An efficient algorithm for generalized polynomial partitioning and its applications

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

    Research output: Chapter in Book/Report/Conference proceedingConference contribution

    Abstract

    In 2015, Guth proved that if S is a collection of n g-dimensional semi-algebraic sets in ℝd 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 ℝd \ Z(P) intersects O(n/Dd−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 ε-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 Rd in O(log n) time, with storage complexity and expected preprocessing time of O(nd+ε). The second is a data structure for answering range search queries with semi-algebraic ranges in O(log n) time, with O(nt+ε) 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 ℝd in O(log2 n) time, with O(nd+ε) storage and expected preprocessing time. The fourth is an efficient algorithm for cutting algebraic planar curves into pseudo-segments.

    Original languageEnglish (US)
    Title of host publication35th International Symposium on Computational Geometry, SoCG 2019
    EditorsGill Barequet, Yusu Wang
    PublisherSchloss Dagstuhl- Leibniz-Zentrum fur Informatik GmbH, Dagstuhl Publishing
    ISBN (Electronic)9783959771047
    DOIs
    StatePublished - Jun 1 2019
    Event35th International Symposium on Computational Geometry, SoCG 2019 - Portland, United States
    Duration: Jun 18 2019Jun 21 2019

    Publication series

    NameLeibniz International Proceedings in Informatics, LIPIcs
    Volume129
    ISSN (Print)1868-8969

    Conference

    Conference35th International Symposium on Computational Geometry, SoCG 2019
    CountryUnited States
    CityPortland
    Period6/18/196/21/19

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    Keywords

    • Polynomial partitioning
    • Quantifier elimination
    • Semi-algebraic range spaces
    • ε-samples

    ASJC Scopus subject areas

    • Software

    Cite this

    Agarwal, P. K., Aronov, B., Ezra, E., & Zahl, J. (2019). An efficient algorithm for generalized polynomial partitioning and its applications. In G. Barequet, & Y. Wang (Eds.), 35th International Symposium on Computational Geometry, SoCG 2019 [5] (Leibniz International Proceedings in Informatics, LIPIcs; Vol. 129). Schloss Dagstuhl- Leibniz-Zentrum fur Informatik GmbH, Dagstuhl Publishing. https://doi.org/10.4230/LIPIcs.SoCG.2019.5