Abstract
We investigate incomplete one-sided variants of binary search trees. The (normed) size of each variant is studied, and convergence to a Gaussian law is proved in each case by asymptotically solving recurrences. These variations are also discussed within the scope of the contraction method with degenerate limit equations. In an incomplete tree the size determines most other parameters of interest, such as the height and the internal path length.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
Arora, S. and Dent, W. (1969). Randomized binary search technique,Communications of the ACM,12, 77–80.
Chassaing, P. and Marchand, R. (2002). Cutting a random tree (and UNION-FIND algorithms) (manuscript).
Devroye, L. (1988). Applications of the theory of records in the study of random trees,Acta Inform.,16, 123–130.
Drmota, M. (2002). The random bisection problem and the distribution of the height of binary search trees (manuscript).
Fill, J., Mahmoud, H. and Szpankowski, W. (1996). On the distribution for the duration of a randomized leader election algorithm,Ann. Appl. Probab.,6, 1260–1283.
Itoh, Y. and Mahmoud, H. (2003). One-sided variations on interval trees,J. Appl. Probab.,40, 1–17.
Kemp, R. (1984).Fundamentals of the Average Case Analysis of Particular Algorithms, Wiley-Teubner, Stuttgart.
Knuth, D. (1998).The Art of Computer Programming, Vol. III: Searching and Sorting, Addison-Wesley, Reading Massachusetts.
Mahmoud, H. (1992).Evolution of Random Search Trees, Wiley, New York.
Mahmound, H. (2000).Sorting: A Distribution Theory, Wiley, New York.
Mahmound, H. and Neininger, R. (2003). Distribution of distances in random binary search trees,Ann. Appl. Probab.,13, 253–276.
Martínez, C., Panholzer, A. and Prodinger, H. (1998). Descendants and ascendants in random search trees,Electronic Journal of Combinatorics,5, R20; 29 pages+Appendix (10 pages).
Meir, A. and Moon, J. (1970). Cutting down random trees,J Austral. Math. Soc.,11, 313–324.
Neininger, R. (2002). On a multivariate contraction method for random recursive structures with applications to Quicksort,Random Structures Algorithms,19, 498–524.
Neininger, R. and Rüschendorf, L. (2002). On the contraction method with degenerate limit equation (manuscript).
Prodinger, H. (1993). How to select a loser,Discrete Math.,120, 149–159.
Rachev, S. and Rüschendorf, L. (1995). Probability metrics and recursive algorithms,Adv. in Appl. Probab.,27, 770–799.
Rösler, U. (1991). A limit theorem for “Quicksort”,RAIRO Inform. Théor. Appl.,25, 85–100.
Rösler, U. (2001). On the analysis of stochastic divide and conquer algorithms,Algorithmica,29, 238–261.
Rösler, U. and Rüschendorf, L. (2001). The contraction method for recursive algorithms,Algorithmica,29, 3–33.
Sibuya, M. and Itoh, Y. (1987). Random sequential bisection and its associated binary tree,Ann. Inst. Statist. Math.,39, 69–84.
Zolotarev, V. M. (1976). Approximation of the distributions of sums of independent random variables with values in infinite-dimensional spaces,Theory Probab. Appl.,21, 721–737.
Author information
Authors and Affiliations
About this article
Cite this article
Mahmoud, H.M. One-sided variations on binary search trees. Ann Inst Stat Math 55, 885–900 (2003). https://doi.org/10.1007/BF02523399
Received:
Revised:
Issue Date:
DOI: https://doi.org/10.1007/BF02523399