How to get close to the median shape

In this paper, we study the problem of L1-fitting a shape to a set of n points in Rd (where d is a fixed constant), where the target is to minimize the sum of distances of the points to the shape, or alternatively the sum of squared distances. We present a general technique for computing a (1 + e)-approximation for such a problem, with running time O(n + poly(logn, 1/e)), where poly(logn, 1/e) is a polynomial of constant degree of logn and 1/e (the power of the polynomial is a function of d). This is a linear time algorithm for a fixed e > 0, and is the first subquadratic algorithm for this problem. Applications of the algorithm include best fitting either a circle, a sphere or a cylinder to a set of points when minimizing the sum of distances (or squared distances) to the respective shape.

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