MapReduce and Streaming Algorithms for Diversity Maximization in Metric Spaces of Bounded Doubling Dimension

Given a dataset of points in a metric space and an integer k, a diversity maximization problem requires determining a subset of k points maximizing some diversity objective measure, e.g., the minimum or the average distance between a pair of points in the subset. Diversity maximization problems are computationally hard, hence only approximate solutions can be hoped for. Although its applications are mostly in massive data analysis, most of the past research on diversity maximization has concentrated on the standard sequential setting. Thus, there is a need for efficient algorithms in computational settings that can handle very large datasets, such as those at the base of the MapReduce and the Streaming models. In this work we provide algorithms for these models in the special case of metric spaces of bounded doubling dimension, which include the important family of Euclidean spaces of constant dimension. Our results show that despite the inherent space-constraints of the two models, for a variety of diversity objective functions, we can achieve efficient MapReduce or Streaming algorithms yielding an (α+ ε)-approximation ratio, for any constant ε > 0, where α the is best approximation ratio achieved by a standard polynomial-time, linear-space sequential algorithm for the same diversity criterion. As for other approaches in the literature, our algorithms revolve upon the determination of a high-quality core-set, that is, a (much) smaller subset of the input dataset which contains a good approximation to the optimal solution for the whole dataset.