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Finding Pairwise Intersections Inside a Query Range

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Algorithms and Data Structures (WADS 2015)

Part of the book series: Lecture Notes in Computer Science ((LNTCS,volume 9214))

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Abstract

We study the following problem: preprocess a set \(\mathcal {O}\) of objects into a data structure that allows us to efficiently report all pairs of objects from \(\mathcal {O}\) that intersect inside an axis-aligned query range \(Q\). We present data structures of size \(O(n\,\mathrm {polylog}\,n)\) and with query time \(O((k+1)\,\mathrm {polylog}\, n)\) time, where k is the number of reported pairs, for two classes of objects in the plane: axis-aligned rectangles and objects with small union complexity. For the 3-dimensional case where the objects and the query range are axis-aligned boxes in \(\mathbb {R}^3\), we present a data structure of size \(O(n\sqrt{n}\, \mathrm {polylog}\, n)\) and query time \(O((\sqrt{n}+k)\,\mathrm {polylog}\,n)\). When the objects and query are fat, we obtain \(O((k+1)\,\mathrm {polylog}\,n)\) query time using \(O(n\,\mathrm {polylog}\,n)\) storage.

M. de Berg and A.D. Mehrabi were supported by the Netherlands Organization for Scientific Research (NWO) under grants 024.002.003 and 612.001.118, respectively. J. Gudmundsson was supported by the Australian Research Council (project numbers FT100100755 and DP150101134)

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Authors and Affiliations

  1. Department of Computer Science, TU Eindhoven, Eindhoven, The Netherlands

    Mark de Berg & Ali D. Mehrabi

  2. School of IT, University of Sydney, Sydney, Australia

    Joachim Gudmundsson

Authors
  1. Mark de Berg
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  2. Joachim Gudmundsson
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  3. Ali D. Mehrabi
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Corresponding author

Correspondence to Ali D. Mehrabi .

Editor information

Editors and Affiliations

  1. Carleton University, Ottawa, Canada

    Frank Dehne

  2. Carleton University, Ottawa, Canada

    Jörg-Rüdiger Sack

  3. University of Victoria, Victoria, Canada

    Ulrike Stege

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© 2015 Springer International Publishing Switzerland

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de Berg, M., Gudmundsson, J., Mehrabi, A.D. (2015). Finding Pairwise Intersections Inside a Query Range. In: Dehne, F., Sack, JR., Stege, U. (eds) Algorithms and Data Structures. WADS 2015. Lecture Notes in Computer Science(), vol 9214. Springer, Cham. https://doi.org/10.1007/978-3-319-21840-3_20

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  • DOI: https://doi.org/10.1007/978-3-319-21840-3_20

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  • Publisher Name: Springer, Cham

  • Print ISBN: 978-3-319-21839-7

  • Online ISBN: 978-3-319-21840-3

  • eBook Packages: Computer ScienceComputer Science (R0)

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