Advertisement

Determining the Convex Hull in Large Multidimensional Databases

  • Christian Böhm
  • Hans-Peter Kriegel
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 2114)

Abstract

Determining the convex hull of a point set is a basic operation for many applications of pattern recognition, image processing, statistics, and data mining. Although the corresponding point sets are often large, the convex hull operation has not been considered much in a database context, and state-of-theart algorithms do not scale well to non main-memory resident data sets. In this paper, we propose two convex hull algorithms which are based on multidimensional index structures such as R-trees. One of them traverses the index depth-first. The other algorithm assigns a priority to each active node (nodes which are not yet accessed but known to the system), which corresponds to the maximum distance of the node region to the tentative convex hull. We show both theoretically as well as experimentally that our algorithms outperform competitive techniques that do not exploit indexes.

Keywords

Convex Hull Online Algorithm Lower Left Corner Isotonic Regression Multidimensional Index 
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.

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. 1.
    Agrawal R., Faloutsos C., Swami A.: Efficient similarity search in sequence databases, Int. Conf. on Found. of Data Organization and Algorithms, 1993.Google Scholar
  2. 2.
    Agrawal R., Imielinski T., Swami A.: Mining Association Rules between Sets of Items in Large Databases, ACM SIGMOD Int. Conf. on Management of Data, 1993.Google Scholar
  3. 3.
    Ankerst M., Kriegel H.-P., Seidl T.: A Multi-Step Approach for Shape Similarity Search in Image Databases, IEEE Transactions on Knowledge and Data Engineering (TKDE), Vol. 10,No. 6, 1998.Google Scholar
  4. 4.
    Akl S.G., Toussaint G.T.: Efficient Convex Hull Algorithms for Pattern Recognition Applications, Int. Joint Conf. on Pattern Recognition, 1978.Google Scholar
  5. 5.
    Börzsönyi S., Kossmann D., Stocker K.: The Skyline Operator, Int. Conf on Data Engineering, 2000.Google Scholar
  6. 6.
    Berchtold S., Böhm C., Kriegel H.-P.: Improving the Query Performance of High-Dimensional Index Structures Using Bulk-Load Operations, Int. Conf. on Extending Database Technology, 1998.Google Scholar
  7. 7.
    Brown K.Q.: Geometric Transformations for Fast Geometric Algorithms, Ph.D. thesis, Dept. of Computer Science, Carnegie Mellon Univ., Dec. 1979a.Google Scholar
  8. 8.
    Bykat A.: Convex Hull of a Finite Set of Points in Two Dimensions, Info. Proc. Lett., No. 7, 1978.Google Scholar
  9. 9.
    Chang Y.-C, Bergman L.D., Castelli V., Li C.-S., Lo M.-L., Smith J. R.: The Onion Technique: Indexing for Linear Optimization Queries, ACM SIGMOD Int. Conf. on Management of Data, 2000.Google Scholar
  10. 10.
    Eddy W.: A New Convex Hull Algorithm for Planar Sets, ACM Trans. Math. Software 3(4), 1977.Google Scholar
  11. 11.
    Ester M., Frommelt A., Kriegel H.-P., Sander J.: Algorithms for Characterization and Trend Detection in Spatial Databases, Int. Conf. on Knowledge Discovery and Data Mining (KDD), 1998.Google Scholar
  12. 12.
    Faloutsos C., Barber R., Flickner M., Hafner J., Niblack W., Petkovic D., Equitz W.: Efficient and Effective Querying by Image Content, Journal of Intelligent Information Systems, Vol. 3, 1994.Google Scholar
  13. 13.
    Freeman H., Shapira R.: Determining the Minimum-Area Encasing Rectangle for an Arbitrary Closed Curve, Comm. ACM, Vol 18,No. 7, 1975.Google Scholar
  14. 14.
    Gaede V., Günther O.: Multidimensional Access Methods, ACM Computing Surveys, 30(2), 1998.Google Scholar
  15. 15.
    Gastwirth J.: On Robust Procedures, Journal Amer. Stat. Ass., Vol 65, 1966.Google Scholar
  16. 16.
    Graham R.L.: An Efficient Algorithm for Determining the Convex Hull of a Finite Planar Set, Info. Proc. Lett., Vol. 1, 1972.Google Scholar
  17. 17.
    Green P.J., Silverman B.W.: Constructing the Convex Hull of a Set of Points in the Plane, Computer Journal, Vol. 22, 1979.Google Scholar
  18. 18.
    Guttman A.: R-trees: A Dynamic Index Structure for Spatial Searching, Proc. ACM SIGMOD Int. Conf. on Management of Data, 1984.Google Scholar
  19. 19.
    Huber P.J.: Robust Statistics: A Review, Ann. Math. Stat., Vol. 43,No. 3, 1972.Google Scholar
  20. 20.
    Jagadish H.V.: A Retrieval Technique for Similar Shapes, ACM SIGMOD Int. Conf. Manag. Data, 1991.Google Scholar
  21. 21.
    Jarvis R.A.: On the Identification of the Convex Hull of a Finite Set of Points, Info. Proc. Lett., Vol. 2, 1973.Google Scholar
  22. 22.
    Knorr E.M., Ng R.T.: Algorithms for Mining Distance-Based Outliers in Large Datasets, Int. Conf. on Very Large Data Bases (VLDB), 1998.Google Scholar
  23. 23.
    Kriegel H.-P., Seidl T.: Approximation-Based Similarity Search for 3-D Surface Segments, GeoInformatica Int. Journal, Vol. 2,No. 2, 1998.Google Scholar
  24. 24.
    Korn F., Sidiropoulos N., Faloutsos C., Siegel E., Protopapas Z.: Fast Nearest Neighbor Search in Medical Image Databases, Int. Conf. on Very Large Data Bases (VLDB), 1996.Google Scholar
  25. 25.
    Mitchell T.M.: Machine Learning, McCraw-Hill, 1997.Google Scholar
  26. 26.
    Preparata F.P.: An Optimal Real Time Algorithm for Planar Convex Hulls, Comm. ACM 22, 1979.Google Scholar
  27. 27.
    Preparata F.P., Shamos M.I.: Computational Geometry, Springer New York, 1985.Google Scholar
  28. 28.
    Rosenfeld A.: Picture Processing by Computers, Academic Press, New York, 1969.Google Scholar
  29. 29.
    Sander J., Ester M., Kriegel H.-P., Xu X.: Density-Based Clustering in Spatial Databases: The Algorithm GDBSCAN and its Applications, Data Mining and Knowledge Discovery, Vol. 2,No. 2, 1998.Google Scholar
  30. 30.
    Shamos M.I.: Computational Geometry, Ph.D. Thesis, Dept. of CS, Yale University, 1978.Google Scholar
  31. 31.
    Seidl T., Kriegel H.-P.: Efficient User-Adaptable Similarity Search in Large Multimedia Databases, Int. Conf. on Very Large Data Bases (VLDB), 1997.Google Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2001

Authors and Affiliations

  • Christian Böhm
    • 1
  • Hans-Peter Kriegel
    • 1
  1. 1.University of MunichMunichGermany

Personalised recommendations