eng
Schloss Dagstuhl – Leibniz-Zentrum für Informatik
Leibniz International Proceedings in Informatics
1868-8969
2022-06-08
8:1
8:16
10.4230/LIPIcs.AofA.2022.8
article
Partial Match Queries in Quad- K-d Trees
Duch, Amalia
1
https://orcid.org/0000-0003-4371-1286
Martínez, Conrado
1
https://orcid.org/0000-0003-1302-9067
Universitat Politècnica de Catalunya, Barcelona, Spain
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.
https://drops.dagstuhl.de/storage/00lipics/lipics-vol225-aofa2022/LIPIcs.AofA.2022.8/LIPIcs.AofA.2022.8.pdf
Quadtree
Partial match queries
Associative queries
Multidimensional search
Analysis of algorithms