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Encyclopedia of Database Systems pp 1239–1244Cite as

Geometric Stream Mining

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Definition

Let P ={p 1, p 2,...} be a stream of points in the metric space (X, L q ). Usually, X = ℝ d or X = {1,...,U}d (discrete case), and L q = L 2 is the Euclidean distance. The set P is called a spatial data stream. Geometric stream mining algorithms compute the (approximate) answer to a geometric question over the subset of P seen so far. For example, the diameter problem asks to maintain the pair of points that are farthest away in the current stream. A more comprehensive list of problems is presented later.

Historical Background

Geometric algorithms in the offline setting have been extensively studied over the past decades. Their applications encompass many fields, such as image processing, robotics, data mining, or VLSI design. For an introduction to computational geometry, refer to the book [8]. On the other hand, research on spatial data streams is a recent development. Shortly after the first results on numeric data streams appeared, a slew of papers argued that in many...

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Recommended Reading

  1. Agarwal P.K., Har-Peled S., and Varadarajan K.R. Approximating extent measures of points. J. ACM, 51(4):606–633, 2004.

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  2. Bagchi A., Chaudhary A., Eppstein D., and Goodrich M.T. Deterministic sampling and range counting in geometric data streams. In Proc. 20th Annual Symposium on Computational Geometry, 2004, pp. 144–151.

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  3. Chan T.M. Faster core-set constructions and data-stream algorithms in fixed dimensions. Comput. Geom., 35(1–2):20–35, 2006.

    Article  MATH  MathSciNet  Google Scholar 

  4. Cormode G., Muthukrishnan S., and Rozenbaum I. Summarizing and mining inverse distributions on data streams via dynamic inverse sampling. In Proc. 31st Int. Conf. on Very Large Data Bases, 2005, pp. 25–36.

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  5. Frahling G., Indyk P., and Sohler C. Sampling in dynamic data streams and applications. In Proc. 21st Annual Symposium on Computational Geometry, 2005, 142–149.

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  6. Indyk P. Algorithms for dynamic geometric problems over data streams. In Proc. 41st Annual ACM Symp. on Theory of Computing, 2004, pp. 373–380.

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  7. Korn F., Muthukrishnan S., and Srivastava D. Reverse nearest neighbor aggregates over data streams. In Proc. 28th Int. Conf. on Very Large Data Bases, 2002, 814–825.

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  8. Preparata F.P. and Shamos M.I. Computational Geometry: An Introduction, 3rd edn. Springer, Berlin Hiedelberg New York, Oct. 1990.

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  9. Vitter J.S. Random sampling with a reservoir. ACM Trans. Math. Software, 11(1)37–57, 1985.

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Author information

Authors and Affiliations

  1. AT&T Labs, Florham Park, NJ, USA

    Cecilia M. Procopiuc

Authors
  1. Cecilia M. Procopiuc
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Editor information

Editors and Affiliations

  1. College of Computing, Georgia Institute of Technology, 266 Ferst Drive, 30332-0765, Atlanta, GA, USA

    LING LIU (Professor) (Professor)

  2. Database Research Group David R. Cheriton School of Computer Science, University of Waterloo, 200 University Avenue West, N2L 3G1, Waterloo, ON, Canada

    M. TAMER ÖZSU (Professor and Director, University Research Chair) (Professor and Director, University Research Chair)

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© 2009 Springer Science+Business Media, LLC

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Procopiuc, C.M. (2009). Geometric Stream Mining. In: LIU, L., ÖZSU, M.T. (eds) Encyclopedia of Database Systems. Springer, Boston, MA. https://doi.org/10.1007/978-0-387-39940-9_180

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