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.AofA.2022.8
URN: urn:nbn:de:0030-drops-160949
URL: http://dagstuhl.sunsite.rwth-aachen.de/volltexte/2022/16094/
Duch, Amalia ;
Martínez, Conrado
Partial Match Queries in Quad- K-d Trees
Abstract
Quad-K-d trees [Bereckzy et al., 2014] are a generalization of several well-known hierarchical K-dimensional data structures. They were introduced to provide a unified framework for the analysis of associative queries and to investigate the trade-offs between the cost of different operations and the memory needs (each node of a quad-K-d tree has arity 2^m for some m, 1 ≤ m ≤ K). Indeed, we consider here partial match - one of the fundamental associative queries - for several families of quad-K-d trees including, among others, relaxed K-d trees and quadtrees. In particular, we prove that the expected cost of a random partial match P̂_n that has s out of K specified coordinates in a random quad-K-d tree of size n is P̂_n ∼ β⋅ n^α where α and β are constants given in terms of K and s as well as additional parameters that characterize the specific family of quad-K-d trees under consideration. Additionally, we derive a precise asymptotic estimate for the main order term of P_{n,?} - the expected cost of a fixed partial match in a random quad-K-d tree of size n. The techniques and procedures used to derive the mentioned costs extend those already successfully applied to derive analogous results in quadtrees and relaxed K-d trees; our results show that the previous results are just particular cases, and states the validity of the conjecture made in [Duch et al., 2016] to a wider variety of multidimensional data structures.
BibTeX - Entry
@InProceedings{duch_et_al:LIPIcs.AofA.2022.8,
author = {Duch, Amalia and Mart{\'\i}nez, Conrado},
title = {{Partial Match Queries in Quad- K-d Trees}},
booktitle = {33rd International Conference on Probabilistic, Combinatorial and Asymptotic Methods for the Analysis of Algorithms (AofA 2022)},
pages = {8:1--8:16},
series = {Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN = {978-3-95977-230-3},
ISSN = {1868-8969},
year = {2022},
volume = {225},
editor = {Ward, Mark Daniel},
publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
address = {Dagstuhl, Germany},
URL = {https://drops.dagstuhl.de/opus/volltexte/2022/16094},
URN = {urn:nbn:de:0030-drops-160949},
doi = {10.4230/LIPIcs.AofA.2022.8},
annote = {Keywords: Quadtree, Partial match queries, Associative queries, Multidimensional search, Analysis of algorithms}
}
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Keywords: |
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Quadtree, Partial match queries, Associative queries, Multidimensional search, Analysis of algorithms |
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Collection: |
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33rd International Conference on Probabilistic, Combinatorial and Asymptotic Methods for the Analysis of Algorithms (AofA 2022) |
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Issue Date: |
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2022 |
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Date of publication: |
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08.06.2022 |
