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Dynamic Geometric Connectivity in the Plane with Constant Query Time

Authors Timothy M. Chan , Zhengcheng Huang

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LIPIcs.SoCG.2024.36.pdf
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Author Details

Timothy M. Chan
  • Department of Computer Science, University of Illinois at Urbana-Champaign, IL, USA
Zhengcheng Huang
  • Department of Computer Science, University of Illinois at Urbana-Champaign, IL, USA

Funding

  • Chan, Timothy M.: Work supported by NSF Grant CCF-2224271.

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Timothy M. Chan and Zhengcheng Huang. Dynamic Geometric Connectivity in the Plane with Constant Query Time. In 40th International Symposium on Computational Geometry (SoCG 2024). Leibniz International Proceedings in Informatics (LIPIcs), Volume 293, pp. 36:1-36:13, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2024)
https://doi.org/10.4230/LIPIcs.SoCG.2024.36

Abstract

We present the first fully dynamic connectivity data structures for geometric intersection graphs achieving constant query time and sublinear amortized update time for many classes of geometric objects in 2D . Our data structures can answer connectivity queries between two objects, as well as "global" connectivity queries (e.g., deciding whether the entire graph is connected). Previously, the data structure by Afshani and Chan (ESA'06) achieved such bounds only in the special case of axis-aligned line segments or rectangles but did not work for arbitrary line segments or disks, whereas the data structures by Chan, Pătraşcu, and Roditty (FOCS'08) worked for more general classes of geometric objects but required n^{Ω(1)} query time and could not handle global connectivity queries. Specifically, we obtain new data structures with O(1) query time and amortized update time near n^{4/5}, n^{7/8}, and n^{20/21} for axis-aligned line segments, disks, and arbitrary line segments respectively. Besides greatly reducing the query time, our data structures also improve the previous update times for axis-aligned line segments by Afshani and Chan (from near n^{10/11} to n^{4/5}) and for disks by Chan, Pătraşcu, and Roditty (from near n^{20/21} to n^{7/8}).

Subject Classification

ACM Subject Classification
  • Theory of computation → Computational geometry
Keywords
  • Connectivity
  • dynamic data structures
  • geometric intersection graphs

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References

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