Pan Xu
Srikanta Tirthapura
Abstract
Space-filling curves have been used in the design of data structures for multidimensional data for many decades. A fundamental quality metric of a space-filling curve is its “clustering number” with respect to a class of queries, which is the average number of contiguous segments on the space-filling curve that a query region can be partitioned into. We present a characterization of the clustering number of a general class of space-filling curves, as well as the first nontrivial lower bounds on the clustering number for any space-filling curve. Our results answer questions that have been open for more than 15 years.
- J. Alber and R. Niedermeier. 2000. On multidimensional curves with hilbert property. Theory Comput. Syst. 33, 4, 295--312.Google Scholar
Cross Ref
- S. Aluru and F. Sevilgen. 1997. Parallel domain decomposition and load balancing using space-filling curves. In Proceedings of the International Conference on High-Performance Computing. 230--235. Google ScholarDigital Library
- T. Asano, D. Ranjan, T. Roos, E. Welzl, and P. Widmayer. 1997. Space-filling curves and their use in the design of geometric data structures. Theor. Comput. Sci. 181, 1, 3--15. Google ScholarDigital Library
- E. Bugnion, T. Roos, R. Wattenhofer, and P. Widmayer. 1997. Space filling curves versus random walks. In Algorithmic Foundations of Geographic Information Systems, Springer, 199--211. Google ScholarDigital Library
- C. Faloutsos. 1986. Multiattribute hashing using gray codes. SIGMOD Rec. 15, 227--238. Google ScholarDigital Library
- C. Faloutsos. 1988. Gray codes for partial match and range queries. IEEE Trans. Softw. Engin. 14, 1381--1393. Google ScholarDigital Library
- R. Gutman. 1999. Space-filling curves in geospatial applications. Dr. Dobb's J. http://www.drdobbs.com/database/space-filling-curves-in-geospatial-appli/184410998.Google Scholar
- H. J. Haverkort. 2011. Recursive tilings and space-filling curves with little fragmentation. J. Comput. Geom. 2, 1, 92--127.Google Scholar
- D. Hilbert. 1891. Uber die stetige abbildung einer linie auf ein flachenstuck. Math. Ann. 38, 459--460.Google ScholarCross Ref
- H. V. Jagadish. 1990. Linear clustering of objects with multiple attributes. In Proceedings of the ACM International Conference on Management of Data (SIGMOD'90). ACM Press, New York, 332--342. Google ScholarDigital Library
- H. V. Jagadish. 1997. Analysis of the hilbert curve for representing two-dimensional space. Inf. Process. Lett. 62, 17--22. Google ScholarDigital Library
- Y. Matias and A. Shamir. 1987. A video scrambling technique based on space filling curves. In Proceedings of the Conference on the Theory and Applications of Cryptographic Techniques (CRYPTO'87). Springer, 398--417. Google ScholarDigital Library
- B. Moon, H. V. Jagadish, C. Faloutsos, and J. H. Saltz. 2001. Analysis of the clustering properties of the hilbert space-filling curve. IEEE Trans. Knowl. Data Engin. 13, 1, 124--141. Google ScholarDigital Library
- G. Morton. 1966. A computer oriented geodetic data base and a new technique in file sequencing. Tech. rep., IBM.Google Scholar
- ORACLE. Oracle spatial and oracle locator. http://www.oracle.com/technetwork/database/options/spatial/overview/introduction/index.html.Google Scholar
- J. A. Orenstein and T. H. Merrett. 1984. A class of data structures for associative searching. In Proceedings of the ACM Symposium on Principles of Database Systems (PODS'84). ACM Press, New York, 181--190. Google ScholarDigital Library
- J. R. Pilkington and S. B. Baden. 1996. Dynamic partitioning of non-uniform structured workloads with space filling curves. IEEE Trans. Parallel Distrib. Syst. 7, 3, 288--300. Google ScholarDigital Library
- M. Warren and J. Salmon. 1993. A parallel hashed-octtree n-body algorithm. In Proceedings of the Supercomputing Conference. IEEE Computer Society/ACM Press, New York. Google ScholarDigital Library
- P. Xu and S. Tirthapura. 2012. On optimality of clustering properties of space filling curves. In Proceedings of the ACM Symposium on Principles of Database Systems (PODS'12). ACM Press, New York, 215--224. Google ScholarDigital Library
Index Terms
- Optimality of Clustering Properties of Space-Filling Curves
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