Random Partial Match in Quad-K-d Trees

Abstract

Quad-K-d trees were introduced by Bereczky et al. [3] as a generalization of several well-known hierarchical multidimensional data structures such as K-d trees and quad trees. One of the interesting features of quad-$K$-d trees is that they provide a unified framework for the analysis of associative queries in hierarchical multidimensional data structures. In this paper we consider partial match, one of the fundamental associative queries, and prove that the expected cost of a random partial match in a random quad-$K$-d tree of size n is of the form $\varTheta (n^\alpha )$, with $0 < \alpha < 1$, for several families of quad-$K$-d trees including, among others, K-d trees and quad trees. We actually give a general result that applies to any family of quad-$K$-d trees where each node has a type that is independent of the type of other nodes. We derive, exploiting Roura’s Continuous Master Theorem, the general equation satisfied by $\alpha$, in terms of the dimension K, the number of specified coordinates s in the partial match query, and also the additional parameters that characterize each of the families of quad-$K$-d trees considered in the paper. We also conduct an experimental study whose results match our theoretical findings; as a by-product we propose an implementation of the partial match search in quad-$K$-d trees in full generality.

This work has been partially supported by funds from the Spanish Ministry for Economy and Competitiveness (MINECO) and the European Union (FEDER funds) under grant COMMAS (ref. TIN2013-46181-C2-1-R) and AGAUR grant SGR 2014:1034 (ALBCOM).