Partial match queries in quad-K-d trees

Abstract

Quad-K-d trees [Bereckzy et al., 2014] are a generalization of several well-known hierarchical Kdimensional 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 Pˆn that has s out of K specified coordinates in a random quad-K-d tree of size n is Pˆn ∼ 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 Pn,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.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.Peer ReviewedPostprint (published version