Discrete & Computational Geometry

, Volume 52, Issue 3, pp 492–514 | Cite as

A Faster Algorithm for Computing Motorcycle Graphs

Article
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Abstract

We present a new algorithm for computing motorcycle graphs that runs in \(O(n^{4/3+\varepsilon })\) time for any \(\varepsilon >0\), improving on all previously known algorithms. The main application of this result is to computing the straight skeleton of a polygon. It allows us to compute the straight skeleton of a non-degenerate polygon with \(h\) holes in \(O(n \sqrt{h+1} \log ^2 n+n^{4/3+\varepsilon })\) expected time. If all input coordinates are \(O(\log n)\)-bit rational numbers, we can compute the straight skeleton of a (possibly degenerate) polygon with \(h\) holes in \(O(n \sqrt{h+1}\log ^3 n)\) expected time. In particular, it means that we can compute the straight skeleton of a simple polygon in \(O(n\log ^3n)\) expected time if all input coordinates are \(O(\log n)\)-bit rationals, while all previously known algorithms have worst-case running time \(\omega (n^{3/2})\).

Keywords

Algorithms design and analysis Motorcycle graph Straight skeleton Medial axis Polygon 

Mathematics Subject Classification

68U05 65D18 68Q25 

Notes

Acknowledgments

Lie Yan was supported by KAUST base funding. We thank the anonymous referees for their helpful comments.

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Copyright information

© Springer Science+Business Media New York 2014

Authors and Affiliations

  1. 1.Visual Computing CenterKing Abdullah University of Science and Technology (KAUST)ThuwalSaudi Arabia
  2. 2.Department of Computer Science and EngineeringThe Hong Kong University of Science and Technology (HKUST)KowloonHong Kong

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