Polygon Queries for Convex Hulls of Points

Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 10976)


We study the following range searching problem: Preprocess a set P of n points in the plane with respect to a set \(\mathcal {O}\) of k orientations in the plane so that given an \(\mathcal {O}\)-oriented convex polygon Q as a query, the convex hull of \(P\cap Q\), and its perimeter and area, can be reported efficiently, where an \(\mathcal {O}\)-oriented polygon is a polygon whose edges have orientations in \(\mathcal {O}\). We present a data structure with \(O(nk^3\log ^2n)\) space and \(O(nk^3\log ^2n)\) construction time, and a query algorithm to compute the perimeter or area of the convex hull of \(P\cap Q\) in \(O(s\log ^2n)\) time for any query \(\mathcal {O}\)-oriented convex s-gon Q. For reporting the convex hull, O(h) is added to the running times of query algorithms, where h is the complexity of the convex hull.


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Authors and Affiliations

  1. 1.Max Planck Institute for InformaticsSaarbrückenGermany
  2. 2.Pohang University of Science and TechnologyPohangKorea

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