Algorithmica

, Volume 1, Issue 1–4, pp 133–162

Fractional cascading: I. A data structuring technique

  • Bernard Chazelle
  • Leonidas J. Guibas
Article

Abstract

In computational geometry many search problems and range queries can be solved by performing an iterative search for the same key in separate ordered lists. In this paper we show that, if these ordered lists can be put in a one-to-one correspondence with the nodes of a graph of degreed so that the iterative search always proceeds along edges of that graph, then we can do much better than the obvious sequence of binary searches. Without expanding the storage by more than a constant factor, we can build a data-structure, called afractional cascading structure, in which all original searches after the first can be carried out at only logd extra cost per search. Several results related to the dynamization of this structure are also presented. A companion paper gives numerous applications of this technique to geometric problems.

Key words

Binary search B-tree Iterative search Multiple look-up Range query Dynamization of data structures 

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References

  1. [BSa]
    J. L. Bentley and J. B. Saxe.Decomposable searching problems I: static to dynamic transformations. J. Algorithms, 1 (1980), 301–358.MATHCrossRefMathSciNetGoogle Scholar
  2. [BKZ]
    P. van Emde Boas, B. Kaas and E. Zijlstra.Design and implementation of an efficient priority queue. Math. Syst. Theory,10 (1977), 99–127.MATHCrossRefGoogle Scholar
  3. [C]
    B. Chazelle.Filtering search: A new approach to query-answering. Proc. 24th Ann. Symp. Found. Comp. Sci. (1983), pp. 122–132. To appear in SIAM J. Comput. (1986).Google Scholar
  4. [CG]
    B. Chazelle and L. J. Guibas.Fractional cascading II: applications. To appear in Algorithmica (1986).Google Scholar
  5. [Co]
    R. Cole.Searching and storing similar lists. Tech. Report No. 88, Courant Inst., New York University, New York, Oct. 1983. To apper in J. Algorithms.Google Scholar
  6. [FMN]
    O. Fries, K. Mehlhorn, and St. Näher.Dynamization of geometric data structures. Proc. 1st ACM Computational Geometry Symposium, 1985, pp. 168–176.Google Scholar
  7. [EGS]
    H. Edelsbrunner, L. J. Guibas, and J. Stolfi.Optimal point location in a monotone subdivision. To appear in SIAM J. Comput. Also DEC/SRC Research Report No. 2, 1984.Google Scholar
  8. [IA]
    H. Imai and T. Asano.Dynamic segment intersection search with applications, Proc. of 25th FOCS Sumposium, 1984, pp. 393–402.Google Scholar
  9. [GT]
    H. N. Gabow and R. E. Tarjan.A linear-time algorithm for a special case of disjoint set union. Proc. of 24th FOCS Symposium, 1983, pp. 246–251.Google Scholar
  10. [O]
    M. H. Overmars.The design of dynamic data structures. PhD Thesis, University of Utrecht, The Netherlands, 1983.Google Scholar
  11. [T]
    R. E. Tarjan.Amortized computational complexity. SIAM J. Alg. Disc. Meth.,6 (2) (April 1985), 306–318.MATHCrossRefMathSciNetGoogle Scholar
  12. [VW]
    V. K. Vaishani and D. Wood.Rectilinear line segment intersection, layered segment trees, and dynamization. J. Algorithms,3 (1982), 160–176.CrossRefMathSciNetGoogle Scholar

Copyright information

© Springer-Verlag New York Inc. 1986

Authors and Affiliations

  • Bernard Chazelle
    • 1
  • Leonidas J. Guibas
    • 2
  1. 1.Brown University and Ecole Normale SupérieureUSA
  2. 2.DEC/SRC and Stanford UniversityUSA
  3. 3.DEC Systems Research CenterPalo AltoUSA

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