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A note on tree realizations of matrices

Published online by Cambridge University Press:  11 October 2007

Alain Hertz  and
Sacha Varone
Alain Hertz
Affiliation:
Département de mathématiques et de génie industriel, École Polytechnique, Montréal, Canada; [email protected]
Sacha Varone
Affiliation:
Haute école de gestion de Genève, Économie d'Entreprise, Genève, Switzerland; [email protected]
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Abstract

It is well known that each tree metric M has a unique realization as a tree, and that this realization minimizes the total length of the edges among all other realizations of M. We extend this result to the class of symmetric matrices M with zero diagonal, positive entries, and such that mij + mkl ≤ max{mik + mjl, mil + mjk} for all distinct i,j,k,l.

Keywords

Matricestree metrics4-point condition
Type
Research Article
Information
RAIRO - Operations Research , Volume 41 , Issue 4 , October 2007 , pp. 361 - 366
Copyright
© EDP Sciences, ROADEF, SMAI, 2007

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References

Bandelt, H.-J., Recognition of tree metrics. SIAM J. Algebr. Discrete Methods 3 (1990) 1–6. CrossRef
J.-P. Barthélémy and A. Guénoche, Trees and proximity representations. John Wiley & Sons Ltd., Chichester (1991).
Buneman, P., A note on metric properties of trees. J. Combin. Theory Ser. B 17 (1974) 48–50. CrossRef
J.C. Culberson and P. Rudnicki, A fast algorithm for constructing trees from distance matrices. In Inf. Process. Lett. 30 (1989) 215–220.
Farach, M., Kannan, S. and Warnow, T., A robust model for finding optimal evolutionary trees. Algorithmica 13 (1995) 155–179. CrossRef
Floyd, R.W., Algorithm 97. Shortest path. Comm. ACM 5 (1962) 345. CrossRef
Hakimi, S.L. and Yau, S.S., Distance matrix of a graph and its realizability. Q. Appl. Math. 22 (1964) 305–317. CrossRef
Patrinos, A.N. and Hakimi, S.L., The distance matrix of a graph and its tree realization. Q. Appl. Math. 30 (1972) 255–269. CrossRef
Simões-Pereira, J.M.S., A note on the tree realizability of a distance matrix. J. Combin. Theory 6 (1969) 303–310. CrossRef
Varone, S.C., Trees related to realizations of distance matrices. Discrete Math. 192 (1998) 337–346. CrossRef