Fully Dynamic $k$-Clustering in $\tilde O(k)$ Update Time

TL;DR

This paper introduces a fully dynamic algorithm for the $k$-median and $k$-means problems with near-optimal update time, supported by theoretical analysis and experimental validation on general metrics.

Contribution

It provides the first $O(1)$-approximate fully dynamic algorithm with $ ilde O(k)$ update time for $k$-median and $k$-means, along with a lower bound for such algorithms.

Findings

Achieves $ ilde O(k)$ amortized update time for dynamic $k$-median and $k$-means.

First experimental comparison of dynamic algorithms on general metrics.

Establishes a lower bound indicating $ ilde ext{O}(k)$ update time is necessary for $O(1)$-approximate algorithms.

Abstract

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.