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DOI: 10.4230/LIPIcs.ICALP.2022.75
URN: urn:nbn:de:0030-drops-164167
URL: https://drops.dagstuhl.de/opus/volltexte/2022/16416/
Huang, Ziyun ;
Xu, Jinhui
In-Range Farthest Point Queries and Related Problem in High Dimensions
Abstract
Range-aggregate query is an important type of queries with numerous applications. It aims to obtain some structural information (defined by an aggregate function F(⋅)) of the points (from a point set P) inside a given query range B. In this paper, we study the range-aggregate query problem in high dimensional space for two aggregate functions: (1) F(P ∩ B) is the farthest point in P ∩ B to a query point q in ℝ^d and (2) F(P ∩ B) is the minimum enclosing ball (MEB) of P ∩ B. For problem (1), called In-Range Farthest Point (IFP) Query, we develop a bi-criteria approximation scheme: For any ε > 0 that specifies the approximation ratio of the farthest distance and any γ > 0 that measures the "fuzziness" of the query range, we show that it is possible to pre-process P into a data structure of size Õ_{ε,γ}(dn^{1+ρ}) in Õ_{ε,γ}(dn^{1+ρ}) time such that given any ℝ^d query ball B and query point q, it outputs in Õ_{ε,γ}(dn^ρ) time a point p that is a (1-ε)-approximation of the farthest point to q among all points lying in a (1+γ)-expansion B(1+γ) of B, where 0 < ρ < 1 is a constant depending on ε and γ and the hidden constants in big-O notations depend only on ε, γ and Polylog(nd). For problem (2), we show that the IFP result can be applied to develop query scheme with similar time and space complexities to achieve a (1+ε)-approximation for MEB. To the best of our knowledge, these are the first theoretical results on such high dimensional range-aggregate query problems. Our results are based on several new techniques, such as multi-scale construction and ball difference range query, which are interesting in their own rights and could be potentially used to solve other range-aggregate problems in high dimensional space.
BibTeX - Entry
@InProceedings{huang_et_al:LIPIcs.ICALP.2022.75,
author = {Huang, Ziyun and Xu, Jinhui},
title = {{In-Range Farthest Point Queries and Related Problem in High Dimensions}},
booktitle = {49th International Colloquium on Automata, Languages, and Programming (ICALP 2022)},
pages = {75:1--75:21},
series = {Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN = {978-3-95977-235-8},
ISSN = {1868-8969},
year = {2022},
volume = {229},
editor = {Boja\'{n}czyk, Miko{\l}aj and Merelli, Emanuela and Woodruff, David P.},
publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
address = {Dagstuhl, Germany},
URL = {https://drops.dagstuhl.de/opus/volltexte/2022/16416},
URN = {urn:nbn:de:0030-drops-164167},
doi = {10.4230/LIPIcs.ICALP.2022.75},
annote = {Keywords: Farthest Point Query, Range Aggregate Query, Minimum Enclosing Ball, Approximation, High Dimensional Space}
}
Keywords: |
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Farthest Point Query, Range Aggregate Query, Minimum Enclosing Ball, Approximation, High Dimensional Space |
Collection: |
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49th International Colloquium on Automata, Languages, and Programming (ICALP 2022) |
Issue Date: |
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2022 |
Date of publication: |
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28.06.2022 |