Clustering motion

Given a set of moving points in R^d, we show that one can cluster them in advance, using a small number of clusters, so that at any point in time this static clustering is competitive with the optimal k-centre clustering of the point-set at this point in time. The advantage of this approach is that it avoids the usage of kinetic data structures and as such it does not need to update the clustering as time passes. To implement this static clustering efficiently, we describe a simple technique for speeding up clustering algorithms, and apply it to achieve faster clustering algorithms for several problems. In particular, we present a linear time algorithm for computing a 2-approximation to the k-centre clustering of a set of n points in R^d. This is a slight improvement over the algorithm of T. Feder and D. Greene (1988), that runs in Θ(n log k) time (which is optimal in the comparison model).