Reliable and Efficient Computational Geometry Via Controlled Perturbation

  • Kurt Mehlhorn
  • Ralf Osbild
  • Michael Sagraloff
Part of the Lecture Notes in Computer Science book series (LNCS, volume 4051)


Most algorithms of computational geometry are designed for the Real-RAM and non-degenerate input. We call such algorithms idealistic. Executing an idealistic algorithm with floating point arithmetic may fail. Controlled perturbation replaces an input x by a random nearby \(\tilde{x}\) in the δ-neighborhood of x and then runs the floating point version of the idealistic algorithm on \(\tilde{x}\). The hope is that this will produce the correct result for \(\tilde{x}\) with constant probability provided that δ is small and the precision L of the floating point system is large enough. We turn this hope into a theorem for a large class of geometric algorithms and describe a general methodology for deriving a relation between δ and L. We exemplify the usefulness of the methodology by examples.


Voronoi Diagram Computational Geometry Delaunay Triangulation Query Point Tubular Neighborhood 
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Copyright information

© Springer-Verlag Berlin Heidelberg 2006

Authors and Affiliations

  • Kurt Mehlhorn
    • 1
  • Ralf Osbild
    • 1
  • Michael Sagraloff
    • 1
  1. 1.Max-Planck-Institut für InformatikSaarbrückenGermany

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