Testing of Clustering
Noga Alon, Seannie Dar, Michal Parnas, Dana Ron · SIAM Journal on Discrete Mathematics · 2003
A set X of points in $\Re^d$ 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 $0 < \epsilon\leq 1$ and for $b' \geq b$. By $\epsilon$-far from being (k,b')-clusterable we mean that more than $\epsilon\cdot|X|$ 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 $1/\epsilon$. Our algorithms can also be used to find approximately good clusterings. Namely, these are clusterings of all but an $\epsilon$-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 to which any given point belongs.