On the optimality of clustering properties of space filling curves

Space filling curves have for long been used in the design of data structures for multidimensional data. 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 non-trivial lower bounds on the clustering number for any space filling curve. Our results also answer an open problem that was posed by Jagadish in 1997.

[1]  Christos Faloutsos,et al.  Gray Codes for Partial Match and Range Queries , 1988, IEEE Trans. Software Eng..

[2]  Christos Faloutsos,et al.  Analysis of the Clustering Properties of the Hilbert Space-Filling Curve , 2001, IEEE Trans. Knowl. Data Eng..

[3]  Rolf Niedermeier,et al.  On Multidimensional Curves with Hilbert Property , 2000, Theory of Computing Systems.

[4]  H. V. Jagadish,et al.  Linear clustering of objects with multiple attributes , 1990, SIGMOD '90.

[5]  Michael S. Warren,et al.  A parallel hashed oct-tree N-body algorithm , 1993, Supercomputing '93. Proceedings.

[6]  Srinivas Aluru,et al.  Parallel domain decomposition and load balancing using space-filling curves , 1997, Proceedings Fourth International Conference on High-Performance Computing.

[7]  Roger Wattenhofer,et al.  Space Filling Curves versus Random Walks , 1996, Algorithmic Foundations of Geographic Information Systems.

[8]  H. V. Jagadish,et al.  Analysis of the Hilbert Curve for Representing Two-Dimensional Space , 1997, Inf. Process. Lett..

[9]  Desh Ranjan,et al.  Space-Filling Curves and Their Use in the Design of Geometric Data Structures , 1997, Theor. Comput. Sci..

[10]  Christos Faloutsos,et al.  Multiattribute hashing using Gray codes , 1986, SIGMOD '86.

[11]  Gerhard Weikum,et al.  ACM Transactions on Database Systems , 2005 .

[12]  Scott B. Baden,et al.  Dynamic Partitioning of Non-Uniform Structured Workloads with Spacefilling Curves , 1996, IEEE Trans. Parallel Distributed Syst..

[13]  Herman J. Haverkort,et al.  Recursive tilings and space-filling curves with little fragmentation , 2010, J. Comput. Geom..

[14]  Yossi Matias,et al.  A Video Scrambling Technique Based On Space Filling Curves , 1987, CRYPTO.

[15]  T. H. Merrett,et al.  A class of data structures for associative searching , 1984, PODS.

[16]  D. Hilbert.  Über die stetige Abbildung einer Linie auf ein Flächenstück , 1935 .