Testing of clustering

A set X of points in R-d is (k, b)-clusterable if X can be partitioned into k subsets (clusters) so that the diameter (alternatively, the radius) of each cluster is at most b. We present algorithms that by sampling from a set X, distinguish between the case that X is (k, b)-clusterable and the case that X is epsilon -far from being (k, b')-clusterable for any given 0 < <epsilon> less than or equal to 1 and for b' greater than or equal to b. In epsilon -far from being (k, b')-clusterable we mean that more than epsilon. \X\ points should be removed from X so that it becomes (Ic, b')-clusterable. We give algorithms for a variety of cost measures that use a sample of size independent of \X\, and polynomial in Ic and 1/epsilon. Our algorithms can also be used to find approximately good clusterings. Namely, these are clusterings of all but an e-fraction of the points in X that have optimal (or close to optimal) cost. The benefit of our algorithms is that they construct an implicit representation of such clusterings in time independent of \X\. That is, without actually having to partition all points in X, the implicit representation can be used to answer queries concerning the cluster any given point belongs to.