### Finding Pairwise Intersections Inside a Query Range

Mark de Berg, Joachim Gudmundsson, Ali D. Mehrabi
<span title="2017-10-31">2017</span> <i title="Springer Nature America, Inc"> <a target="_blank" rel="noopener" href="https://fatcat.wiki/container/qhi3z76be5c5xeihac5cyiid3m" style="color: black;">Algorithmica</a> </i> &nbsp;
We study the following problem: preprocess a set O of objects into a data structure that allows us to efficiently report all pairs of objects from O that intersect inside an axis-aligned query range Q . We present data structures of size O ( n · polylog n ) and with query time O ( ( k + 1 ) · polylog n ) time, where k is the number of reported pairs, for two classes of objects in R 2 : axis-aligned rectangles and objects with small union complexity. For the 3-dimensional case where the objects
more &raquo; ... nd the query range are axis-aligned boxes in R 3 , we present a data structure of size O ( n n · polylog n ) and query time O ( ( n + k ) · polylog n ) . When the objects and query are fat, we obtain O ( ( k + 1 ) · polylog n ) query time using O ( n · polylog n ) storage.
<span class="external-identifiers"> <a target="_blank" rel="external noopener noreferrer" href="https://doi.org/10.1007/s00453-017-0384-3">doi:10.1007/s00453-017-0384-3</a> <a target="_blank" rel="external noopener" href="https://www.ncbi.nlm.nih.gov/pubmed/30956379">pmid:30956379</a> <a target="_blank" rel="external noopener" href="https://pubmed.ncbi.nlm.nih.gov/PMC6428404/">pmcid:PMC6428404</a> <a target="_blank" rel="external noopener" href="https://fatcat.wiki/release/bxt5dpw7ozbvdbfgzdp457oobi">fatcat:bxt5dpw7ozbvdbfgzdp457oobi</a> </span>
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