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Dynamic Unit-Disk Range Reporting

Authors Haitao Wang , Yiming Zhao

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LIPIcs.STACS.2025.76.pdf
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Haitao Wang
  • Kahlert School of Computing, University of Utah, Salt Lake City, UT, USA
Yiming Zhao
  • Department of Computer Sciences, Metropolitan State University of Denver, CO, USA

Funding

This research was supported in part by NSF under Grant CCF-2300356.

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Haitao Wang and Yiming Zhao. Dynamic Unit-Disk Range Reporting. In 42nd International Symposium on Theoretical Aspects of Computer Science (STACS 2025). Leibniz International Proceedings in Informatics (LIPIcs), Volume 327, pp. 76:1-76:19, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2025) https://doi.org/10.4230/LIPIcs.STACS.2025.76

Abstract

For a set P of n points in the plane and a value r > 0, the unit-disk range reporting problem is to construct a data structure so that given any query disk of radius r, all points of P in the disk can be reported efficiently. We consider the dynamic version of the problem where point insertions and deletions of P are allowed. The previous best method provides a data structure of O(n log n) space that supports O(log^{3+ε} n) amortized insertion time, O(log^{5+ε} n) amortized deletion time, and O(log² n/log log n+k) query time, where ε is an arbitrarily small positive constant and k is the output size. In this paper, we improve the query time to O(log n+k) while keeping other complexities the same as before. A key ingredient of our approach is a shallow cutting algorithm for circular arcs, which may be interesting in its own right. A related problem that can also be solved by our techniques is the dynamic unit-disk range emptiness queries: Given a query unit disk, we wish to determine whether the disk contains a point of P. The best previous work can maintain P in a data structure of O(n) space that supports O(log² n) amortized insertion time, O(log⁴n) amortized deletion time, and O(log² n) query time. Our new data structure also uses O(n) space but can support each update in O(log^{1+ε} n) amortized time and support each query in O(log n) time.

Subject Classification

ACM Subject Classification
  • Theory of computation → Computational geometry
  • Theory of computation → Design and analysis of algorithms
Keywords
  • Unit disks
  • range reporting
  • range emptiness
  • alpha-hulls
  • dynamic data structures
  • shallow cuttings

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