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Accurate MapReduce Algorithms for k-Median and k-Means in General Metric Spaces

Authors Alessio Mazzetto, Andrea Pietracaprina, Geppino Pucci

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Author Details

Alessio Mazzetto
  • Department of Computer Science, Brown University, Providence, USA
Andrea Pietracaprina
  • Department of Information Engineering, University of Padova, Padova, Italy
Geppino Pucci
  • Department of Information Engineering, University of Padova, Padova, Italy

Funding

This work was supported, in part, by the University of Padova under grant SID2017 and by MIUR, the Italian Ministry of Education, University and Research, under grant PRIN AHeAD: efficient Algorithms for HArnessing networked Data and grant L. 232 "Dipartimenti di Eccellenza’’.

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Alessio Mazzetto, Andrea Pietracaprina, and Geppino Pucci. Accurate MapReduce Algorithms for k-Median and k-Means in General Metric Spaces. In 30th International Symposium on Algorithms and Computation (ISAAC 2019). Leibniz International Proceedings in Informatics (LIPIcs), Volume 149, pp. 34:1-34:16, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2019) https://doi.org/10.4230/LIPIcs.ISAAC.2019.34

Abstract

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-median and k-means variants which, given a set P of points from a metric space and a parameter k<|P|, require to identify a set S of k centers minimizing, respectively, the sum of the distances and of the squared distances of all points in P from their closest centers. Our specific focus is on general metric spaces, for which it is reasonable to require that the centers belong to the input set (i.e., S subseteq P). We present coreset-based 3-round distributed approximation algorithms for the above problems using the MapReduce computational model. The algorithms are rather simple and obliviously adapt to the intrinsic complexity of the dataset, captured by the doubling dimension D of the metric space. Remarkably, the algorithms attain approximation ratios that can be made arbitrarily close to those achievable by the best known polynomial-time sequential approximations, and they are very space efficient for small D, requiring local memory sizes substantially sublinear in the input size. To the best of our knowledge, no previous distributed approaches were able to attain similar quality-performance guarantees in general metric spaces.

Subject Classification

ACM Subject Classification
  • Theory of computation → Approximation algorithms analysis
  • Theory of computation → Facility location and clustering
  • Theory of computation → MapReduce algorithms
Keywords
  • Clustering
  • k-median
  • k-means
  • MapReduce
  • Coreset

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