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Algorithms for Halfplane Coverage and Related Problems

Authors Haitao Wang , Jie Xue

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LIPIcs.SoCG.2024.79.pdf
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Author Details

Haitao Wang
  • Kahlert School of Computing, University of Utah, Salt Lake City, UT, USA
Jie Xue
  • Department of Computer Science, New York University Shanghai, China

Funding

  • Wang, Haitao: Supported in part by NSF under Grant CCF-2300356.

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Haitao Wang and Jie Xue. Algorithms for Halfplane Coverage and Related Problems. In 40th International Symposium on Computational Geometry (SoCG 2024). Leibniz International Proceedings in Informatics (LIPIcs), Volume 293, pp. 79:1-79:15, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2024)
https://doi.org/10.4230/LIPIcs.SoCG.2024.79

Abstract

Given in the plane a set of points and a set of halfplanes, we consider the problem of computing a smallest subset of halfplanes whose union covers all points. In this paper, we present an O(n^{4/3}log^{5/3}nlog^{O(1)}log n)-time algorithm for the problem, where n is the total number of all points and halfplanes. This improves the previously best algorithm of n^{10/3}2^{O(log^*n)} time by roughly a quadratic factor. For the special case where all halfplanes are lower ones, our algorithm runs in O(nlog n) time, which improves the previously best algorithm of n^{4/3}2^{O(log^*n)} time and matches an Ω(nlog n) lower bound. Further, our techniques can be extended to solve a star-shaped polygon coverage problem in O(nlog n) time, which in turn leads to an O(nlog n)-time algorithm for computing an instance-optimal ε-kernel of a set of n points in the plane. Agarwal and Har-Peled presented an O(nklog n)-time algorithm for this problem in SoCG 2023, where k is the size of the ε-kernel; they also raised an open question whether the problem can be solved in O(nlog n) time. Our result thus answers the open question affirmatively.

Subject Classification

ACM Subject Classification
  • Theory of computation → Computational geometry
  • Theory of computation → Design and analysis of algorithms
Keywords
  • halfplane coverage
  • circular coverage
  • star-shaped polygon coverage
  • ε-kernels

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