License: 
Creative Commons Attribution 3.0 Unported license (CC BY 3.0)
When quoting this document, please refer to the following
DOI: 10.4230/LIPIcs.SoCG.2019.5
URN: urn:nbn:de:0030-drops-104096
URL: http://dagstuhl.sunsite.rwth-aachen.de/volltexte/2019/10409/
Agarwal, Pankaj K. ;
Aronov, Boris ;
Ezra, Esther ;
Zahl, Joshua
An Efficient Algorithm for Generalized Polynomial Partitioning and Its Applications
Abstract
In 2015, Guth proved that if S is a collection of n g-dimensional semi-algebraic sets in R^d and if D >= 1 is an integer, then there is a d-variate polynomial P of degree at most D so that each connected component of R^d \ Z(P) intersects O(n/D^{d-g}) sets from S. Such a polynomial is called a generalized partitioning polynomial. We present a randomized algorithm that computes such polynomials efficiently - the expected running time of our algorithm is linear in |S|. Our approach exploits the technique of quantifier elimination combined with that of epsilon-samples.
We present four applications of our result. The first is a data structure for answering point-enclosure queries among a family of semi-algebraic sets in R^d in O(log n) time, with storage complexity and expected preprocessing time of O(n^{d+epsilon}). The second is a data structure for answering range search queries with semi-algebraic ranges in O(log n) time, with O(n^{t+epsilon}) storage and expected preprocessing time, where t > 0 is an integer that depends on d and the description complexity of the ranges. The third is a data structure for answering vertical ray-shooting queries among semi-algebraic sets in R^{d} in O(log^2 n) time, with O(n^{d+epsilon}) storage and expected preprocessing time. The fourth is an efficient algorithm for cutting algebraic planar curves into pseudo-segments.
BibTeX - Entry
@InProceedings{agarwal_et_al:LIPIcs:2019:10409,
author = {Pankaj K. Agarwal and Boris Aronov and Esther Ezra and Joshua Zahl},
title = {{An Efficient Algorithm for Generalized Polynomial Partitioning and Its Applications}},
booktitle = {35th International Symposium on Computational Geometry (SoCG 2019)},
pages = {5:1--5:14},
series = {Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN = {978-3-95977-104-7},
ISSN = {1868-8969},
year = {2019},
volume = {129},
editor = {Gill Barequet and Yusu Wang},
publisher = {Schloss Dagstuhl--Leibniz-Zentrum fuer Informatik},
address = {Dagstuhl, Germany},
URL = {http://drops.dagstuhl.de/opus/volltexte/2019/10409},
URN = {urn:nbn:de:0030-drops-104096},
doi = {10.4230/LIPIcs.SoCG.2019.5},
annote = {Keywords: Polynomial partitioning, quantifier elimination, semi-algebraic range spaces, epsilon-samples}
}
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Keywords: |
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Polynomial partitioning, quantifier elimination, semi-algebraic range spaces, epsilon-samples |
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Collection: |
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35th International Symposium on Computational Geometry (SoCG 2019) |
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Issue Date: |
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2019 |
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Date of publication: |
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11.06.2019 |
