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The lower bound on opt( U) given in Lemma B.10 holds for ρ -metric spaces with no modifications. By making the appropriate modifications to the proof of Theorem C.1, we can extend this theorem to In particular, we can obtain a proof of Theorem A.5 by taking the proof of Theorem C.1 and adding extra ρ factors whenever the triangle inequality is applied. We first prove Lemma B.1, which shows that the sizes of the sets U By Lemma B.2, we get that Henceforth, we fix some positive ξ and sufficiently large α such that Lemma B.3 holds. By now applying Lemma B.4 it follows that The following lemma is proven in [25]. Lemma B.1, the third inequality follows from Lemma B.7, and the fourth inequality follows from the The second inequality follows from Lemma B.8, the third inequality from averaging and the choice Proof of Lemma 3.3: It follows that with probability at least 1 e Hence, by Theorem D.1, we must have that O (poly( k)) query time must have Ω( k) amortized update time.

Clustering is a fundamental problem in unsupervised learning with several practical applications. In clustering, one is interested in partitioning elements into different groups (i.e.

The metric $k$-center clustering problem with $z$ outliers, also known as $(k,z)$-center clustering, involves clustering a given point set $P$ in a metric space $(M,d)$ using at most $k$ balls, minimizing the maximum ball radius while excluding up to $z$ points from the clustering. This problem holds fundamental significance in various domains such as machine learning, data mining, and database systems.This paper addresses the fully dynamic version of the problem, where the point set undergoes continuous updates (insertions and deletions) over time. The objective is to maintain an approximate $(k,z)$-center clustering with efficient update times. We propose a novel fully dynamic algorithm that maintains a $(4+\epsilon)$-approximate solution to the $(k,z)$-center clustering problem that covers all but at most $(1+\epsilon)z$ points at any time in the sequence with probability $1-k/e^{\Omega(\log k)}$. The algorithm achieves an expected amortized update time of $\mathcal{O}(\epsilon^{-2} k^6\log(k) \log(\Delta))$, and is applicable to general metric spaces. Our dynamic algorithm presents a significant improvement over the recent dynamic $(14+\epsilon)$-approximation algorithm by Chan, Lattanzi, Sozio, and Wang for this problem.

We consider classic clustering problems in fully dynamic data streams, where data elements can be both inserted and deleted. In this context, several parameters are of importance: (1) the quality of the solution after each insertion or deletion, (2) the time it takes to update the solution, and (3) how different consecutive solutions are. The question of obtaining efficient algorithms in this context for facility location, $k$-median and $k$-means has been raised in a recent paper by Hubert-Chan et al. [WWW'18] and also appears as a natural follow-up on the online model with recourse studied by Lattanzi and Vassilvitskii [ICML'17] (i.e.: in insertion-only streams). In this paper, we focus on general metric spaces and mainly on the facility location problem. We give an arguably simple algorithm that maintains a constant factor approximation, with $O(n\log n)$ update time, and total recourse $O(n)$. This improves over the naive algorithm which consists in recomputing a solution at each time step and that can take up to $O(n^2)$ update time, and $O(n^2)$ total recourse. These bounds are nearly optimal: in general metric space, inserting a point take $O(n)$ times to describe the distances to other points, and we give a simple lower bound of $O(n)$ for the recourse. Moreover, we generalize this result for the $k$-medians and $k$-means problems: our algorithm maintains a constant factor approximation in time $\widetilde{O}(n+k^2)$. We complement our analysis with experiments showing that the cost of the solution maintained by our algorithm at any time $t$ is very close to the cost of a solution obtained by quickly recomputing a solution from scratch at time $t$ while having a much better running time.

We present a $O(1)$-approximate fully dynamic algorithm for the $k$-median and $k$-means problems on metric spaces with amortized update time $\tilde O(k)$ and worst-case query time $\tilde O(k^2)$. We complement our theoretical analysis with the first in-depth experimental study for the dynamic $k$-median problem on general metrics, focusing on comparing our dynamic algorithm to the current state-of-the-art by Henzinger and Kale [ESA'20]. Finally, we also provide a lower bound for dynamic $k$-median which shows that any $O(1)$-approximate algorithm with $\tilde O(\text{poly}(k))$ query time must have $\tilde \Omega(k)$ amortized update time, even in the incremental setting.