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: https://drops.dagstuhl.de/opus/volltexte/2022/16094/
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Duch, Amalia ; Martínez, Conrado

Partial Match Queries in Quad- K-d Trees

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LIPIcs-AofA-2022-8.pdf (0.8 MB)


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,𝐪} - 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}
}

Keywords: Quadtree, Partial match queries, Associative queries, Multidimensional search, Analysis of algorithms
Collection: 33rd International Conference on Probabilistic, Combinatorial and Asymptotic Methods for the Analysis of Algorithms (AofA 2022)
Issue Date: 2022
Date of publication: 08.06.2022


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