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Creative Commons Attribution 3.0 Unported license (CC BY 3.0)
When quoting this document, please refer to the following
DOI: 10.4230/LIPIcs.ICALP.2018.31
URN: urn:nbn:de:0030-drops-90352
URL: https://drops.dagstuhl.de/opus/volltexte/2018/9035/
Chan, Timothy M. ;
Nekrich, Yakov ;
Rahul, Saladi ;
Tsakalidis, Konstantinos
Orthogonal Point Location and Rectangle Stabbing Queries in 3-d
Abstract
In this work, we present a collection of new results on two fundamental problems in geometric data structures: orthogonal point location and rectangle stabbing.
- Orthogonal point location. We give the first linear-space data structure that supports 3-d point location queries on n disjoint axis-aligned boxes with optimal O(log n) query time in the (arithmetic) pointer machine model. This improves the previous O(log^{3/2} n) bound of Rahul [SODA 2015]. We similarly obtain the first linear-space data structure in the I/O model with optimal query cost, and also the first linear-space data structure in the word RAM model with sub-logarithmic query time.
- Rectangle stabbing. We give the first linear-space data structure that supports 3-d 4-sided and 5-sided rectangle stabbing queries in optimal O(log_wn+k) time in the word RAM model. We similarly obtain the first optimal data structure for the closely related problem of 2-d top-k rectangle stabbing in the word RAM model, and also improved results for 3-d 6-sided rectangle stabbing.
For point location, our solution is simpler than previous methods, and is based on an interesting variant of the van Emde Boas recursion, applied in a round-robin fashion over the dimensions, combined with bit-packing techniques. For rectangle stabbing, our solution is a variant of Alstrup, Brodal, and Rauhe's grid-based recursive technique (FOCS 2000), combined with a number of new ideas.
BibTeX - Entry
@InProceedings{chan_et_al:LIPIcs:2018:9035,
author = {Timothy M. Chan and Yakov Nekrich and Saladi Rahul and Konstantinos Tsakalidis},
title = {{Orthogonal Point Location and Rectangle Stabbing Queries in 3-d}},
booktitle = {45th International Colloquium on Automata, Languages, and Programming (ICALP 2018)},
pages = {31:1--31:14},
series = {Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN = {978-3-95977-076-7},
ISSN = {1868-8969},
year = {2018},
volume = {107},
editor = {Ioannis Chatzigiannakis and Christos Kaklamanis and D{\'a}niel Marx and Donald Sannella},
publisher = {Schloss Dagstuhl--Leibniz-Zentrum fuer Informatik},
address = {Dagstuhl, Germany},
URL = {http://drops.dagstuhl.de/opus/volltexte/2018/9035},
URN = {urn:nbn:de:0030-drops-90352},
doi = {10.4230/LIPIcs.ICALP.2018.31},
annote = {Keywords: geometric data structures, orthogonal point location, rectangle stabbing, pointer machines, I/O model, word RAM model}
}
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Keywords: |
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geometric data structures, orthogonal point location, rectangle stabbing, pointer machines, I/O model, word RAM model |
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Collection: |
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45th International Colloquium on Automata, Languages, and Programming (ICALP 2018) |
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Issue Date: |
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2018 |
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Date of publication: |
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04.07.2018 |
