Recognizing Treelike k-Dissimilarities
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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.
Keywordsk-dissimilarity Phylogenetic tree Dissimilarity Metric 4-point condition Ultrametric condition Equidistant tree
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- 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
- FELSENSTEIN, J. (2003), Inferring Phylogenies, Sunderland, Massachusetts: Sinauer Associates.Google 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
- 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