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

Computing Delaunay triangulations in R involves evaluating the so-called in_sphere predicate that determines if a point x lies inside, on or outside the sphere circumscribing d+ 1 points p0, . . . , pd. This predicate reduces to evaluating the sign of a multivariate polynomial of degree d + 2 in the coordinates of the points x, p0, . . . , pd. Despite much progress on exact geometric computing, the fact that the degree of the polynomial increases with d makes the evaluation of the sign of such a polynomial problematic except in very low dimensions. In this paper, we propose a new 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 = R. 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 Lovász 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.