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EXTREMAL WEIGHTED PATH LENGTHS IN RANDOM BINARY SEARCH TREES

Published online by Cambridge University Press:  15 December 2006

Rafik Aguech ,
Nabil Lasmar  and
Hosam Mahmoud
Rafik Aguech
Affiliation:
Département de Mathématiques, Faculté des Sciences de Monastir, 5019 Monastir, Tunisia, E-mail: rafikaguech@ipeit.rnu.tn
Nabil Lasmar
Affiliation:
Département de mathématiques, Institut préparatoire aux études d'ingénieurs de Tunis, IPEIT, Tunis, Tunisia, E-mail: nabillasmar@yahoo.fr
Hosam Mahmoud
Affiliation:
Department of Statistics, The George Washington University, Washington, DC 20052, E-mail: hosam@gwu.edu
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Abstract

We consider weighted path lengths to the extremal leaves in a random binary search tree. When linearly scaled, the weighted path length to the minimal label has Dickman's infinitely divisible distribution as a limit. By contrast, the weighted path length to the maximal label needs to be centered and scaled to converge to a standard normal variate in distribution. The exercise shows that path lengths associated with different ranks exhibit different behaviors depending on the rank. However, the majority of the ranks have a weighted path length with average behavior similar to that of the weighted path to the maximal node.

Information

Type
Research Article
Information
Probability in the Engineering and Informational Sciences , Volume 21 , Issue 1 , January 2007 , pp. 133 - 141
Copyright
© 2007 Cambridge University Press

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References

REFERENCES

Billingsley, P. (1995). Probability and measure. New York: Wiley.
Devroye, L. (1986). A note on the height of binary search trees. Journal of the ACM 33: 489–498.Google Scholar
Devroye, L. (1987). Branching processes in the analysis of the height of trees. Acta Informatica 24: 277–298.Google Scholar
Devroye, L. (1988). Applications of the theory of records in the study of random trees. Acta Informatica 26: 123–130.Google Scholar
Devroye, L. & Neininger, R. (2004). Distances and finger search in random binary search trees. SIAM Journal on Computing 33: 647–658.Google Scholar
Drmota, M. (2001). An analytic approach to the height of binary search trees. Algorithmica 29: 89–119.Google Scholar
Drmota, M. (2002). The variance of the height of binary search trees. Theoretical Computer Science 270: 913–919.Google Scholar
Hwang, H. & Tsai, T. (2002). Quickselect and Dickman function. Combinatorics, Probability and Computing 11: 353–371.Google Scholar
Kemp, R. (1984). Fundamentals of the average case analysis of particular algorithms. Wiley-Teubner Series in Computer Science. New York: Wiley.
Knuth, D. (1998). The art of computer programming. Vol. 3: Sorting and Searching, 2nd ed. Reading, MA: Addison-Wesley.
Mahmoud, H. (1992). Evolution of random search trees. New York: Wiley.
Mahmoud, H. (2000). Sorting: A distribution theory. New York: Wiley.CrossRef
Mahmoud, H., Modarres, R., & Smythe, R. (1995). Analysis of Quickselect: An algorithm for order statistics. RAIRO: Theoretical Informatics and Its Applications 29: 255–276.Google Scholar
Mahmoud, H. & Pittel, B. (1984). On the most probable shape of a search tree grown from a random permutation. SIAM Journal on Algebraic and Discrete Methods 5: 69–81.Google Scholar
Pittel, B. (1984). On growing random binary trees. Journal of Mathematical Analysis and Its Applications 103: 461–480.Google Scholar
Reed, B. (2003). The height of a random binary search tree. Journal of the Association for Computing Machinery 50: 306–332.Google Scholar
Robson, J. (1979). The height of binary search trees. The Australian Computer Journal 11: 151–153.Google Scholar