Optimal Dynamic Range Searching inNon-replicating Index Structures

  • K. V. Ravi Kanth
  • Ambuj Singh
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 1540)


In this paper, we examine the complexity of multi-dimensional range searching in non-replicating index structures. Such nonreplicating structures achieve low storage costs and fast update times due to lack of multiple copies. We first obtain a lower bound for range searching in non-replicating structures. Assuming a simple tree structure model of an index, we prove that the worst-case time for a query retrieving t out of n data items is Ω(n/b)(d-1)/d + t/b), where d is the data dimensionality and b is the capacity of index nodes. We then propose a new index structure, called the O-tree, that achieves this query time in dynamic environments. Updates are supported in O(logb n) amortized time and exact match queries in O(logb n) worst-case time. This structure improves the query time of the best known non-replicating structure, the divided k-d tree, and is optimal for both queries and updates in non-replicating tree structures.


Data Item Index Structure Range Query Point Query Query Time 
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.


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. Ben75.
    J. L. Bentley. Multi-dimensional binary search trees used for associative searching. Communications of the ACM, 18:509–517, 1975.zbMATHCrossRefGoogle Scholar
  2. BKK96.
    S. Berchtold, D. A. Keim, and H. P. Kriegel. The X-tree: An index structure for high dimensional data. In Proc. Int. Conf. on Very Large Data Bases, pages 28–39, 1996.Google Scholar
  3. BKSS90.
    N. Beckmann, H. Kriegel, R. Schneider, and B. Seeger. The R* tree: An efficient and robust access method for points and rectangles. In Proc. ACM SIGMOD Int. Conf. on Management of Data, pages 322–331, May 23–25 1990.Google Scholar
  4. Cha90.
    B. Chazelle. Lower bounds for orthogonal range searching: I. The reporting case. Journal of the ACM, 37(2):200–212, April 1990.zbMATHCrossRefMathSciNetGoogle Scholar
  5. DRSS96.
    A. A. Diwan, S. Rane, S. Seshadri, and M. Sudarshan. Clustering techniques for minimizing external path length. Proc. Int. Conf. on Very Large Data Bases, pages 342–353, 1996.Google Scholar
  6. Fre87.
    M. W. Freeston. The BANG file: A new kind of grid file. In Proc. ACM SIGMOD Int. Conf. on Management of Data, pages 260–269, 1987.Google Scholar
  7. Fre95.
    M. W. Freeston. A general solution of the n-dimensional B-tree problem. Proc. ACM SIGMOD Int. Conf. on Management of Data, pages 80–92, May 1995.Google Scholar
  8. GI97.
    R. Grossi and F. Italiano. Efficient splitting and merging algorithms for order-decomposable searching problems. In International Colloquium on Automata, Languages and Programming (ICALP ‘97), pages 605–615, July 1997.Google Scholar
  9. Gut84.
    A. Guttman. R-trees: A dynamic index structure for spatial searching. In Proc. ACM SIGMOD Int. Conf. on Management of Data, pages 47–57, June 1984.Google Scholar
  10. HKP97.
    J. M. Hellerstein, E. Koutsoupias, and C. H. Papadimitriou. On the analysis of indexing schemes. In Proc. ACM Symp. on Principles of Database Systems, pages 249–256, Tucson, Arizona, May 1997.Google Scholar
  11. KS97.
    N. Katayama and S. Satoh. The SR-tree: An index structure for high-dimensional nearest-neighbor queries. In Proc. ACM SIGMOD Int. Conf. on Management of Data, pages 369–380, May 1997.Google Scholar
  12. KT98.
    E. Koutsoupias and D. S. Taylor. Tight bounds for 2-dimensional indexing schemes. In Proc. ACM Symp. on Principles of Database Systems, pages 52–58, Seattle, Washington, June 1998.Google Scholar
  13. LS90.
    D. B. Lomet and B. Salzberg. The hB-tree: A multiattribute indexing method with good guaranteed performance. Proc. ACM Transactions on Database Systems, 15(4):625–658, 1990.CrossRefGoogle Scholar
  14. Meh84.
    K. Mehlhorn. Data Structures and Algorithms 3: Multidimensional Searching and Computational Geometry. Springer-Verlag, 1984.Google Scholar
  15. Ove83.
    M. Overmars. The design of dynamic data structures. Lecture Notes in Computer Science 156, 1983.Google Scholar
  16. OvL81.
    M. Overmars and J. van Leeuwen. Worst-case optimal insertion and deletion methods for decomposable searching problems. Information Processing Letters, 12(4):168–172, 1981.zbMATHCrossRefGoogle Scholar
  17. Rav98.
    K. V. Ravi Kanth. Indexing multi-dimensional data. In Ph.D. thesis, University of California, Santa Barbara, April 1998.Google Scholar
  18. Rob81.
    J. T. Robinson. The kdb-tree: A search structure for large multi-dimensional dynamic indexes. In Proc. ACM SIGMOD Int. Conf. on Management of Data, pages 10–18, 1981.Google Scholar
  19. Sam89.
    H. Samet. The design and analysis of spatial data structures. Addison-Wesley, 1989.Google Scholar
  20. SM98.
    V. Samoladas and D. P. Miranker. A lower bound theorem for indexing schemes and its application to multidimensional range queries. In Proc. ACM Symp. on Principles of Database Systems, pages 44–51, Seattle, Washington, June 1998.Google Scholar
  21. SR95.
    S. Subramanian and S. Ramaswamy. The P-range tree: A new data structure for range searching in secondary memory. In Proc. ACM-SIAM Symposium on Discrete Algorithms, pages 378–387, 1995.Google Scholar
  22. Tar79.
    R. E. Tarjan. A class of algorithms which require non-linear time to maintain disjoint sets. Journal of Computer and System Sciences, 18:110–127, 1979.zbMATHCrossRefMathSciNetGoogle Scholar
  23. Vai89.
    P. M. Vaidya. Space-time tradeoffs for orthogonal range queries. SIAM Journal of Computing, 18(4):748–758, 1989.zbMATHCrossRefMathSciNetGoogle Scholar
  24. vKO91.
    M. van Kreveld and M. Overmars. The divided k-d tree. Algorithmica, 6:840–858, 1991.zbMATHCrossRefMathSciNetGoogle Scholar
  25. WJ96.
    D. White and R. Jain. Similarity indexing with the SS-tree. In Proc. Int. Conf. on Data Engineering, pages 516–523, 1996.Google Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 1999

Authors and Affiliations

  • K. V. Ravi Kanth
    • 1
  • Ambuj Singh
    • 2
  1. 1.Oracle NEDCNashuaUSA
  2. 2.Department of Computer ScienceUniversity of CaliforniaSanta BarbaraUSA

Personalised recommendations