We present a new and simple randomized algorithm for constructing the Delaunay triangulation using nearest neighbor graphs for point location. Under suitable assumptions, it runs in linear expected time for points in the plane with polynomially bounded spread, i.e., if the ratio between the largest and smallest pointwise distance is polynomially bounded. This also holds for point sets with bounded spread in higher dimensions as long as the expected complexity of the Delaunay triangulation of a sample of the points is linear in the sample size.
The \emph{Delaunay graph} of a point set $P \subseteq \mathbb{R}^2$ is the plane graph with the vertex-set $P$ and the edge-set that contains $\{p,p'\}$ if there exists a disc whose intersection with $P$ is exactly $\{p,p'\}$. Accordingly, a triangulated graph $G$ is \emph{Delaunay realizable} if there exists a triangulation of the Delaunay graph of some $P \subseteq \mathbb{R}^2$, called a \emph{Delaunay triangulation} of $P$, that is isomorphic to $G$. The objective of \textsc{Delaunay Realization} is to compute a point set $P \subseteq \mathbb{R}^2$ that realizes a given graph $G$ (if such …
Computing the Delaunay triangulation (DT) of a given point set in $\mathbb{R}^D$ is one of the fundamental operations in computational geometry. Recently, Funke and Sanders (2017) presented a divide-and-conquer DT algorithm that merges two partial triangulations by re-triangulating a small subset of their vertices - the border vertices - and combining the three triangulations efficiently via parallel hash table lookups. The input point division should therefore yield roughly equal-sized partitions for good load-balancing and also result in a small number of border vertices for fast merging. In…
We propose an algorithm to create a 3-colorable Delaunay Triangulation. The input of the problem we are trying to solve is a set X of n twodimensional points. The output is a 3-colorable two-dimensional Delaunay triangulation T for X U Y , where Y is a set of m new points. We want to m be as few as possible.
We study metrics that assess how close a triangulation is to being a Delaunay triangulation, for use in contexts where a good triangulation is desired but constraints (e.g., maximum degree) prevent the use of the Delaunay triangulation itself. Our near-Delaunay metrics derive from common Delaunay properties and satisfy a basic set of design criteria, such as being invariant under similarity transformations. We compare the metrics, showing that each can make different judgments as to which triangulation is closer to Delaunay. We also present a preliminary experiment, showing how optimizing for …
We present a self-contained short proof of the seminal result of Dillencourt (SoCG 1987 and DCG 1990) that Delaunay triangulations, of planar point sets in general position, are 1-tough. An important implication of this result is that Delaunay triangulations have perfect matchings. Another implication of our result is a proof of the conjecture of Aichholzer et al. (2010) that at least $n$ points are required to block any $n$-vertex Delaunay triangulation
Delaunay triangulations of a point set in the Euclidean plane are ubiquitous in a number of computational sciences, including computational geometry. Delaunay triangulations are not well defined as soon as 4 or more points are concyclic but since it is not a generic situation, this difficulty is usually handled by using a (symbolic or explicit) perturbation. As an alternative, we propose to define a canonical triangulation for a set of concyclic points by using a max-min angle characterization of Delaunay triangulations. This point of view leads to a well defined and unique triangulation as lo…
The problem of computing the exact stretch factor (i.e., the tight bound on the worst case stretch factor) of a Delaunay triangulation is one of the longstanding open problems in computational geometry. Over the years, a series of upper and lower bounds on the exact stretch factor have been obtained but the gap between them is still large. An alternative approach to solving the problem is to develop techniques for computing the exact stretch factor of ``easier'' types of Delaunay triangulations, in particular those defined using regular-polygons instead of a circle. Tight bounds exist for Dela…
University of Ljubljana（卢布尔雅那大学）；Institute of Mathematics, Physics and Mechanics（数学、物理和力学研究所）；Siebel School of Computing and Data Science, University of Illinois at Urbana-Champaign（伊利诺伊大学厄巴纳-香槟分校赛博计算与数据科学学院）；Department of Computer Science, City St George’s, University of London（伦敦大学城市圣乔治学院计算机科学系）
A 3D Sweep Hull Algorithm for computing Convex Hulls and Delaunay Triangulation
本文提出了一种新的计算三维点集的凸包的O（nlog（n））算法。该算法首先对（x，y，z）中的点进行排序，然后使用约束条件将排序后的点递增地添加到凸包中，该约束条件是添加到包中的每个新点都可以“看到”至少一个与最后添加的点接触的小平面。减少了添加新点的搜索时间。该算法属于扫掠船体算法家族。虽然对于一般情况比q壳慢，但对于所有点都在3D壳上的病理情况（如德洛内三角剖分的情况），它显著优于q壳。该算法被命名为“Newton Apple Wrapper算法”，并已在GPL下以C++发布。关键词：德洛内三角剖分，三维凸包。
Motivated by recent work on Delaunay triangulations of hyperbolic surfaces, we consider the minimal number of vertices of such triangulations. First, we will show that every hyperbolic surface of genus $g$ has a simplicial Delaunay triangulation with $O(g)$ vertices, where edges are given by distance paths. Then, we will construct a class of hyperbolic surfaces for which the order of this bound is optimal. Finally, to give a general lower bound, we will show that the $Ω(\sqrt{g})$ lower bound for the number of vertices of a simplicial triangulation of a topological surface of genus $g$ is tigh…
A New Approach to Output-Sensitive Voronoi Diagrams and Delaunay Triangulations
我们描述了一个新的算法计算的沃罗诺伊图的一组$n$点在常维欧氏空间。我们的算法的运行时间是$O(f \log n \log Δ)$，其中$f$是沃罗诺伊图的输出复杂度，$Δ$是输入的扩展，最大与最小成对距离的比率。尽管该算法及其分析简单，但它改进了所有具有多项式扩展和近线性输出大小的输入的现有技术。关键思想是首先使用Voronoi细化网格生成的思想构建输入点的超集的沃罗诺伊图。然后，以一种简单的方式去除额外的点，允许总工作在输出复杂度方面有界，从而产生输出敏感界。移除仅涉及局部翻转，并且受到动力学数据结构的启发。
Let $P$ be a collection of $n$ points moving along pseudo-algebraic trajectories in the plane. One of the hardest open problems in combinatorial and computational geometry is to obtain a nearly quadratic upper bound, or at least a subcubic bound, on the maximum number of discrete changes that the Delaunay triangulation $\DT(P)$ of $P$ experiences during the motion of the points of $P$. In this paper we obtain an upper bound of $O(n^{2+\eps})$, for any $\eps>0$, under the assumptions that (i) any four points can be co-circular at most twice, and (ii) either no triple of points can be collinear …
On the Embeddability of Delaunay Triangulations in Anisotropic, Normed, and Bregman Spaces
给定一个二维空间赋予的发散函数是凸的第一个参数，连续可微的第二个，并满足适当的正则性条件在Voronoi顶点，我们表明，自由度（没有断开的Voronoi地区）是足以确保Voronoi边和顶点也连接，和对偶是一个简单的平面图。然后，我们证明了直边对偶的无顶点沃罗诺伊图（网站作为第一个参数的分歧）总是一个嵌入的三角剖分。我们的证明所涵盖的分歧是Bregman分歧，各向异性分歧，以及严格凸$\mathcal{C}^1$规范（包括具有$1< p < \infty$的$L_p$规范）的所有距离。虽然Bregman图的{第一类}是简单的仿射图，和他们的（{加权}德洛内三角剖分）总是嵌入的，我们证明了无Bregman图的\emph{second kind}的总是嵌入的。
Delaunay protection is a measure of how far a Delaunay triangulation is from being degenerate. In this short paper we study the protection properties and other quality measures of the Delaunay triangulations of a family of lattices that is obtained by distorting the integer grid in $\mathbb{R}^d$. We show that the quality measures of this family are maximized for a certain distortion parameter, and that for this parameter, the lattice is isometric to the permutahedral lattice, which is a well-known object in discrete geometry.
The Bolza surface can be seen as the quotient of the hyperbolic plane, represented by the Poincaré disk model, under the action of the group generated by the hyperbolic isometries identifying opposite sides of a regular octagon centered at the origin. We consider generalized Bolza surfaces $\mathbb{M}_g$, where the octagon is replaced by a regular $4g$-gon, leading to a genus $g$ surface. We propose an extension of Bowyer's algorithm to these surfaces. In particular, we compute the value of the systole of $\mathbb{M}_g$. We also propose algorithms computing small sets of points on $\mathbb{M}_…
Randomized HyperSteiner: A Stochastic Delaunay Triangulation Heuristic for the Hyperbolic Steiner Minimal Tree
School of Electrical Engineering and Computer Science, KTH Royal Institute of Technology（KTH皇家理工学院电气工程与计算机科学学院）；Amsterdam Machine Learning Lab (AMLab), University of Amsterdam（阿姆斯特丹大学阿姆斯特丹机器学习实验室）；School of Engineering Sciences, KTH Royal Institute of Technology（KTH皇家理工学院工程科学学院）；Leiden Institute of Advanced Computer Science, Universiteit Leiden（莱顿大学莱顿高级计算机科学研究所）
Given two distinct point sets $P$ and $Q$ in the plane, we say that $Q$ \emph{blocks} $P$ if no two points of $P$ are adjacent in any Delaunay triangulation of $P\cup Q$. Aichholzer et al. (2013) showed that any set $P$ of $n$ points in general position can be blocked by $\frac{3}{2}n$ points and that every set $P$ of $n$ points in convex position can be blocked by $\frac{5}{4}n$ points. Moreover, they conjectured that, if $P$ is in convex position, $n$ blocking points are sufficient and necessary. The necessity was recently shown by Biniaz (2021) who proved that every point set in general pos…
