Computing Delaunay triangulations in Rd 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=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 Lov\'asz 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.Comment: 24 page