 92-XX, 93-XX, 94-XX) / 60Gxx Stochastic processes / 60G15 Gaussian processes
68-XX COMPUTER SCIENCE (For papers involving machine computations and programs in a specific mathematical area, see Section {04 in that areag 68-00 General reference works (handbooks, dictionaries, bibliographies, etc.) / 68Pxx Theory of data / 68P10 Searching and sorting
68-XX COMPUTER SCIENCE (For papers involving machine computations and programs in a specific mathematical area, see Section {04 in that areag 68-00 General reference works (handbooks, dictionaries, bibliographies, etc.) / 68Qxx Theory of computing / 68Q25 Analysis of algorithms and problem complexity [See also 68W40]
Sammlungen:Universitätspublikationen
Licence (German):License LogoCreative Commons - Namensnennung 3.0