Analysis of n-Dimensional Quadtrees using the Hausdorff Fractal Dimension.

@inproceedings{DBLP:conf/vldb/FaloutsosG96,
  author    = {Christos Faloutsos and
               Volker Gaede},
  editor    = {T. M. Vijayaraman and
               Alejandro P. Buchmann and
               C. Mohan and
               Nandlal L. Sarda},
  title     = {Analysis of n-Dimensional Quadtrees using the Hausdorff Fractal
               Dimension},
  booktitle = {VLDB'96, Proceedings of 22th International Conference on Very
               Large Data Bases, September 3-6, 1996, Mumbai (Bombay), India},
  publisher = {Morgan Kaufmann},
  year      = {1996},
  isbn      = {1-55860-382-4},
  pages     = {40-50},
  ee        = {db/conf/vldb/FaloutsosG96.html},
  crossref  = {DBLP:conf/vldb/96},
  bibsource = {DBLP, http://dblp.uni-trier.de}
}

Abstract

There is mounting evidence [Man77, Sch91] that real datasets are statistically self-similar, and thus, `fractal'. This is an important insight since it permits a compact statistical description of spatial datasets; subsequently, as we show, it also forms the basis for the theoretical analysis of spatial access methods, without using the typical, but unrealistic, uniformity assumption.

In this paper, we focus on the estimation of the number of number of quadtree blocks that a real, spatial dataset will require. Using the the well-known Hausdorff fractal dimension, we derive some closed formulas which allow us to predict the number of quadtree blocks, given some few parameters. Using our formulas, it is possible to predict the space overhead and the response time of linear quadtrees/z-orderings [OM88], which are widely used in practice. In order to verify our analytical model, we performed an extensive experimental investigation using several real datasets coming from different domains. In these experiments, we found that our analytical model agrees well with our experiments as well as with older empirical observations on 2-d [Gae95b] and 3-d [ACF+94] data.

Copyright © 1996 by the VLDB Endowment. Permission to copy without fee all or part of this material is granted provided that the copies are not made or distributed for direct commercial advantage, the VLDB copyright notice and the title of the publication and its date appear, and notice is given that copying is by the permission of the Very Large Data Base Endowment. To copy otherwise, or to republish, requires a fee and/or special permission from the Endowment.

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References

[ACF+94]

Manish Arya, William F. Cody, Christos Faloutsos, Joel E. Richardson, Arthur Toya: QBISM: Extending a DBMS to Support 3D Medical Images. ICDE 1994: 314-325 BibTeX

[BB82]

...

[BF95]

Alberto Belussi, Christos Faloutsos: Estimating the Selectivity of Spatial Queries Using the `Correlation' Fractal Dimension. VLDB 1995: 299-310 BibTeX

[BKSS90]

Norbert Beckmann, Hans-Peter Kriegel, Ralf Schneider, Bernhard Seeger: The R*-Tree: An Efficient and Robust Access Method for Points and Rectangles. SIGMOD Conference 1990: 322-331 BibTeX

[Dye82]

...

[Fal92]

...

[FJM94]

Christos Faloutsos, H. V. Jagadish, Yannis Manolopoulos: Analysis of the n-Dimensional Quadtree Decomposition for Arbitrary Hyperectangles. IEEE Trans. Knowl. Data Eng. 9(3): 373-383(1997) BibTeX

[FK94]

Christos Faloutsos, Ibrahim Kamel: Beyond Uniformity and Independence: Analysis of R-trees Using the Concept of Fractal Dimension. PODS 1994: 4-13 BibTeX

[FRM94]

Christos Faloutsos, M. Ranganathan, Yannis Manolopoulos: Fast Subsequence Matching in Time-Series Databases. SIGMOD Conference 1994: 419-429 BibTeX

[Gae95a]

Volker Gaede: Geometric Information Makes Spatial Query Processing More Efficient. ACM-GIS 1995: 45-52 BibTeX

[Gae95b]

Volker Gaede: Optimal Redundancy in Spatial Database Systems. SSD 1995: 96-116 BibTeX

[Gar82]

Irene Gargantini: An Effective Way to Represent Quadtrees. Commun. ACM 25(12): 905-910(1982) BibTeX

[GR94]

Volker Gaede, Wolf-Fritz Riekert: Spatial Access Methods and Query Processing in the Object-Oriented GIS GODOT. AGDM 1994: 40- BibTeX

[Gut84]

Antonin Guttman: R-Trees: A Dynamic Index Structure for Spatial Searching. SIGMOD Conference 1984: 47-57 BibTeX

[Güt94]

Ralf Hartmut Güting: An Introduction to Spatial Database Systems. VLDB J. 3(4): 357-399(1994) BibTeX

[HS79]

...

[Jag91]

H. V. Jagadish: A Retrieval Technique for Similar Shapes. SIGMOD Conference 1991: 208-217 BibTeX

[Kli71]

...

[Man77]

Benoit Mandelbrot: Fractal Geometry of Nature. W. H. Freeman 1977
BibTeX

[OM84]

Jack A. Orenstein, T. H. Merrett: A Class of Data Structures for Associative Searching. PODS 1984: 181-190 BibTeX

[OM88]

Jack A. Orenstein, Frank Manola: PROBE Spatial Data Modeling and Query Processing in an Image Database Application. IEEE Trans. Software Eng. 14(5): 611-629(1988) BibTeX

[Ore89]

Jack A. Orenstein: Redundancy in Spatial Databases. SIGMOD Conference 1989: 295-305 BibTeX

[Sch91]

Manfred Schroeder: Fractals, Chaos, Power Laws: Minutes From an Infinite Paradise. W. H. Freeman 1991
BibTeX

[Sha88]

...

[SRF87]

Timos K. Sellis, Nick Roussopoulos, Christos Faloutsos: The R+-Tree: A Dynamic Index for Multi-Dimensional Objects. VLDB 1987: 507-518 BibTeX

[SS85]

...

[SSN87]

...

[VM96]

...

[Whi81]

...

Referenced by

1. Kevin S. Beyer, Jonathan Goldstein, Raghu Ramakrishnan, Uri Shaft: When Is ''Nearest Neighbor'' Meaningful? ICDT 1999: 217-235
2. Volker Gaede, Oliver Günther: Multidimensional Access Methods. ACM Comput. Surv. 30(2): 170-231(1998)
3. Geraldo Zimbrao, Jano Moreira de Souza: A Raster Approximation For Processing of Spatial Joins. VLDB 1998: 558-569
4. Joseph M. Hellerstein, Elias Koutsoupias, Christos H. Papadimitriou: On the Analysis of Indexing Schemes. PODS 1997: 249-256
5. Stefan Berchtold, Christian Böhm, Daniel A. Keim, Hans-Peter Kriegel: A Cost Model For Nearest Neighbor Search in High-Dimensional Data Space. PODS 1997: 78-86