Duals of Orphan-Free Anisotropic Voronoi Diagrams are Triangulations

A New Approach to Output-Sensitive Voronoi Diagrams and Delaunay Triangulations

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 $O(f \log n \log Δ)$ where $f$ is the output complexity of the Voronoi diagram and $Δ$ is the spread of the input, the ratio of largest to smallest pairwise distances. Despite the simplicity of the algorithm and its analysis, it improves on the state of the art for all inputs with polynomial spread and near-linear output size. The key idea is to first build the Voronoi diagram of a superset of the input points using ideas from Vorono…

Time-Space Trade-offs for Triangulations and Voronoi Diagrams

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 nearest neighbor in $S$. Classically, both structures can be computed in $O(n \log n)$ time and $O(n)$ space. We study the situation when the available workspace is limited: given a parameter $s \in \{1, \dots, n\}$, an $s$-workspace algorithm has read-only access to an input array with the points from $S$ in arbitrary order, and it may use only $O(s)$ additional…

Constructing Intrinsic Delaunay Triangulations from the Dual of Geodesic Voronoi Diagrams

Intrinsic Delaunay triangulation (IDT) is a fundamental data structure in computational geometry and computer graphics. However, except for some theoretical results, such as existence and uniqueness, little progress has been made towards computing IDT on simplicial surfaces. To date the only way for constructing IDTs is the edge-flipping algorithm, which iteratively flips the non-Delaunay edge to be locally Delaunay. Although the algorithm is conceptually simple and guarantees to stop in finite steps, it has no known time complexity. Moreover, the edge-flipping algorithm may produce non-regula…

Kinetic Voronoi Diagrams and Delaunay Triangulations under Polygonal Distance Functions

Let $P$ be a set of $n$ points and $Q$ a convex $k$-gon in ${\mathbb R}^2$. We analyze in detail the topological (or discrete) changes in the structure of the Voronoi diagram and the Delaunay triangulation of $P$, under the convex distance function defined by $Q$, as the points of $P$ move along prespecified continuous trajectories. Assuming that each point of $P$ moves along an algebraic trajectory of bounded degree, we establish an upper bound of $O(k^4nλ_r(n))$ on the number of topological changes experienced by the diagrams throughout the motion; here $λ_r(n)$ is the maximum length of an $…

The hyperbolic Voronoi diagram in arbitrary dimension

We show that in the Klein projective ball model of hyperbolic space, the hyperbolic Voronoi diagram is affine and amounts to clip a corresponding power diagram, requiring however algebraic arithmetic. By considering the lesser-known Beltrami hemisphere model of hyperbolic geometry, we overcome the arithmetic limitations of Klein construction. Finally, we characterize the bisectors and geodesics in the other Poincar\' e upper half-space, the Poincaré ball, and the Lorentz hyperboloid models, and discusses on degenerate cases for which the dual hyperbolic Delaunay complex is not a triangulation.

Hyperbolic Voronoi diagrams made easy

We present a simple framework to compute hyperbolic Voronoi diagrams of finite point sets as affine diagrams. We prove that bisectors in Klein's non-conformal disk model are hyperplanes that can be interpreted as power bisectors of Euclidean balls. Therefore our method simply consists in computing an equivalent clipped power diagram followed by a mapping transformation depending on the selected representation of the hyperbolic space (e.g., Poincaré conformal disk or upper-plane representations). We discuss on extensions of this approach to weighted and $k$-order diagrams, and describe their du…

Voronoi图——面向高维Voronoi图的改进光线投射与积分方案

Voronoi Graph -- Improved raycasting and integration schemes for high dimensional Voronoi diagrams

针对高维Voronoi图计算难题，本文提出更精确快速的改进顶点搜索法与深度优先图遍历算法，可完整计算高维Voronoi图，计算时间优于或持平qHull等经典算法，还拓展了光线投射算法用于Voronoi胞体的蒙特卡洛积分近似。

Abstract Voronoi-like Graphs: Extending Delaunay's Theorem and Applications

Any system of bisectors (in the sense of abstract Voronoi diagrams) defines an arrangement of simple curves in the plane. We define Voronoi-like graphs on such an arrangement, which are graphs whose vertices are locally Voronoi. A vertex $v$ is called locally Voronoi, if $v$ and its incident edges appear in the Voronoi diagram of three sites. In a so-called admissible bisector system, where Voronoi regions are connected and cover the plane, we prove that any Voronoi-like graph is indeed an abstract Voronoi diagram. The result can be seen as an abstract dual version of Delaunay's theorem on (lo…

On Voronoi diagrams and dual Delaunay complexes on the information-geometric Cauchy manifolds

We study the Voronoi diagrams of a finite set of Cauchy distributions and their dual complexes from the viewpoint of information geometry by considering the Fisher-Rao distance, the Kullback-Leibler divergence, the chi square divergence, and a flat divergence derived from Tsallis' quadratic entropy related to the conformal flattening of the Fisher-Rao curved geometry. We prove that the Voronoi diagrams of the Fisher-Rao distance, the chi square divergence, and the Kullback-Leibler divergences all coincide with a hyperbolic Voronoi diagram on the corresponding Cauchy location-scale parameters, …

MadVoro：分布式内存系统中Voronoi图的并行构建

MadVoro: Parallel Construction of Voronoi Diagrams in Distributed Memory Systems

本文提出了一种在分布式内存系统中并行构建Voronoi图的新算法MadVoro，利用Delaunay三角剖分作为中间步骤，并实现了精确构建和负载平衡，运行时间优于现有框架。

Incremental Voronoi Diagrams

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 sites are added. We define a general update operation for planar graphs modeling the incremental construction of several variants of Voronoi diagrams as well as the incremental construction of an intersection of halfspaces in $\mathbb{R}^3$. We show that the amortized number of edge insertions and removals needed to add a new site is $O(\sqrt{n})$. A matching $Ω…

良好中心三角剖分

Well-Centered Triangulation

本文提出了一种基于优化的启发式算法，通过最小化代价函数并移动内部顶点，将二维或三维三角剖分转化为良好中心三角剖分，并证明了其最优性质及与Delaunay和最小最大角三角剖分的关系。