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.

翻译：摘要： 我们提出了一种针对度量空间上 $k$-中位数和 $k$-均值问题的 $O(1)$-近似全动态算法，其均摊更新时间为 $\tilde O(k)$，最坏情况查询时间为 $\tilde O(k^2)$。我们通过首次针对一般度量空间上动态 $k$-中位数问题的深入实验研究，补充了理论分析，重点是将我们的动态算法与 Henzinger 和 Kale [ESA'20] 提出的当前最先进方法进行了比较。最后，我们还给出了动态 $k$-中位数问题的一个下界，表明任何具有 $\tilde O(\text{poly}(k))$ 查询时间的 $O(1)$-近似算法，即使在增量设置下，也必须具有 $\tilde \Omega(k)$ 的均摊更新时间。