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 ε-far from being (k,b´)-clusterable for any given 0<ε⩽1 and for b´⩾b. In ε-far from being (k,b´)-clusterable we mean that more than ε.|X| points should be removed from X so that it becomes (k,b´)-clusterable. We give algorithms for a variety of cost measures that use a sample of size independent of |X|, and polynomial in k and 1/ε. Our algorithms can also be used to find approximately good clusterings. Namely, these are clusterings of all but an ε-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