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A hyperplane Incidence problem with applications to counting distances

Part of the Lecture Notes in Computer Science book series (LNCS,volume 450)

Abstract

This paper proves an O(m 2/3 n 2/3+m+n) upper bound on the number of incidences between m points and n hyperplanes in four dimensions, assuming all points lie on one side of each hyperplane and the points and hyperplanes satisfy certain natural general position conditions. This result has application to various three-dimensional combinatorial distance problems. For example, it implies the same upper bound for the number of bichromatic minimum distance pairs in a set of m blue and n red points in three-dimensional space. This improves the best previous bound for this problem.

Keywords

  • Delaunay Triangulation
  • Blue Point
  • Neighbor Pair
  • High Vertex
  • Euclidean Minimum Span Tree

These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

Research of the first author was supported by the National Science Foundation under grant CCR-8714565. Work of the second author was supported by Office of Naval Research Grants DCR-83-20085 and CCR-89-01484, and by grants from the U.S.-Israeli Binational Science Foundation, the NCRD — the Israeli National Council for Research and Development, and the Fund for Basic Research in Electronics, Computers and Communication administered by the Israeli Academy of Sciences.

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© 1990 Springer-Verlag Berlin Heidelberg

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Edelsbrunner, H., Sharir, M. (1990). A hyperplane Incidence problem with applications to counting distances. In: Asano, T., Ibaraki, T., Imai, H., Nishizeki, T. (eds) Algorithms. SIGAL 1990. Lecture Notes in Computer Science, vol 450. Springer, Berlin, Heidelberg. https://doi.org/10.1007/3-540-52921-7_91

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  • DOI: https://doi.org/10.1007/3-540-52921-7_91

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  • Publisher Name: Springer, Berlin, Heidelberg

  • Print ISBN: 978-3-540-52921-7

  • Online ISBN: 978-3-540-47177-6

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