Abstract
Given a set S of weighted points in the plane, we consider two problems dealing with planar lines in ℝ2 under the weighted Euclidean distance: (1) Preprocess S into a data structure that efficiently finds a nearest point among S of a query “line”. (2) Find an optimal “line” that maximizes the minimum of the weighted distance to any point of S. We introduce a unified approach to both problems based on a new geometric transformation that maps lines in the plane into points in a line space. It turns out that our transformation, together with its target space, well describes the proximity relations between given weighted points S and every planar line in ℝ2. We define a Voronoi-like tessellation on the line space and investigate its geometric, combinatorial, and computational properties. As its applications, we obtain several new results on the two problems.
This work is dedicated to our advisor, Professor Kyung-Yong Chwa, on the occasion of his honorable retirement.
Work by S.W.Bae was supported by National Research Foundation of Korea(NRF) grant funded by the Korea government(MEST) (No. 2010-0005974). Work by C.-S.Shin was supported by National Research Foundation of Korea(NRF) grant funded by the Korea government(MEST) (No. 2010-0016416), and the HUFS Research Fund.
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Bae, S.W., Shin, CS. (2010). The Onion Diagram: A Voronoi-Like Tessellation of a Planar Line Space and Its Applications. In: Cheong, O., Chwa, KY., Park, K. (eds) Algorithms and Computation. ISAAC 2010. Lecture Notes in Computer Science, vol 6507. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-17514-5_20
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DOI: https://doi.org/10.1007/978-3-642-17514-5_20
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