# Recognizing Treelike *k*-Dissimilarities

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## 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.

## Keywords

*k*-dissimilarity Phylogenetic tree Dissimilarity Metric 4-point condition Ultrametric condition Equidistant tree

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## References

- BANDELT, H.-J. (1990), “Recognition of Tree Metrics,”
*SIAM Journal on Discrete Mathematics, 3*, 1–6.MathSciNetMATHCrossRefGoogle Scholar - BANDELT, H.-J., and DRESS, A.W.M. (1994), “An Order-Theoretic Framework for Overlapping Clustering,”
*Discrete Mathematics, 136*, 21–37.MathSciNetMATHCrossRefGoogle Scholar - BOCCI, C., and COOLS, F. (2009), “A Tropical Interpretation of
*m*-Dissimilarity Maps,”*Applied Mathematics and Computation, 212*, 349–356.MathSciNetMATHCrossRefGoogle Scholar - BUNEMAN, P. (1971), “The Recovery of Trees fromMeasures of Dissimilarity,” in
*Mathematics in the Archaeological and Historical Sciences*, eds. D.G. Kendall and P. Tautu, Edinburgh: Edinburgh University Press, pp. 387–395.Google Scholar - CHEPOI, V., and FICHET, B. (2007), “A Note on Three-Way Dissimilarities and Their Relationship with Two-Way Dissimilarities,” in
*Selected Contributions in Data Analysis and Classification*, ed. P. Brito et al., Berlin: Springer, pp. 465–475.Google Scholar - CULBERSON, J., and RUDNICKI, P. (1989), “A Fast Algorithm for Constructing Trees from Distance Matrices,”
*Information Processing Letters, 30*, 215–220.MathSciNetMATHCrossRefGoogle Scholar - DE SOETE, G. (1983), “A Least Squares Algorithm for Fitting Additive Trees to Proximity Data,”
*Psychometrika, 48*, 621–626.CrossRefGoogle Scholar - DEZA, M.-M., and ROSENBERG, I.G. (2000), “
*n*-Semimetrics,”*European Journal of Combinatorics, 21*, 797–806.MathSciNetMATHCrossRefGoogle Scholar - DRESS, A.W.M., HUBER, K.T., KOOLEN, J., MOULTON, V., and SPILLNER, A. (2011),
*Basic Phylogenetic Combinatorics*, Cambridge: Cambridge University Press.CrossRefGoogle Scholar - DRESS, A.W.M., and STEEL,M. (2007), “Phylogenetic Diversity over an Abelian Group,”
*Annals of Combinatorics, 11*, 143–160.MathSciNetMATHCrossRefGoogle Scholar - FAITH, D.P. (1992), “Conservation Evaluation and Phylogenetic Diversity,”
*Biological Conservation, 61*, 1–10.CrossRefGoogle Scholar - FELSENSTEIN, J. (2003),
*Inferring Phylogenies*, Sunderland, Massachusetts: Sinauer Associates.Google Scholar - GORDON, A.D. (1987), “A Review of Hierarchical Classification,”
*Journal of the Royal Statistical Society. Series A. General, 150*, 119–137.MathSciNetMATHCrossRefGoogle Scholar - GRISHIN, N. (1999), “A Novel Approach to Phylogeny Reconstruction from Protein Sequences,”
*Journal of Molecular Evolution, 48*, 264–273.CrossRefGoogle Scholar - HAYASHI, C. (1972), “Two Dimensional Quatification Based on the Measure of Dissimilarity Among Three Elements,”
*Annals of the Institute of Statistical Mathematics, 24*, 251–257.MathSciNetMATHCrossRefGoogle Scholar - HEISER,W.J., and BENNANI, M. (1997), “Triadic Distance Models: Axiomatization and Least Squares Representation,”
*Journal of Mathematical Psychology, 41*, 189–206.MathSciNetMATHCrossRefGoogle Scholar - JOLY, S., and LE CALVÉ, G. (1995), “Three-Way Distances,”
*Journal of Classification, 12*, 191–205.MathSciNetMATHCrossRefGoogle Scholar - LEVY, D., YOSHIDA, R., and PACHTER, L. (2006), “Beyond Pairwise Distances: Neighbor-Joining with Phylogenetic Diversity Estimates,”
*Molecular Biology and Evolution, 23*, 491–498.CrossRefGoogle Scholar - PACHTER, L., and SPEYER, D. (2004), “Reconstructing Trees from Subtree Weights,”
*Applied Mathematics Letters, 17*, 615–621.MathSciNetMATHCrossRefGoogle Scholar - RUBEI, E. (2011), “Sets of Double and Triple Weights of Trees,”
*Annals of Combinatorics, 15*, 723–734.MathSciNetMATHCrossRefGoogle Scholar - SCHRIJVER, A. (1986),
*Theory of Linear and Integer Programming*, Wiley-Interscience Series in Discrete Mathematics, Chichester: John Wiley & Sons Ltd.Google Scholar - SEMPLE, C., and STEEL, M. (2003),
*Phylogenetics*(Vol. 24),*Oxford Lecture Series in Mathematics and Its Applications*, Oxford: Oxford University Press.Google Scholar - SMOLENSKII, Y.A. (1962), “A Method for the Linear Recording of Graphs,”
*U.S.S.R. Computational Mathematics and Mathematical Physics, 2*, 396–397.Google Scholar - STEEL, M. (2005), “Phylogenetic Diversity and the Greedy Algorithm,”
*Systematic Biology, 54*, 527–529.CrossRefGoogle Scholar - WARRENS, M.J. (2010), “
*n*-Way Metrics,”*Journal of Classification, 27*, 173–190.MathSciNetCrossRefGoogle Scholar - ZARETSKY, K. (1965), “Reconstruction of a Tree from the Distances Between Its Pendant Vertices,”
*Uspekhi Matematicheskikh Nauk (Russian Mathematical Surveys), 20*, 90–92.Google Scholar