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

Quad- 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 2ᵐ⁽ˣ⁾ 2 m ( x ) for some m ( x ), 1≤ m(x)≤ 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 P̂ₙ 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 P̂ₙ~ β · n^α P ^ n ∼ β · n α , where α α and β 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 Pn,q P n , q of a fixed partial match with query 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.