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Approximate Nearest-Neighbor Search for Line Segments

Authors Ahmed Abdelkader , David M. Mount

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Author Details

Ahmed Abdelkader
  • Oden Institute for Computational Engineering and Sciences, The University of Texas at Austin, TX, USA
David M. Mount
  • Department of Computer Science and Institute of Advanced Computer Studies, University of Maryland, College Park, MD, USA

Funding

Supported by NSF grant CCF-1618866 and an Ann G. Wylie Dissertation Fellowship.

Acknowledgements

The first author’s work was conducted in part at the University of Maryland.

Cite AsGet BibTex

Ahmed Abdelkader and David M. Mount. Approximate Nearest-Neighbor Search for Line Segments. In 37th International Symposium on Computational Geometry (SoCG 2021). Leibniz International Proceedings in Informatics (LIPIcs), Volume 189, pp. 4:1-4:15, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2021)
https://doi.org/10.4230/LIPIcs.SoCG.2021.4

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 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.

Subject Classification

ACM Subject Classification
  • Theory of computation → Computational geometry
Keywords
  • Approximate nearest-neighbor searching
  • Approximate Voronoi diagrams
  • Line segments
  • Macbeath regions

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