Skip to main content Accessibility help
×
Home

On Tail Bounds for Random Recursive Trees

  • Götz Olaf Munsonius (a1)

Abstract

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.

    • Send article to Kindle

      To send this article to your Kindle, first ensure no-reply@cambridge.org is added to your Approved Personal Document E-mail List under your Personal Document Settings on the Manage Your Content and Devices page of your Amazon account. Then enter the ‘name’ part of your Kindle email address below. Find out more about sending to your Kindle. Find out more about sending to your Kindle.

      Note you can select to send to either the @free.kindle.com or @kindle.com variations. ‘@free.kindle.com’ emails are free but can only be sent to your device when it is connected to wi-fi. ‘@kindle.com’ emails can be delivered even when you are not connected to wi-fi, but note that service fees apply.

      Find out more about the Kindle Personal Document Service.

      On Tail Bounds for Random Recursive Trees
      Available formats
      ×

      Send article to Dropbox

      To send this article to your Dropbox account, please select one or more formats and confirm that you agree to abide by our usage policies. If this is the first time you use this feature, you will be asked to authorise Cambridge Core to connect with your <service> account. Find out more about sending content to Dropbox.

      On Tail Bounds for Random Recursive Trees
      Available formats
      ×

      Send article to Google Drive

      To send this article to your Google Drive account, please select one or more formats and confirm that you agree to abide by our usage policies. If this is the first time you use this feature, you will be asked to authorise Cambridge Core to connect with your <service> account. Find out more about sending content to Google Drive.

      On Tail Bounds for Random Recursive Trees
      Available formats
      ×

Copyright

Corresponding author

Postal address: Institute of Mathematics, Goethe University Frankfurt, 60054 Frankfurt am Main, Germany. Email address: munsonius@math.uni-frankfurt.de

References

Hide All
[1] Ali Khan, T. and Neininger, R. (2004). Probabilistic analysis for randomized game tree evaluation. In Mathematics and Computer Science. III, Birkhäuser, Basel, pp. 163174.
[2] 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. 279289.
[3] 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. 2448.
[4] Broutin, N. and Devroye, L. (2006). Large deviations for the weighted height of an extended class of trees. Algorithmica 46, 271297.
[5] Broutin, N., Devroye, L., McLeish, E. and de la Salle, M. (2008). The height of increasing trees. Random Structures Algorithms 32, 494518.
[6] Fill, J. A. and Janson, S. (2001). Approximating the limiting Quicksort distribution. Random Structures Algorithms 19, 376406.
[7] 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, 403416.
[8] Janson, S. and Chassaing, P. (2004). The center of mass of the ISE and the Wiener index of trees. Electron. Commun. Prob. 9, 178187.
[9] Knessl, C. and Szpankowski, W. (1999). Quicksort algorithm again revisited. Discrete Math. Theoret. Comput. Sci. 3, 4364.
[10] Lindvall, T. (1992). Lectures on the Coupling Method. John Wiley, New York.
[11] McDiarmid, C. J. H. and Hayward, R. B. (1996). Large deviations for Quicksort. J. Algorithms 21, 476507.
[12] 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/.
[13] 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. 527548.
[14] Pittel, B. (1994). Note on the heights of random recursive trees and random m-ary search trees. Random Structures Algorithms 5, 337347.
[15] Rösler, U. (1991). A limit theorem for “Quicksort”. RAIRO Inf. Théor. Appl. 25, 85100.
[16] Rüschendorf, L. and Schopp, E.-M. (2007). Exponential bounds and tails for additive random recursive sequences. Discrete Math. Theoret. Comput. Sci. 9, 333352.

Keywords

MSC classification

Related content

Powered by UNSILO

On Tail Bounds for Random Recursive Trees

  • Götz Olaf Munsonius (a1)

Metrics

Full text views

Total number of HTML views: 0
Total number of PDF views: 0 *
Loading metrics...

Abstract views

Total abstract views: 0 *
Loading metrics...

* Views captured on Cambridge Core between <date>. This data will be updated every 24 hours.

Usage data cannot currently be displayed.