Approximate Nearest-Neighbor Search for Line Segments

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 $\mathbb{R}<sup>d$, for constant dimension $d$. Given a set $S$ of $n$ disjoint line segments in $\mathbb{R}<sup>d$ and an error parameter $\varepsilon &gt; 0$, 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.