Approximate Nearest-Neighbor Search for Line Segments

TL;DR

This paper introduces a data structure for approximate nearest-neighbor search among line segments in fixed-dimensional space, enabling efficient queries with controlled error, which is a novel extension beyond point set methods.

Contribution

It proposes the first data structure for approximate nearest-neighbor search specifically for line segments in fixed dimensions, with explicit storage and query time bounds.

Findings

Achieves query time of O(log(max(n,Δ)/ε))

Uses storage of O((n^2/ε^d) log(Δ/ε))

Employs anisotropic space covering aligned with segment orientations

Abstract

Approximate nearest-neighbor search is a fundamental algorithmic problem that continues to inspire study due its essential role in numerous contexts. In contrast to most prior work, which has focused on point sets, we consider nearest-neighbor queries against a set of line segments in $\mathbb{R}^d$, for constant dimension $d$. Given a set $S$ of $n$ disjoint line segments in $\mathbb{R}^d$ and an error parameter $\varepsilon > 0$, the objective is to build a data structure such that for any query point $q$, it is possible to return a line segment whose Euclidean distance from $q$ is at most $(1+\varepsilon)$ times the distance from $q$ to its nearest line segment. We present a data structure for this problem with storage $O((n^2/\varepsilon^{d}) \log (\Delta/\varepsilon))$ and query time $O(\log (\max(n,\Delta)/\varepsilon))$, where $\Delta$ is the spread of the set of segments $S$. Our approach is based on a covering of space by anisotropic elements, which align themselves according to the orientations of nearby segments.