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

    Fingerprint

    Polynomials
    Data structures
    Enclosures

    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

    An efficient algorithm for generalized polynomial partitioning and its applications. / Agarwal, Pankaj K.; Aronov, Boris; Ezra, Esther; Zahl, Joshua.

    35th International Symposium on Computational Geometry, SoCG 2019. ed. / Gill Barequet; Yusu Wang. Schloss Dagstuhl- Leibniz-Zentrum fur Informatik GmbH, Dagstuhl Publishing, 2019. 5 (Leibniz International Proceedings in Informatics, LIPIcs; Vol. 129).

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

    Agarwal, PK, 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, 35th International Symposium on Computational Geometry, SoCG 2019, Portland, United States, 6/18/19. https://doi.org/10.4230/LIPIcs.SoCG.2019.5
    Agarwal PK, Aronov B, Ezra E, Zahl J. An efficient algorithm for generalized polynomial partitioning and its applications. In Barequet G, Wang Y, editors, 35th International Symposium on Computational Geometry, SoCG 2019. Schloss Dagstuhl- Leibniz-Zentrum fur Informatik GmbH, Dagstuhl Publishing. 2019. 5. (Leibniz International Proceedings in Informatics, LIPIcs). https://doi.org/10.4230/LIPIcs.SoCG.2019.5
    Agarwal, Pankaj K. ; Aronov, Boris ; Ezra, Esther ; Zahl, Joshua. / An efficient algorithm for generalized polynomial partitioning and its applications. 35th International Symposium on Computational Geometry, SoCG 2019. editor / Gill Barequet ; Yusu Wang. Schloss Dagstuhl- Leibniz-Zentrum fur Informatik GmbH, Dagstuhl Publishing, 2019. (Leibniz International Proceedings in Informatics, LIPIcs).
    @inproceedings{e8b87e9598f24c4a988eb01016051827,
    title = "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 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.",
    keywords = "Polynomial partitioning, Quantifier elimination, Semi-algebraic range spaces, ε-samples",
    author = "Agarwal, {Pankaj K.} and Boris Aronov and Esther Ezra and Joshua Zahl",
    year = "2019",
    month = "6",
    day = "1",
    doi = "10.4230/LIPIcs.SoCG.2019.5",
    language = "English (US)",
    series = "Leibniz International Proceedings in Informatics, LIPIcs",
    publisher = "Schloss Dagstuhl- Leibniz-Zentrum fur Informatik GmbH, Dagstuhl Publishing",
    editor = "Gill Barequet and Yusu Wang",
    booktitle = "35th International Symposium on Computational Geometry, SoCG 2019",

    }

    TY - GEN

    T1 - An efficient algorithm for generalized polynomial partitioning and its applications

    AU - Agarwal, Pankaj K.

    AU - Aronov, Boris

    AU - Ezra, Esther

    AU - Zahl, Joshua

    PY - 2019/6/1

    Y1 - 2019/6/1

    N2 - 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.

    AB - 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.

    KW - Polynomial partitioning

    KW - Quantifier elimination

    KW - Semi-algebraic range spaces

    KW - ε-samples

    UR - http://www.scopus.com/inward/record.url?scp=85066829557&partnerID=8YFLogxK

    UR - http://www.scopus.com/inward/citedby.url?scp=85066829557&partnerID=8YFLogxK

    U2 - 10.4230/LIPIcs.SoCG.2019.5

    DO - 10.4230/LIPIcs.SoCG.2019.5

    M3 - Conference contribution

    AN - SCOPUS:85066829557

    T3 - Leibniz International Proceedings in Informatics, LIPIcs

    BT - 35th International Symposium on Computational Geometry, SoCG 2019

    A2 - Barequet, Gill

    A2 - Wang, Yusu

    PB - Schloss Dagstuhl- Leibniz-Zentrum fur Informatik GmbH, Dagstuhl Publishing

    ER -