GEOMETRIC STREAMING ALGORITHM WITH A SORTING PRIMITIVE

Abstract

We solve several fundamental geometric problems under a new streaming model recently proposed by Ruhl et al.2,15In this model, in one pass the input stream can be scanned to generate an output stream or be sorted based on a user-defined comparator; all intermediate streams must be of size O(n). We obtain the following geometric results for any fixed constant ϵ > 0:• We can construct 2D convex hulls in O(1) passes with O(nϵ) extra space.• We can construct 3D convex hulls in O(1) expected number of passes with O(nϵ) extra space.• We can construct a triangulation of a simple polygon in O(1) expected number of passes with O(nϵ) extra space, where n is the number of vertices on the polygon.• We can report all k intersections of a set of 2D line segments in O(1) passes with O(nϵ) extra space, if an intermediate stream of size O(n + k) is allowed.We also consider a weaker model, where we do not have the sorting primitive but are allowed to choose a scan direction for every scan pass. Here we can construct a 2D convex hull from an x-ordered point set in O(1) passes with O(nϵ) extra space.