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Deterministic Edge Weights in Increasing Tree Families

Published online by Cambridge University Press:  22 June 2009

MARKUS KUBA  and
STEPHAN WAGNER
MARKUS KUBA
Affiliation:
Institut für Diskrete Mathematik und Geometrie, Technische Universität Wien, Wiedner Hauptstr. 8-10/104, 1040 Wien, Austria (e-mail: kuba@dmg.tuwien.ac.at)
STEPHAN WAGNER
Affiliation:
Department of Mathematical Sciences, Stellenbosch University, Private Bag X1, Matieland 7602, South Africa (e-mail: swagner@sun.ac.za)
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Abstract

In this work we study edge weights for two specific families of increasing trees, which include binary increasing trees and plane-oriented recursive trees as special instances, where plane-oriented recursive trees serve as a combinatorial model of scale-free random trees given by the m = 1 case of the Barabási–Albert model. An edge e = (k, l), connecting the nodes labelled k and l, respectively, in an increasing tree, is associated with the weight we = |k − l|. We are interested in the distribution of the number of edges with a fixed edge weight j in a random generalized plane-oriented recursive tree or random d-ary increasing tree. We provide exact formulas for expectation and variance and prove a normal limit law for this quantity. A combinatorial approach is also presented and applied to a related parameter, the maximum edge weight.

Type
Paper
Information
Combinatorics, Probability and Computing , Volume 19 , Issue 1 , January 2010 , pp. 99 - 119
Copyright
Copyright © Cambridge University Press 2009

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References

[1]Albert, R., Jeong, H. and Barabási, A.-L. (1999) The diameter of the world wide web. Nature 401 130–131.CrossRefGoogle Scholar
[2]Barabási, A.-L. and Albert, R. (1999) Emergence of scaling in random networks. Science 286 509–512.CrossRefGoogle ScholarPubMed
[3]Barbour, A. D. and Eagleson, G. K. (1985) Multiple comparisons and sums of dissociated random variables. Adv. Appl. Probab. 17 147–162.CrossRefGoogle Scholar
[4]Barbour, A. D., Karoński, M. and Ruciński, A. (1989) A central limit theorem for decomposable random variables with applications to random graphs. J. Combin. Theory Ser. B 47 125–145.CrossRefGoogle Scholar
[5]Bergeron, F., Flajolet, P. and Salvy, B. (1992) Varieties of Increasing Trees, Vol. 581 of Lecture Notes in Computer Science, Springer, pp. 24–48.Google Scholar
[6]Bergeron, F., Labelle, G. and Leroux, P. (1998) Combinatorial Species and Tree-Like Structures, Vol. 67 of Encyclopedia of Mathematics and its Applications. Cambridge University Press, Cambridge.Google Scholar
[7]Bollobás, B. and Riordan, O. (2003) Mathematical results on scale-free random graphs. In Handbook of Graphs and Networks (Bornholdt, S. and Schuster, H. G., eds), Wiley-VCH, Weinheim, pp. 1–34.Google Scholar
[8]Bollobás, B. and Riordan, O. (2004) Shortest paths and load scaling in scale-free trees. Phys. Rev. E 69 #036114.CrossRefGoogle ScholarPubMed
[9]Dobrow, R. and Smythe, R. (1996) Poisson approximations for functionals of random trees. Random Struct. Alg. 9 79–92.3.0.CO;2-8>CrossRefGoogle Scholar
[10]Hoare, C. A. R. (1962) Quicksort. Comput. J. 5 1015.CrossRefGoogle Scholar
[11]Hwang, H.-K. (1998) On convergence rates in the central limit theorems for combinatorial structures. Europ. J. Combin. 19 329–343.CrossRefGoogle Scholar
[12]Janson, S. (2005) Asymptotic degree distribution in random recursive trees. Random Struct. Alg. 26 69–83.CrossRefGoogle Scholar
[13]Kuba, M. and Panholzer, A. (2008) On edge-weighted recursive trees and inversions in random permutations. Discrete Math. 308 529–540.CrossRefGoogle Scholar
[14]Panholzer, A. and Prodinger, H. (2007) The level of nodes in increasing trees revisited. Random Struct. Alg. 31 203–226.Google Scholar
[15]Vitter, J. and Flajolet, P. (1990) Average Case Analysis of Algorithms and Data Structures. In Handbook of Theoretical Computer Science, Elsevier, Amsterdam, pp. 431–524.Google Scholar