Polygon Queries for Convex Hulls of Points
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We study the following range searching problem: Preprocess a set P of n points in the plane with respect to a set O of k orientations for a constant, in the plane so that given an O-oriented convex polygon Q, the convex hull of P∩ Q can be computed efficiently, where an O-oriented polygon is a polygon whose edges have orientations in O. We present a data structure with O(nk^3log^2n) space and O(nk^3log^2n) construction time, and an O(h+slog^2 n)-time query algorithm for any query O-oriented convex s-gon Q, where h is the complexity of the convex hull. Also, we can compute the perimeter or area of the convex hull of P∩ Q in O(slog^2n) time using the data structure.
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