# Geometric Stream Mining

**DOI:**https://doi.org/10.1007/978-1-4899-7993-3_180-2

## 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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- 1.Agarwal PK, Har-Peled S, Varadarajan KR. Approximating extent measures of points. J ACM. 2004;51(4):606–33.MathSciNetCrossRefMATHGoogle Scholar
- 2.Bagchi A, Chaudhary A, Eppstein D, Goodrich MT. Deterministic sampling and range counting in geometric data streams. In: Proceedings of the 20th annual symposium on computational geometry. 2004.p. 144–51.Google Scholar
- 3.Chan TM. Faster core-set constructions and data-stream algorithms in fixed dimensions. Comput Geom. 2006;35(1–2):20–35.MathSciNetCrossRefMATHGoogle Scholar
- 4.Cormode G, Muthukrishnan S, Rozenbaum I Summarizing and mining inverse distributions on data streams via dynamic inverse sampling. In: Proceedings of the 31st international conference on very large data bases. 2005. p. 25–36.Google Scholar
- 5.Frahling G, Indyk P, Sohler C Sampling in dynamic data streams and applications. In: Proceedings of the 21st annual symposium on computational geometry. 2005. 142–9.Google Scholar
- 6.Indyk P. Algorithms for dynamic geometric problems over data streams. In: Proceedings of the 41st annual ACM symposium on theory of computing. 2004. p. 373–80.Google Scholar
- 7.Korn F, Muthukrishnan S, Srivastava D. Reverse nearest neighbor aggregates over data streams. In: Proceedings 28th international conference on very large data bases. 2002. p. 814–25.Google Scholar
- 8.Preparata FP, Shamos MI. Computational geometry: an introduction. 3rd ed. Berlin Hiedelberg New York: Springer; 1990.MATHGoogle Scholar
- 9.Vitter JS. Random sampling with a reservoir. ACM Trans Math Software. 1985;11(1):37–57.MathSciNetCrossRefMATHGoogle Scholar