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Circle-Segment Intersection Queries in Connected Geometric Graphs

Authors Peyman Afshani , Yannick Bosch , Sabine Storandt

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Subject Classification

ACM Subject Classification
  • Theory of computation → Computational geometry
  • Theory of computation → Data structures design and analysis
Keywords
  • Intersection data structure
  • Graph partitioning
  • Dobkin-Kirkpatrick hierarchy

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Abstract

In this paper, we study the problem of efficiently reporting all intersections between a given set of line segments in the plane and a query circle, focusing on the case where the segments form the edges of a connected geometric graph. While previous data structures for circle-segment intersection queries on general segment sets incur high space or query time costs, we exploit the connectivity of the input to obtain significantly improved performance. In fact, we propose a new circle-segment intersection data structure that can be constructed in 𝒪((n + C) log³ n) time and space on connected graphs with n edges and C edge crossings. It answers intersection queries in 𝒪(k log³ n) time, where k denotes the output size. Our method relies on the construction of efficient circle-graph intersection oracles as well as a novel linear-time algorithm to partition the edges of the graph into balanced, connected components, which might be of independent interest. In a proof-of-concept experimental study on real-world road networks, we show that our novel data structure also performs well in practice. Even on networks with millions of edges, the construction time is within minutes and queries are answered in a few milliseconds.

Cite As Get BibTex

Peyman Afshani, Yannick Bosch, and Sabine Storandt. Circle-Segment Intersection Queries in Connected Geometric Graphs. In 36th International Symposium on Algorithms and Computation (ISAAC 2025). Leibniz International Proceedings in Informatics (LIPIcs), Volume 359, pp. 3:1-3:16, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2025) https://doi.org/10.4230/LIPIcs.ISAAC.2025.3

Author Details

Peyman Afshani
  • Aarhus University, Denmark
Yannick Bosch
  • University of Konstanz, Germany
Sabine Storandt
  • University of Konstanz, Germany

Acknowledgements

This work was initiated at Dagstuhl Seminar 25191 "Adaptive and Scalable Data Structures" (2025-05-04 - 2025-05-09) https://www.dagstuhl.de/25191. We thank Schloss Dagstuhl -– Leibniz-Zentrum für Informatik for their support.

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