Discrete & Computational Geometry

, Volume 51, Issue 2, pp 462–484 | Cite as

Covering Paths for Planar Point Sets

  • Adrian Dumitrescu
  • Dániel Gerbner
  • Balázs Keszegh
  • Csaba D. Tóth
Article

Abstract

Given n points in the plane, a covering path is a polygonal path that visits all the points. If no three points are collinear, every covering path requires at least n/2 segments, and n−1 straight line segments obviously suffice even if the covering path is required to be noncrossing. We show that every set of n points in the plane admits a (possibly self-crossing) covering path consisting of n/2+O(n/logn) straight line segments. If the path is required to be noncrossing, we prove that (1−ε)n straight line segments suffice for a small constant ε>0, and we exhibit n-element point sets that require at least 5n/9−O(1) segments in every such path. Further, the analogous question for noncrossing covering trees is considered and similar bounds are obtained. Finally, it is shown that computing a noncrossing covering path for n points in the plane requires Ω(nlogn) time in the worst case.

Keywords

Covering path Covering tree Noncrossing graph 

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Copyright information

© Springer Science+Business Media New York 2013

Authors and Affiliations

  • Adrian Dumitrescu
    • 1
  • Dániel Gerbner
    • 2
  • Balázs Keszegh
    • 2
  • Csaba D. Tóth
    • 3
    • 4
  1. 1.Computer ScienceUniversity of WisconsinMilwaukeeUSA
  2. 2.Alfréd Rényi Institute of MathematicsHungarian Academy of SciencesBudapestHungary
  3. 3.Department of MathematicsCalifornia State University NorthridgeLos AngelesUSA
  4. 4.Department of Mathematics and StatisticsUniversity of CalgaryCalgaryCanada

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