Algorithms for Computing Closest Points for Segments

Author Haitao Wang

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

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Haitao Wang. Algorithms for Computing Closest Points for Segments. In 41st International Symposium on Theoretical Aspects of Computer Science (STACS 2024). Leibniz International Proceedings in Informatics (LIPIcs), Volume 289, pp. 58:1-58:17, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2024)


Given a set P of n points and a set S of n segments in the plane, we consider the problem of computing for each segment of S its closest point in P. The previously best algorithm solves the problem in n^{4/3}2^{O(log^*n)} time [Bespamyatnikh, 2003] and a lower bound (under a somewhat restricted model) Ω(n^{4/3}) has also been proved. In this paper, we present an O(n^{4/3}) time algorithm and thus solve the problem optimally (under the restricted model). In addition, we also present data structures for solving the online version of the problem, i.e., given a query segment (or a line as a special case), find its closest point in P. Our new results improve the previous work.

Subject Classification

ACM Subject Classification
  • Theory of computation → Design and analysis of algorithms
  • Theory of computation → Computational geometry
  • Closest points
  • Voronoi diagrams
  • Segment dragging queries
  • Hopcroft’s problem
  • Algebraic decision tree model


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