On the expected cost of partial match queries in random Quad-K-d trees
Quad-K-d trees introduced by Bereckzy et al. (In: Proceedings of the 11th Latin merican Theoretical Informatics Conference (LATIN). Lecture Notes in Computer Science, vol. 8392, pp. 743–754, 2014) are a generalization of several well-known hierarchical multidimensional data structures. They provide...
| Autores: | , |
|---|---|
| Tipo de recurso: | artículo |
| Fecha de publicación: | 2024 |
| País: | España |
| Institución: | Universitat Politècnica de Catalunya (UPC) |
| Repositorio: | UPCommons. Portal del coneixement obert de la UPC |
| Idioma: | inglés |
| OAI Identifier: | oai:upcommons.upc.edu:2117/405893 |
| Acceso en línea: | https://hdl.handle.net/2117/405893 https://dx.doi.org/10.1007/s44007-024-00090-5 |
| Access Level: | acceso abierto |
| Palabra clave: | Algorithms Quadtrees K-d trees Partial match queries Associative queries Multidimensional search Analysis of algorithms Algorismes Àrees temàtiques de la UPC::Informàtica::Informàtica teòrica::Algorísmica i teoria de la complexitat |
| Sumario: | Quad-K-d trees introduced by Bereckzy et al. (In: Proceedings of the 11th Latin merican Theoretical Informatics Conference (LATIN). Lecture Notes in Computer Science, vol. 8392, pp. 743–754, 2014) are a generalization of several well-known hierarchical multidimensional data structures. They provide a unified framework for the analysis of associative queries, and they are specially suitable to investigate the trade-offs between the cost of different operations and thememory needs (each node x of a quad-K-d tree has arity 2m(x) for somem(x), 1 ≤ m(x) ≤ 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 ˆPn of a random partial match query that has s out of K specified coordinates in a random quad-K-d tree of size n is ˆPn ∼ β · nα, 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 the expected cost Pn,q of a fixed partial match with query q in a random quad-K-d tree of size n. The techniques used to derive the mentioned costs are those already applied successfully to derive analogous results in quadtrees and relaxed K-d trees; our results show that the previous results are just particular cases and prove the validity of the conjecture made in Duch et al. (In: Proceedings of the 12th Latin American Theoretical Informatics Conference (LATIN). Lecture Notes in Computer Science, vol. 9644, pp. 376–389, 2016) for a wider variety of multidimensional data structures. |
|---|