Abstract
This paper examines the distances between vertices in a rooted k-tree, for a fixed k, by exhibiting a correspondence with a variety of trees that can be specified in terms of combinatorial specifications. Studying these trees via generating functions, we show a Rayleigh limiting distribution for expected distances between pairs of vertices in a random k-tree: in a k-tree on n vertices, the proportion of vertices at distance \(d = x\sqrt{n}\) from a random vertex is asymptotic to \(\frac{c_k^2 x}{\sqrt{n}}\exp({-\frac{c_k^2 x^2}{2}})\), where c k = k H k .
Access provided by Autonomous University of Puebla. Download to read the full chapter text
Chapter PDF
Similar content being viewed by others
References
Arnborg, S., Proskurowski, A.: Linear time algorithms for NP-hard problems restricted to partial k-trees. Discrete Applied Mathematics 23(1), 11–24 (1989)
Beineke, L.W., Pippert, R.E.: The number of labeled k-dimensional trees. Journal of Combinatorial Theory 6(2), 200–205 (1969)
Bodini, O., Darrasse, A., Soria, M.: Distances in random apollonian network structures. In: FPSAC 2008. DMTCS Proceedings, pp. 307–318 (2008)
Drmota, M.: Random Trees: An Interplay between Combinatorics and Probability. Springer, Heidelberg (2009)
Flajolet, P., Sedgewick, R.: Analytic Combinatorics. Cambridge University Press, Cambridge (2009)
Fowler, T., Gessel, I., Labelle, G., Leroux, P.: The specification of 2-trees. Advances in Applied Mathematics 28(2), 145–168 (2002)
Ibarra, L.: The clique-separator graph for chordal graphs and subclasses of chordal graphs. In: Symposium on Discrete Mathematics, Nashville, TN (2004)
Klawe, M.M., Corneil, D.G., Proskurowski, A.: Isomorphism testing in hookup classes. SIAM Journal on Algebraic and Discrete Methods 3(2), 260–274 (1982)
Labelle, G., Lamathe, C., Leroux, P.: Labelled and unlabelled enumeration of k-gonal 2-trees. Journal of Combinatorial Theory, Series A 106(2), 193–219 (2004)
Markenzon, L., Justel, C.M., Paciornik, N.: Subclasses of k-trees: Characterization and recognition. Discrete Applied Mathematics 154(5), 818–825 (2006)
Meir, A., Moon, J.W.: On the altitude of nodes in random trees. Canadian Journal of Mathematics 30, 997–1015 (1978)
Moon, J.W.: The number of labeled k-trees. Journal of Combinatorial Theory 6(2), 196–199 (1969)
Proskurowski, A.: K-trees: representation and distances. In: Congressus Numerantium, vol. 29, pp. 785–794. Utilitas Mathematica (1980)
Proskurowski, A.: Recursive graphs, recursive labelings and shortest paths. SIAM Journal on Computing 10(2), 391–397 (1981)
Rose, D.J.: On simple characterizations of k-trees. Discrete Mathematics 7(3-4), 317–322 (1974)
Author information
Authors and Affiliations
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 2009 Springer-Verlag Berlin Heidelberg
About this paper
Cite this paper
Darrasse, A., Soria, M. (2009). Limiting Distribution for Distances in k-Trees. In: Fiala, J., Kratochvíl, J., Miller, M. (eds) Combinatorial Algorithms. IWOCA 2009. Lecture Notes in Computer Science, vol 5874. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-10217-2_19
Download citation
DOI: https://doi.org/10.1007/978-3-642-10217-2_19
Publisher Name: Springer, Berlin, Heidelberg
Print ISBN: 978-3-642-10216-5
Online ISBN: 978-3-642-10217-2
eBook Packages: Computer ScienceComputer Science (R0)