摘要：We consider the problem of center-based clustering in low-dimensional Euclidean spaces under the perturbation stability assumption. An instance is α-stable if the underlying optimal clustering continues to remain optimal even when all pairwise distances are arbitrarily perturbed by a factor of at most α. Our main contribution is in presenting efficient exact algorithms for α-stable clustering instances whose running times depend near-linearly on the size of the data set when α ⥠2 â^S3. For k-center and k-means problems, our algorithms also achieve polynomial dependence on the number of clusters, k, when α ⥠2 â^S3 ε for any constant ε > 0 in any fixed dimension. For k-median, our algorithms have polynomial dependence on k for α > 5 in any fixed dimension; and for α ⥠2 â^S3 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.
关键词：clustering; stability; local search; dynamic programming; coreset; polyhedral metric; trapezoid decomposition; range query
