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Data Structures for Halfplane Proximity Queries and Incremental Voronoi Diagrams

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LATIN 2006: Theoretical Informatics (LATIN 2006)

Part of the book series: Lecture Notes in Computer Science ((LNTCS,volume 3887))

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Abstract

We consider preprocessing a set S of n points in the plane that are in convex position into a data structure supporting queries of the following form: given a point q and a directed line ℓ in the plane, report the point of S that is farthest from (or, alternatively, nearest to) the point q subject to being to the left of line ℓ. We present two data structures for this problem. The first data structure uses O(n 1 + ε) space and preprocessing time, and answers queries in O(21/ε log n) time. The second data structure uses O(n log3 n) space and polynomial preprocessing time, and answers queries in O(log n) time. These are the first solutions to the problem with O(log n) query time and o(n 2) space.

In the process of developing the second data structure, we develop a new representation of nearest-point and farthest-point Voronoi diagrams of points in convex position. This representation supports insertion of new points in counterclockwise order using only O(log n) amortized pointer changes, subject to supporting O(log n)-time point-location queries, even though every such update may make Θ(n) combinatorial changes to the Voronoi diagram. This data structure is the first demonstration that deterministically and incrementally constructed Voronoi diagrams can be maintained in o(n) pointer changes per operation while keeping O(log n)-time point-location queries.

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Author information

Authors and Affiliations

  1. Department of CIS, Polytechnic University, Brooklyn, NY, USA

    Boris Aronov & John Iacono

  2. School of Computer Science, Carleton University, Ottawa, ON, Canada

    Prosenjit Bose & Michiel Smid

  3. Computer Science and Artificial Intelligence Lab, MIT, Cambridge, MA, USA

    Erik D. Demaine

  4. National ICT Australia, Sydney, Australia

    Joachim Gudmundsson

  5. Départment d’ Informatique, Université Libre de Bruxelles, Brussels, Belgium

    Stefan Langerman

Authors
  1. Boris Aronov
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  2. Prosenjit Bose
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  3. Erik D. Demaine
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  4. Joachim Gudmundsson
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  5. John Iacono
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  6. Stefan Langerman
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  7. Michiel Smid
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Editor information

Editors and Affiliations

  1. School of Business, Universidad Adolfo Ibáñez, Chile

    José R. Correa

  2. Dept. of Computer Science, University of Chile, Blanco Encalada 2120, 3er piso, Santiago, Chile

    Alejandro Hevia

  3. Dept. Ing. Matemática & Ctr. de Modelamiento Matemático, UMI 2807 U. Chile–CNRS, Chile

    Marcos Kiwi

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© 2006 Springer-Verlag Berlin Heidelberg

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Aronov, B. et al. (2006). Data Structures for Halfplane Proximity Queries and Incremental Voronoi Diagrams. In: Correa, J.R., Hevia, A., Kiwi, M. (eds) LATIN 2006: Theoretical Informatics. LATIN 2006. Lecture Notes in Computer Science, vol 3887. Springer, Berlin, Heidelberg. https://doi.org/10.1007/11682462_12

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  • DOI: https://doi.org/10.1007/11682462_12

  • Publisher Name: Springer, Berlin, Heidelberg

  • Print ISBN: 978-3-540-32755-4

  • Online ISBN: 978-3-540-32756-1

  • eBook Packages: Computer ScienceComputer Science (R0)

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