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 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 \(\Omega(\Delta_{p}^2)\).
M. Roeloffzen and B. Speckmann were supported by the Netherlands’ Organisation for Scientific Research (NWO) under project no. 600.065.120 and 639.022.707, respectively. B. Aronov has been supported by grant No. 2006/194 from the U.S.-Israel Binational Science Foundation, by NSF Grants CCF-08-30691, CCF-11-17336, and CCF-12-18791, and by NSA MSP Grant H98230-10-1-0210.
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Aronov, B., de Berg, M., Roeloffzen, M., Speckmann, B. (2013). Distance-Sensitive Planar Point Location. In: Dehne, F., Solis-Oba, R., Sack, JR. (eds) Algorithms and Data Structures. WADS 2013. Lecture Notes in Computer Science, vol 8037. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-40104-6_5
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DOI: https://doi.org/10.1007/978-3-642-40104-6_5
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