Partial Match Queries in Two-Dimensional Quadtrees : a Probabilistic Approach

TL;DR

This paper investigates the average cost of partial match queries in random 2D quadtrees using probabilistic methods, including fragmentation theory and Markov chain coupling, to compute the limiting behavior.

Contribution

It introduces a probabilistic approach employing fragmentation theory and Markov chain coupling to analyze query costs in 2D quadtrees, providing explicit limit calculations.

Findings

Mean query cost converges to a fixed point.

Method applies fragmentation theory to spatial data structures.

Provides explicit formulas for asymptotic behavior.

Abstract

We analyze the mean cost of the partial match queries in random two-dimensional quadtrees. The method is based on fragmentation theory. The convergence is guaranteed by a coupling argument of Markov chains, whereas the value of the limit is computed as the fixed point of an integral equation.