Priority Range Trees

TL;DR

The paper introduces a priority range tree data structure that efficiently handles orthogonal range reporting queries on prioritized points, enabling fast retrieval of points based on rank and weight in logarithmic time.

Contribution

It presents a novel data structure, the priority range tree, optimized for prioritized range queries, with specific time complexities and space bounds for three- and four-sided queries.

Findings

Supports fast reporting of points with rank at least a threshold

Efficiently reports top-ranked points within a range

Achieves optimal space complexity for three-sided queries

Abstract

We describe a data structure, called a priority range tree, which accommodates fast orthogonal range reporting queries on prioritized points. Let $S$ be a set of $n$ points in the plane, where each point $p$ in $S$ is assigned a weight $w(p)$ that is polynomial in $n$, and define the rank of $p$ to be $r(p)=\lfloor \log w(p) \rfloor$. Then the priority range tree can be used to report all points in a three- or four-sided query range $R$ with rank at least $\lfloor \log w \rfloor$ in time $O(\log W/w + k)$, and report $k$ highest-rank points in $R$ in time $O(\log\log n + \log W/w' + k)$, where $W=\sum_{p\in S}{w(p)}$, $w'$ is the smallest weight of any point reported, and $k$ is the output size. All times assume the standard RAM model of computation. If the query range of interest is three sided, then the priority range tree occupies $O(n)$ space, otherwise $O(n\log n)$ space is used to answer four-sided queries. These queries are motivated by the Weber--Fechner Law, which states that humans perceive and interpret data on a logarithmic scale.