The Onion Diagram: A Voronoi-Like Tessellation of a Planar Line Space and Its Applications

Abstract

Given a set S of weighted points in the plane, we consider two problems dealing with lines in R 2 under the weighted Euclidean distance: (1) Preprocess S into a data structure that efficiently finds a nearest point among S to a query line. (2) Find an optimal line that maximizes the minimum of the weighted distance to any point of S. The latter problem is also known as the obnoxious line location problem. We introduce a unified approach to both problems based on a new geometric transformation that maps lines in the plane to 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 line in R 2 . We define a Voronoi-like tessellation on the line space and investigate its geometric, combinatorial, and computational properties. As its direct applications, we obtain several new algorithmic results on the two problems. We also show that our approach extends to weighted line segments and weighted polygons.