The problem of searching for a key in many ordered lists arises frequently in computational geometry. Chazelle and Guibas recently introduced fractional cascading as a general technique for solving this type of problem. In this paper we show that fractional cascading also supports insertions into and deletions from the lists efficiently. More specifically, we show that a search for a key inn lists takes timeO(logN +n log logN) and an insertion or deletion takes timeO(log logN). HereN is the total size of all lists. If only insertions or deletions have to be supported theO(log logN) factor reduces toO(1). As an application we show that queries, insertions, and deletions into segment trees or range trees can be supported in timeO(logn log logn), whenn is the number of segments (points).
Key wordsComputational geometry Linear lists Dynamic data structures Amortized complexity
Unable to display preview. Download preview PDF.
- [B77]J. L. Bentley: Solutions to Klee's Rectangle Problem, unpublished manuscript, Department of Computer Science, Carnegie-Mellon University, 1977.Google Scholar
- [DSST]J. R. Driscoll, N. Sarnak, D. D. Sleator, R. E. Tarjan: Making Data Structures Persistent,J. Comput. System Sci, to appear.Google Scholar
- [FMN85]O. Fries, K. Mehlhorn, St. Näher: Dynamization of Geometric Data Structures,Proc. ACM Symposium on Computational Geometry, 1985, 168–176.Google Scholar
- [L78]G. S. Luecker: A Data Structure for Orthogonal Range Queries,Proc. 19th FOCS, 1978, 28–34.Google Scholar
- [M84a]K. Mehlhorn:Data Structures and Algorithms, Vol. 1, Springer-Verlag, Berlin, 1984.Google Scholar
- [M84b]Ibid..Google Scholar
- [M84c]Ibid..Google Scholar
- [M86]K. Mehlhorn:Datenstrukturen und Algorithmen 1, Teubner, 1986.Google Scholar
- [MNA87]K. Mehlhorn, S. Näher, H. Alt: A Lower Bound on the Complexity of the Union-Split-Find Problem,Proc. 13th ICALP, 1987, 479–488.Google Scholar
- [N87]S. Näher: Dynamic Fractional Cascading oder die Verwaltung vieler linearer Listen, Dissertation, University des Saarlandes, Saarbrücken, 1987.Google Scholar
- [PS85]F. P. Preparata, M. I. Shamos:Computational Geometry, An Introduction, Springer-Verlag, Berlin, 1985.Google Scholar
- [W78]D. E. Willard: New Data Structures for Orthogonal Range Queries, Technical Report, Harvard University, 1978.Google Scholar
- [W85]D. E. Willard: New Data Structures for Orthogonal Queries,SIAM J. Comput., 1985, 232–253.Google Scholar