On computing Voronoi diagrams for sorted point sets

We show that the Voronoi diagram of a nite 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 nite sequence of points in the plane which gives the sorted order of the points with respect to a single direction requires (n logn) operations in the algebraic decision tree model. As a corollary from the rst result, we show that the bounded Voronoi diagrams of simple n-vertex polygons which can be eeciently cut into the so called monotone histograms can be computed in o(n log n) time.