# On Coresets for Clustering in Small Dimensional Euclidean Spaces

## TL;DR

This paper studies small coresets for k-Median clustering in low-dimensional Euclidean spaces, providing improved bounds, new lower bounds, and the first separation results between k=1 and k=2 in 1D.

## Contribution

It offers improved coreset size bounds for small dimensions, establishes new lower bounds, and demonstrates a novel separation between 1-Median and 2-Median in 1D.

## Key findings

- Improved coreset bounds for small dimensions.
- New lower bounds for coreset sizes.
- First known separation between 1-Median and 2-Median in 1D.

## Abstract

We consider the problem of constructing small coresets for $k$-Median in Euclidean spaces. Given a large set of data points $P\subset \mathbb{R}^d$, a coreset is a much smaller set $S\subset \mathbb{R}^d$, so that the $k$-Median costs of any $k$ centers w.r.t. $P$ and $S$ are close. Existing literature mainly focuses on the high-dimension case and there has been great success in obtaining dimension-independent bounds, whereas the case for small $d$ is largely unexplored. Considering many applications of Euclidean clustering algorithms are in small dimensions and the lack of systematic studies in the current literature, this paper investigates coresets for $k$-Median in small dimensions. For small $d$, a natural question is whether existing near-optimal dimension-independent bounds can be significantly improved. We provide affirmative answers to this question for a range of parameters. Moreover, new lower bound results are also proved, which are the highest for small $d$. In particular, we completely settle the coreset size bound for $1$-d $k$-Median (up to log factors). Interestingly, our results imply a strong separation between $1$-d $1$-Median and $1$-d $2$-Median. As far as we know, this is the first such separation between $k=1$ and $k=2$ in any dimension.