Optimal Fully Dynamic k-Center Clustering for Adaptive and Oblivious Adversaries
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In fully dynamic clustering problems, a clustering of a given data set in a metric space must be maintained while it is modified through insertions and deletions of individual points. In this paper, we resolve the complexity of fully dynamic k-center clustering against both adaptive and oblivious adversaries. Against oblivious adversaries, we present the first algorithm for fully dynamic k-center in an arbitrary metric space that maintains an optimal (2+ϵ)-approximation in O(k ·polylog(n,Δ)) amortized update time. Here, n is an upper bound on the number of active points at any time, and Δ is the aspect ratio of the metric space. Previously, the best known amortized update time was O(k^2·polylog(n,Δ)), and is due to Chan, Gourqin, and Sozio (2018). Moreover, we demonstrate that our runtime is optimal up to polylog(n,Δ) factors. In fact, we prove that even offline algorithms for k-clustering tasks in arbitrary metric spaces, including k-medians, k-means, and k-center, must make at least Ω(n k) distance queries to achieve any non-trivial approximation factor. This implies a lower bound of Ω(k) which holds even for the insertions-only setting. We also show deterministic lower and upper bounds for adaptive adversaries, demonstrate that an update time sublinear in k is possible against oblivious adversaries for metric spaces which admit locally sensitive hash functions (LSH) and give the first fully dynamic O(1)-approximation algorithms for the closely related k-sum-of-radii and k-sum-of-diameter problems.
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