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Characterizing Cycle-Complete Dissimilarities in Terms of Associated Indexed 2-Hierarchies

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Book cover Combinatorial Optimization and Applications (COCOA 2018)

Part of the book series: Lecture Notes in Computer Science ((LNTCS,volume 11346))

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Abstract

2-ultrametrics are a generalization of the ultrametrics and it is known that there is a one-to-one correspondence between the set of 2-ultrametrics and the set of indexed 2-hierarchies (which are a generalization of indexed hierarchies). Cycle-complete dissimilarities, recently introduced by Trudeau, are a generalization of ultrametrics and form a subset of the 2-ultrametrics; therefore the set of cycle-complete dissimilarities corresponds to a subset of the indexed 2-hierarchies. In this study, we characterize this subset as the set of indexed acyclic 2-hierarchies, which in turn allows us to characterize the cycle-complete dissimilarities. In addition, we present an O\((n^2\log n)\) time algorithm that, given an arbitrary cycle-complete dissimilarities of order n, finds the corresponding indexed acyclic 2-hierarchy.

This work was supported by JSPS KAKENHI Grant Number 18K11180.

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Acknowledgments

The authors are grateful to the anonymous referees for useful comments which improved the presentation of the original version of this paper.

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Authors and Affiliations

  1. Faculty of Engineering, Shizuoka University, Hamamatsu, Shizuoka, 432-8561, Japan

    Kazutoshi Ando

  2. Graduate School of Integrated Science and Technology, Shizuoka University, Hamamatsu, Shizuoka, 432-8561, Japan

    Kazuya Shoji

Authors
  1. Kazutoshi Ando
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  2. Kazuya Shoji
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Corresponding author

Correspondence to Kazutoshi Ando .

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Editors and Affiliations

  1. Department of Computer Science, Kennesaw State University, Marietta, GA, USA

    Donghyun Kim

  2. Department of Mathematics and Computer Science, North Carolina Central University, Durham, NC, USA

    R. N. Uma

  3. Department of Computer Science, Georgia State University, Atlanta, GA, USA

    Alexander Zelikovsky

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Ando, K., Shoji, K. (2018). Characterizing Cycle-Complete Dissimilarities in Terms of Associated Indexed 2-Hierarchies. In: Kim, D., Uma, R., Zelikovsky, A. (eds) Combinatorial Optimization and Applications. COCOA 2018. Lecture Notes in Computer Science(), vol 11346. Springer, Cham. https://doi.org/10.1007/978-3-030-04651-4_43

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  • DOI: https://doi.org/10.1007/978-3-030-04651-4_43

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  • Publisher Name: Springer, Cham

  • Print ISBN: 978-3-030-04650-7

  • Online ISBN: 978-3-030-04651-4

  • eBook Packages: Computer ScienceComputer Science (R0)

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