Hostname: page-component-89b8bd64d-shngb Total loading time: 0 Render date: 2026-05-06T20:40:29.735Z Has data issue: false hasContentIssue false
  • English
  • Français

ON CLIMBING TRIES

Published online by Cambridge University Press:  18 December 2007

Costas Christophi  and
Hosam Mahmoud
Costas Christophi
Affiliation:
Cyprus International Institute for the Environment and Public Health in Association with Harvard School of Public Health, 1105, Nicosia, Cyprus Biostatistics Center, The George Washington University, Rockville, MD 20852 E-mail: cchristophi@cyprusinstitute.org
Hosam Mahmoud
Affiliation:
Department of Statistics, The George Washington UniversityWashington, DC 20052 E-mail: hosam@gwu.edu
Get access Rights & Permissions [Opens in a new window]

Abstract

To sample a typical key in a “trie,” an appropriate climbing might consider generating random edges in the same manner as the data are generated. In the absence of the probability generating the keys, an uninformed random choice among the children still provides an alternative. We are also interested in extremal sampling, achieved by following a leftmost (or a rightmost) path. Each of these climbing strategies always generates a key, but one that might not necessarily be in the database. We investigate the altitude of the position at which climbing is terminated. Analytical techniques, including poissonization and the Mellin transform, are used for the accurate calculation of moments. In all strategies, the mean is always logarithmic. For typical and uninformed climbing, the variance is bounded in unbiased tries but grows logarithmically in biased tries. Consequently, in the biased case, one can find appropriate centering and scaling to produce a limit distribution for these two climbing strategies; the limit is normal. For extremal climbing, the variance is always bounded for both biased and unbiased cases, and no nontrivial limit exists under any scaling.

Information

Type
Research Article
Information
Probability in the Engineering and Informational Sciences , Volume 22 , Issue 1 , January 2008 , pp. 133 - 149
Copyright
Copyright © Cambridge University Press 2008

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

1.Christophi, C. & Mahmoud, H. (2005). The oscillatory distribution of distances in random tries. Annals of Applied Probability 15: 1536–1564.CrossRefGoogle Scholar
2.De La Briandais, R. (1959). File searching using variable length keys. In Proceedings of the Western Joint Computer Conference, AFIPS, San Francisco, pp. 295–298.CrossRefGoogle Scholar
3.Fagin, R., Nievergelt, J., Pippenger, N., & Strong, H. (1979). Extendible hashing: A fast access method for dynamic files. ACM Transactions on Database Systems 4: 315–344.CrossRefGoogle Scholar
4.Flajolet, P., Gourdon, X., & Dumas, P. (1995). Mellin transform and asymptotic harmonic sums. Theoretical Computer Science 144: 3–58.CrossRefGoogle Scholar
5.Fredkin, E. (1960). Trie memory. Communications of the ACM 3: 490–499.CrossRefGoogle Scholar
6.Jacquet, P. (1989). Contribution de l'Analyse d'Algorithmes a l'Evaluation de Protocoles de Communication. Thèse Universitè de Paris Sud-Orsay, Paris, France.Google Scholar
7.Jacquet, P. & Szpankowski, W. (1998). Analytical depoissonization and its applications. Theoretical Computer Science 201: 1–62.CrossRefGoogle Scholar
8.Knuth, D. (1998). The art of computer programming, Vol. 3: Sorting and searching, 2nd ed.Reading, MA: Addison-Wesley.Google Scholar
9.Kuipers, L. & Niederreiter, H. (1974). Uniform distribution of sequences. New York: Wiley.Google Scholar
10.Moon, J. (1970). Climbing random trees. Aequationes Mathematicae 5: 68–74.CrossRefGoogle Scholar
11.Meir, A. & Moon, J. (1975). Climbing certain types of rooted trees I. In Proceedings of the Fifth British Combinatorial Conference, pp. 461–469.Google Scholar
12.Meir, A. & Moon, J. (1978). Climbing certain types of rooted trees II. Acta Mathematica Academia Scientiarum Hungaricae 31: 43–54.CrossRefGoogle Scholar
13.Panholzer, A. (2005). The climbing depth of random trees. Random Structures and Algorithms 26: 84–109.CrossRefGoogle Scholar
14.Schachinger, W. (1993). Beiträge zur Analyse von Datenstrukturen zur Digitalen Suche. Dissertation, Technische Universität Wien, Vienna.Google Scholar
15.Szpankowski, W. (2001). Average case analysis of algorithms on sequences. New York: Wiley.CrossRefGoogle Scholar