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Intersecting line segments, ray shooting, and other applications of geometric partitioning techniques

  • Leonidas Guibas
  • Mark Overmars
  • Micha Sharir
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 318)

Abstract

We present a variety of applications of certain techniques, based on partition trees, that were originally developed for range searching problems. Our results are obtained by enhancing and extending these techniques, and include: (i) An O(n4/3+δ+k)-time (for any δ>0), O(n)-space randomized algorithm for finding all k intersections of n line segments in the plane (we can count the number of these intersections in O(n4/3+δ) time and linear space). (ii) Preprocessing a collection of n (possibly intersecting) segments in the plane so that, given any query ray, we can find quickly the first segment it hits. Other applications concern “implicit” point location, hidden surface removal in three dimensions, polygon placement queries, and problems involving overlapping planar maps. We also present several efficient algorithms involving the analysis of the connectivity and other useful properties of arrangements of line segments.

Keywords

Line Segment Computational Geometry Query Time Simple Polygon Partition 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.

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Copyright information

© Springer-Verlag Berlin Heidelberg 1988

Authors and Affiliations

  • Leonidas Guibas
    • 1
    • 2
  • Mark Overmars
    • 3
  • Micha Sharir
    • 4
    • 5
  1. 1.Computer Science DepartmentStanford UniversityUSA
  2. 2.DEC Systems Research CenterUSA
  3. 3.Department of Computer ScienceUniversity of UtrechtThe Netherlands
  4. 4.Courant Institute of Mathematical SciencesNew York UniversityUSA
  5. 5.School of Mathematical SciencesTel Aviv UniversityIsrael

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