An Efficient Massively Parallel Constant-Factor Approximation Algorithm for the $k$-Means Problem

TL;DR

This paper introduces a massively parallel algorithm that efficiently approximates the $k$-means problem within a constant factor in fewer than logarithmic rounds, using nearly linear global memory, advancing parallel clustering methods.

Contribution

It presents the first constant-factor approximation algorithm for the general $k$-means problem in o(log n) MPC rounds with nearly linear memory, building on facility location approximations with LMP properties.

Findings

Achieves $O(rac{ ext{loglog n} imes ext{logloglog n}})$ round complexity.

Uses $O(n^{ ext{small } ext{constant}})$ bits per machine, with nearly linear total memory.

Provides a constant-factor approximation with LMP property for facility location.

Abstract

In this paper, we present an efficient massively parallel approximation algorithm for the $k$-means problem. Specifically, we provide an MPC algorithm that computes a constant-factor approximation to an arbitrary $k$-means instance in $O(\log\log n \cdot \log\log\log n)$ rounds. The algorithm uses $O(n^\sigma)$ bits of memory per machine, where $\sigma > 0$ is a constant that can be made arbitrarily small. The global memory usage is $O(n^{1+\varepsilon})$ bits for an arbitrarily small constant $\varepsilon > 0$, and is thus only slightly superlinear. Recently, Czumaj, Gao, Jiang, Krauthgamer, and Vesel\'{y} showed that a constant-factor bicriteria approximation can be computed in $O(1)$ rounds in the MPC model. However, our algorithm is the first constant-factor approximation for the general $k$-means problem that runs in $o(\log n)$ rounds in the MPC model. Our approach builds upon the foundational framework of Jain and Vazirani. The core component of our algorithm is a constant-factor approximation for the related facility location problem. While such an approximation was already achieved in constant time in the work of Czumaj et al.\ mentioned above, our version additionally satisfies the so-called Lagrangian Multiplier Preserving (LMP) property. This property enables the transformation of a facility location approximation into a comparably good $k$-means approximation.