Abstract
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).
Similar content being viewed by others
References
J. L. Bentley: Solutions to Klee's Rectangle Problem, unpublished manuscript, Department of Computer Science, Carnegie-Mellon University, 1977.
J. L. Bentley: Decomposable Searching Problems,Inform. Process. Lett. 8, 1979, 244–251.
N. Blum, K. Mehlhorn: On the Average Number of Rebalancing Operations in Weight-Balanced Trees,Theoret. Comput. Sci 11, 1980, 303–320.
B. Chazelle, L. Guibas: Fractional Cascading: I, A Data Structuring Technique; II, Applications,Algorithmica 1, 1986, 133–191.
J. R. Driscoll, N. Sarnak, D. D. Sleator, R. E. Tarjan: Making Data Structures Persistent,J. Comput. System Sci, to appear.
H. Edelsbrunner, L. Guibas, I. Stolfi: Optimal Point Location in a Monotone Subdivision,SIAM J. Comput. 15, 1986, 317–340.
P. van Emde Boas, R. Kaas, E. Zijlstra: Design and Implementation of an Efficient Priority Queue,Math. Systems Theory 10, 1977, 99–127.
O. Fries, K. Mehlhorn, St. Näher: Dynamization of Geometric Data Structures,Proc. ACM Symposium on Computational Geometry, 1985, 168–176.
H. N. Gabow, R. E. Tarjan: A Linear-Time Algorithm for a Special Case of Disjoint Set Union,J. Comput. System Sci 30, 1985, 209–221.
R. H. Güting: Fast Dynamic Intersection Searching in a Set of Isothetic Line Segments,Inform. Process. Lett. 21, 1985, 165–171.
S. Huddleston, K. Mehlhorn: A New Representation for Linear Lists,Acta Inform. 17, 1982, 157–184.
T. Imai, T. Asano: Dynamic Orthogonal Segment Intersection Search,J. Algorithms 8, 1987, 1–18.
W. Lipski: Finding a Manhattan Path and Related Problems,Networks 13, 1983, 399–409.
W. Lipski: AnO(n logn) Manhattan Path Algorithm,Inform. Process. Lett. 19, 1984, 99–102.
G. S. Luecker: A Data Structure for Orthogonal Range Queries,Proc. 19th FOCS, 1978, 28–34.
K. Mehlhorn:Data Structures and Algorithms, Vol. 1, Springer-Verlag, Berlin, 1984.
Ibid..
Ibid..
K. Mehlhorn:Datenstrukturen und Algorithmen 1, Teubner, 1986.
K. Mehlhorn, S. Näher, H. Alt: A Lower Bound on the Complexity of the Union-Split-Find Problem,Proc. 13th ICALP, 1987, 479–488.
S. Näher: Dynamic Fractional Cascading oder die Verwaltung vieler linearer Listen, Dissertation, University des Saarlandes, Saarbrücken, 1987.
F. P. Preparata, M. I. Shamos:Computational Geometry, An Introduction, Springer-Verlag, Berlin, 1985.
R. E. Tarjan: Amortized Computational Complexity,SIAM J. Algebraic Discrete Methods 6, 1985, 306–318.
A. K. Tsakalidis: Maintaining Order in a Generalized Linked List,Acta Inform. 21, 1984, 101–112.
V. K. Vaishnavi, D. Wood: Rectilinear Line Segment Intersection, Layered Segment Trees and Dynamization,J. Algorithms,3, 1982, 160–176.
D. E. Willard: New Data Structures for Orthogonal Range Queries, Technical Report, Harvard University, 1978.
D. E. Willard: New Data Structures for Orthogonal Queries,SIAM J. Comput., 1985, 232–253.
D. E. Willard, G. S. Luecker: Adding Range Restriction Capability to Dynamic Data Structures,J. Assoc. Comput. Mach. 32, 1985, 597–617.
Author information
Authors and Affiliations
Additional information
Communicated by D. T. Lee.
This research was supported by the Deutsche Forschungsgemeinschaft under Grants Me 620/6-1 and SFB 124, Teilprojekt B2. A preliminary version of this research was presented at the ACM Symposium on Computational Geometry, Baltimore, 1985.
Rights and permissions
About this article
Cite this article
Mehlhorn, K., Näher, S. Dynamic fractional cascading. Algorithmica 5, 215–241 (1990). https://doi.org/10.1007/BF01840386
Received:
Revised:
Issue Date:
DOI: https://doi.org/10.1007/BF01840386