A probabilistic approach to reducing the algebraic complexity of computing Delaunay triangulations

Computing Delaunay triangulations in Rdinvolves evaluating the so-called in\_sphere predicate thatdetermines if a point x lies inside, on or outside the spherecircumscribing d+1 points p0,...,pd. This predicate reduces to evaluating thesign of a multivariate polynomial of degree d+2in the coordinatesof the points x, p0,... pd. Despite much progress on exactgeometric computing, the fact that the degree of the polynomialincreases with d makes the evaluation of the sign of such a polynomial problematic except in very low dimensions. In this paper, we propose anew approach that is based on the witness complex, a weak form of the Delaunay complex introduced by Carlsson and de Silva. The witness complex wit (L,W) is defined from two sets L and W in some metric space X: a finite set of points L on which the complex is built, and a set W of witnesses that serves as an approximation of X. A fundamental result of de Silva states that wit (L,W)= del (L) if W=X=Rd. In this paper, we give conditions on L that ensure that the witness complex and the Delaunay triangulation coincide when W is a finite set, and we introduce a new perturbation scheme to compute a perturbed set L' close to L such that del (L')= wit (L', W). Our perturbation algorithm is a geometric application of the Moser-Tardos constructive proof of the Lovàsz local lemma. The only numerical operations we use are (squared) distance comparisons (i.e., predicates of degree 2). The time-complexity of the algorithm is sublinear in |W|. Interestingly, although the algorithm does not compute any measure of simplex quality, a lower bound on the thickness of the output simplices can be guaranteed