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Clustering Under Perturbation Stability in Near-Linear Time

Authors Pankaj K. Agarwal, Hsien-Chih Chang, Kamesh Munagala, Erin Taylor, Emo Welzl



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Author Details

Pankaj K. Agarwal
  • Department of Computer Science, Duke University, Durham, NC, USA
Hsien-Chih Chang
  • Department of Computer Science, Dartmouth College, Hanover, NH, USA
Kamesh Munagala
  • Department of Computer Science, Duke University, Durham, NC, USA
Erin Taylor
  • Department of Computer Science, Duke University, Durham, NC, USA
Emo Welzl
  • Department of Computer Science, ETH Zürich, Switzerland

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Pankaj K. Agarwal, Hsien-Chih Chang, Kamesh Munagala, Erin Taylor, and Emo Welzl. Clustering Under Perturbation Stability in Near-Linear Time. In 40th IARCS Annual Conference on Foundations of Software Technology and Theoretical Computer Science (FSTTCS 2020). Leibniz International Proceedings in Informatics (LIPIcs), Volume 182, pp. 8:1-8:16, Schloss Dagstuhl - Leibniz-Zentrum für Informatik (2020)
https://doi.org/10.4230/LIPIcs.FSTTCS.2020.8

Abstract

We consider the problem of center-based clustering in low-dimensional Euclidean spaces under the perturbation stability assumption. An instance is α-stable if the underlying optimal clustering continues to remain optimal even when all pairwise distances are arbitrarily perturbed by a factor of at most α. Our main contribution is in presenting efficient exact algorithms for α-stable clustering instances whose running times depend near-linearly on the size of the data set when α ≥ 2 + √3. For k-center and k-means problems, our algorithms also achieve polynomial dependence on the number of clusters, k, when α ≥ 2 + √3 + ε for any constant ε > 0 in any fixed dimension. For k-median, our algorithms have polynomial dependence on k for α > 5 in any fixed dimension; and for α ≥ 2 + √3 in two dimensions. Our algorithms are simple, and only require applying techniques such as local search or dynamic programming to a suitably modified metric space, combined with careful choice of data structures.

Subject Classification

ACM Subject Classification
  • Theory of computation → Design and analysis of algorithms
Keywords
  • clustering
  • stability
  • local search
  • dynamic programming
  • coreset
  • polyhedral metric
  • trapezoid decomposition
  • range query

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