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Affine and Projective Tree Metric Theorems

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Abstract

The tree metric theorem provides a combinatorial four-point condition that characterizes dissimilarity maps derived from pairwise compatible split systems. A related weaker four point condition characterizes dissimilarity maps derived from circular split systems known as Kalmanson metrics. The tree metric theorem was first discovered in the context of phylogenetics and forms the basis of many tree reconstruction algorithms, whereas Kalmanson metrics were first considered by computer scientists, and are notable in that they are a non-trivial class of metrics for which the traveling salesman problem is tractable. We present a unifying framework for these theorems based on combinatorial structures that are used for graph planarity testing. These are (projective) PC-trees, and their affine analogs, PQ-trees. In the projective case, we generalize a number of concepts from clustering theory, including hierarchies, pyramids, ultrametrics, and Robinsonian matrices, and the theorems that relate them. As with tree metrics and ultrametrics, the link between PC-trees and PQ-trees is established via the Gromov product.

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Authors and Affiliations

  1. Department of Mathematics, University of California, 970 Evans Hall #3840, Berkeley, CA, 94720-3840, USA

    Aaron Kleinman

  2. Department of Mathematics, Courant Institute of Mathematical Sciences, 251 Mercer Street, New York, NY, 10012, USA

    Matan Harel

  3. Departments of Mathematics and Molecular & Cell Biology, University of California, 142 LSA #3200, Berkeley, CA, 94720-3840, USA

    Lior Pachter

Authors
  1. Aaron Kleinman
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  2. Matan Harel
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  3. Lior Pachter
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Correspondence to Lior Pachter.

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Kleinman, A., Harel, M. & Pachter, L. Affine and Projective Tree Metric Theorems. Ann. Comb. 17, 205–228 (2013). https://doi.org/10.1007/s00026-012-0173-2

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  • DOI: https://doi.org/10.1007/s00026-012-0173-2

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