Hostname: page-component-857557d7f7-ktsnh Total loading time: 0 Render date: 2025-12-10T18:11:43.759Z Has data issue: false hasContentIssue false
  • English
  • Français

Central limit theorems for additive functionals of patricia tries

Part of: Theory of data Algorithms - Computer Science Limit theorems

Published online by Cambridge University Press:  09 December 2025

Jasper Ischebeck Open the ORCID record for Jasper Ischebeck [Opens in a new window]
Jasper Ischebeck*
Affiliation:
Goethe University Frankfurt
*
*Postal address: Robert-Mayer-Straße 10, 60325 Frankfurt am Main, Germany. Email address: ischebec@math.unifrankfurt.de
Get access Rights & Permissions [Opens in a new window]

Abstract

General additive functionals of patricia tries are studied asymptotically in a probabilistic model with independent, identically distributed letters from a finite alphabet. Asymptotic normality is shown after normalization together with asymptotic expansions of the moments. There are two regimes depending on the algebraic structure of the letter probabilities, with and without oscillations in the expansion of moments. As applications firstly the proportion of fringe trees of patricia tries with k keys is studied, which is oscillating around $(1-\rho(k))/(2H)k(k-1)$, where H denotes the source entropy and $\rho(k)$ is exponentially decreasing. The oscillations are identified explicitly. Secondly, the independence number of patricia tries and of tries is considered. The general results for additive functions also apply, where a leading constant is numerically approximated. The results extend work of Janson on tries by relating additive functionals on patricia tries to additive functionals on tries.

Keywords

Fringe treeadditive functionalpatricia trielimit theoremindependence number

MSC classification

Primary: 60F05: Central limit and other weak theorems
Secondary: 68P10: Searching and sorting 68P05: Data structures 68W40: Analysis of algorithms

Information

Type
Original Article
Information
Journal of Applied Probability , First View , pp. 1 - 14
Check for updates
Copyright
© The Author(s), 2025. Published by Cambridge University Press on behalf of Applied Probability Trust

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.)

Article purchase

Temporarily unavailable

References

Aldous, D. J. (1991). Asymptotic fringe distributions for general families of random trees. Ann. Appl. Prob. 1, 228–266.10.1214/aoap/1177005936CrossRefGoogle Scholar
Bourdon, J. (2001). Size and path length of Patricia tries: Dynamical sources context. Random Structures Algorithms 19, 289–315.10.1002/rsa.10015CrossRefGoogle Scholar
Clément, J., Flajolet, P. and Vallée, B. (2001). Dynamical sources in information theory: A general analysis of trie structures. Algorithmica 29, 307–369.10.1007/BF02679623CrossRefGoogle Scholar
Drmota, M. (2009). Random Trees: An Interplay between Combinatorics and Probability. Springer, Vienna.10.1007/978-3-211-75357-6CrossRefGoogle Scholar
Fuchs, M., Hwang, H.-K. and Zacharovas, V. (2014). An analytic approach to the asymptotic variance of trie statistics and related structures. Theoret. Comput. Sci. 527, 1–36.10.1016/j.tcs.2014.01.024CrossRefGoogle Scholar
Ischebeck, J. (2022). Central limit theorems for fringe trees in patricia tries. Master’s thesis, Universitätsbibliothek Johann Christian Senckenberg.Google Scholar
Jacquet, P. and Szpankowski, W. (2015). Algorithms and data structures. In Analytic Pattern Matching: From DNA to Twitter, pp. 135–153. Cambridge University Press, Cambridge. 10.1017/CBO9780511843204.009CrossRefGoogle Scholar
Janson, S. (2016). Asymptotic normality of fringe subtrees and additive functionals in conditioned Galton–Watson trees. Random Structures Algorithms 48, 57–101.10.1002/rsa.20568CrossRefGoogle Scholar
Janson, S. (2020). On the independence number of some random trees. Electron. Commun. Prob. 25, 1–14.10.1214/20-ECP345CrossRefGoogle Scholar
Janson, S. (2022). Central limit theorems for additive functionals and fringe trees in tries. Electron. J. Prob. 27, 1–63.10.1214/22-EJP776CrossRefGoogle Scholar
Knuth, D. (1973). The Art of Computer Programming, Vol. 3, Sorting and Searching. Addison-Wesley, Boston.Google Scholar
Morrison, D. R. (1968). PATRICIA: Practical algorithm to retrieve information coded in alphanumeric. J. Assoc. Comput. Mach. 15, 514–534.10.1145/321479.321481CrossRefGoogle Scholar
Szpankowski, W. (1990). Patricia tries again revisited. J. Assoc. Comput. Mach. 37, 691–711.10.1145/96559.214080CrossRefGoogle Scholar
Wagner, S. (2015). Central limit theorems for additive tree parameters with small toll functions. Combinatorics Prob. Comput. 24, 329–353.10.1017/S0963548314000443CrossRefGoogle Scholar