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

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 $p_0,\dots ,p_d$. This predicate reduces to evaluating the sign of a multivariate polynomial of degree $d+2$ in the coordinates of the points $x,p_0,\dots ,p_d$. 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 $L^{\prime }$ close to $L$ such that $\mathrm{Del}(L^{\prime })=\mathrm{wit}(L^{\prime },W)$. 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.

24 pages