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.
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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)
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)
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)
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)
Huddleston, S., Mehlhorn, K.: A new data structure for representing sorted lists. Acta Informatica 17, 157–184 (1982)
Jacobsen, L.: Search trees with local rules. PhD thesis, Department of Mathematics and Computer Science, University of Southern Denmark (2001)
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)
Larsen, K.S.: Relaxed multi-way trees with group updates. Journal of Computer and System Sciences, 66(4), 657–670 (2003)
Mehlhorn, K.: Sorting and Searching, vol. 1 of Data Structures and Algorithms. Springer-Verlag (1984)
Mehlhorn, K. Tsakalidis, A.: An amortized analysis of insertions into AVL-trees. SIAM Journal on Computing 15(1), 22–33 (1986)
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)
Raman, R.: Eliminating amortization: on data structures with guaranteed response time. PhD thesis, Department of Computer Science, University of Rochester, Rochester, New York (1992)
Sarnak, N.: Persistent data structures. PhD thesis, Department of Computer Science, New York University, New York (1986)
Tsakalidis, A. K.: Rebalancing operations for deletions in avl-trees. R.A.I.R.O. Informatique Théorique 19(4), 323–329 (1985)
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Supported in part by the Danish Natural Science Research Council (SNF) and in part by the Future and Emerging Technologies programme of the EU under contract number IST-1999-14186 (ALCOM-FT). A preliminary version of this paper appeared in the Seventh Italian Conference on Theoretical Computer Science, Lecture Notes in Computer Science, vol. 2202, pages 293–311, Springer-Verlag, 2001.
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Jacobsen, L., Larsen, K.S. Exponentially decreasing number of operations in balanced trees. Acta Informatica 42, 57–78 (2005). https://doi.org/10.1007/s00236-005-0173-3
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DOI: https://doi.org/10.1007/s00236-005-0173-3