Trajectory clustering of points in R

The problem of clustering a set of points moving on the line consists of the following: given positive integers n and k, and the initial position and velocity of n points, find an optimal k-clustering of the points. We consider two classical quality measures for the clustering: minimizing the sum of the clusters diameters and minimizing the maximum diameter of a cluster. For the former, we present a polynomial-time algorithm under some assumptions and, for the latter, a (2.71 + ε)-approximation.