Distributed Algorithms for Euclidean Clustering

TL;DR

This paper develops improved distributed algorithms for constructing small, accurate coresets for Euclidean clustering problems, reducing communication costs and achieving near-optimal bounds in different models.

Contribution

It introduces new protocols for distributed Euclidean clustering that significantly lower communication complexity while providing $(1+ ext{epsilon})$-approximate coresets, improving upon prior methods.

Findings

Achieved $(1+ ext{epsilon})$-coresets with reduced communication in coordinator model.

Extended techniques to the blackboard model with further communication reduction.

Matched lower bounds up to polylogarithmic factors for distributed coreset construction.

Abstract

We study the problem of constructing $(1+\varepsilon)$-coresets for Euclidean $(k,z)$-clustering in the distributed setting, where $n$ data points are partitioned across $s$ sites. We focus on two prominent communication models: the coordinator model and the blackboard model. In the coordinator model, we design a protocol that achieves a $(1+\varepsilon)$-strong coreset with total communication complexity $\tilde{O}\left(sk + \frac{dk}{\min(\varepsilon^4,\varepsilon^{2+z})} + dk\log(n\Delta)\right)$ bits, improving upon prior work (Chen et al., NeurIPS 2016) by eliminating the need to communicate explicit point coordinates in-the-clear across all servers. In the blackboard model, we further reduce the communication complexity to $\tilde{O}\left(s\log(n\Delta) + dk\log(n\Delta) + \frac{dk}{\min(\varepsilon^4,\varepsilon^{2+z})}\right)$ bits, achieving better bounds than previous approaches while upgrading from constant-factor to $(1+\varepsilon)$-approximation guarantees. Our techniques combine new strategies for constant-factor approximation with efficient coreset constructions and compact encoding schemes, leading to optimal protocols that match both the communication costs of the best-known offline coreset constructions and existing lower bounds (Chen et al., NeurIPS 2016, Huang et. al., STOC 2024), up to polylogarithmic factors.