On the Cost of Fixed Partial Match Queries in K-d Trees

Abstract

Partial match queries constitute the most basic type of associative queries in multidimensional data structures such as $$K$$K-d trees or quadtrees. Given a query $$\mathbf {q}=(q_0,\ldots ,q_{K-1})$$q=(q0,?,qK-1) where s of the coordinates are specified and $$K-s$$K-s are left unspecified ($$q_i=*$$qi=?), a partial match search returns the subset of data points $$\mathbf {x}=(x_0,\ldots ,x_{K-1})$$x=(x0,?,xK-1) in the data structure that match the given query, that is, the data points such that $$x_i=q_i$$xi=qi whenever $$q_i\not =*$$qi??. There exists a wealth of results about the cost of partial match searches in many different multidimensional data structures, but most of these results deal with random queries. Only recently a few papers have begun to investigate the cost of partial match queries with a fixed query $$\mathbf {q}$$q. This paper represents a new contribution in this direction, giving a detailed asymptotic estimate of the expected cost $$P_{{n},\mathbf {q}}$$Pn,q for a given fixed query $$\mathbf {q}$$q. From previous results on the cost of partial matches with a fixed query and the ones presented here, a deeper understanding is emerging, uncovering the following functional shape for $$P_{{n},\mathbf {q}}$$Pn,q$$\begin{aligned} P_{{n},\mathbf {q}} = \nu \cdot \left( \prod _{i:q_i\text { is specified}}\, q_i(1-q_i)\right) ^{\alpha /2}\cdot n^\alpha + \text {l.o.t.} \end{aligned}$$Pn,q=?·?i:qiis specifiedqi(1-qi)?/2·n?+l.o.t.(l.o.t. lower order terms, throughout this work) in many multidimensional data structures, which differ only in the exponent $$\alpha $$? and the constant $$\nu $$?, both dependent on s and K, and, for some data structures, on the whole pattern of specified and unspecified coordinates in $$\mathbf {q}$$q as well. Although it is tempting to conjecture that this functional shape is universal, we have shown experimentally that it seems not to be true for a variant of $$K$$K-d trees called squarish $$K$$K-d trees.