Approximate Nearest-Neighbor Search for Line Segments
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 ℝ^d, for constant dimension d. Given a set S of n disjoint line segments in ℝ^d and an error parameter ε > 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+ε) times the distance from q to its nearest line segment. We present a data structure for this problem with storage O((n²/ε^d) log (Δ/ε)) and query time O(log (max(n,Δ)/ε)), where Δ 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.
Approximate nearest-neighbor searching
Approximate Voronoi diagrams
Line segments
Macbeath regions
Theory of computation~Computational geometry
4:1-4:15
Regular Paper
Supported by NSF grant CCF-1618866 and an Ann G. Wylie Dissertation Fellowship.
https://arxiv.org/abs/2103.16071
The first author’s work was conducted in part at the University of Maryland.
Ahmed
Abdelkader
Ahmed Abdelkader
Oden Institute for Computational Engineering and Sciences, The University of Texas at Austin, TX, USA
https://orcid.org/0000-0002-6749-1807
David M.
Mount
David M. Mount
Department of Computer Science and Institute of Advanced Computer Studies, University of Maryland, College Park, MD, USA
https://orcid.org/0000-0002-3290-8932
10.4230/LIPIcs.SoCG.2021.4
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Ahmed Abdelkader and David M. Mount
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