Improved analysis of D2-sampling based PTAS for k-means and other Clustering problems

TL;DR

This paper improves the theoretical analysis of a D^2-sampling based PTAS for k-means clustering, reducing its running time from exponential in k^2 to exponential in k, making it more efficient.

Contribution

The authors provide a tighter analysis that significantly reduces the running time of the existing PTAS for k-means clustering.

Findings

Running time improved from $O(nd imes 2^{ ilde{O}(k^2/\epsilon)})$ to $O(nd imes 2^{ ilde{O}(k/\epsilon)})$.

Analysis enhances understanding of D^2-sampling efficiency.

Potential for faster clustering algorithms in high-dimensional data.

Abstract

We give an improved analysis of the simple $D^2$-sampling based PTAS for the $k$-means clustering problem given by Jaiswal, Kumar, and Sen (Algorithmica, 2013). The improvement on the running time is from $O\left(nd \cdot 2^{\tilde{O}(k^2/\epsilon)}\right)$ to $O\left(nd \cdot 2^{\tilde{O}(k/\epsilon)}\right)$.