Abstract

We show that the Voronoi diagram of a finite sequence of points in the plane which gives sorted order of the points with respect to two perpendicular directions can be computed in linear time. In contrast, we observe that the problem of computing the Voronoi diagram of a finite sequence of points in the plane which gives the sorted order of the points with respect to a single direction requires\Omega\Gamma n log n) operations in the algebraic decision tree model. As a corollary from the first result, we show that the bounded Voronoi diagrams of simple n-vertex polygons which can be efficiently cut into the so called monotone histograms can be computed in o(n log n) time. 1. Introduction Consider a set S of n points (sites), S = s 1 ; :::; s n , in the Euclidean plane. For i = 1; :::; n, let R(s i ) be the region of all points p in the plane which are closer to s i than to any other site in S. The region R(s i ) can be seen as the common intersection of the n \Gamma 1 half-planes in...

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