Expected behaviour analysis of AVL trees
In this paper we improve previous bounds on expected measures of AVL trees by using fringe analysis. A new way of handling larger tree collections that are not closed is presented. An inherent difficulty posed by the transformations necessary to keep the AVL tree balanced makes its analysis difficult when using fringe analysis methods. We derive a technique to cope with this difficulty obtaining the exact solution for fringe parameters even when unknown probabilities are involved. We show that the probability of a rotation in an insertion is between 0.37 and 0.73, that the fraction of balanced nodes is between 0.56 and 0.78, and that the expected number of comparisons in a search seems to be at most 12% more than in the complete balanced tree.
KeywordsExpected Number Unknown Probability Random Insertion Tree Collection Fringe Analysis
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- [AVL62]G.M. Adel'son-Vel'skii and E.M. Landis. An algorithm for the organization of information. Dokladi Akademia Nauk SSSR, 146(2):263–266, 1962. English translation in Soviet Math. Doklay 3, 1962, 1259–1263.Google Scholar
- [BYG89]R.A. Baeza-Yates and G.H. Gonnet. Solving matrix recurrences with applications. Technical Report CS-89-16, Department of Computer Science, University of Waterloo, May 1989.Google Scholar
- [Knu69]D.E. Knuth. The Art of Computer Programming: Fundamental Algorithms, volume 1. Addison-Wesley, Reading, Mass., 1969.Google Scholar
- [Knu73]D.E. Knuth. The Art of Computer Programming: Sorting and Searching, volume 3. Addison-Wesley, Reading, Mass., 1973.Google Scholar
- [Meh79]K. Mehlhorn. A partial analysis of height-balanced trees. Technical Report Report A 79/13, Universitat de Saarlandes, Saarbrucken, West Germany, 1979.Google Scholar
- [Meh82]Kurt Mehlhorn. A partial analysis of height-balanced trees under random insertions and deletions. SIAM J on Computing, 11(4):748–760, Nov 1982.Google Scholar
- [MT86]Kurt Mehlhorn and A. Tsakalidis. An amortized analysis of insertions into AVL-trees. SIAM J on Computing, 15(1):22–33, Feb 1986.Google Scholar
- [OS76]Th. Ottmann and H.W. Six. Eine neue klasse von ausgeglichenen binarbaumen. Augewandte Informartik, 9:395–400, 1976.Google Scholar
- [OW80]Thomas Ottmann and Derick Wood. 1-2 brother trees or AVL trees revisited. Computer Journal, 23(3):248–255, Aug 1980.Google Scholar
- [Ziv82]N. Ziviani. The Fringe Analysis of Search Trees. PhD thesis, Department of Computer Science, University of Waterloo, 1982.Google Scholar
- [ZT82]N. Ziviani and F.W. Tompa. A look at symmetric binary B-trees. Infor, 20(2):65–81, May 1982.Google Scholar