Partial Match Queries in Quad- K-d Trees
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.
Quadtree
Partial match queries
Associative queries
Multidimensional search
Analysis of algorithms
Theory of computation~Data structures design and analysis
Theory of computation~Design and analysis of algorithms
8:1-8:16
Regular Paper
This work has been supported by funds from the MOTION Project (Project PID2020-112581GB-C21) of the Spanish Ministery of Science and Innovation MCIN/AEI/10.13039/ 501100011033.
Amalia
Duch
Amalia Duch
Universitat Politècnica de Catalunya, Barcelona, Spain
https://orcid.org/0000-0003-4371-1286
Conrado
Martínez
Conrado Martínez
Universitat Politècnica de Catalunya, Barcelona, Spain
https://orcid.org/0000-0003-1302-9067
10.4230/LIPIcs.AofA.2022.8
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Amalia Duch and Conrado Martínez
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