Abstract
The class of height balanced 2–3 trees is defined. This class properly contains the class of 2–3 trees. Insertion and deletion algorithms for these trees having 0 (logn) performance are provided. This paper thus, effectively demonstrates that “height balancing” can be usefully applied to classes of trees other than binary trees, which is a contribution to the solution of one of Knuth's problems.
Zusammenfassung
Es wird die Klasse der höhenbalancierten 2–3 Bäume eingeführt. Diese Klasse umfaßt die Klasse der 2–3 Bäume echt. Einfüge- und Entferne-Algorithmen mit 0 (logn) Zeitkomplexität werden für diese Bäume angegeben. Mit dieser Arbeit wird gezeigt, daß das Konzept des Höhenbalancierens vorteilhaft auf Klassen von nichtbinären Bäumen angewandt werden kann. Dies stellt einen Beitrag zur Lösung eines Problems von Knuth dar.
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References
Adelson-Velskij, G. M., Landis, E. M.: An algorithm for the organization of information. (Russian.) Doklady Akad. Nauk, SSSR146, 263–266 (1962).
Aho, A. V., Hopcroft, J. E., Ullman, J. D.: The design and analysis of computer algorithms. Reading: Addison-Wesley 1974.
Bayer, R.: BinaryB-trees for virtual memory, Proceedings of 1971 ACM SIGFIDET Workshop on Data Description, Access and Control (Codd, E. F., Dean, A. L., eds.), p. 219–235. San Diego: 1971.
Bayer, R., McCreight, E. M.: Organization and maintenance of large ordered indexes. Acta Informatica1, 173–189 (1972).
Bayer, R.: Symmetric binaryB-trees; Data structures and maintenance algorithms. Acta Informatica1, 290–306 (1972).
Foster, C. C.: A generalization of AVL trees. Comm. ACM16, 513–517 (1973).
Hirschberg, D. S.: An insertion technique for one-sided height-balanced trees. Comm. ACM19, 471–473 (1976).
Knuth, D. E.: The art of computer programming, Vol. III: Sorting and searching. Reading: Addison-Wesley 1973.
Mauer, H. A., Wood, D.: Zur Manipulation von Zahlenmengen. Angewandte Informatik4, 143–149 (1976).
Miller, R. E., Pippenger, N., Rosenberg, A. L., Snyder, L.: Optimal 2–3 trees, in: Proceedings of a conference on Theoretical Computer Science, August 15–17, 1977, University of Waterloo, Waterloo, Ontario, Canada, p. 30–35.
Nievergelt, J., Reingold, E. M.: Binary trees of bounded balance. SIAM Journal of Computing2, 33–43 (1973).
Ottmann, Th., Six, H. W.: Eine neue Klasse von ausgeglichenen Binärbäumen, Angewandte Informatik9, 395–400 (1976).
Ottmann, Th., Wood, D.: Deletion in one-sided height-balanced search trees. Int. J. of Computer Mathematics6, 265–271 (1978).
Ottmann, Th., Six, H. W., Wood, D.: Right brother trees. Comm. ACM21, 769–776 (1978).
Ottmann, Th., Six, H. W., Wood, D.: On the correspondence between AVL trees and brother trees. Computer Science Technical Report 77-CS-12, Department of Applied Mathematics, McMaster University, Hamilton (1977).
Ottmann, Th., Wood, D.: 1–2 Brother Trees. Computer Journal (1978, to appear).
Räihö, K. J.: An 0 (logn) insertion algorithm for one-sided height-balanced binary search trees, Department of Computer Science, University of Helsinki, Finland, Report A-1977-9.
Rosenberg, A. L., Snyder, L.: Minimal comparison 2,3-trees. SIAM J. of Computing (1978, to appear).
Zweben, S. H., McDonald, M. A.: An optimal method for deletion in one-sided height-balanced trees. Comm. ACM21, 441–445 (1978).
Zweben, S. H.: An optimal insertion method for one-sided height-balanced trees, Technical Report, Department of Computer and Information Science, Ohio State University, Columbus, Ohio (1977).
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Work supported partially by a Natural Sciences and Engineering Research Council of Canada Grant No. A-7700 and partially by the German Academic Exchange Service under Nato Research Grant No. 430/402/584/8.
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Vaishnavi, V.K., Kriegel, H.P. & Wood, D. Height balanced 2–3 trees. Computing 21, 195–211 (1979). https://doi.org/10.1007/BF02253053
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DOI: https://doi.org/10.1007/BF02253053