Line Intersection Searching Amid Unit Balls in 3-Space

Pankaj K. Agarwal; Esther Ezra

doi:10.4230/LIPIcs.SoCG.2023.5

Line Intersection Searching Amid Unit Balls in 3-Space

Pankaj K. Agarwal
, Esther Ezra

Department of Computer Science

Duke University

Research output: Chapter in Book/Report/Conference proceeding › Conference contribution › peer-review

1 Scopus citations

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 (Equation presented) 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.

Original language	English
Title of host publication	39th International Symposium on Computational Geometry, SoCG 2023
Editors	Erin W. Chambers, Joachim Gudmundsson
Publisher	Schloss Dagstuhl- Leibniz-Zentrum fur Informatik GmbH, Dagstuhl Publishing
ISBN (Electronic)	9783959772730
DOIs	https://doi.org/10.4230/LIPIcs.SoCG.2023.5
State	Published - 1 Jun 2023
Event	39th International Symposium on Computational Geometry, SoCG 2023 - Dallas, United States Duration: 12 Jun 2023 → 15 Jun 2023

Publication series

Name	Leibniz International Proceedings in Informatics, LIPIcs
Volume	258
ISSN (Print)	1868-8969

Conference

Conference	39th International Symposium on Computational Geometry, SoCG 2023
Country/Territory	United States
City	Dallas
Period	12/06/23 → 15/06/23

Bibliographical note

Publisher Copyright:
© Pankaj K. Agarwal and Esther Ezra; licensed under Creative Commons License CC-BY 4.0.

Funding

Work partially supported by NSF grants IIS-18-14493, CCF-20-07556,

Funders	Funder number
National Science Foundation	CCF-20-07556, IIS-18-14493

Keywords

Intersection searching
cylindrical range searching
partition trees
union of cylinders

Access to Document

10.4230/LIPIcs.SoCG.2023.5

Cite this

Agarwal, P. K., & Ezra, E. (2023). Line Intersection Searching Amid Unit Balls in 3-Space. In E. W. Chambers, & J. Gudmundsson (Eds.), 39th International Symposium on Computational Geometry, SoCG 2023 Article 5 (Leibniz International Proceedings in Informatics, LIPIcs; Vol. 258). Schloss Dagstuhl- Leibniz-Zentrum fur Informatik GmbH, Dagstuhl Publishing. https://doi.org/10.4230/LIPIcs.SoCG.2023.5

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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 (Equation presented) 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(nε), 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.",

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author = "Agarwal, \{Pankaj K.\} and Esther Ezra",

note = "Publisher Copyright: {\textcopyright} Pankaj K. Agarwal and Esther Ezra; licensed under Creative Commons License CC-BY 4.0.; 39th International Symposium on Computational Geometry, SoCG 2023 ; Conference date: 12-06-2023 Through 15-06-2023",

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Agarwal, PK & Ezra, E 2023, Line Intersection Searching Amid Unit Balls in 3-Space. in EW Chambers & J Gudmundsson (eds), 39th International Symposium on Computational Geometry, SoCG 2023., 5, Leibniz International Proceedings in Informatics, LIPIcs, vol. 258, Schloss Dagstuhl- Leibniz-Zentrum fur Informatik GmbH, Dagstuhl Publishing, 39th International Symposium on Computational Geometry, SoCG 2023, Dallas, United States, 12/06/23. https://doi.org/10.4230/LIPIcs.SoCG.2023.5

Line Intersection Searching Amid Unit Balls in 3-Space. / Agarwal, Pankaj K.; Ezra, Esther.
39th International Symposium on Computational Geometry, SoCG 2023. ed. / Erin W. Chambers; Joachim Gudmundsson. Schloss Dagstuhl- Leibniz-Zentrum fur Informatik GmbH, Dagstuhl Publishing, 2023. 5 (Leibniz International Proceedings in Informatics, LIPIcs; Vol. 258).

Research output: Chapter in Book/Report/Conference proceeding › Conference contribution › peer-review

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N2 - 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 (Equation presented) 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(nε), 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.

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KW - partition trees

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Line Intersection Searching Amid Unit Balls in 3-Space

Abstract

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Bibliographical note

Funding

Keywords

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