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Recognizing Treelike k-Dissimilarities

Author

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  • Sven Herrmann
  • Katharina Huber
  • Vincent Moulton
  • Andreas Spillner

Abstract

A k-dissimilarity D on a finite set X, |X| ≥ k, is a map from the set of size k subsets of X to the real numbers. Such maps naturally arise from edgeweighted trees T with leaf-set X: Given a subset Y of X of size k, D(Y ) is defined to be the total length of the smallest subtree of T with leaf-set Y . In case k = 2, it is well-known that 2-dissimilarities arising in this way can be characterized by the so-called “4-point condition”. However, in case k > 2 Pachter and Speyer ( 2004 ) recently posed the following question: Given an arbitrary k-dissimilarity, how do we test whether this map comes from a tree? In this paper, we provide an answer to this question, showing that for k ≥ 3 a k-dissimilarity on a set X arises from a tree if and only if its restriction to every 2 k-element subset of X arises from some tree, and that 2 k is the least possible subset size to ensure that this is the case. As a corollary, we show that there exists a polynomial-time algorithm to determine when a k-dissimilarity arises from a tree. We also give a 6-point condition for determining when a 3-dissimilarity arises from a tree, that is similar to the aforementioned 4-point condition. Copyright Springer Science+Business Media, LLC 2012

Suggested Citation

  • Sven Herrmann & Katharina Huber & Vincent Moulton & Andreas Spillner, 2012. "Recognizing Treelike k-Dissimilarities," Journal of Classification, Springer;The Classification Society, vol. 29(3), pages 321-340, October.
    • Handle: RePEc:spr:jclass:v:29:y:2012:i:3:p:321-340
    DOI: 10.1007/s00357-012-9115-2
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    References listed on IDEAS

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    1. Geert Soete, 1983. "A least squares algorithm for fitting additive trees to proximity data," Psychometrika, Springer;The Psychometric Society, vol. 48(4), pages 621-626, December.
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    Cited by:

    1. Douglas L. Steinley, 2019. "Editorial: Journal of Classification Vol. 36-3," Journal of Classification, Springer;The Classification Society, vol. 36(3), pages 393-396, October.
    2. Katharina T. Huber & Vincent Moulton & Guillaume E. Scholz, 2019. "Three-Way Symbolic Tree-Maps and Ultrametrics," Journal of Classification, Springer;The Classification Society, vol. 36(3), pages 513-540, October.
    3. Elena Rubei, 2013. "On the weights of simple paths in weighted complete graphs," Indian Journal of Pure and Applied Mathematics, Springer, vol. 44(4), pages 511-525, August.
    4. Elena Rubei, 2016. "Weighted Graphs with Distances in Given Ranges," Journal of Classification, Springer;The Classification Society, vol. 33(2), pages 282-297, July.
    5. Agnese Baldisserri & Elena Rubei, 2018. "Treelike Families of Multiweights," Journal of Classification, Springer;The Classification Society, vol. 35(2), pages 367-390, July.
    6. Simone Calamai, 2020. "Moduli Space of Families of Positive (n − 1)-Weights," Journal of Classification, Springer;The Classification Society, vol. 37(2), pages 317-327, July.

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