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A Sum of Squares Theorem for Visibility Complexes and Applications

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Discrete and Computational Geometry

Part of the book series: Algorithms and Combinatorics ((AC,volume 25))

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Abstract

We present a new method to implement in constant amortized time the flip operation of the so-called Greedy Flip Algorithm, an optimal algorithm to compute the visibility complex of a collection of pairwise disjoint bounded convex sets of constant complexity (disks). The method uses simple data structures and only the left-turn predicate for disks; it relies, among other things, on a sum of squares like theorem for visibility complexes stated and proved in this paper. (The sum of squares theorem for a simple arrangement of lines states that the average value of the square of the number of vertices of a face of the arrangement is bounded by a constant.)

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Authors and Affiliations

  1. Department of Computer Science, Ecole Normale Supérieure, UMR 8548 (CNRS), Paris, France

    Pierre Angelier & Michel Pocchiola

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  1. Pierre Angelier
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  2. Michel Pocchiola
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Editors and Affiliations

  1. Department of Computer and Information Science, Polytechnic University, Six Metro Tech Center, Brooklyn, NY, 11201, USA

    Boris Aronov

  2. School of Mathematics, Georgia Institute of Technology, Atlanta, GA, 30332-0160, USA

    Saugata Basu

  3. City College and Courant Institute, 251 Mercer St., New York, NY, 10012, USA

    János Pach

  4. School of Computer Science, Tel Aviv University, Tel Aviv, 69978, Israel

    Micha Sharir

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Angelier, P., Pocchiola, M. (2003). A Sum of Squares Theorem for Visibility Complexes and Applications. In: Aronov, B., Basu, S., Pach, J., Sharir, M. (eds) Discrete and Computational Geometry. Algorithms and Combinatorics, vol 25. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-55566-4_5

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  • DOI: https://doi.org/10.1007/978-3-642-55566-4_5

  • Publisher Name: Springer, Berlin, Heidelberg

  • Print ISBN: 978-3-642-62442-1

  • Online ISBN: 978-3-642-55566-4

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