Binary search trees: How low can you go?
We prove that no algorithm for balanced binary search trees performing insertions and deletions in amortized time O(f(n)) can guarantee a height smaller than [log(n + 1) + 1/f(n)] for all n. We improve the existing upper bound to [log(n + 1) + log2 (f(n))/f(n)], thus almost matching our lower bound. We also improve the existing upper bound for worst case algorithms, and give a lower bound for the semi-dynamic case.
KeywordsLeft Endpoint Binary Search Tree Empty Slot Optimal Height Amortize Time
Unable to display preview. Download preview PDF.
- 1.G. M. Adel'son-Vel'skii and E. M. Landis. An Algorithm for the Organisation of Information. Dokl. Akad. Nauk SSSR, 146:263–266, 1962. In Russian. English translation in Soviet Math. Dokl., 3:1259–1263, 1962.Google Scholar
- 2.A. Andersson. Optimal bounds on the dictionary problem. In Proc. Symp. on Optimal Algorithms, Varna, volume 401 of LNCS, pages 106–114. Springer-Verlag, 1989.Google Scholar
- 3.A. Andersson. Efficient Search Trees. PhD thesis, Department of Computer Science, Lund University, Sweden, 1990.Google Scholar
- 4.A. Andersson, C. Icking, R. Klein, and T. Ottmann. Binary search trees of almost optimal height. Acta Informatica, 28:165–178, 1990.Google Scholar
- 5.A. Andersson and T. W. Lai. Fast updating of well-balanced trees. In SWAT'90, volume 447 of LNCS, pages 111–121. Springer-Verlag, 1990.Google Scholar
- 6.A. Andersson and T. W. Lai. Comparison-efficient and write-optimal searching and sorting. In ISA'91, volume 557 of LNCS, pages 273–282. Springer-Verlag, 1991.Google Scholar
- 7.P. F. Dietz and R. Raman. A constant update time finger search tree. Information Processing Letters, 52, 1994.Google Scholar
- 8.P. F. Dietz, J. I. Seiferas, and J. Zhang. A tight lower bound for on-line monotonie list labeling. In SWAT'94, volume 824 of LNCS, pages 131–142. Springer-Verlag, 1994.Google Scholar
- 9.P. F. Dietz and D. D. Sleator. Two algorithms for maintaining order in a list. In 19th STOC, pages 365–372, 1987.Google Scholar
- 10.P. F. Dietz and J. Zhang. Lower bounds for monotonic list labeling. In SWAT'90, volume 447 of LNCS, pages 173–180. Springer-Verlag, 1990.Google Scholar
- 11.R. Fleischer. A simple balanced search tree with O(1) worst-case update time. In ISSAC'93, volume 762 of LNCS, pages 139–146. Springer-Verlag, 1993.Google Scholar
- 12.L. J. Guibas and R. Sedgewick. A Dichromatic Framework for Balanced Trees. In 19th FOCS, pages 8–21, 1978.Google Scholar
- 13.T. Lai. Efficient Maintenance of Binary Search Trees. PhD thesis, Department of Computer Science, University of Waterloo, Canada, 1990.Google Scholar
- 14.T. Lai and D. Wood. Updating almost complete trees or one level makes all the difference. In STACS'90, volume 415 of LNCS, pages 188–194. Springer-Verlag, 1990.Google Scholar
- 16.H. A. Maurer, T. Ottmann, and H.-W. Six. Implementing dictionaries using binary trees of very small height. Information Processing Letters, 5, 1976.Google Scholar
- 17.M. H. Overmars. The Design of Dynamic Data Structures. Springer, Berlin, 1983.Google Scholar
- 18.J. Zhang. Density Control and On-Line Labeling Problems. PhD thesis, Department of Computer Science, University of Rochester, New York, 1993.Google Scholar