Advertisement

Acta Informatica

, Volume 42, Issue 1, pp 57–78 | Cite as

Exponentially decreasing number of operations in balanced trees

  • Lars Jacobsen
  • Kim Skak Larsen
Original Paper
  • 43 Downloads

Abstract

While many tree-like structures have been proven to support amortized constant number of operations after updates, considerably fewer structures have been proven to support the more general exponentially decreasing number of operations with respect to distance from the update. In addition, all existing proofs of exponentially decreasing operations are tailor-made for specific structures. We provide the first formalization of conditions under which amortized constant number of operations imply exponentially decreasing number of operations. Since our proof is constructive, we obtain the constants involved immediately. Moreover, we develop a number of techniques to improve these constants.

Keywords

Information System Operating System Data Structure Communication Network Information Theory 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. 1.
    Adel'son-Vel'skii, G.M., Landis, E.M.: An algorithm for the organisation of information. Doklady Akadamii Nauk SSSR. 146, 263–266 (1962) In Russian. English translation in Soviet Math. Doklady, 3, 1259-1263 (1962)Google Scholar
  2. 2.
    Andersson, A., Fagerberg, R., Larsen, K.S.: Balanced binary search trees. In: Dinesh P. Mehta, Sartaj Sahni (eds.), Handbook of Data Structures and Applications, Chapman & Hall/CRC Computer & Information Science Series, pp. 10–1–10–28. CRC Press (2005)Google Scholar
  3. 3.
    Dietz, P.F., Raman, R.: Persistence, amortization and randomization. In: Proceedings of the Second Annual ACM-SIAM Symposium on Discrete Algorithms, pp 78–88, (1991)Google Scholar
  4. 4.
    Driscoll, J.R., Sarnak, N., Sleator, D.D., Tarjan, R.E.: Making data structures persistent. Journal of Computer and System Sciences 38, 86–124 (1989)CrossRefMathSciNetGoogle Scholar
  5. 5.
    Huddleston, S., Mehlhorn, K.: A new data structure for representing sorted lists. Acta Informatica 17, 157–184 (1982)CrossRefMathSciNetGoogle Scholar
  6. 6.
    Jacobsen, L.: Search trees with local rules. PhD thesis, Department of Mathematics and Computer Science, University of Southern Denmark (2001)Google Scholar
  7. 7.
    Jacobsen, L., Larsen, K.S., Nielsen, M.N.: On the existence and construction of non-extreme \((a,b)\)-trees. Information Processing Letters, 84(2), 69–73 (2002)Google Scholar
  8. 8.
    Larsen, K.S.: Relaxed multi-way trees with group updates. Journal of Computer and System Sciences, 66(4), 657–670 (2003)CrossRefzbMATHMathSciNetGoogle Scholar
  9. 9.
    Mehlhorn, K.: Sorting and Searching, vol. 1 of Data Structures and Algorithms. Springer-Verlag (1984)Google Scholar
  10. 10.
    Mehlhorn, K. Tsakalidis, A.: An amortized analysis of insertions into AVL-trees. SIAM Journal on Computing 15(1), 22–33 (1986)CrossRefMathSciNetGoogle Scholar
  11. 11.
    Overmars, M. H.: Searching in the past ii: general transforms. Technical Report RUU-CS-81-9. Department of Computer Science, University of Utrecht, The Netherlands (1981)Google Scholar
  12. 12.
    Raman, R.: Eliminating amortization: on data structures with guaranteed response time. PhD thesis, Department of Computer Science, University of Rochester, Rochester, New York (1992)Google Scholar
  13. 13.
    Sarnak, N.: Persistent data structures. PhD thesis, Department of Computer Science, New York University, New York (1986)Google Scholar
  14. 14.
    Tsakalidis, A. K.: Rebalancing operations for deletions in avl-trees. R.A.I.R.O. Informatique Théorique 19(4), 323–329 (1985)zbMATHMathSciNetGoogle Scholar

Copyright information

© Springer-Verlag 2005

Authors and Affiliations

  1. 1.Systematic Software Engineering A/SAarhusDenmark
  2. 2.Department of Mathematics and Computer ScienceUniversity of Southern DenmarkOdense MDenmark

Personalised recommendations