Clustering Under Perturbation Stability in Near-Linear Time

We consider the problem of center-based clustering in low-dimensional Euclidean spaces under the perturbation stability assumption. An instance is ?-stable if the underlying optimal clustering continues to remain optimal even when all pairwise distances are arbitrarily perturbed by a factor of at most ?. Our main contribution is in presenting efficient exact algorithms for ?-stable clustering instances whose running times depend near-linearly on the size of the data set when ? ? 2 + ?3. For k-center and k-means problems, our algorithms also achieve polynomial dependence on the number of clusters, k, when ? ? 2 + ?3 + ? for any constant ? > 0 in any fixed dimension. For k-median, our algorithms have polynomial dependence on k for ? > 5 in any fixed dimension; and for ? ? 2 + ?3 in two dimensions. Our algorithms are simple, and only require applying techniques such as local search or dynamic programming to a suitably modified metric space, combined with careful choice of data structures.