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Counting Unit Circular Arc Intersections

Author Haitao Wang

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LIPIcs.STACS.2026.81.pdf
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  • 18 pages

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Subject Classification

ACM Subject Classification
  • Theory of computation → Computational geometry
  • Theory of computation → Design and analysis of algorithms
Keywords
  • circular arc intersections
  • unit circles
  • arrangements
  • cuttings
  • segment intersections

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Abstract

Given a set of n circular arcs of the same radius in the plane, we consider the problem of computing the number of intersections among the arcs. The problem was studied before and the previously best algorithm solves the problem in O(n^{4/3+ε}) time [Agarwal, Pellegrini, and Sharir, SIAM J. Comput., 1993], for any constant ε > 0. No progress has been made on the problem for more than 30 years. We present a new algorithm of O(n^{4/3}log^{16/3} n) time and improve it to O(n^{1+ε}+K^{1/3}n^{2/3}((n²)/(n+K))^{ε}log^{16/3}n) time for small K, where K is the number of intersections of all arcs.

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Haitao Wang. Counting Unit Circular Arc Intersections. In 43rd International Symposium on Theoretical Aspects of Computer Science (STACS 2026). Leibniz International Proceedings in Informatics (LIPIcs), Volume 364, pp. 81:1-81:18, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2026) https://doi.org/10.4230/LIPIcs.STACS.2026.81

Author Details

Haitao Wang
  • Kahlert School of Computing, University of Utah, Salt Lake City, UT, USA

References

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