When quoting this document, please refer to the following
DOI: 10.4230/LIPIcs.ISAAC.2017.60
URN: urn:nbn:de:0030-drops-82417
URL: http://drops.dagstuhl.de/opus/volltexte/2017/8241/
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### Finding Pairwise Intersections of Rectangles in a Query Rectangle

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### Abstract

We consider the following problem: Preprocess a set S of n axis-parallel boxes in \mathbb{R}^d so that given a query of an axis-parallel box Q in \mathbb{R}^d, the pairs of boxes of S whose intersection intersects the query box can be reported efficiently. For the case that d=2, we present a data structure of size O(n\log n) supporting O(\log n+k) query time, where k is the size of the output. This improves the previously best known result by de Berg et al. which requires O(\log n\log^* n+ k\log n) query time using O(n\log n) space.There has been no known result for this problem for higher dimensions, except that for d=3, the best known data structure supports O(\sqrt{n}+k\log^2\log^* n) query time using O(n\sqrt {n}\log n) space. For a constant d>2, we present a data structure supporting O(n^{1-\delta}\log^{d-1} n + k \polylog n) query time for any constant 1/d\leq\delta<1.The size of the data structure is O(n^{\delta d}\log n) if 1/d\leq\delta<1/2, or O(n^{\delta d-2\delta+1}) if 1/2\leq \delta<1.

### BibTeX - Entry

@InProceedings{oh_et_al:LIPIcs:2017:8241,
author =	{Eunjin Oh and Hee-Kap Ahn},
title =	{{Finding Pairwise Intersections of Rectangles in a Query Rectangle}},
booktitle =	{28th International Symposium on Algorithms and Computation (ISAAC 2017)},
pages =	{60:1--60:12},
series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN =	{978-3-95977-054-5},
ISSN =	{1868-8969},
year =	{2017},
volume =	{92},
editor =	{Yoshio Okamoto and Takeshi Tokuyama},
publisher =	{Schloss Dagstuhl--Leibniz-Zentrum fuer Informatik},