Clustering Motion

Given a set of moving points in ^d, we show how to cluster them in advance, using a small number of clusters, so that at any time this static clustering is competitive with the optimal k-center clustering at that time. The advantage of this approach is that it avoids updating the clustering as time passes. We also show how to maintain this static clustering efficiently under insertions and deletions. 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-center clustering of a set of n points in ^d. This slightly improves the algorithm of Feder and Greene, that runs in (n log k) time (which is optimal in the algebraic decision tree model).