A note on tree realizations of matrices

Alain Hertz; Sacha Varone

RAIRO - Operations Research (2007)

  • Volume: 41, Issue: 4, page 361-366
  • ISSN: 0399-0559

Abstract

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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.

How to cite

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Hertz, Alain, and Varone, Sacha. "A note on tree realizations of matrices." RAIRO - Operations Research 41.4 (2007): 361-366. <http://eudml.org/doc/250136>.

@article{Hertz2007,
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. },
author = {Hertz, Alain, Varone, Sacha},
journal = {RAIRO - Operations Research},
keywords = {Matrices; tree metrics; 4-point condition; matrices},
language = {eng},
month = {10},
number = {4},
pages = {361-366},
publisher = {EDP Sciences},
title = {A note on tree realizations of matrices},
url = {http://eudml.org/doc/250136},
volume = {41},
year = {2007},
}

TY - JOUR
AU - Hertz, Alain
AU - Varone, Sacha
TI - A note on tree realizations of matrices
JO - RAIRO - Operations Research
DA - 2007/10//
PB - EDP Sciences
VL - 41
IS - 4
SP - 361
EP - 366
AB - 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.
LA - eng
KW - Matrices; tree metrics; 4-point condition; matrices
UR - http://eudml.org/doc/250136
ER -

References

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  1. H.-J. Bandelt, Recognition of tree metrics. SIAM J. Algebr. Discrete Methods3 (1990) 1–6.  Zbl0687.05017
  2. J.-P. Barthélémy and A. Guénoche, Trees and proximity representations. John Wiley & Sons Ltd., Chichester (1991).  Zbl0790.05021
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  4. J.C. Culberson and P. Rudnicki, A fast algorithm for constructing trees from distance matrices. In Inf. Process. Lett.30 (1989) 215–220.  Zbl0665.68052
  5. M. Farach, S. Kannan and T. Warnow, A robust model for finding optimal evolutionary trees. Algorithmica13 (1995) 155–179.  Zbl0831.92019
  6. R.W. Floyd, Algorithm 97. Shortest path. Comm. ACM5 (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.  Zbl0125.11804
  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.  Zbl0293.05103
  9. J.M.S. Simões-Pereira, A note on the tree realizability of a distance matrix. J. Combin. Theory6 (1969) 303–310.  Zbl0177.26903
  10. S.C. Varone, Trees related to realizations of distance matrices. Discrete Math.192 (1998) 337–346.  Zbl0955.05070

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