Complexity analysis of random convex hulls

Rémy Thomasse 1
1 GEOMETRICA - Geometric computing
CRISAM - Inria Sophia Antipolis - Méditerranée , Inria Saclay - Ile de France
Abstract : In this thesis, we give some new results about the average size of convex hulls made of points chosen in a convex body. This size is known when the points are chosen uniformly (and independently) in a convex polytope or in a "smooth" enough convex body. This average size is also known if the points are independently chosen according to a centered Gaussian distribution. In the first part of this thesis, we introduce a technique that will give new results when the points are chosen arbitrarily in a convex body, and then noised by some random perturbations. This kind of analysis, called smoothed analysis, has been initially developed by Spielman and Teng in their study of the simplex algorithm. For an arbitrary set of point in a ball, we obtain a lower and a upper bound for this smoothed complexity, in the case of uniform perturbation in a ball (in arbitrary dimension) and in the case of Gaussian perturbations in dimension 2. The asymptotic behavior of the expected size of the convex hull of uniformly random points in a convex body is polynomial for a "smooth" body and polylogarithmic for a polytope. In the second part, we construct a convex body so that the expected size of the convex hull of points uniformly chosen in that body oscillates between these two behaviors when the number of points increases. In the last part, we present an algorithm to generate efficiently a random convex hull made of points chosen uniformly and independently in a disk. We also compute its average time and space complexity. This algorithm can generate a random convex hull without explicitly generating all the points. It has been implemented in C++ and integrated in the CGAL library.
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Rémy Thomasse. Complexity analysis of random convex hulls. Other [cs.OH]. Université Nice Sophia Antipolis, 2015. English. ⟨NNT : 2015NICE4116⟩. ⟨tel-01252937v2⟩

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