# Linear-Time Algorithms for Geometric Graphs with Sublinearly Many Edge Crossings

@article{Eppstein2010LinearTimeAF,
title={Linear-Time Algorithms for Geometric Graphs with Sublinearly Many Edge Crossings},
author={David Eppstein and Michael T. Goodrich and Darren Strash},
journal={SIAM J. Comput.},
year={2010},
volume={39},
pages={3814-3829}
}
• Published 4 December 2008
• Mathematics, Computer Science
• SIAM J. Comput.
We provide linear-time algorithms for geometric graphs with sublinearly many edge crossings. That is, we provide algorithms running in $O(n)$ time on connected geometric graphs having $n$ vertices and $k$ pairwise crossings, where $k$ is smaller than $n$ by an iterated logarithmic factor. Specific problems that we study include Voronoi diagrams and single-source shortest paths. Our algorithms all run in linear time in the standard comparison-based computational model; hence, we make no…
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