Near-Optimal Clustering in the $k$-machine model

TL;DR

This paper introduces near-optimal algorithms for three clustering problems in the $k$-machine model, achieving constant-factor approximations with optimal round complexity, and establishes matching lower bounds, marking the first such results in this model.

Contribution

It presents the first constant-factor approximation algorithms for clustering in the $k$-machine model with optimal round complexity, and proves matching lower bounds, advancing large-scale distributed clustering theory.

Findings

Algorithms achieve $O(1)$-factor approximation in $ ilde{O}(n/k)$ rounds.

Lower bounds show $ ilde{ ext{Omega}}(n/k)$ rounds are necessary for polynomial-factor approximations.

Main technical contribution: learning a small portion of the input metric suffices for approximation.

Abstract

The clustering problem, in its many variants, has numerous applications in operations research and computer science (e.g., in applications in bioinformatics, image processing, social network analysis, etc.). As sizes of data sets have grown rapidly, researchers have focused on designing algorithms for clustering problems in models of computation suited for large-scale computation such as MapReduce, Pregel, and streaming models. The $k$-machine model (Klauck et al., SODA 2015) is a simple, message-passing model for large-scale distributed graph processing. This paper considers three of the most prominent examples of clustering problems: the uncapacitated facility location problem, the $p$-median problem, and the $p$-center problem and presents $O(1)$-factor approximation algorithms for these problems running in $\tilde{O}(n/k)$ rounds in the $k$-machine model. These algorithms are optimal up to polylogarithmic factors because this paper also shows $\tilde{\Omega}(n/k)$ lower bounds for obtaining polynomial-factor approximation algorithms for these problems. These are the first results for clustering problems in the $k$-machine model. We assume that the metric provided as input for these clustering problems in only implicitly provided, as an edge-weighted graph and in a nutshell, our main technical contribution is to show that constant-factor approximation algorithms for all three clustering problems can be obtained by learning only a small portion of the input metric.