Streaming algorithms for computing coresets and $k$-median clustering in the Hamming space

2026-08-25

AI Summary

本文提出了一种在Hamming空间中计算连续k-中位数聚类问题的(1+ε)近似算法，该算法具有FPT运行时间，并可在流式处理中实现。

Abstract

Clustering is one of the most fundamental tools in data analysis, allowing large datasets to be summarized by a small number of representative points. Given a metric space $(\mathcal{X}, \mathbb{d})$ and a set $S$ of $n$ points in this space, the continuous $k$-median clustering problem asks to find a set $C$ of $k$ points that minimizes the objective function $\sum_{s\in S} \mathbb{d}(s,C)$. When $\mathcal{X} = Σ^\ell$ is the set of strings of length $\ell$ and $\mathbb{d}$ is the Hamming distance, the continuous $k$-median clustering problem is known to be W[1]-hard when parameterized by $k$. In this work, we present the first $(1+\varepsilon)$-approximation algorithm for this problem with FPT runtime $2^{\mathrm{poly}(\varepsilon^{-1},k)} \cdot n\ell \mathrm{polylog} \; n$. An additional feature of the algorithm is that it can be implemented in streaming, requiring only $\tilde{O}_\varepsilon(\ell k + k^2)$ space. As an auxiliary tool of independent interest, we show the first streaming algorithm for computing an $\varepsilon$-coreset for continuous $k$-median clustering under the Hamming

Problem

• continuous k-median clustering
• Hamming space
• streaming algorithms

Innovation

• (1+ε)-approximation algorithm
• FPT runtime
• streaming algorithm
• ε-coreset
• Hamming space