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

TL;DR

This work addresses maintaining high-quality clustering under fully dynamic updates by constructing and maintaining a compact $\varepsilon$-coreset that preserves clustering costs for all $k$-center solutions. The authors develop a dynamic coreset framework built on a refined merge-and-reduce strategy, achieving near-optimal amortized update times $\tilde{O}(k)$ in general metrics and $\tilde{O}(d)$ in $\mathbb{R}^d$, with flexible query times that scale as $\tilde{O}(k^2)$ (or $\tilde{O}(kd)$) depending on the approximation factor. A key contribution is a coreset algorithm that supports insertions and deletions with an amortized cost of $\frac{T(n,k,\varepsilon)}{k}$, where $T$ is the static clustering time, and its integration into the merge-and-reduce framework yields fast dynamic updates. The paper also establishes Euclidean-space implications and two conjectures suggesting potential near-optimality for both update and query times in that setting, highlighting practical significance for real-time clustering on evolving data streams.

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 $\tilde O(kd)$.

Figures (1)

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Theorems & Definitions (18)

• Theorem 1
• Corollary 2
• Lemma 4
• Lemma 5
• Lemma 6
• proof
• proof : Proof of \ref{['thm:main']}
• proof
• Lemma 8
• proof