A Note on Indexing Point Sets for Approximate Bottleneck Distance Queries

TL;DR

This paper introduces a trie-based data structure for efficiently approximating nearest neighbor searches among point sets based on the bottleneck distance, enabling faster queries in planar point set collections.

Contribution

The work presents a novel trie-based data structure that approximates nearest bottleneck distance queries among planar point sets, improving query efficiency.

Findings

Finds a 6-approximate nearest neighbor in O(-log(d_B) * n) time

Applicable to point sets in [0,1]^2 under L_infinity norm

Techniques extend to other norms

Abstract

The {\em bottleneck distance} is a natural measure of the distance between two finite point sets of equal cardinality, defined as the minimum over all bijections between the point sets of the maximum distance between any pair of points put in correspondence by the bijection. In this work, we consider the problem of building a data structure $\mathbb{D}$ that indexes a collection of $m$ planar point sets (of varying sizes) and supports nearest bottleneck distance queries: given a query point set $Q$ of size $n$, we would like to find the point set(s) $P \in \mathbb{D}$ of size $n$ that are closest in terms of bottleneck distance. Without loss of generality, we assume that all point sets belong to the unit box $[0,1]^2$ in the plane and focus on the $L_\infty$ norm, although the techniques can also be used for other norms. The main contribution is a {\em trie}-based data structure finds a $6$-approximate nearest neighbor in $O(-\lg(d_B(\mathbb{D},Q)) n)$ time, where $d_B(\mathbb{D},Q)$ is the minimum bottleneck distance from $Q$ to any point set in $\mathbb{D}$.