Faster Clustering via Preprocessing

TL;DR

This paper introduces fast algorithms for clustering queries in metric spaces with low doubling dimension, utilizing preprocessing to significantly reduce query times for objectives like p-center and p-median.

Contribution

It presents novel preprocessing-based algorithms that enable near-linear query times for clustering in low doubling dimension metric spaces, improving efficiency over previous methods.

Findings

Query time is near-linear in the size of the query set.

Preprocessing reduces dependence on the total number of points.

Algorithms work for standard clustering objectives like p-center and p-median.

Abstract

We examine the efficiency of clustering a set of points, when the encompassing metric space may be preprocessed in advance. In computational problems of this genre, there is a first stage of preprocessing, whose input is a collection of points $M$; the next stage receives as input a query set $Q\subset M$, and should report a clustering of $Q$ according to some objective, such as 1-median, in which case the answer is a point $a\in M$ minimizing $\sum_{q\in Q} d_M(a,q)$. We design fast algorithms that approximately solve such problems under standard clustering objectives like $p$-center and $p$-median, when the metric $M$ has low doubling dimension. By leveraging the preprocessing stage, our algorithms achieve query time that is near-linear in the query size $n=|Q|$, and is (almost) independent of the total number of points $m=|M|$.