Near Linear Time Approximation Schemes for Clustering of Partially Doubling Metrics

TL;DR

This paper develops near linear time approximation algorithms for clustering in metrics where either the set of points or the set of centers has bounded doubling dimension, extending previous work to more general cases.

Contribution

It generalizes existing near linear time approximation algorithms to cases where only one of the point sets or centers has bounded doubling dimension, including applications to Fréchet distance clustering.

Findings

First near linear time approximation for (k,ℓ)-median under discrete Fréchet distance.

Introduces a novel complexity reduction for time series clustering.

Extends algorithms to metric facility location problem.

Abstract

Given a finite metric space $(X\cup Y, \mathbf{d})$ the $k$-median problem is to find a set of $k$ centers $C\subseteq Y$ that minimizes $\sum_{p\in X} \min_{c\in C} \mathbf{d}(p,c)$. In general metrics, the best polynomial time algorithm computes a $(2+\epsilon)$-approximation for arbitrary $\epsilon>0$ (Cohen-Addad et al. STOC 2025). However, if the metric is doubling, a near linear time $(1+\epsilon)$-approximation algorithm is known (Cohen-Addad et al. J. ACM 2021). We show that the $(1+\epsilon)$-approximation algorithm can be generalized to the case when either $X$ or $Y$ has bounded doubling dimension (but the other set not). The case when $X$ is doubling is motivated by the assumption that even though $X$ is part of a high-dimensional space, it may be that it is close to a low-dimensional structure. The case when $Y$ is doubling is motivated by specific clustering problems where the centers are low-dimensional. Specifically, our work in this setting implies the first near linear time approximation algorithm for the $(k,\ell)$-median problem under discrete Fr\'echet distance when $\ell$ is constant. We further introduce a novel complexity reduction for time series of real values that leads to a similar result for the case of discrete Fr\'echet distance. In order to solve the case when $Y$ has a bounded doubling dimension, we introduce a dimension reduction that replaces points from $X$ by sets of points in $Y$. To solve the case when $X$ has a bounded doubling dimension, we generalize Talwar's decomposition (Talwar STOC 2004) to our setting. The running time of our algorithms is $2^{2^t} \tilde O(n+m)$ where $t=O(\mathrm{ddim} \log \frac{\mathrm{ddim}}{\epsilon})$ and where $\mathrm{ddim}$ is the doubling dimension of $X$ (resp.\ $Y$). The results also extend to the metric facility location problem.