Product Range Search Problem

TL;DR

This paper introduces two new data structures for approximate product range search in doubling metric spaces, leveraging greedy trees to efficiently handle intersection queries involving two different metrics.

Contribution

It proposes novel data structures based on greedy trees for efficient approximate product range searches in doubling metrics, generalizing existing range trees.

Findings

Both data structures efficiently answer approximate product range queries.

The first structure generalizes range trees using greedy trees.

The second structure constructs a greedy tree on the product metric.

Abstract

Given a metric space, a standard metric range search, given a query $(q,r)$, finds all points within distance $r$ of the point $q$. Suppose now we have two different metrics $d_1$ and $d_2$. A product range query $(q, r_1, r_2)$ is a point $q$ and two radii $r_1$ and $r_2$. The output is all points within distance $r_1$ of $q$ with respect to $d_1$ and all points within $r_2$ of $q$ with respect to $d_2$. In other words, it is the intersection of two searches. We present two data structures for approximate product range search in doubling metrics. Both data structures use a net-tree variant, the greedy tree. The greedy tree is a data structure that can efficiently answer approximate range searches in doubling metrics. The first data structure is a generalization of the range tree from computational geometry using greedy trees rather than binary trees. The second data structure is a single greedy tree constructed on the product induced by the two metrics.