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Tight upper bounds for the discrepancy of half-spaces

Author:

J. MatoušekAuthors Info & Claims

Discrete & Computational Geometry, Volume 13, Issue 3-4

Pages 593 - 601

https://doi.org/10.1007/BF02574066

Published: 01 December 1995 Publication History

Abstract

We show that the discrepancy of anyn-point setP in the Euclideand-space with respect to half-spaces is bounded byCdn1/2 1/2d, that is, a mapping :P { 1,1} exists such that, for any half-space, |Σp P (p)|≤Cdn1/2-1/2d. In fact, the result holds for arbitrary set systems as long as theprimal shatter function isO(md). This matches known lower bounds, improving previous upper bounds by a $$\sqrt {\log n} $$ factor.

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Cited By

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Phillips JZheng Y(2015)Subsampling in Smoothed Range SpacesProceedings of the 26th International Conference on Algorithmic Learning Theory - Volume 935510.1007/978-3-319-24486-0_15(224-238)Online publication date: 4-Oct-2015
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cover image Discrete & Computational Geometry

Discrete & Computational Geometry Volume 13, Issue 3-4

June 1995

382 pages

ISSN:0179-5376

Issue’s Table of Contents

Copyright © Copyright © 1995 Springer-Verlag New York Inc.

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Springer-Verlag

Berlin, Heidelberg

Publication History

Published: 01 December 1995

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Cited By

Dutta KEzra EGhosh A(2016)Two Proofs for Shallow PackingsDiscrete & Computational Geometry10.1007/s00454-016-9824-056:4(910-939)Online publication date: 1-Dec-2016
https://dl.acm.org/doi/10.1007/s00454-016-9824-0
Phillips JZheng Y(2015)Subsampling in Smoothed Range SpacesProceedings of the 26th International Conference on Algorithmic Learning Theory - Volume 935510.1007/978-3-319-24486-0_15(224-238)Online publication date: 4-Oct-2015
https://dl.acm.org/doi/10.1007/978-3-319-24486-0_15
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https://dl.acm.org/doi/10.1145/2500128
Muthukrishnan SNikolov AKarloff HPitassi T(2012)Optimal private halfspace counting via discrepancyProceedings of the forty-fourth annual ACM symposium on Theory of computing10.1145/2213977.2214090(1285-1292)Online publication date: 19-May-2012
https://dl.acm.org/doi/10.1145/2213977.2214090
Joshi SKommaraji RPhillips JVenkatasubramanian SHurtado Fvan Kreveld M(2011)Comparing distributions and shapes using the kernel distanceProceedings of the twenty-seventh annual symposium on Computational geometry10.1145/1998196.1998204(47-56)Online publication date: 13-Jun-2011
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Doerr B(2006)Non-independent randomized rounding and coloringDiscrete Applied Mathematics10.5555/1141038.1705264154:4(650-659)Online publication date: 15-Mar-2006
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Suri STóth CZhou YSnoeyink JBoissonnat J(2004)Range counting over multidimensional data streamsProceedings of the twentieth annual symposium on Computational geometry10.1145/997817.997844(160-169)Online publication date: 8-Jun-2004
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