Fast approximate $\ell$-center clustering in high dimensional spaces

TL;DR

This paper introduces efficient approximation algorithms for high-dimensional $\,\ell$-center clustering problems using randomized dimension reduction, achieving faster runtimes and better approximation guarantees in Euclidean and Hamming spaces.

Contribution

The paper develops a general dimension reduction technique that improves the efficiency of approximation algorithms for high-dimensional $\,\ell$-center clustering, including variants with outliers.

Findings

Achieves $(2+\epsilon)$-approximation algorithms faster than existing methods.

Provides a speed-up for $O(1)$-approximation algorithms with outliers.

Reduces dependency of runtime on high dimension in clustering algorithms.

Abstract

We study the design of efficient approximation algorithms for the $\ell$-center clustering and minimum-diameter $\ell$-clustering problems in high dimensional Euclidean and Hamming spaces. Our main tool is randomized dimension reduction. First, we present a general method of reducing the dependency of the running time of a hypothetical algorithm for the $\ell$-center problem in a high dimensional Euclidean space on the dimension size. Utilizing in part this method, we provide $(2+\epsilon)$- approximation algorithms for the $\ell$-center clustering and minimum-diameter $\ell$-clustering problems in Euclidean and Hamming spaces that are substantially faster than the known $2$-approximation ones when both $\ell$ and the dimension are super-logarithmic. Next, we apply the general method to the recent fast approximation algorithms with higher approximation guarantees for the $\ell$-center clustering problem in a high dimensional Euclidean space. Finally, we provide a speed-up of the known $O(1)$-approximation method for the generalization of the $\ell$-center clustering problem to include $z$ outliers (i.e., $z$ input points can be ignored while computing the maximum distance of an input point to a center) in high dimensional Euclidean and Hamming spaces.