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Encyclopedia of Algorithms pp 1027–1030Cite as

Well Separated Pair Decomposition

2003; Gao, Zhang

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Keywords and Synonyms

Proximity algorithms for growth‐restricted metrics      

Problem Definition

Well‐separated pair decomposition, introduced by Callahan and Kosaraju [3], has found numerous applications in solving proximity problems for points in the Euclidean space. A pair of point sets (A, B) is c-well‐separated if the distance between A and B is at least c times the diameters of both A and B. A well‐separated pair decomposition of a point set consists of a set of well‐separated pairs that “cover” all the pairs of distinct points, i. e., any two distinct points belong to the different sets of some pair. Callahan and Kosaraju [3] showed that for any point set in a Euclidean space and for any constant \( { c\geq 1 } \), there always exists a c-well‐separated pair decomposition (c-WSPD) with linearly many pairs. This fact has been very useful for obtaining nearly linear time algorithms for many problems, such as computing k-nearest neighbors, N‑body potential fields, geometric spanners,...

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Notes

  1. 1.

    A metric induced by a graph (with positive edge weights) is the shortest-path distance metric of the graph.

  2. 2.

    Select an arbitrary node v and compute the shortest-path tree rooted at v. Suppose that the furthest node from v is distance D away. Then the diameter of the graph is no longer than 2D, by triangle inequality.

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Author information

Authors and Affiliations

  1. Department of Computer Science, Stony Brook University, Stony Brook, NY, USA

    Jie Gao

  2. HP Labs, Palo Alto, CA, USA

    Li Zhang

Authors
  1. Jie Gao
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  2. Li Zhang
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Editor information

Editors and Affiliations

  1. Department of Electrical Engineering and Computer ScienceMcCormick School of Engineering and Applied Science, Northwestern University, Evanston, IL, 60208, USA

    Ming-Yang Kao Professor of Computer Science

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© 2008 Springer-Verlag

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Gao, J., Zhang, L. (2008). Well Separated Pair Decomposition. In: Kao, MY. (eds) Encyclopedia of Algorithms. Springer, Boston, MA. https://doi.org/10.1007/978-0-387-30162-4_479

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