License: 
Creative Commons Attribution 4.0 International license (CC BY 4.0)
When quoting this document, please refer to the following
DOI: 10.4230/LIPIcs.ICALP.2023.7
URN: urn:nbn:de:0030-drops-180590
URL: http://dagstuhl.sunsite.rwth-aachen.de/volltexte/2023/18059/
Afshani, Peyman ;
Cheng, Pingan ;
Basu Roy, Aniket ;
Wei, Zhewei
On Range Summary Queries
Abstract
We study the query version of the approximate heavy hitter and quantile problems. In the former problem, the input is a parameter ε and a set P of n points in ℝ^d where each point is assigned a color from a set C, and the goal is to build a structure such that given any geometric range γ, we can efficiently find a list of approximate heavy hitters in γ∩P, i.e., colors that appear at least ε |γ∩P| times in γ∩P, as well as their frequencies with an additive error of ε |γ∩P|. In the latter problem, each point is assigned a weight from a totally ordered universe and the query must output a sequence S of 1+1/ε weights such that the i-th weight in S has approximate rank iε|γ∩P|, meaning, rank iε|γ∩P| up to an additive error of ε|γ∩P|. Previously, optimal results were only known in 1D [Wei and Yi, 2011] but a few sub-optimal methods were available in higher dimensions [Peyman Afshani and Zhewei Wei, 2017; Pankaj K. Agarwal et al., 2012].
We study the problems for two important classes of geometric ranges: 3D halfspace and 3D dominance queries. It is known that many other important queries can be reduced to these two, e.g., 1D interval stabbing or interval containment, 2D three-sided queries, 2D circular as well as 2D k-nearest neighbors queries. We consider the real RAM model of computation where integer registers of size w bits, w = Θ(log n), are also available. For dominance queries, we show optimal solutions for both heavy hitter and quantile problems: using linear space, we can answer both queries in time O(log n + 1/ε). Note that as the output size is 1/ε, after investing the initial O(log n) searching time, our structure takes on average O(1) time to find a heavy hitter or a quantile! For more general halfspace heavy hitter queries, the same optimal query time can be achieved by increasing the space by an extra log_w(1/ε) (resp. log log_w(1/ε)) factor in 3D (resp. 2D). By spending extra log^O(1)(1/ε) factors in both time and space, we can also support quantile queries.
We remark that it is hopeless to achieve a similar query bound for dimensions 4 or higher unless significant advances are made in the data structure side of theory of geometric approximations.
BibTeX - Entry
@InProceedings{afshani_et_al:LIPIcs.ICALP.2023.7,
author = {Afshani, Peyman and Cheng, Pingan and Basu Roy, Aniket and Wei, Zhewei},
title = {{On Range Summary Queries}},
booktitle = {50th International Colloquium on Automata, Languages, and Programming (ICALP 2023)},
pages = {7:1--7:17},
series = {Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN = {978-3-95977-278-5},
ISSN = {1868-8969},
year = {2023},
volume = {261},
editor = {Etessami, Kousha and Feige, Uriel and Puppis, Gabriele},
publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
address = {Dagstuhl, Germany},
URL = {https://drops.dagstuhl.de/opus/volltexte/2023/18059},
URN = {urn:nbn:de:0030-drops-180590},
doi = {10.4230/LIPIcs.ICALP.2023.7},
annote = {Keywords: Computational Geometry, Range Searching, Random Sampling, Geometric Approximation, Data Structures and Algorithms}
}
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Keywords: |
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Computational Geometry, Range Searching, Random Sampling, Geometric Approximation, Data Structures and Algorithms |
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Collection: |
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50th International Colloquium on Automata, Languages, and Programming (ICALP 2023) |
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Issue Date: |
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2023 |
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Date of publication: |
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05.07.2023 |
