NDLI: Multilevel Polynomial Partitions and Simplified Range Searching

Content Provider	Springer Nature Link
Author	Matoušek, Jiří Patáková, Zuzana
Copyright Year	2015
Abstract	The polynomial partitioning method of Guth and Katz ( arXiv:1011.4105 ) has numerous applications in discrete and computational geometry. It partitions a given n-point set $$P\subset {\mathbb {R}}^d$$ using the zero set Z(f) of a suitable d-variate polynomial f. Applications of this result are often complicated by the problem, “What should be done with the points of P lying within Z(f)?” A natural approach is to partition these points with another polynomial and continue further in a similar manner. So far this has been pursued with limited success—several authors managed to construct and apply a second partitioning polynomial, but further progress has been prevented by technical obstacles. We provide a polynomial partitioning method with up to d polynomials in dimension d, which allows for a complete decomposition of the given point set. We apply it to obtain a new algorithm for the semialgebraic range searching problem. Our algorithm has running time bounds similar to a recent algorithm by Agarwal et al. (SIAM J Comput 42:2039–2062, 2013), but it is simpler both conceptually and technically. While this paper has been in preparation, Basu and Sombra, as well as Fox, Pach, Sheffer, Suk, and Zahl, obtained results concerning polynomial partitions which overlap with ours to some extent.
Starting Page	22
Ending Page	41
Page Count	20
File Format	PDF
ISSN	01795376
Journal	Discrete & Computational Geometry
Volume Number	54
Issue Number	1
e-ISSN	14320444
Language	English
Publisher	Springer US
Publisher Date	2015-05-22
Publisher Place	New York
Access Restriction	One Nation One Subscription (ONOS)
Subject Keyword	Combinatorics Computational Mathematics and Numerical Analysis
Content Type	Text
Resource Type	Article
Subject	Discrete Mathematics and Combinatorics Theoretical Computer Science Computational Theory and Mathematics Geometry and Topology

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1	Ministry of Education (GoI), Department of Higher Education	Sanctioning Authority	https://www.education.gov.in/ict-initiatives
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3	National Digital Library of India Office, Indian Institute of Technology Kharagpur	The administrative and infrastructural headquarters of the project	Dr. B. Sutradhar bsutra@ndl.gov.in
4	Project PI / Joint PI	Principal Investigator and Joint Principal Investigators of the project	Dr. B. Sutradhar bsutra@ndl.gov.in Prof. Saswat Chakrabarti will be added soon
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