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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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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