WADS 2013: Algorithms and Data Structures pp 49-60

# Distance-Sensitive Planar Point Location

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

## Abstract

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 Δp 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 $$\Omega(\Delta_{p}^2)$$.

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© 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