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UBC Theses and Dissertations

Output-sensitive construction of convex hulls Chan, Timothy Moon-Yew


The construction of the convex hull of a finite point set in a low-dimensional Euclidean space is a fundamental problem in computational geometry. This thesis investigates efficient algorithms for the convex hull problem, where complexity is measured as a function of both the size of the input point set and the size of the output polytope. Two new, simple, optimal, output-sensitive algorithms are presented in two dimensions and a simple, optimal, output-sensitive algorithm is presented in three dimensions. In four dimensions, we give the first output-sensitive algorithm that is within a poly logarithmic factor of optimal. In higher fixed dimensions, we obtain an algorithm that is optimal for sufficiently small output sizes and is faster than previous methods for sublinear output sizes; this result is further improved in even dimensions. Although the focus of the thesis is on the convex hull problem, applications of our techniques to many related problems in computational geometry are also explored, including the computation of Voronoi diagrams, extreme points, convex layers, levels in arrangements, and envelopes of line segments, as well as problems relating to ray shooting and linear programming.

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