Abstract
Stratified trees form a family of classes of search trees of special interest because of their generality: they include symmetric binary B-trees, halfbalanced trees, and red-black trees, among others. Moreover, stratified trees can be used as a basis for relaxed rebalancing in a very elegant way. The purpose of this paper is to study the rebalancing cost of stratified trees after update operations. The operations considered are the usual insert and delete operations and also bulk insertion, in which a number of keys are inserted into the same place in the tree. Our results indicate that when insertions, deletions, and bulk insertions are applied in an arbitrary order, the amortized rebalancing cost for single insertions and deletions is constant, and for bulk insertions O(logm), where m is the size of the bulk. The latter is also a bound on the structural changes due to a bulk insertion in the worst case.
This is a preview of subscription content, log in via an institution to check access.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
Preview
Unable to display preview. Download preview PDF.
References
G. M. Adel’son-Vel’skii and E. M. Landis. An algorithm for the organisation of information. Dokl. Akad. Nauk SSSR 146 (1962), 263–266 (in Russian); English Translation in Soviet. Math. 3, 1259-1262.
R. Bayer: Symmetric binary B-trees: Data structure and maintenance algorithms. Acta Informatica 1 (1972), 290–306.
T. H. Cormen, C. E. Leiserson, R. L. Rivest, and C. Stein: Introduction to Algorithms, Second Edition, The MIT Press, Cambridge, Massachusetts, 2001.
E. M. McCreight: Priority search trees. SIAM Journal on Computing 14 (1985), 257–276.
L. J. Guibas and R. Sedgewick: A dichromatic framework for balanaced trees. In: Proceedings of the 19th Annual IEEE Synposium on Foundations of Computer Science, Ann Arbor, pp. 8–21, 1978.
Chr. Icking, R. Klein, and Th. Ottmann: Priority search trees in secondary memory. In: Graphtheoretic Concepts in Computer Science (WG’ 87). Lecture Notes in Computer Science 314 (Springer-Verlag), pp. 84–93.
C. Jermaine, A. Datta, and E. Omiecinski. A novel index supporting high volume data warehouse insertion. In: Proceedings of the 25th International Conference on Very Large Databases. Morgan Kaufmann Publishers, 1999, pp. 235–246.
T.-W. Kuo, C-H. Wei, and K.-Y. Lam. Real-time data access control on B-tree index structures. In: Proceedings of the 15th International Conference on Data Engineering. IEEE Computer Society, 1999, pp. 458–467.
T. W. Lai and D. Wood: Adaptive heuristics for binary search trees and constant linkage cost. SIAM Journal on Computing 27:6 (1998), 1564–1591.
K. S. Larsen. Relaxed multi-way trees with group updates. In: Proceedings of the 20th ACM SIGMOD-SIGACT-SIGART Symposium on principles of Database Systems. ACMPress, 2001, pp. 93–101. To appear in Journal of Computer and System Sciences.
J. van Leeuwen and M. H. Overmars: Stratified balanced search trees. Acta Informatica 18 (1983), 345–359.
L. Malmi and E. Soisalon-Soininen. Group updates for relaxed height-balanced trees. In: Proceedings of the 18th ACM SIGMOD-SIGACT-SIGART Symposium on Principles of Database Systems. ACM Press, 1999, pp. 358–367.
K. Mehlhorn: Data Strucures and Algorithms, Vol. 1: Sorting and Searching, Springer-Verlag, 1986.
O. Nurmi, E. Soisalon-Soininen, and D. Wood: Concurrency control in database structures with relaxed balance. In: Proceedings of the Sixth ACM Symposium on Principles of Database Systems, pp. 170–176, 1987.
H. J. Olivie: A new class of balanced search trees: Half-balanced search trees. RAIRO Informatique Theorique 16 (1982), 51–71.
Th. Ottmann, and E. Soisalon-Soininen: Relaxed balancing made simple. Institut für Informatik, Universität Freiburg, Technical Report 71, 1995.
Th. Ottmann and P. Widmayer: Algorithmen und Datenstrukturen, 4. Auflage, Spektrum-Verlag, Heidelberg, 2002.
Th. Ottmann and D. Wood: Updating binary trees with constant linkage cost. International Journal of Foundations of Computer Science 3:4 (1992), 479–501.
K. Pollari-Malmi: Batch updates and concurrency control in in B-trees. Ph.D.Thesis, Helsinki University of Technology, Department of Computer Science and Engineering, Report A38/02, 2002.
K. Pollari-Malmi, E. Soisalon-Soininen, and T. Ylönen: Concurrency control in B-trees with batch updates. IEEE Transactions on Knowledge and Data Engineering 8 (1996), 975–984.
N. Sarnak and R. E. Tarjan: Planar point location using persistent search trees. Communications of the ACM 29 (1986), 669–679.
H.-W. Six and D. Wood: Counting and reporting insersections of d-ranges. IEEE Transactions on Computers, C–31 (1982), 181–187.
D. D. Sleator and R. E. Tarjan: Self-adjusting binary search trees. Journal of the ACM 32 (1985), 652–686.
R. E. Tarjan. Amortized computational complexity. SIAM Journal on Algebraic and Discrete Methods 6 (1985), 306–318.
Author information
Authors and Affiliations
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 2003 Springer-Verlag Berlin Heidelberg
About this chapter
Cite this chapter
Soisalon-Soininen, E., Widmayer, P. (2003). Single and Bulk Updates in Stratified Trees: An Amortized andWorst-Case Analysis. In: Klein, R., Six, HW., Wegner, L. (eds) Computer Science in Perspective. Lecture Notes in Computer Science, vol 2598. Springer, Berlin, Heidelberg. https://doi.org/10.1007/3-540-36477-3_21
Download citation
DOI: https://doi.org/10.1007/3-540-36477-3_21
Published:
Publisher Name: Springer, Berlin, Heidelberg
Print ISBN: 978-3-540-00579-7
Online ISBN: 978-3-540-36477-1
eBook Packages: Springer Book Archive