Efficient algorithm for generalized polynomial partitioning and its applications

Pankaj K. Agarwaly, Boris Aronov, Esther Ezrax, Joshua Zahl

Research output: Contribution to journalArticlepeer-review

24 Scopus citations

Abstract

In 2015, Guth proved that if S is a collection of n g-dimensional semialgebraic 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 also present an extension of our construction to multilevel polynomial partitioning for semialgebraic sets in Rd. We present five applications of our result. The first is a data structure for answering point-enclosure queries among a family of semialgebraic 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-searching queries with semialgebraic ranges in Rd 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 semialgebraic 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 curves in R2 into pseudosegments. The fifth application is for eliminating depth cycles among triangles in R3, where we show a nearly optimal algorithm to cut n pairwise disjoint nonvertical triangles in R3 into pieces that form a depth order.

Original languageEnglish
Pages (from-to)760-787
Number of pages28
JournalSIAM Journal on Computing
Volume50
Issue number2
DOIs
StatePublished - 2021

Bibliographical note

Publisher Copyright:
© 2021 Society for Industrial and Applied Mathematics.

Funding

∗Received by the editors June 26, 2019; accepted for publication (in revised form) January 21, 2021; published electronically April 15, 2021. A preliminary version of this work appeared in the Proceedings of the 35th International Symposium on Computational Geometry [2]. https://doi.org/10.1137/19M1268550 Funding: The first author 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. The second author was supported by NSF grants CCF-12-18791 and CCF-15-40656 and by grant 2014/170 from the US-Israel Binational Science Foundation. The third author was supported by NSF CAREER grant CCF:AF 1553354 and by grant 824/17 from the Israel Science Foundation. The fourth author was supported by a NSERC Discovery grant.

FundersFunder number
National Science FoundationCCF-15-46392, CCF-15-13816, IIS-14-08846
Army Research OfficeW911NF-15-1-0408
Bonfils-Stanton Foundation2012/229
Natural Sciences and Engineering Research Council of Canada
United States-Israel Binational Science Foundation824/17, AF 1553354, 2014/170, CCF-12-18791, CCF-15-40656
Israel Science Foundation

    Keywords

    • Polynomial partitioning
    • Quantifier elimination
    • Semialgebraic range spaces
    • ϵ-samples

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