Algorithmica

, Volume 5, Issue 1–4, pp 215–241

Dynamic fractional cascading

  • Kurt Mehlhorn
  • Stefan Näher
Article

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).

Key words

Computational geometry Linear lists Dynamic data structures Amortized complexity 

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References

  1. [B77]
    J. L. Bentley: Solutions to Klee's Rectangle Problem, unpublished manuscript, Department of Computer Science, Carnegie-Mellon University, 1977.Google Scholar
  2. [B79]
    J. L. Bentley: Decomposable Searching Problems,Inform. Process. Lett. 8, 1979, 244–251.MATHCrossRefMathSciNetGoogle Scholar
  3. [BM80]
    N. Blum, K. Mehlhorn: On the Average Number of Rebalancing Operations in Weight-Balanced Trees,Theoret. Comput. Sci 11, 1980, 303–320.MATHCrossRefMathSciNetGoogle Scholar
  4. [CG86]
    B. Chazelle, L. Guibas: Fractional Cascading: I, A Data Structuring Technique; II, Applications,Algorithmica 1, 1986, 133–191.MATHCrossRefMathSciNetGoogle Scholar
  5. [DSST]
    J. R. Driscoll, N. Sarnak, D. D. Sleator, R. E. Tarjan: Making Data Structures Persistent,J. Comput. System Sci, to appear.Google Scholar
  6. [EGS86]
    H. Edelsbrunner, L. Guibas, I. Stolfi: Optimal Point Location in a Monotone Subdivision,SIAM J. Comput. 15, 1986, 317–340.MATHCrossRefMathSciNetGoogle Scholar
  7. [EKZ77]
    P. van Emde Boas, R. Kaas, E. Zijlstra: Design and Implementation of an Efficient Priority Queue,Math. Systems Theory 10, 1977, 99–127.MATHCrossRefGoogle Scholar
  8. [FMN85]
    O. Fries, K. Mehlhorn, St. Näher: Dynamization of Geometric Data Structures,Proc. ACM Symposium on Computational Geometry, 1985, 168–176.Google Scholar
  9. [GT85]
    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.MATHCrossRefMathSciNetGoogle Scholar
  10. [G85]
    R. H. Güting: Fast Dynamic Intersection Searching in a Set of Isothetic Line Segments,Inform. Process. Lett. 21, 1985, 165–171.MATHCrossRefMathSciNetGoogle Scholar
  11. [HM82]
    S. Huddleston, K. Mehlhorn: A New Representation for Linear Lists,Acta Inform. 17, 1982, 157–184.MATHCrossRefMathSciNetGoogle Scholar
  12. [IA87]
    T. Imai, T. Asano: Dynamic Orthogonal Segment Intersection Search,J. Algorithms 8, 1987, 1–18.MATHCrossRefMathSciNetGoogle Scholar
  13. [L83]
    W. Lipski: Finding a Manhattan Path and Related Problems,Networks 13, 1983, 399–409.MATHCrossRefMathSciNetGoogle Scholar
  14. [L84]
    W. Lipski: AnO(n logn) Manhattan Path Algorithm,Inform. Process. Lett. 19, 1984, 99–102.MATHCrossRefMathSciNetGoogle Scholar
  15. [L78]
    G. S. Luecker: A Data Structure for Orthogonal Range Queries,Proc. 19th FOCS, 1978, 28–34.Google Scholar
  16. [M84a]
    K. Mehlhorn:Data Structures and Algorithms, Vol. 1, Springer-Verlag, Berlin, 1984.Google Scholar
  17. [M84b]
  18. [M84c]
  19. [M86]
    K. Mehlhorn:Datenstrukturen und Algorithmen 1, Teubner, 1986.Google Scholar
  20. [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
  21. [N87]
    S. Näher: Dynamic Fractional Cascading oder die Verwaltung vieler linearer Listen, Dissertation, University des Saarlandes, Saarbrücken, 1987.Google Scholar
  22. [PS85]
    F. P. Preparata, M. I. Shamos:Computational Geometry, An Introduction, Springer-Verlag, Berlin, 1985.Google Scholar
  23. [T85]
    R. E. Tarjan: Amortized Computational Complexity,SIAM J. Algebraic Discrete Methods 6, 1985, 306–318.MATHCrossRefMathSciNetGoogle Scholar
  24. [T84]
    A. K. Tsakalidis: Maintaining Order in a Generalized Linked List,Acta Inform. 21, 1984, 101–112.MATHCrossRefMathSciNetGoogle Scholar
  25. [VW82]
    V. K. Vaishnavi, D. Wood: Rectilinear Line Segment Intersection, Layered Segment Trees and Dynamization,J. Algorithms,3, 1982, 160–176.MATHCrossRefMathSciNetGoogle Scholar
  26. [W78]
    D. E. Willard: New Data Structures for Orthogonal Range Queries, Technical Report, Harvard University, 1978.Google Scholar
  27. [W85]
    D. E. Willard: New Data Structures for Orthogonal Queries,SIAM J. Comput., 1985, 232–253.Google Scholar
  28. [WL85]
    D. E. Willard, G. S. Luecker: Adding Range Restriction Capability to Dynamic Data Structures,J. Assoc. Comput. Mach. 32, 1985, 597–617.MATHMathSciNetGoogle Scholar

Copyright information

© Springer-Verlag New York Inc. 1990

Authors and Affiliations

  • Kurt Mehlhorn
    • 1
  • Stefan Näher
    • 1
  1. 1.Universität des SaarlandesSaarbrückenFederal Republic of Germany

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