A note on tree realizations of matrices

Hertz, Alain; Varone, Sacha

doi:10.1051/ro:2007028

Hertz, Alain ; Varone, Sacha

RAIRO - Operations Research - Recherche Opérationnelle, Tome 41 (2007) no. 4, pp. 361-366.

Résumé

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 $m_{i j} + m_{k l} \leq m a x {m_{i k} + m_{j l}, m_{i l} + m_{j k}}$ for all distinct $i, j, k, l$ .

DOI : 10.1051/ro:2007028

Classification : 05C50, 05B20, 68R10, 68U99
Mots clés : matrices, tree metrics, 4-point condition

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     title = {A note on tree realizations of matrices},
     journal = {RAIRO - Operations Research - Recherche Op\'erationnelle},
     pages = {361--366},
     publisher = {EDP-Sciences},
     volume = {41},
     number = {4},
     year = {2007},
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     mrnumber = {2361290},
     language = {en},
     url = {http://www.numdam.org/articles/10.1051/ro:2007028/}
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Hertz, Alain; Varone, Sacha. A note on tree realizations of matrices. RAIRO - Operations Research - Recherche Opérationnelle, Tome 41 (2007) no. 4, pp. 361-366. doi : 10.1051/ro:2007028. http://www.numdam.org/articles/10.1051/ro:2007028/

Bibliographie
Cité par

[1] H.-J. Bandelt, Recognition of tree metrics. SIAM J. Algebr. Discrete Methods 3 (1990) 1-6. | Zbl

[2] J.-P. Barthélémy and A. Guénoche, Trees and proximity representations. John Wiley & Sons Ltd., Chichester (1991). | MR | Zbl

[3] P. Buneman, A note on metric properties of trees. J. Combin. Theory Ser. B 17 (1974) 48-50. | Zbl

[4] J.C. Culberson and P. Rudnicki, A fast algorithm for constructing trees from distance matrices. In Inf. Process. Lett. 30 (1989) 215-220. | Zbl

[5] M. Farach, S. Kannan and T. Warnow, A robust model for finding optimal evolutionary trees. Algorithmica 13 (1995) 155-179. | Zbl

[6] R.W. Floyd, Algorithm 97. Shortest path. Comm. ACM 5 (1962) 345.

[7] S.L. Hakimi and S.S. Yau, Distance matrix of a graph and its realizability. Q. Appl. Math. 22 (1964) 305-317. | Zbl

[8] A.N. Patrinos and S.L. Hakimi, The distance matrix of a graph and its tree realization. Q. Appl. Math. 30 (1972) 255-269. | Zbl

[9] J.M.S. Simões-Pereira, A note on the tree realizability of a distance matrix. J. Combin. Theory 6 (1969) 303-310. | Zbl

[10] S.C. Varone, Trees related to realizations of distance matrices. Discrete Math. 192 (1998) 337-346. | Zbl

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