Approximate Range Queries for Clustering

TL;DR

This paper develops data structures and algorithms for efficiently computing approximate solutions to range-based clustering problems in high-dimensional Euclidean spaces, enabling fast $(1+ ext{epsilon})$-approximations for queries.

Contribution

It introduces novel data structures and algorithms for approximate range clustering, specifically for $k$-median, $k$-means, and $k$-center problems, in high-dimensional spaces.

Findings

Efficient $(1+ ext{epsilon})$-approximate algorithms for range clustering.

Data structures supporting fast query times for high-dimensional data.

Applicable to $k$-median, $k$-means, and $k$-center clustering problems.

Abstract

We study the approximate range searching for three variants of the clustering problem with a set $P$ of $n$ points in $d$-dimensional Euclidean space and axis-parallel rectangular range queries: the $k$-median, $k$-means, and $k$-center range-clustering query problems. We present data structures and query algorithms that compute $(1+\varepsilon)$-approximations to the optimal clusterings of $P\cap Q$ efficiently for a query consisting of an orthogonal range $Q$, an integer $k$, and a value $\varepsilon>0$.