Internship proposal

On the one hand, classical analysis of geometric algorithms is done in the worst case. The worst case is often very pessimistic and need to designed carefully pathological data set to be reached. On the other hand, analysis of geometric algorithms can also be done under probabilistic hypotheses on the data distribution. Unfortunately, common hypotheses, such as Poisson distribution or uniform distribution, are often too optimistic compared to real world data. In 2004, Spielman and Teng [8] introduced a new technique called smoothed analysis to interpolate between this two situations. A particular data set is analyzed under the hypothesis of some level of noise , then the worst case on all data set of size n is taken. The complexity obtained depends on n and . Spielman and Teng proved that the simplex algorithm for linear programming, whose worst case is exponential in n, decreases quickly with removing the exponential behavior if is a reasonable function of n. Recently, we proposed a smoothed analysis of the size of the convex hull [4]. As an example of result, in a lemma, we prove that if each vertex of the regular ngon is perturbed by a uniform noise in a disk of radius 2 [ 1 n2 , 1] the complexity of the convex hull is O(n 4 8 ) and thus decreasing with .