On the Expected Cost of Partial Match Queries in Random Quad-K-d Trees

Abstract

AbstractQuad-$$K$$ 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 the memory needs (each node x of a quad-$$K$$ K -d tree has arity $$2^{m(x)}$$ 2 m ( x ) for some m(x), $$1\le m(x)\le K$$ 1 ≤ m ( x ) ≤ K ). Indeed, we consider here partial match—one of the fundamental associative queries for several families of quad-$$K$$ K -d trees including, among others, relaxed K-d trees and quadtrees. In particular, we prove that the expected cost $$\hat{P}_{n}$$ P ^ n of a random partial match query that has s out of K specified coordinates in a random quad-$$K$$ K -d tree of size n is $$\hat{P}_{n}\sim \beta \cdot n^\alpha $$ P ^ n ∼ β · n α , where $$\alpha $$ α and $$\beta $$ β are constants given in terms of K and s as well as additional parameters that characterize the specific family of quad-$$K$$ K -d trees under consideration. Additionally, we derive a precise asymptotic estimate for the main order term of the expected cost $$P_{n,\textbf{q}}$$ P n , q of a fixed partial match with query $$\textbf{q}$$ q in a random quad-$$K$$ 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.