How to get close to the median shape

In this paper, we study the problem of L1-fitting a shape to a set of point, 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 + ε)approximation for such a problem, with running time O(n + poly(log n, 1/ε)), where poly(log n, 1/ε) is polynomial of constant degree of log n and 1/ε. This is a linear time algorithm for a fixed ε > 0, and it 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.

• Type: Conference
• Year: 2006
• Where: COMPGEOM
• Authors: Sariel Har-Peled
• Keywords: COMPGEOM 2006, Linear Time Algorithm, Squared Distances, first Subquadratic Algorithm