Complexity of the delaunay triangulation of points on surfaces the smooth case
Proceedings of the nineteenth annual symposium on Computational geometry
Smoothed analysis of algorithms: Why the simplex algorithm usually takes polynomial time
Journal of the ACM (JACM)
Improved Bounds on the Union Complexity of Fat Objects
Discrete & Computational Geometry
Visibility maps of realistic terrains have linear smoothed complexity
Proceedings of the twenty-fifth annual symposium on Computational geometry
Smoothed analysis: an attempt to explain the behavior of algorithms in practice
Communications of the ACM - A View of Parallel Computing
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Average-case analysis of data-structures or algorithms is commonly used in computational geometry when the, more classical, worst-case analysis is deemed overly pessimistic. Since these analyses are often intricate, the models of random geometric data that can be handled are often simplistic and far from "realistic inputs". We present a new simple scheme for the analysis of geometric structures. While this scheme only produces results up to a polylog factor, it is much simpler to apply than the classical techniques and therefore succeeds in analyzing new input distributions related to smoothed complexity analysis. We illustrate our method on two classical structures: convex hulls and Delaunay triangulations. Specifically, we give short and elementary proofs of the classical results that n points uniformly distributed in a ball in Rd have a convex hull and a Delaunay triangulation of respective expected complexities ~Θ(n(d-1)/(d+1)) and ~Θn. We then prove that if we start with n points well-spread on a sphere, e.g. an (ε,κ)-sample of that sphere, and perturb that sample by moving each point randomly and uniformly within distance at most δ of its initial position, then the expected complexity of the convex hull of the resulting point set is ~Θ(√n)1-1/d1/(√[4]δ)d-1/d}.