Partial match queries in random quadtrees

TL;DR

This paper analyzes the cost of partial match queries in random quadtrees and k-d trees, providing precise estimates and limit distributions for query costs as data size grows.

Contribution

It introduces a new approach analyzing fixed query costs to derive detailed asymptotic behavior and limit processes for partial match query costs in multidimensional trees.

Findings

Expected query cost scales as n^β with explicit constants.

Variance and limit distribution of fixed query costs are characterized.

Maximum query cost over [0,1] scales as γ n^β.

Abstract

We consider the problem of recovering items matching a partially specified pattern in multidimensional trees (quad trees and k-d trees). We assume the traditional model where the data consist of independent and uniform points in the unit square. For this model, in a structure on $n$ points, it is known that the number of nodes $C_n(\xi)$ to visit in order to report the items matching an independent and uniformly on $[0,1]$ random query $\xi$ satisfies $\Ec{C_n(\xi)}\sim \kappa n^{\beta}$, where $\kappa$ and $\beta$ are explicit constants. We develop an approach based on the analysis of the cost $C_n(x)$ of any fixed query $x\in [0,1]$, and give precise estimates for the variance and limit distribution of the cost $C_n(x)$. Our results permit to describe a limit process for the costs $C_n(x)$ as $x$ varies in $[0,1]$; one of the consequences is that $E{\max_{x\in [0,1]} C_n(x)} \sim \gamma n^\beta$.