Line Intersection Searching Amid Unit Balls in 3-Space

Agarwal, Pankaj K.; Ezra, Esther

doi:10.4230/LIPIcs.SoCG.2023.5

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Document Identifiers

DOI: 10.4230/LIPIcs.SoCG.2023.5
URN: urn:nbn:de:0030-drops-178559

Author Details

Pankaj K. Agarwal

Department of Computer Science, Duke University, Durham, NC, USA

Esther Ezra

School of Computer Science, Bar Ilan University, Ramat Gan, Israel

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Pankaj K. Agarwal and Esther Ezra. Line Intersection Searching Amid Unit Balls in 3-Space. In 39th International Symposium on Computational Geometry (SoCG 2023). Leibniz International Proceedings in Informatics (LIPIcs), Volume 258, pp. 5:1-5:14, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2023)
https://doi.org/10.4230/LIPIcs.SoCG.2023.5

Abstract

Let ℬ be a set of n unit balls in ℝ³. We present a linear-size data structure for storing ℬ that can determine in O^*(n^{1/2}) time whether a query line intersects any ball of ℬ and report all k such balls in additional O(k) time. The data structure can be constructed in O(n log n) time. (The O^*(⋅) notation hides subpolynomial factors, e.g., of the form O(n^ε), for arbitrarily small ε > 0, and their coefficients which depend on ε.) We also consider the dual problem: Let ℒ be a set of n lines in ℝ³. We preprocess ℒ, in O^*(n²) time, into a data structure of size O^*(n²) that can determine in O^*(1) time whether a query unit ball intersects any line of ℒ, or report all k such lines in additional O(k) time.

Subject Classification

ACM Subject Classification

Theory of computation
Theory of computation → Computational geometry

Keywords

Intersection searching
cylindrical range searching
partition trees
union of cylinders

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References

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