Distance-Sensitive Planar Point Location

  • Boris Aronov
  • Mark de Berg
  • Marcel Roeloffzen
  • Bettina Speckmann
Part of the Lecture Notes in Computer Science book series (LNCS, volume 8037)


Let \(\mathcal{S}\) be a connected planar polygonal subdivision with n edges and of total area 1. We present a data structure for point location in \(\mathcal{S}\) where queries with points far away from any region boundary are answered faster. More precisely, we show that point location queries can be answered in time \(O(1+\min(\log \frac{1}{\Delta_{p}}, \log n))\), where Δ p is the distance of the query point p to the boundary of the region containing p. Our structure is based on the following result: any simple polygon P can be decomposed into a linear number of convex quadrilaterals with the following property: for any point p ∈ P, the quadrilateral containing p has area \(\Omega(\Delta_{p}^2)\).


Medial Axis Query Point Query Time Steiner Point Simple Polygon 
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Copyright information

© Springer-Verlag Berlin Heidelberg 2013

Authors and Affiliations

  • Boris Aronov
    • 1
  • Mark de Berg
    • 2
  • Marcel Roeloffzen
    • 2
  • Bettina Speckmann
    • 2
  1. 1.Dept. of Computer Science and EngineeringPolytechnic Institute of NYUUSA
  2. 2.Dept. of Computer ScienceTU EindhovenThe Netherlands

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