相关论文: Predicates of the 3D Apollonius Diagram

In this paper we study one of the fundamental predicates required for the construction of the 3D Apollonius diagram (also known as the 3D Additively Weighted Voronoi diagram), namely the EdgeConflict predicate: given five sites $S_i,…

In this paper we study the most-demanding predicate for computing the Euclidean Voronoi diagram of axes-aligned line segments, namely the Incircle predicate. Our contribution is two-fold: firstly, we describe, in algorithmic terms, how to…

Voronoi diagrams are a fundamental geometric data structure for obtaining proximity relations. We consider collections of axis-aligned orthogonal polyhedra in two and three-dimensional space under the max-norm, which is a particularly…

We study the amortized number of combinatorial changes (edge insertions and removals) needed to update the graph structure of the Voronoi diagram $\mathcal{V}(S)$ (and several variants thereof) of a set $S$ of $n$ sites in the plane as…

Computing the Voronoi diagram of mixed geometric objects in $R^3$ is challenging due to the high cost of exact geometric predicates via Cylindrical Algebraic Decomposition (CAD). We propose an efficient exact verification framework that…

We study algorithms and combinatorial complexity bounds for \emph{stable-matching Voronoi diagrams}, where a set, $S$, of $n$ point sites in the plane determines a stable matching between the points in $\mathbb{R}^2$ and the sites in $S$…

Voronoi and related diagrams have technological applications, for example, in motion planning and surface reconstruction, and also find significant use in materials science, molecular biology, and crystallography. Apollonius diagrams…

Given a set of $n$ sites from $\mathbb{R}^d$, each having some positive weight factor, the Multiplicatively Weighted Voronoi Diagram is a subdivision of space that associates each cell to the site whose weighted Euclidean distance is…

We present an explicit and efficient construction of additively weighted Voronoi diagrams on planar graphs. Let $G$ be a planar graph with $n$ vertices and $b$ sites that lie on a constant number of faces. We show how to preprocess $G$ in…

We present the first optimal randomized algorithm for constructing the order-$k$ Voronoi diagram of $n$ points in two dimensions. The expected running time is $O(n\log n + nk)$, which improves the previous, two-decades-old result of Ramos…

In this paper, we provide an $O(n \mathrm{polylog} n)$ bound on the expected complexity of the randomly weighted Voronoi diagram of a set of $n$ sites in the plane, where the sites can be either points, interior-disjoint convex sets, or…

We describe a new algorithm for computing the Voronoi diagram of a set of $n$ points in constant-dimensional Euclidean space. The running time of our algorithm is $O(f \log n \log \Delta)$ where $f$ is the output complexity of the Voronoi…

A data structure is presented that explicitly maintains the graph of a Voronoi diagram of $N$ point sites in the plane or the dual graph of a convex hull of points in three dimensions while allowing insertions of new sites/points. Our…

Let $S$ be a planar $n$-point set. A triangulation for $S$ is a maximal plane straight-line graph with vertex set $S$. The Voronoi diagram for $S$ is the subdivision of the plane into cells such that all points in a cell have the same…

We consider the Voronoi diagram of lines in $\mathbb{R}^3$ under the Euclidean metric, and give a full classification of its structure in the base case of four lines in general position. We first show that the number of vertices in the…

In this paper, we propose to compute Voronoi diagrams over mesh surfaces driven by an arbitrary geodesic distance solver, assuming that the input is a triangle mesh as well as a collection of sites $P=\{p_i\}_{i=1}^m$ on the surface. We…

Given two point sets in the plane, we study the minimization of the bottleneck distance between a point set B and an equally-sized subset of a point set A under translations. We relate this problem to a Voronoi-type diagram and derive…

Surprisingly, the order-$k$ Voronoi diagram of line segments had received no attention in the computational-geometry literature. It illustrates properties surprisingly different from its counterpart for points; for example, a single…

We present subquadratic algorithms in the algebraic decision-tree model for several \textsc{3Sum}-hard geometric problems, all of which can be reduced to the following question: Given two sets $A$, $B$, each consisting of $n$ pairwise…

We present a simple wavefront-like approach for computing multiplicatively weighted Voronoi diagrams of points and straight-line segments in the Euclidean plane. If the input sites may be assumed to be randomly weighted points then the use…