MapReduce algorithms for robust center-based clustering in doubling metrics

Abstract

Clustering is a pivotal primitive for unsupervised learning and data analysis. A popular variant is the $(k,\ell )$-clustering problem, where, given a pointset P from a metric space, one must determine a subset S of k centers minimizing the sum of the ℓ-th powers of the distances of points in P from their closest centers. This formulation covers the well-studied k-median ($\ell =1$) and k-means ($\ell =2$) clustering problems. A more general variant, introduced to deal with noisy pointsets, features a further parameter z and allows up to z points of P (outliers) to be disregarded when computing the sum. We present a distributed coreset-based 3-round approximation algorithm for the $(k,\ell )$-clustering problem with z outliers, using MapReduce as a computational model. An important feature of our algorithm is that it obliviously adapts to the intrinsic complexity of the dataset, captured by its doubling dimension D. Remarkably, for $D=O\left(1\right)$, our algorithm requires sublinear local memory per reducer, and yields a solution whose approximation ratio is an additive term $O\left(\gamma \right)$ away from the one achievable by the best known sequential (possibly bicriteria) algorithm, where γ can be made arbitrarily small. To the best of our knowledge, no previous distributed approaches were able to attain similar quality-performance tradeoffs for metrics with constant doubling dimension.