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An Optimal Algorithm for Computing Many Faces in Line Arrangements

Author Haitao Wang

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LIPIcs.SoCG.2026.95.pdf
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  • 14 pages

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Subject Classification

ACM Subject Classification
  • Theory of computation → Computational geometry
  • Theory of computation → Design and analysis of algorithms
Keywords
  • Many faces
  • line arrangements
  • cuttings
  • Γ-algorithms
  • decision tree complexities

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Abstract

Given a set of m points and a set of n lines in the plane, we consider the classical problem of computing the faces of the arrangement of the lines that contain at least one point. We present an algorithm of O(m^{2/3} n^{2/3} + (n+m)log n) time for the problem. We also prove that this matches the lower bound under the algebraic decision tree model and thus our algorithm is optimal. In particular, when m = n, the runtime is O(n^{4/3}), which matches the worst case combinatorial complexity Ω(n^{4/3}) of all output faces. This is the first optimal algorithm since the problem was first studied more than three decades ago [Edelsbrunner, Guibas, and Sharir, SoCG 1988].

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Haitao Wang. An Optimal Algorithm for Computing Many Faces in Line Arrangements. In 42nd International Symposium on Computational Geometry (SoCG 2026). Leibniz International Proceedings in Informatics (LIPIcs), Volume 367, pp. 95:1-95:14, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2026) https://doi.org/10.4230/LIPIcs.SoCG.2026.95

Author Details

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

References

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