Abstract
Computing Delaunay triangulations in $\mathbb{R}^d$ 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 $p_0,\ldots ,p_d$. This
predicate reduces to evaluating the sign of a multivariate polynomial of degree
$d+2$ in the coordinates of the points $x, \, p_0,\, \ldots,\, p_d$. 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
$\mathrm{Wit} (L,W)$ 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 $\mathrm{Wit}(L,W)=\mathrm{Del} (L)$ if $W=X=\mathbb{R}^d$.
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 $\mathrm{Del} (L')= \mathrm{wit} (L', W)$. 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.
