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Semialgebraic Range Stabbing, Ray Shooting, and Intersection Counting in the Plane

Authors Timothy M. Chan , Pingan Cheng , Da Wei Zheng

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Author Details

Timothy M. Chan
  • University of Illinois Urbana-Champaign, Urbana, IL, USA
Pingan Cheng
  • Aarhus University, Denmark
Da Wei Zheng
  • University of Illinois Urbana-Champaign, Urbana, IL, USA

Funding

  • Chan, Timothy M.: Work supported by NSF Grant CCF-2224271.
  • Cheng, Pingan: Work supported by DFF (Det Frie Forskningsråd) of Danish Council for Independent Research under grant ID DFF–7014–00404 and STIBOFONDENs IT-rejsestipendier til ph.d.-studerende during the author’s visit to UIUC.

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Timothy M. Chan, Pingan Cheng, and Da Wei Zheng. Semialgebraic Range Stabbing, Ray Shooting, and Intersection Counting in the Plane. In 40th International Symposium on Computational Geometry (SoCG 2024). Leibniz International Proceedings in Informatics (LIPIcs), Volume 293, pp. 33:1-33:15, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2024) https://doi.org/10.4230/LIPIcs.SoCG.2024.33

Abstract

Polynomial partitioning techniques have recently led to improved geometric data structures for a variety of fundamental problems related to semialgebraic range searching and intersection searching in 3D and higher dimensions (e.g., see [Agarwal, Aronov, Ezra, and Zahl, SoCG 2019; Ezra and Sharir, SoCG 2021; Agarwal, Aronov, Ezra, Katz, and Sharir, SoCG 2022]). They have also led to improved algorithms for offline versions of semialgebraic range searching in 2D, via lens-cutting [Sharir and Zahl (2017)]. In this paper, we show that these techniques can yield new data structures for a number of other 2D problems even for online queries:  
1) Semialgebraic range stabbing. We present a data structure for n semialgebraic ranges in 2D of constant description complexity with O(n^{3/2+ε}) preprocessing time and space, so that we can count the number of ranges containing a query point in O(n^{1/4+ε}) time, for an arbitrarily small constant ε > 0. (The query time bound is likely close to tight for this space bound.)
2) Ray shooting amid algebraic arcs. We present a data structure for n algebraic arcs in 2D of constant description complexity with O(n^{3/2+ε}) preprocessing time and space, so that we can find the first arc hit by a query (straight-line) ray in O(n^{1/4+ε}) time. (The query bound is again likely close to tight for this space bound, and they improve a result by Ezra and Sharir with near n^{3/2} space and near √n query time.)
3) Intersection counting amid algebraic arcs. We present a data structure for n algebraic arcs in 2D of constant description complexity with O(n^{3/2+ε}) preprocessing time and space, so that we can count the number of intersection points with a query algebraic arc of constant description complexity in O(n^{1/2+ε}) time. In particular, this implies an O(n^{3/2+ε})-time algorithm for counting intersections between two sets of n algebraic arcs in 2D. (This generalizes a classical O(n^{3/2+ε})-time algorithm for circular arcs by Agarwal and Sharir from SoCG 1991.)

Subject Classification

ACM Subject Classification
  • Theory of computation → Computational geometry
Keywords
  • Computational geometry
  • range searching
  • intersection searching
  • semialgebraic sets
  • data structures
  • polynomial partitioning

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