Skip to main content
SpringerLink
Log in

Dynamic fractional cascading

  • Published:
Algorithmica Aims and scope Submit manuscript

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).

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  1. J. L. Bentley: Solutions to Klee's Rectangle Problem, unpublished manuscript, Department of Computer Science, Carnegie-Mellon University, 1977.

  2. J. L. Bentley: Decomposable Searching Problems,Inform. Process. Lett. 8, 1979, 244–251.

    Article  MATH  MathSciNet  Google Scholar 

  3. N. Blum, K. Mehlhorn: On the Average Number of Rebalancing Operations in Weight-Balanced Trees,Theoret. Comput. Sci 11, 1980, 303–320.

    Article  MATH  MathSciNet  Google Scholar 

  4. B. Chazelle, L. Guibas: Fractional Cascading: I, A Data Structuring Technique; II, Applications,Algorithmica 1, 1986, 133–191.

    Article  MATH  MathSciNet  Google Scholar 

  5. J. R. Driscoll, N. Sarnak, D. D. Sleator, R. E. Tarjan: Making Data Structures Persistent,J. Comput. System Sci, to appear.

  6. H. Edelsbrunner, L. Guibas, I. Stolfi: Optimal Point Location in a Monotone Subdivision,SIAM J. Comput. 15, 1986, 317–340.

    Article  MATH  MathSciNet  Google Scholar 

  7. P. van Emde Boas, R. Kaas, E. Zijlstra: Design and Implementation of an Efficient Priority Queue,Math. Systems Theory 10, 1977, 99–127.

    Article  MATH  Google Scholar 

  8. O. Fries, K. Mehlhorn, St. Näher: Dynamization of Geometric Data Structures,Proc. ACM Symposium on Computational Geometry, 1985, 168–176.

  9. 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.

    Article  MATH  MathSciNet  Google Scholar 

  10. R. H. Güting: Fast Dynamic Intersection Searching in a Set of Isothetic Line Segments,Inform. Process. Lett. 21, 1985, 165–171.

    Article  MATH  MathSciNet  Google Scholar 

  11. S. Huddleston, K. Mehlhorn: A New Representation for Linear Lists,Acta Inform. 17, 1982, 157–184.

    Article  MATH  MathSciNet  Google Scholar 

  12. T. Imai, T. Asano: Dynamic Orthogonal Segment Intersection Search,J. Algorithms 8, 1987, 1–18.

    Article  MATH  MathSciNet  Google Scholar 

  13. W. Lipski: Finding a Manhattan Path and Related Problems,Networks 13, 1983, 399–409.

    Article  MATH  MathSciNet  Google Scholar 

  14. W. Lipski: AnO(n logn) Manhattan Path Algorithm,Inform. Process. Lett. 19, 1984, 99–102.

    Article  MATH  MathSciNet  Google Scholar 

  15. G. S. Luecker: A Data Structure for Orthogonal Range Queries,Proc. 19th FOCS, 1978, 28–34.

  16. K. Mehlhorn:Data Structures and Algorithms, Vol. 1, Springer-Verlag, Berlin, 1984.

    Google Scholar 

  17. Ibid..

    Google Scholar 

  18. Ibid..

    Google Scholar 

  19. K. Mehlhorn:Datenstrukturen und Algorithmen 1, Teubner, 1986.

  20. K. Mehlhorn, S. Näher, H. Alt: A Lower Bound on the Complexity of the Union-Split-Find Problem,Proc. 13th ICALP, 1987, 479–488.

  21. S. Näher: Dynamic Fractional Cascading oder die Verwaltung vieler linearer Listen, Dissertation, University des Saarlandes, Saarbrücken, 1987.

    Google Scholar 

  22. F. P. Preparata, M. I. Shamos:Computational Geometry, An Introduction, Springer-Verlag, Berlin, 1985.

    Google Scholar 

  23. R. E. Tarjan: Amortized Computational Complexity,SIAM J. Algebraic Discrete Methods 6, 1985, 306–318.

    Article  MATH  MathSciNet  Google Scholar 

  24. A. K. Tsakalidis: Maintaining Order in a Generalized Linked List,Acta Inform. 21, 1984, 101–112.

    Article  MATH  MathSciNet  Google Scholar 

  25. V. K. Vaishnavi, D. Wood: Rectilinear Line Segment Intersection, Layered Segment Trees and Dynamization,J. Algorithms,3, 1982, 160–176.

    Article  MATH  MathSciNet  Google Scholar 

  26. D. E. Willard: New Data Structures for Orthogonal Range Queries, Technical Report, Harvard University, 1978.

  27. D. E. Willard: New Data Structures for Orthogonal Queries,SIAM J. Comput., 1985, 232–253.

  28. D. E. Willard, G. S. Luecker: Adding Range Restriction Capability to Dynamic Data Structures,J. Assoc. Comput. Mach. 32, 1985, 597–617.

    MATH  MathSciNet  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Universität des Saarlandes, Fachbereich 10, D-6600, Saarbrücken, Federal Republic of Germany

    Kurt Mehlhorn & Stefan Näher

Authors
  1. Kurt Mehlhorn
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Stefan Näher
    View author publications

    You can also search for this author in PubMed Google Scholar

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

Reprints 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

Download citation

  • Received:

  • Revised:

  • Issue Date:

  • DOI: https://doi.org/10.1007/BF01840386

Key words

Search

Navigation