Reduced memory space for multi-dimensional search trees (extended abstract)

  • E. Dan Willard
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 182)


Our main result is that space O[N(log N / log log N)k−1] suffices for either doing dynamic k-dimensional aggregate orthogonal range queries in time O(logkN) on a set of N records, or for arbitrary ∈>0 to do static aggregate queries in time O(logk−1+εN). This result improves upon the memory space used by slightly more than one dozen previous authors by a factor O[(log log N)k−1], and it has applications to rectangle intersection problems, VLSI-design, relational data bases, and queries about the past.


Memory Space Range Query Retrieval Time Critical Node Auxiliary Field 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.


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  1. [AS81]
    Z. Avid and E. Shamir, A direct solution to range search and related problems for product regions, Proc. 22nd Ann. Symp. on Foundations of Computer Science (1981), pp. 123–126.Google Scholar
  2. [Be75]
    J.L. Bentley, Multidimensional binary search trees used for associative searching, Comm. of ACM 18 (1975), pp. 509–517.Google Scholar
  3. [Be80]
    —, Multidimensional divide-and-conquer, Comm. of ACM 23 (1980), pp. 214–228.Google Scholar
  4. [BM80]
    J.L. Bentley and H.A. Maurer, Efficient worst-case data structures for range searching Acta Inf. 13 (1980), pp. 155–168.Google Scholar
  5. [BS77]
    J.L. Bentley and M.I. Shamos, A problem in multi-variate statistics: algorithm, data structure and applications, 15-th Allerton Conf. on Comm., Contr., and Comp. (1977), p. 193–201.Google Scholar
  6. [BS80]
    J.L. Bentley and J.B. Saxe, Decomposable searching problems #1: static to dynamic transformations, J. Alg. (1980), pp. 301–358.Google Scholar
  7. [Ch83]
    B. Chazelle, Filter Search a New Approach to Query Processing, 24th IEEE Symp. on Foundations of Computer Science, 1983, pp. 122–132.Google Scholar
  8. [DM80]
    D. Dobkin and J.I. Munro, Efficient use of the past, Proc. 21st Ann. Sym. on Foundations of Computer Science, (1980), pp. 200–206.Google Scholar
  9. [Ed81]
    H. Edelsbrunner, A note on dynamic range searching, Bulletin of EATCS, 15 (1981), pp. 34–40.Google Scholar
  10. [EO81]
    H. Edelsbrunner and M.H. Overmars, On the equivalence of some rectangle search problems, Inf. Proc. Letters, 14(1982), pp. 124–127.Google Scholar
  11. [EO83]
    H. Edelsbrunner and M.H. Overmars, Batch dynamic solutions for decomposable search problems, U. Graz F118, 1983.Google Scholar
  12. [Fr81]
    M.L. Fredman, A lower bound on the complexity of orthogonal range queries, Journal of ACM 28 (1981), pp. 696–706.Google Scholar
  13. [GBT84]
    H. Gabow, J. Bentley and R. Tarjan, Scaling and Related Techniques for Geometry, 16th ACM STOC Symp. (1984), pp. 135–143.Google Scholar
  14. [Lu78]
    G.S. Lueker, A data structure for orthogonal range queries, Proc. 19th FOCS (1978), pp. 28–34.Google Scholar
  15. [Lu79]
    —, A transformation for adding range restriction capability to dynamic data structures for decomposable searching problems, Technical Report #129, Department of Information and Computer Science, University of California, Irvine (1979).Google Scholar
  16. [LW80]
    D.T. Lee and C.K. Wong, Quintary tree: a file structure for multidimensional database systems, ACM TODS 5 (1980), pp. 339–347.Google Scholar
  17. [LW82]
    G.S. Lueker and D.E. Willard, A data structure for dynamic range queries, Inf. Proc. Letters 15 (1982), pp. 209–213.Google Scholar
  18. [Mc81]
    E.M. McCreight, Priority search trees, Xerox report CSL-81-5, Xerox PARC, Palo Alto (1981).Google Scholar
  19. [Me84]
    K. Mehlhorn, Data Structures and Algorithms (Volume 3), published by Springer-Verlag, 1984.Google Scholar
  20. [Wi78a]
    D.E. Willard, Predicate-oriented database search algorithms, Ph.D. thesis, Harvard University, 1978. Also in Outstanding Dissertations in Computer Science, Garland Publishing, New York, 1979.Google Scholar
  21. [Wi78b]
    —, New data structures for orthogonal queries. First draft was Technical Report TR-22-78, Center for Research in Computing Technology, Harvard University (1978). Second draft appeared in 1982 Allerton Conf. on Comm., Contr. and Comp. pp. 462–471. Third draft of this paper is [Wi85] to appear in SIAM J. on Comp., in Feb. 1985.Google Scholar
  22. [Wi83a]
    —, Log-logarithmic worst-case range queries are possible in space O(N), Information Processing Letters, 17 (1984), pp. 81–84.Google Scholar
  23. [Wi83b]
    —, Predicate retrieval theory, 21st Allerton Conf. on Comm. Contr. and Comp. (1983), pp. 663–674.Google Scholar
  24. [Wi84a]
    —, Efficient Processing of Relational Calculus Expressions Using Range Query Theory, Proc. of 1984 ACM SIGMOD Conference, pp. 160–172.Google Scholar
  25. [Wi84b]
    —, Sampling Algorithms for Differentiable Batch Retrieval Problems, Proc. of ICALP-1984, published by Springer-Verlag, pp. 514–526.Google Scholar
  26. [Wi84c]
    —, Multi-dimensional search trees that provide new types of memory reduction, SUNY-Albany Technical Report TR 84-5, 1984.Google Scholar
  27. [Wi85]
    —, New data structure for orthogonal range queries, SIAM Journal on Computing, Feb. 1985.Google Scholar
  28. [WL85]
    D.E. Willard and G.S. Lueker, Adding range restriction capability to dynamic data structures, submitted for publication.Google Scholar
  29. [Ya82]
    A.C. Yao, On the complexity of maintaining partial sums, 14th ACM STOC Symp. (1982), p. 128–135.Google Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 1984

Authors and Affiliations

  • E. Dan Willard
    • 1
  1. 1.State University of New York at AlbanyAlbany

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