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: https://drops.dagstuhl.de/opus/volltexte/2019/10409/
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Agarwal, Pankaj K. ; Aronov, Boris ; Ezra, Esther ; Zahl, Joshua

An Efficient Algorithm for Generalized Polynomial Partitioning and Its Applications

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LIPIcs-SoCG-2019-5.pdf (0.5 MB)


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}
}

Keywords: Polynomial partitioning, quantifier elimination, semi-algebraic range spaces, epsilon-samples
Collection: 35th International Symposium on Computational Geometry (SoCG 2019)
Issue Date: 2019
Date of publication: 11.06.2019


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