Hostname: page-component-78c5997874-s2hrs Total loading time: 0 Render date: 2024-11-10T17:45:05.779Z Has data issue: false hasContentIssue false
  • English
  • Français

Combinatorial Analysis of Growth Models for Series-Parallel Networks

Part of: Combinatorics Graph theory

Published online by Cambridge University Press:  14 August 2018

MARKUS KUBA  and
ALOIS PANHOLZER
MARKUS KUBA
Affiliation:
Institute of Applied Mathematics and Natural Sciences, University of Applied Sciences – Technikum Wien, Höchstädtplatz 5, 1200 Wien, Austria (e-mail: kuba@technikum-wien.at)
ALOIS PANHOLZER
Affiliation:
Institut für Diskrete Mathematik und Geometrie, Technische Universität Wien, Wiedner Hauptstraße 8-10/104, 1040 Wien, Austria (e-mail: Alois.Panholzer@tuwien.ac.at)
Get access Rights & Permissions [Opens in a new window]

Abstract

We give combinatorial descriptions of two stochastic growth models for series-parallel networks introduced by Hosam Mahmoud by encoding the growth process via recursive tree structures. Using decompositions of the tree structures and applying analytic combinatorics methods allows a study of quantities in the corresponding series-parallel networks. For both models we obtain limiting distribution results for the degree of the poles and the length of a random source-to-sink path, and furthermore we get asymptotic results for the expected number of source-to-sink paths. Moreover, we introduce generalizations of these stochastic models by encoding the growth process of the networks via further important increasing tree structures.

MSC classification

Primary: 05C80: Random graphs
Secondary: 05A15: Exact enumeration problems, generating functions 05A16: Asymptotic enumeration
Type
Paper
Information
Combinatorics, Probability and Computing , Volume 28 , Special Issue 4: Analysis of Algorithms , July 2019 , pp. 574 - 599
Copyright
Copyright © Cambridge University Press 2018 

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

Footnotes

†

The second author was supported by the Austrian Science Foundation FWF, grant P25337-N23.

References

Bodini, O., Dien, M., Fontaine, X., Genitrini, A. and Hwang, H.-K. (2016) Increasing diamonds. In LATIN 2016: Theoretical Informatics (Kranakis, E. et al., eds), Vol. 9644 of Lecture Notes in Computer Science, Springer, pp. 207–219.CrossRefGoogle Scholar
Brandstädt, A., Le, V. B. and Spinrad, J. (1999) Graph Classes: A Survey, Vol. 3 of SIAM Monographs on Discrete Mathematics and Applications, SIAM.CrossRefGoogle Scholar
Chern, H.-H., Fernández-Camacho, M.-I., Hwang, H.-K. and Martínez, C. (2014) Psi-series method for equality of random trees and quadratic convolution recurrences. Random Struct. Alg. 44 67–108.CrossRefGoogle Scholar
Dobrow, R. and Fill, J. (1999) Total path length for random recursive trees. Combin. Probab. Comput. 8 31–333.CrossRefGoogle Scholar
Drmota, M., Fuchs, M. and Lee, Y.-W. (2016) Stochastic analysis of the extra clustering model for animal grouping. J. Math. Biol. 73 123–159.CrossRefGoogle ScholarPubMed
Drmota, M., Giménez, O. and Noy, M. (2010) Vertices of given degree in series-parallel graphs. Random Struct. Alg. 36 273–314.CrossRefGoogle Scholar
Flajolet, P. and Sedgewick, R. (2009) Analytic Combinatorics, Cambridge University Press.CrossRefGoogle Scholar
van der Hofstad, R., Hooghiemstra, G. and Van Mieghem, P. (2002) On the covariance of the level sizes in random recursive trees. Random Struct. Alg. 20 519–539.CrossRefGoogle Scholar
Janson, S. (2010) Moments of Gamma type and the Brownian supremum process area. Probab. Surv. 7 1–52.CrossRefGoogle Scholar
Kuba, M. and Panholzer, A. (2010) A combinatorial approach to the analysis of bucket recursive trees. Theoret. Comput. Sci. 411 3255–3273.CrossRefGoogle Scholar
Loève, M. (1977) Probability Theory I, fourth edition, Graduate Texts in Mathematics, Springer.Google Scholar
Mahmoud, H. (2009) Pólya Urn Models, Texts in Statistical Science series, CRC Press.Google Scholar
Mahmoud, H. (2013) Some node degree properties of series-parallel graphs evolving under a stochastic growth model. Probab. Engng Inform. Sci. 27 297–307.CrossRefGoogle Scholar
Mahmoud, H. (2014) Some properties of binary series-parallel graphs. Probab. Engng Inform. Sci. 28 565–572.CrossRefGoogle Scholar
Mahmoud, H. and Smythe, R. (1995) Probabilistic analysis of bucket recursive trees. Theoret. Comput. Sci. 144 221–249.CrossRefGoogle Scholar
Neininger, R. and Rüschendorf, L. (2004) A general limit theorem for recursive algorithms and combinatorial structures, Ann. Appl. Probab. 14 378–418.Google Scholar
Neininger, R. and Sulzbach, H. (2015) On a functional contraction method. Ann. Probab. 43 1777–1822.CrossRefGoogle Scholar
Panholzer, A. (2003) Analysis of multiple quickselect variants. Theoret. Comput. Sci. 302 45–91.CrossRefGoogle Scholar
Panholzer, A. (2004) The distribution of the size of the ancestor-tree and of the induced spanning subtree for random trees. Random Struct. Alg. 25 179–207.CrossRefGoogle Scholar
Panholzer, A. and Prodinger, H. (2007) Level of nodes in increasing trees revisited. Random Struct. Alg. 31 203–226.CrossRefGoogle Scholar