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.
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References
Agrawal R., Faloutsos C., Swami A.: Efficient similarity search in sequence databases, Int. Conf. on Found. of Data Organization and Algorithms, 1993.
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.
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.
Akl S.G., Toussaint G.T.: Efficient Convex Hull Algorithms for Pattern Recognition Applications, Int. Joint Conf. on Pattern Recognition, 1978.
Börzsönyi S., Kossmann D., Stocker K.: The Skyline Operator, Int. Conf on Data Engineering, 2000.
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.
Brown K.Q.: Geometric Transformations for Fast Geometric Algorithms, Ph.D. thesis, Dept. of Computer Science, Carnegie Mellon Univ., Dec. 1979a.
Bykat A.: Convex Hull of a Finite Set of Points in Two Dimensions, Info. Proc. Lett., No. 7, 1978.
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.
Eddy W.: A New Convex Hull Algorithm for Planar Sets, ACM Trans. Math. Software 3(4), 1977.
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.
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.
Freeman H., Shapira R.: Determining the Minimum-Area Encasing Rectangle for an Arbitrary Closed Curve, Comm. ACM, Vol 18,No. 7, 1975.
Gaede V., Günther O.: Multidimensional Access Methods, ACM Computing Surveys, 30(2), 1998.
Gastwirth J.: On Robust Procedures, Journal Amer. Stat. Ass., Vol 65, 1966.
Graham R.L.: An Efficient Algorithm for Determining the Convex Hull of a Finite Planar Set, Info. Proc. Lett., Vol. 1, 1972.
Green P.J., Silverman B.W.: Constructing the Convex Hull of a Set of Points in the Plane, Computer Journal, Vol. 22, 1979.
Guttman A.: R-trees: A Dynamic Index Structure for Spatial Searching, Proc. ACM SIGMOD Int. Conf. on Management of Data, 1984.
Huber P.J.: Robust Statistics: A Review, Ann. Math. Stat., Vol. 43,No. 3, 1972.
Jagadish H.V.: A Retrieval Technique for Similar Shapes, ACM SIGMOD Int. Conf. Manag. Data, 1991.
Jarvis R.A.: On the Identification of the Convex Hull of a Finite Set of Points, Info. Proc. Lett., Vol. 2, 1973.
Knorr E.M., Ng R.T.: Algorithms for Mining Distance-Based Outliers in Large Datasets, Int. Conf. on Very Large Data Bases (VLDB), 1998.
Kriegel H.-P., Seidl T.: Approximation-Based Similarity Search for 3-D Surface Segments, GeoInformatica Int. Journal, Vol. 2,No. 2, 1998.
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.
Mitchell T.M.: Machine Learning, McCraw-Hill, 1997.
Preparata F.P.: An Optimal Real Time Algorithm for Planar Convex Hulls, Comm. ACM 22, 1979.
Preparata F.P., Shamos M.I.: Computational Geometry, Springer New York, 1985.
Rosenfeld A.: Picture Processing by Computers, Academic Press, New York, 1969.
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.
Shamos M.I.: Computational Geometry, Ph.D. Thesis, Dept. of CS, Yale University, 1978.
Seidl T., Kriegel H.-P.: Efficient User-Adaptable Similarity Search in Large Multimedia Databases, Int. Conf. on Very Large Data Bases (VLDB), 1997.
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Böhm, C., Kriegel, HP. (2001). Determining the Convex Hull in Large Multidimensional Databases. In: Kambayashi, Y., Winiwarter, W., Arikawa, M. (eds) Data Warehousing and Knowledge Discovery. DaWaK 2001. Lecture Notes in Computer Science, vol 2114. Springer, Berlin, Heidelberg. https://doi.org/10.1007/3-540-44801-2_29
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DOI: https://doi.org/10.1007/3-540-44801-2_29
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