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Euclidean Voronoi Diagrams of 3D Spheres: Their Construction and Related Problems from Biochemistry

  • Conference paper
Mathematics of Surfaces XI

Part of the book series: Lecture Notes in Computer Science ((LNTCS,volume 3604))

Abstract

Voronoi diagrams have several important applications in science and engineering. While the properties and algorithms for the ordinary Voronoi diagrams of point sets have been well-known, their counterparts for a set of spheres have not been sufficiently studied.

In this paper, we present properties and two algorithms for Voronoi diagrams of 3D spheres based on the Euclidean distance from the surface of spheres. Starting from a valid initial Voronoi vertex, the edge-tracing algorithm follows Voronoi edges until the construction is completed. The region-expansion algorithm constructs the desired diagram by successively expanding the Voronoi region of each sphere, one after another, via a series of topology operations, starting from the ordinary Voronoi diagram for the centres of spheres.

In the worst-case, the edge-tracing algorithm takes O(mn) time, and the region-expansion algorithm takes O(n 3 log n) time, where m and n are the numbers of edges and spheres, respectively. It should, however, be noted that the worst-case time complexity for both algorithms reduce to O(n 2) for proteins since the number of immediate neighbor atoms for an atom is constant. Adapting appropriate filtering techniques to reduce search space, the expected time complexities can even reduce to linear.

Then, we show how such a Voronoi diagram can be used for solving various important geometric problems in biological systems by illustrating two examples: the computation of surfaces defined on a protein, and the extraction and characterization of interaction interfaces between multiple proteins.

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Author information

Authors and Affiliations

  1. Department of Industrial Engineering, Hanyang University, 17 Haengdang-dong, Seongdong-gu, Seoul, 133-791, South Korea

    Deok-Soo Kim

  2. Voronoi Diagram Research Center, Hanyang University, 17 Haengdang-dong, Seongdong-gu, Seoul, 133-791, South Korea

    Donguk Kim & Youngsong Cho

Authors
  1. Deok-Soo Kim
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  2. Donguk Kim
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  3. Youngsong Cho
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Editor information

Editors and Affiliations

  1. School of Computer Science, Cardiff University, Cardiff, UK

    Ralph Martin

  2. Department of Computer Science, Loughborough University, LE11 3TU, Leicestershire, UK

    Helmut Bez

  3. Numerical Geometry Ltd., UK

    Malcolm Sabin

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© 2005 Springer-Verlag Berlin Heidelberg

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Kim, DS., Kim, D., Cho, Y. (2005). Euclidean Voronoi Diagrams of 3D Spheres: Their Construction and Related Problems from Biochemistry. In: Martin, R., Bez, H., Sabin, M. (eds) Mathematics of Surfaces XI. Lecture Notes in Computer Science, vol 3604. Springer, Berlin, Heidelberg. https://doi.org/10.1007/11537908_16

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  • DOI: https://doi.org/10.1007/11537908_16

  • Publisher Name: Springer, Berlin, Heidelberg

  • Print ISBN: 978-3-540-28225-9

  • Online ISBN: 978-3-540-31835-4

  • eBook Packages: Computer ScienceComputer Science (R0)

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