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Online Unit Clustering in Higher Dimensions

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Approximation and Online Algorithms (WAOA 2017)

Part of the book series: Lecture Notes in Computer Science ((LNTCS,volume 10787))

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Abstract

We revisit the online Unit Clustering problem in higher dimensions: Given a set of n points in \(\mathbb {R}^d\), that arrive one by one, partition the points into clusters (subsets) of diameter at most one, so as to minimize the number of clusters used. In this paper, we work in \(\mathbb {R}^d\) using the \(L_\infty \) norm. We show that the competitive ratio of any algorithm (deterministic or randomized) for this problem must depend on the dimension d. This resolves an open problem raised by Epstein and van Stee (WAOA 2008). We also give a randomized online algorithm with competitive ratio \(O(d^2)\) for Unit Clustering of integer points (i.e., points in \(\mathbb {Z}^d\), \(d\in \mathbb {N}\), under \(L_{\infty }\) norm). We complement these results with some additional lower bounds for related problems in higher dimensions.

Research supported in part by the NSF awards CCF-1422311 and CCF-1423615.

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Author information

Authors and Affiliations

  1. University of Wisconsin–Milwaukee, Milwaukee, WI, USA

    Adrian Dumitrescu

  2. California State University, Northridge, Los Angeles, CA, USA

    Csaba D. Tóth

Authors
  1. Adrian Dumitrescu
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  2. Csaba D. Tóth
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Correspondence to Csaba D. Tóth .

Editor information

Editors and Affiliations

  1. The University of Western Ontario, London, Ontario, Canada

    Roberto Solis-Oba

  2. German University of Technology in Oman, Muscat, Oman

    Rudolf Fleischer

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Dumitrescu, A., Tóth, C.D. (2018). Online Unit Clustering in Higher Dimensions. In: Solis-Oba, R., Fleischer, R. (eds) Approximation and Online Algorithms. WAOA 2017. Lecture Notes in Computer Science(), vol 10787. Springer, Cham. https://doi.org/10.1007/978-3-319-89441-6_18

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  • DOI: https://doi.org/10.1007/978-3-319-89441-6_18

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  • Publisher Name: Springer, Cham

  • Print ISBN: 978-3-319-89440-9

  • Online ISBN: 978-3-319-89441-6

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