Coresets for $k$-Means and $k$-Median Clustering and their Applications
Sariel Har-Peled, Soham Mazumdar · arXiv (Cornell University) · 2018
$\renewcommand{\Re}{{\rm I\!\hspace{-0.025em} R}} ewcommand{\eps}{\varepsilon} ewcommand{\Coreset}{\mathcal{S}} $ In this paper, we show the existence of small coresets for the problems of computing $k$-median and $k$-means clustering for points in low dimension. In other words, we show that given a point set $P$ in $\Re^d$, one can compute a weighted set $\Coreset \subseteq P$, of size $O(k \eps^{-d} \log{n})$, such that one can compute the $k$-median/means clustering on $\Coreset$ instead of on $P$, and get an $(1+\eps)$-approximation. As a result, we improve the fastest known algorithms for $(1+\eps)$-approximate $k$-means and $k$-median clustering. Our algorithms have linear running time for a fixed $k$ and $\eps$. In addition, we can maintain the $(1+\eps)$-approximate $k$-median or $k$-means clustering of a stream when points are being only inserted, using polylogarithmic space and update time.