Hostname: page-component-78c5997874-94fs2 Total loading time: 0 Render date: 2024-11-05T04:15:41.487Z Has data issue: false hasContentIssue false
  • English
  • Français

On Tail Bounds for Random Recursive Trees

Part of: Distribution theory - Probability Combinatorial probability Graph theory

Published online by Cambridge University Press:  04 February 2016

Götz Olaf Munsonius
Götz Olaf Munsonius*
Affiliation:
Goethe University Frankfurt
*
∗ Postal address: Institute of Mathematics, Goethe University Frankfurt, 60054 Frankfurt am Main, Germany. Email address: [email protected]
Rights & Permissions [Opens in a new window]

Abstract

Core share and HTML view are not available for this content. However, as you have access to this content, a full PDF is available via the ‘Save PDF’ action button.

We consider a multivariate distributional recursion of sum type, as arises in the probabilistic analysis of algorithms and random trees. We prove an upper tail bound for the solution using Chernoff's bounding technique by estimating the Laplace transform. The problem is traced back to the corresponding problem for binary search trees by stochastic domination. The result obtained is applied to the internal path length and Wiener index of random b-ary recursive trees with weighted edges and random linear recursive trees. Finally, lower tail bounds for the Wiener index of these trees are given.

Keywords

Random treeprobabilistic analysis of algorithmstail boundpath lengthWiener index

MSC classification

Primary: 60C05: Combinatorial probability 05C05: Trees 05C80: Random graphs 60E15: Inequalities; stochastic orderings
Type
Research Article
Information
Journal of Applied Probability , Volume 49 , Issue 2 , June 2012 , pp. 566 - 581
Copyright
© Applied Probability Trust 

References

Ali Khan, T. and Neininger, R. (2004). Probabilistic analysis for randomized game tree evaluation. In Mathematics and Computer Science. III, Birkhäuser, Basel, pp. 163–174.Google Scholar
Ali Khan, T. and Neininger, R. (2007). Tail bounds for the Wiener index of random trees. In 2007 Conf. Analysis of Algorithms (AofA '07 Discrete Math. Theoret. Comput. Sci. Proc. AH), Assoc. Discrete Math. Theoret. Comput. Sci., Nancy, pp. 279–289.Google Scholar
Bergeron, F., Flajolet, P. and Salvy, B. (1992). Varieties of increasing trees. In CAAP '92 (Rennes, 1992; Lecture Notes Comput. Sci. 581), ed. Raoult, J.-C., Springer, Berlin, pp. 24–48.Google Scholar
Broutin, N. and Devroye, L. (2006). Large deviations for the weighted height of an extended class of trees. Algorithmica 46, 271–297.Google Scholar
Broutin, N., Devroye, L., McLeish, E. and de la Salle, M. (2008). The height of increasing trees. Random Structures Algorithms 32, 494–518.Google Scholar
Fill, J. A. and Janson, S. (2001). Approximating the limiting Quicksort distribution. Random Structures Algorithms 19, 376–406.Google Scholar
Fill, J. A. and Janson, S. (2009). Precise logarithmic asymptotics for the right tails of some limit random variables for random trees. Ann. Combinatorics 12, 403–416.CrossRefGoogle Scholar
Janson, S. and Chassaing, P. (2004). The center of mass of the ISE and the Wiener index of trees. Electron. Commun. Prob. 9, 178–187.Google Scholar
Knessl, C. and Szpankowski, W. (1999). Quicksort algorithm again revisited. Discrete Math. Theoret. Comput. Sci. 3, 43–64.Google Scholar
Lindvall, T. (1992). Lectures on the Coupling Method. John Wiley, New York.Google Scholar
McDiarmid, C. J. H. and Hayward, R. B. (1996). Large deviations for Quicksort. J. Algorithms 21, 476–507.Google Scholar
Munsonius, G. O. (2010). Limit theorems for functionals of recursive trees. , University of Freiburg, Germany. Available at http://www.freidok.uni-freiburg.de/volltexte/7472/.Google Scholar
Munsonius, G. O. (2010). The total Steiner k-distance for b-ary recursive trees and linear recursive trees. In 21st Internat. Conf. Probabilistic, Combinatorial, and Asymptotic Methods in the Analysis of Algorithms (AofA '10 Discrete Math. Theoret. Comput. Sci. Proc. AH), Assoc. Discrete Math. Theoret. Comput. Sci., Nancy, pp. 527–548.Google Scholar
Pittel, B. (1994). Note on the heights of random recursive trees and random m-ary search trees. Random Structures Algorithms 5, 337–347.Google Scholar
Rösler, U. (1991). A limit theorem for “Quicksort”. RAIRO Inf. Théor. Appl. 25, 85–100.CrossRefGoogle Scholar
Rüschendorf, L. and Schopp, E.-M. (2007). Exponential bounds and tails for additive random recursive sequences. Discrete Math. Theoret. Comput. Sci. 9, 333–352.Google Scholar