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An Efficient Algorithm for Generalized Polynomial Partitioning and Its Applications

Authors Pankaj K. Agarwal, Boris Aronov , Esther Ezra, Joshua Zahl

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ACM Subject Classification
  • Mathematics of computing → Combinatorial algorithms
  • Theory of computation → Randomness, geometry and discrete structures
Keywords
  • Polynomial partitioning
  • quantifier elimination
  • semi-algebraic range spaces
  • epsilon-samples

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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.

Cite As Get BibTex

Pankaj K. Agarwal, Boris Aronov, Esther Ezra, and Joshua Zahl. An Efficient Algorithm for Generalized Polynomial Partitioning and Its Applications. In 35th International Symposium on Computational Geometry (SoCG 2019). Leibniz International Proceedings in Informatics (LIPIcs), Volume 129, pp. 5:1-5:14, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2019) https://doi.org/10.4230/LIPIcs.SoCG.2019.5

Author Details

Pankaj K. Agarwal
  • Department of Computer Science, Duke University, Box 90129, Durham, NC 27708-0129 USA
Boris Aronov
  • Department of Computer Science and Engineering, Tandon School of Engineering, New York University, Brooklyn, NY 11201, USA
Esther Ezra
  • Department of Computer Science, Bar-Ilan University, Ramat Gan, Israel
  • School of Math, Georgia Institute of Technology, Atlanta, Georgia 30332, USA
Joshua Zahl
  • Department of Mathematics, University of British Columbia, Vancouver, BC V6T 1Z2, Canada

Funding

  • Agarwal, Pankaj K.: P. Agarwal was supported by NSF under grants CCF-15-13816, CCF-15-46392, and IIS-14-08846, by an ARO grant W911NF-15-1-0408, and by BSF Grant 2012/229 from the U.S.-Israel Binational Science Foundation.
  • Aronov, Boris: B. Aronov was supported by NSF grants CCF-12-18791 and CCF-15-40656, and by grant 2014/170 from the US-Israel Binational Science Foundation.
  • Ezra, Esther: E. Ezra was supported by NSF CAREER under grant CCF:AF 1553354 and by Grant 824/17 from the Israel Science Foundation.
  • Zahl, Joshua: J. Zahl was supported by an NSERC Discovery grant.

References

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  2. Pankaj K. Agarwal, Jivr'i Matouvsek, and Micha Sharir. On Range Searching with Semialgebraic Sets. II. SIAM J. Comput., 42(6):2039-2062, 2013. URL: http://dx.doi.org/10.1137/120890855.
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