Linear-Time Approximation Scheme for k-Means Clustering of Axis-Parallel Affine Subspaces

Authors Kyungjin Cho, Eunjin Oh

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Kyungjin Cho
  • POSTECH, Pohang, South Korea
Eunjin Oh
  • POSTECH, Poahng, South Korea

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Kyungjin Cho and Eunjin Oh. Linear-Time Approximation Scheme for k-Means Clustering of Axis-Parallel Affine Subspaces. In 32nd International Symposium on Algorithms and Computation (ISAAC 2021). Leibniz International Proceedings in Informatics (LIPIcs), Volume 212, pp. 46:1-46:13, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2021)


In this paper, we present a linear-time approximation scheme for k-means clustering of incomplete data points in d-dimensional Euclidean space. An incomplete data point with Δ > 0 unspecified entries is represented as an axis-parallel affine subspace of dimension Δ. The distance between two incomplete data points is defined as the Euclidean distance between two closest points in the axis-parallel affine subspaces corresponding to the data points. We present an algorithm for k-means clustering of axis-parallel affine subspaces of dimension Δ that yields an (1+ε)-approximate solution in O(nd) time. The constants hidden behind O(⋅) depend only on Δ, ε and k. This improves the O(n² d)-time algorithm by Eiben et al. [SODA'21] by a factor of n.

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ACM Subject Classification
  • Theory of computation → Computational geometry
  • k-means clustering
  • affine subspaces


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