Fully Dynamic k-Means Coreset in Near-Optimal Update Time

TL;DR

This paper introduces a near-optimal dynamic algorithm for maintaining coresets in k-means clustering, enabling efficient updates and queries in high-dimensional and general metric spaces.

Contribution

It presents a novel fully dynamic coreset construction with near-optimal update time, improving efficiency for k-means and k-median clustering in various metric spaces.

Findings

Near-optimal update time of  O(k) in general metrics

Efficient algorithms for Euclidean space with  O(d) update time

Reduced query time for approximate solutions

Abstract

We study in this paper the problem of maintaining a solution to $k$-median and $k$-means clustering in a fully dynamic setting. To do so, we present an algorithm to efficiently maintain a coreset, a compressed version of the dataset, that allows easy computation of a clustering solution at query time. Our coreset algorithm has near-optimal update time of $\tilde O(k)$ in general metric spaces, which reduces to $\tilde O(d)$ in the Euclidean space $\mathbb{R}^d$. The query time is $O(k^2)$ in general metrics, and $O(kd)$ in $\mathbb{R}^d$. To maintain a constant-factor approximation for $k$-median and $k$-means clustering in Euclidean space, this directly leads to an algorithm update time $\tilde O(d)$, and query time $\tilde O(kd + k^2)$. To maintain a $O(polylog~k)$-approximation, the query time is reduced to $\tilde O(kd)$.