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An Algorithm for Computing Cutpoints in Finite Metric Spaces

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Abstract

The theory of the tight span, a cell complex that can be associated to every metric D, offers a unifying view on existing approaches for analyzing distance data, in particular for decomposing a metric D into a sum of simpler metrics as well as for representing it by certain specific edge-weighted graphs, often referred to as realizations of D. Many of these approaches involve the explicit or implicit computation of the so-called cutpoints of (the tight span of) D, such as the algorithm for computing the “building blocks” of optimal realizations of D recently presented by A. Hertz and S. Varone. The main result of this paper is an algorithm for computing the set of these cutpoints for a metric D on a finite set with n elements in O(n3) time. As a direct consequence, this improves the run time of the aforementioned O(n6)-algorithm by Hertz and Varone by “three orders of magnitude”.

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Author information

Authors and Affiliations

  1. CAS-MPG Partner Institute and Key Lab for Computational Biology (PICB), Shanghai, China

    Andreas Dress

  2. University of East Anglia, East Anglia, UK

    Katharina T. Huber & Vincent Moulton

  3. Pohang Mathematics Institute and POSTECH, Pohang-si, South Korea

    Jacobus Koolen

  4. University of Greifswald, Deutschland, Germany

    Andreas Spillner

Authors
  1. Andreas Dress
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  2. Katharina T. Huber
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  3. Jacobus Koolen
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  4. Vincent Moulton
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  5. Andreas Spillner
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Additional information

Authors Moulton and Spillner were supported by the Engineering and Physical Sciences Research Council [grant number EP/D068800/1]. A. Dress thanks the Chinese Academy of Sciences, the Max-Planck-Gesellschaft, and the German BMBF for their support, as well as the Warwick Institute for Advanced Study where, during two wonderful weeks, the basic outline of this paper was conceived. Huber and Koolen thank the Royal Society for their support in the context of a International Joint Project grant. Koolen was also partially supported by the Priority Research Centers Program through the National Research Foundation of Korea (NRF) funded by the Ministry of Education, Science and Technology (Grant #: 2009-0094069). We would also like to thank the anonymous referees for their helpful comments on earlier versions of this paper.

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Dress, A., Huber, K.T., Koolen, J. et al. An Algorithm for Computing Cutpoints in Finite Metric Spaces. J Classif 27, 158–172 (2010). https://doi.org/10.1007/s00357-010-9055-7

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  • DOI: https://doi.org/10.1007/s00357-010-9055-7

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