Center-based clustering is a fundamental primitive for data analysis and becomes very challenging for large datasets. In this paper, we focus on the popular k -center variant which, given a set S of points from some metric space and a parameter k < | S |, requires to identify a subset of k centers in S minimizing the maximum distance of any point of S from its closest center. A more general formulation, introduced to deal with noisy datasets, features a further parameter z and allows up to z points of S (outliers) to be disregarded when computing the maximum distance from the centers. We present coreset-based 2-round MapReduce algorithms for the above two formulations of the problem, and a 1-pass Streaming algorithm for the case with outliers. For any fixed &#1013; > 0, the algorithms yield solutions whose approximation ratios are a mere additive term &#1013; away from those achievable by the best known polynomial-time sequential algorithms, a result that substantially improves upon the state of the art. Our algorithms are rather simple and adapt to the intrinsic complexity of the dataset, captured by the doubling dimension D of the metric space. Specifically, our analysis shows that the algorithms become very space-efficient for the important case of small (constant) D . These theoretical results are complemented with a set of experiments on real-world and synthetic datasets of up to over a billion points, which show that our algorithms yield better quality solutions over the state of the art while featuring excellent scalability, and that they also lend themselves to sequential implementations much faster than existing ones.

An uncertain graph 𝒢 = (V, E, p : E → (0, 1]) can be viewed as a probability space whose outcomes (referred to as possible worlds ) are subgraphs of 𝒢 where any edge e ε E occurs with probability p ( e ), independently of the other edges. These graphs naturally arise in many application domains where data management systems are required to cope with uncertainty in interrelated data, such as computational biology, social network analysis, network reliability, and privacy enforcement, among the others. For this reason, it is important to devise fundamental querying and mining primitives for uncertain graphs. This paper contributes to this endeavor with the development of novel strategies for clustering uncertain graphs. Specifically, given an uncertain graph 𝒢 and an integer k , we aim at partitioning its nodes into k clusters, each featuring a distinguished center node, so to maximize the minimum/average connection probability of any node to its cluster's center, in a random possible world. We assess the NP-hardness of maximizing the minimum connection probability, even in the presence of an oracle for the connection probabilities, and develop efficient approximation algorithms for both problems and some useful variants. Unlike previous works in the literature, our algorithms feature provable approximation guarantees and are capable to keep the granularity of the returned clustering under control. Our theoretical findings are complemented with several experiments that compare our algorithms against some relevant competitors, with respect to both running-time and quality of the returned clusterings.

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 two points in the subset. Diversity maximization is computationally hard, hence only approximate solutions can be hoped for. Although its applications are mainly in massive data analysis, most of the past research on diversity maximization focused on the sequential setting. In this work we present space and pass/round-efficient diversity maximization algorithms for the Streaming and MapReduce models and analyze their approximation guarantees for the relevant class of metric spaces of bounded doubling dimension. Like other approaches in the literature, our algorithms rely on the determination of high-quality core-sets, i.e., (much) smaller subsets of the input which contain good approximations to the optimal solution for the whole input. For a variety of diversity objective functions, our algorithms attain an ( α + ε )-approximation ratio, for any constant ε > 0, where α is the best approximation ratio achieved by a polynomial-time, linear-space sequential algorithm for the same diversity objective. This improves substantially over the approximation ratios attainable in Streaming and MapReduce by state-of-the-art algorithms for general metric spaces. We provide extensive experimental evidence of the effectiveness of our algorithms on both real world and synthetic datasets, scaling up to over a billion points.

We develop a novel parallel decomposition strategy for unweighted, undirected graphs, based on growing disjoint connected clusters from batches of centers progressively selected from yet uncovered nodes. With respect to similar previous decompositions, our strategy exercises a tighter control on both the number of clusters and their maximum radius. We present two important applications of our parallel graph decomposition: (1) $k$-center clustering approximation; and (2) diameter approximation. In both cases, we obtain algorithms which feature a polylogarithmic approximation factor and are amenable to a distributed implementation that is geared for massive (long-diameter) graphs. The total space needed for the computation is linear in the problem size, and the parallel depth is substantially sublinear in the diameter for graphs with low doubling dimension. To the best of our knowledge, ours are the first parallel approximations for these problems which achieve sub-diameter parallel time, for a relevant class of graphs, using only linear space. Besides the theoretical guarantees, our algorithms allow for a very simple implementation on clustered architectures: we report on extensive experiments which demonstrate their effectiveness and efficiency on large graphs as compared to alternative known approaches.