On the average performance of fixed partial match queries in random relaxed K-d trees

Abstract

We obtain an explicit formula for the expected cost of a fixed partial match query in a random relaxed K-d tree, that is, the expected cost for a query of the form q = (q0 , q1 , . . . , qK-1 ) with 0 < s < K specified coordinates with values z0 , . . . , zs-1 ¿ (0, 1). This is a generalization of previous results in the literature for s = 1. Qualitatively similar results hold for standard K-d trees and we conjecture that this also holds for other multidimensional tree structures such as quadtrees and K-d tries.