Massively Parallel and Dynamic Algorithms for Minimum Size Clustering

TL;DR

This paper introduces scalable parallel and dynamic algorithms for the $r$-gather clustering problem, achieving constant-factor approximations with efficient update times in metric spaces and Euclidean settings.

Contribution

It presents the first scalable MPC algorithm and a dynamic algorithm for $r$-gather clustering with provable approximation guarantees and efficient complexity.

Findings

MPC algorithm computes $O(1)$-approximate solutions in $O( ext{polylog } n)$ rounds.

Dynamic algorithm maintains $O(1)$-approximate solutions with polylogarithmic update and query times.

Algorithms are scalable and handle high-dimensional Euclidean and general metric spaces.

Abstract

In this paper, we study the $r$-gather problem, a natural formulation of minimum-size clustering in metric spaces. The goal of $r$-gather is to partition $n$ points into clusters such that each cluster has size at least $r$, and the maximum radius of the clusters is minimized. This additional constraint completely changes the algorithmic nature of the problem, and many clustering techniques fail. Also previous dynamic and parallel algorithms do not achieve desirable complexity. We propose algorithms both in the Massively Parallel Computation (MPC) model and in the dynamic setting. Our MPC algorithm handles input points from the Euclidean space $\mathbb{R}^d$. It computes an $O(1)$-approximate solution of $r$-gather in $O(\log^{\varepsilon} n)$ rounds using total space $O(n^{1+\gamma}\cdot d)$ for arbitrarily small constants $\varepsilon,\gamma > 0$. In addition our algorithm is fully scalable, i.e., there is no lower bound on the memory per machine. Our dynamic algorithm maintains an $O(1)$-approximate $r$-gather solution under insertions/deletions of points in a metric space with doubling dimension $d$. The update time is $r \cdot 2^{O(d)}\cdot \log^{O(1)}\Delta$ and the query time is $2^{O(d)}\cdot \log^{O(1)}\Delta$, where $\Delta$ is the ratio between the largest and the smallest distance.