A Probabilistic Approach to Reducing Algebraic Complexity of Delaunay Triangulations

Jean-Daniel Boissonnat (1) , Ramsay Dyer (2) , Arijit Ghosh (3)

1 GEOMETRICA - Geometric computing
2 Bernoulli Institute for Mathematics and Computer Science and Artificial Intelligence
3 Algorithms and Complexity

We propose algorithms to compute the Delaunay triangulation of a point set L using only (squared) distance comparisons (i.e., predicates of degree 2). Our approach is based on the witness complex, a weak form of the Delaunay complex introduced by Carlsson and de Silva. We give conditions that ensure that the witness complex and the Delaunay triangulation coincide and we introduce a new perturbation scheme to compute a perturbed set L′ close to L such that the Delaunay triangulation and the witness complex coincide. Our perturbation algorithm is a geometric application of the Moser-Tardos constructive proof of the Lovász local lemma.

Mots clés: Delaunay complex; witness complex; relaxed Delaunay complex; distance and incircle predicates; simplex quality; Lovász local lemma

Domaines: Géométrie algorithmique [cs.CG]

Jean-Daniel Boissonnat, Ramsay Dyer, Arijit Ghosh. A Probabilistic Approach to Reducing Algebraic Complexity of Delaunay Triangulations. Algorithms - ESA 2015, Sep 2015, Patras, Greece. pp.595-606, ⟨10.1007/978-3-662-48350-3_50⟩. ⟨hal-01213070⟩