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Randomized Incremental Construction of Delaunay Triangulations of Nice Point Sets

Authors Jean-Daniel Boissonnat, Olivier Devillers , Kunal Dutta , Marc Glisse

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Author Details

Jean-Daniel Boissonnat
  • INRIA Sophia-Antipolis, Université Côte d’Azur, Nice, France
Olivier Devillers
  • INRIA, CNRS, Loria, Université de Lorraine, Nancy, France
Kunal Dutta
  • INRIA Sophia-Antipolis, Université Côte d’Azur, Nice, France
Marc Glisse
  • INRIA, Université Paris-Saclay, France

Funding

This work has been partially supported by the European Research Council under Advanced Grant 339025 GUDHI (Algorithmic Foundations of Geometric Understanding in Higher Dimensions) and by grant ANR-17-CE40-0017 of the French National Research Agency (ANR project ASPAG).

Acknowledgements

The authors would like to acknowledge the referees for their helpful comments which helped to improve the presentation of the paper, in particular a simpler proof of Lemma 9.

Cite As Get BibTex

Jean-Daniel Boissonnat, Olivier Devillers, Kunal Dutta, and Marc Glisse. Randomized Incremental Construction of Delaunay Triangulations of Nice Point Sets. In 27th Annual European Symposium on Algorithms (ESA 2019). Leibniz International Proceedings in Informatics (LIPIcs), Volume 144, pp. 22:1-22:13, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2019) https://doi.org/10.4230/LIPIcs.ESA.2019.22

Abstract

Randomized incremental construction (RIC) is one of the most important paradigms for building geometric data structures. Clarkson and Shor developed a general theory that led to numerous algorithms that are both simple and efficient in theory and in practice. 
Randomized incremental constructions are most of the time space and time optimal in the worst-case, as exemplified by the construction of convex hulls, Delaunay triangulations and arrangements of line segments. However, the worst-case scenario occurs rarely in practice and we would like to understand how RIC behaves when the input is nice in the sense that the associated output is significantly smaller than in the worst-case. For example, it is known that the Delaunay triangulations of nicely distributed points on polyhedral surfaces in E^3 has linear complexity, as opposed to a worst-case quadratic complexity. The standard analysis does not provide accurate bounds on the complexity of such cases and we aim at establishing such bounds in this paper. More precisely, we will show that, in the case of nicely distributed points on polyhedral surfaces, the complexity of the usual RIC is O(n log n), which is optimal. In other words, without any modification, RIC nicely adapts to good cases of practical value.
Our proofs also work for some other notions of nicely distributed point sets, such as (epsilon, kappa)-samples. Along the way, we prove a probabilistic lemma for sampling without replacement, which may be of independent interest.

Subject Classification

ACM Subject Classification
  • Theory of computation → Computational geometry
Keywords
  • Randomized incremental construction
  • Delaunay triangulations
  • Voronoi diagrams
  • polyhedral surfaces
  • probabilistic analysis

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References

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