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Reporting and Counting Intersections Between Two Sets of Line Segments

  • Harry G. Mairson
  • Jorge Stolfi
Part of the NATO ASI Series book series (volume 40)

Abstract

We consider the problem of computing all intersections between two sets S and T of line segments in the plane, where no two segments in S (similarly, T) intersect. We present an asymptotically optimal algorithm which reports all those intersections in O(n log n + k) time and O(n) space, where n is the total number of line segments, and k is the number of intersections. Our algorithm works also for general arcs of single-valued curves, within the same time bounds. Applications include the intersection of two polygons and the “merge” of two planar maps in time O(n log n + m) and space O(n + m), where n and m are the number of input and output edges, respectively. In the case of straight line segments, a simple modification allows us to count the number of intersections (without reporting them) in time O((n +√ nk)log n) = O(n 1.5log n).

Keywords

Intersection problems polygon intersection counting intersections segments planar maps plane sweep computational geometry algorithms 

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Copyright information

© Springer-Verlag Berlin Heidelberg 1988

Authors and Affiliations

  • Harry G. Mairson
    • 1
  • Jorge Stolfi
    • 2
  1. 1.Department of Computer ScienceBrandeis UniversityWalthamUSA
  2. 2.Stanford University and DEC Systems Research CenterPalo AltoUSA

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