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The Union of Colorful Simplices Spanned by a Colored Point Set

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Combinatorial Optimization and Applications (COCOA 2010)

Part of the book series: Lecture Notes in Computer Science ((LNTCS,volume 6508))

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Abstract

A simplex spanned by a colored point set in Euclidean d-space is colorful if all vertices have distinct colors. The union of all full-dimensional colorful simplices spanned by a colored point set is called the colorful union. We show that for every d ∈ ℕ, the maximum combinatorial complexity of the colorful union of n colored points in ℝd is between \(\Omega(n^{(d-1)^2})\) and \(O(n^{(d-1)^2}\log n)\). For d = 2, the upper bound is known to be O(n), and for d = 3 we present an upper bound of O(n 4 α(n)), where α(·) is the extremely slowly growing inverse Ackermann function. We also prove several structural properties of the colorful union. In particular, we show that the boundary of the colorful union is covered by O(n d − 1) hyperplanes, and the colorful union is the union of d + 1 star-shaped polyhedra. These properties lead to efficient data structures for point inclusion queries in the colorful union.

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Author information

Authors and Affiliations

  1. Institut für Mathematsche Logik und Grundlagenforschung, Universität Münster, Germany

    André Schulz

  2. Department of Mathematics and Statistics, University of Calgary, AB, Canada

    Csaba D. Tóth

Authors
  1. André Schulz
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  2. Csaba D. Tóth
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Editor information

Editors and Affiliations

  1. Department of Computer Science, University of Texas at Dallas, 75080, Richardson, TX, USA

    Weili Wu  & Ovidiu Daescu  & 

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Schulz, A., Tóth, C.D. (2010). The Union of Colorful Simplices Spanned by a Colored Point Set. In: Wu, W., Daescu, O. (eds) Combinatorial Optimization and Applications. COCOA 2010. Lecture Notes in Computer Science, vol 6508. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-17458-2_27

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  • DOI: https://doi.org/10.1007/978-3-642-17458-2_27

  • Publisher Name: Springer, Berlin, Heidelberg

  • Print ISBN: 978-3-642-17457-5

  • Online ISBN: 978-3-642-17458-2

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