Theta graphs are important geometric graphs that have many applications, including wireless networking, motion planning, real-time animation, and minimum-spanning tree construction. We give closed form expressions for the average degree of theta graphs of a homogeneous Poisson point process over the plane. We then show that essentially the same bounds---with vanishing error terms---hold for theta graphs of finite sets of points that are uniformly distributed in a square. Finally, we show that the number of edges in a theta graph of points uniformly distributed in a square is concentrated around its expected value.
Bern al @cite_0 study the maximum degree of Delaunay triangulations of random point sets. Devroye al study the maximum degree of Gabriel graphs @cite_17 of random point sets. Arkin al @cite_14 study the length of the longest edge in Delaunay triangulations of random point sets.
@cite_0 @cite_14 @cite_17
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This paper presents a kinetic data structure (KDS) for maintenance of the Euclidean minimum spanning tree (EMST) on a set of moving points in 2-dimensional space. For a set of n points moving in the plane we build a KDS of size O(n) in O(nlogn) preprocessing time by which the EMST is maintained efficiently during the motion. This is done by applying the required changes to the combinatorial structure of the EMST which is changed in discrete timestamps. We assume that the motion of the points, i.e. x and y coordinates of the points, are defined by algebraic functions of constant maximum degree. In terms of the KDS performance parameters, our KDS is responsive, local, and compact. The presented KDS is based on monitoring changes of the Delaunay triangulation of the points and edge-length changes of the edges of the current Delaunay triangulation. Several localized routing protocols (see Bose, P. and Morin, P., Proc. 10th Annual Int. Symp. on Algorithms and Computation ISAAC, 1999) guarantee the delivery of packets when the underlying network topology is the Delaunay triangulation of all wireless nodes. However, it is expensive to construct the Delaunay triangulation in a distributed manner. Given a set of wireless nodes, we more accurately model the network as a unit-disk graph, UDG, in which a link between two nodes exists only if the distance between them is at most the maximum transmission range. Given a graph H, a spanning subgraph G of H is a t-spanner if the length of the shortest path connecting any two points in G is no more than t times the length of the shortest path connecting the two points in H. We present a novel localized networking protocol that constructs a planar 2.5-spanner of UDG, called the localized Delaunay triangulation, as network topology. It contains all edges that are in both the UDG and the Delaunay triangulation of all wireless nodes. Our experiments show that the delivery rates of existing localized routing protocols are increased when localized Delaunay triangulation is used instead of several previously proposed topologies. The total communication cost of our networking protocol is O(n log n) bits. Moreover, the computation cost of each node u is O(d sub u log d sub u ), where d sub u is the number of 1-hop neighbors of u in UDG. We give an expected-case analysis of Delaunay triangulations. To avoid edge effects we consider a unit-intensity Poisson process in Euclidean d-space, and then limit attention to the portion of the triangulation within a cube of side n1 d. For d equal to two, we calculate the expected maximum edge length, the expected minimum and maximum angles, and the average aspect ratio of a triangle. We also show that in any fixed dimension the expected maximum vertex degree is Θ(log n loglog n). Altogether our results provide some measure of the suitability of the Delaunay triangulation for certain applications, such as interpolation and mesh generation. An assumption of nearly all algorithms in computational geometry is that the input points are given precisely, so it is interesting to ask what is the value of imprecise information about points. We show how to preprocess a set of n disjoint unit disks in the plane in O(nlogn) time so that if one point per disk is specified with precise coordinates, the Delaunay triangulation can be computed in linear time. From the Delaunay, one can obtain the Gabriel graph and a Euclidean minimum spanning tree; it is interesting to note the roles that these two structures play in our algorithm to quickly compute the Delaunay. In a geometric bottleneck shortest path problem, we are given a set S of n points in the plane, and want to answer queries of the following type: given two points p and q of S and a real number L, compute (or approximate) a shortest path between p and q in the subgraph of the complete graph on S consisting of all edges whose lengths are less than or equal to L. We present efficient algorithms for answering several query problems of this type. Our solutions are based on Euclidean minimum spanning trees, spanners, and the Delaunay triangulation. A result of independent interest is the following. For any two points p and q of S, there is a path between p and q in the Delaunay triangulation, whose length is less than or equal to 2π (3 cos(π 6)) times the Euclidean distance |pq| between p and q, and all of whose edges have length at most |pq|. Given a set S of points in the plane, there is a triangulation of S such that a path found within this triangulation has length bounded by a constant times the straight-line distance between the endpoints of the path. Specifically, for any two points a and b of S there is a path along edges of the triangulation with length less than √10 times |ab|, where |ab| is the straight-line Euclidean distance between a and b. Thus, a shortest path in this planar graph is less than about 3 times longer than the corresponding straight-line distance. The triangulation that has this property is the L 1 metric Delaunay triangulation for the set 5. This result can be applied to motion planning in the plane. Given a source, a destination, and a set of polygonal obstacles of size n, an O(n) size data structure can be used to find a reasonable approximation to the shortest path between the source and the destination in O(n log n) time. Motivated by low energy consumption in geographic routing in wireless networks, there has been recent interest in determining bounds on the length of edges in the Delaunay graph of randomly distributed points. Asymptotic results are known for random networks in planar domains. In this paper, we obtain upper and lower bounds that hold with parametric probability in any dimension, for points distributed uniformly at random in domains with and without boundary. The results obtained are asymptotically tight for all relevant values of such probability and constant number of dimensions, and show that the overhead produced by boundary nodes in the plane holds also for higher dimensions. To our knowledge, this is the first comprehensive study on the lengths of long edges in Delaunay graphs Motivated by applications of Gabriel graphs and Yao graphs in wireless ad-hoc networks, we show that the maximum degree of a random Gabriel graph or Yao graph defined on n points drawn uniformly at random from a unit square grows as ?(log n log log n) in probability. Urban reconstruction from a video captured by a surveying vehicle constitutes a core module of automated mapping. When computational power represents a limited resource and, a detailed map is not the primary goal, the reconstruction can be performed incrementally, from a monocular video, carving a 3D Delaunay triangulation of sparse points; this allows online incremental mapping for tasks such as traversability analysis or obstacle avoidance. To exploit the sharp edges of urban landscape, we propose to use a Delaunay triangulation of Edge-Points, which are the 3D points corresponding to image edges. These points constrain the edges of the 3D Delaunay triangulation to real-world edges. Besides the use of the Edge-Points, a second contribution of this paper is the Inverse Cone Heuristic that preemptively avoids the creation of artifacts in the reconstructed manifold surface. We force the reconstruction of a manifold surface since it makes it possible to apply computer graphics or photometric refinement algorithms to the output mesh. We evaluated our approach on four real sequences of the public available KITTI dataset by comparing the incremental reconstruction against Velodyne measurements. Given a background graph with n vertices, the binary censored block model assumes that vertices are partitioned into two clusters, and every edge is labeled independently at random with labels drawn from Bern(1 − e) if two endpoints are in the same cluster, or from Bern(e) otherwise, where e ∈ [0; 1 2] is a fixed constant. For Erdős-Renyi graphs with edge probability p = a log n n and fixed a, we show that the semidefinite programming relaxation of the maximum likelihood estimator achieves the optimal threshold equation for exactly recovering the partition from the labeled graph with probability tending to one as n → ∞. For random regular graphs with degree scaling as a log n, we show that the semidefinite programming relaxation also achieves the optimal recovery threshold aD(Bern(1 2)∥Bern(e)) > 1, where D denotes the Kullback-Leibler divergence. A generalization of the convex hull of a finite set of points in the plane is introduced and analyzed. This generalization leads to a family of straight-line graphs, " -shapes," which seem to capture the intuitive notions of "fine shape" and "crude shape" of point sets. It is shown that a-shapes are subgraphs of the closest point or furthest point Delaunay triangulation. Relying on this result an optimal O(n n) algorithm that constructs -shapes is developed. Abstract The relative neighbourhood graph (RNG) of a set of n points on the plane is defined. The ability of the RNG to extract a perceptually meaningful structure from the set of points is briefly discussed and compared to that of two other graph structures: the minimal spanning tree (MST) and the Delaunay (Voronoi) triangulation (DT). It is shown that the RNG is a superset of the MST and a subset of the DT. Two algorithms for obtaining the RNG of n points on the plane are presented. One algorithm runs in 0(n2) time and the other runs in 0(n3) time but works also for the d-dimensional case. Finally, several open problems concerning the RNG in several areas such as geometric complexity, computational perception, and geometric probability, are outlined. SUMMARY In contrast to most triangulation algorithms which implicitly assume that triangulation point locations are fixed, 'Laplacian' smoothing focuses on moving point locations to improve triangulation. Laplacian smoothing is attractive for its simplicity but it does require an existing triangulation. In this paper the effect of Laplacian smoothing on Delaunay triangulations is explored. It will become clear that constraining Laplacian smoothing to maintain a Delaunay triangulation measurably improves Laplacian smoothing. An early reference to the use of Laplacian smoothing is to be found in a papel by Buell and Bush.' They surveyed the use of Laplace's equation to generate two-dimensional meshes, and described an equipotential method2 which extends finite differences of Laplace's equation on rectangular grids to differences on triangular grids. On triangular grids, the weighted difference equations The convex hull of a set of points is the smallest convex set that contains the points. This article presents a practical convex hull algorithm that combines the two-dimensional Quickhull algorithm with the general-dimension Beneath-Beyond Algorithm. It is similar to the randomized, incremental algorithms for convex hull and delaunay triangulation. We provide empirical evidence that the algorithm runs faster when the input contains nonextreme points and that it used less memory. computational geometry algorithms have traditionally assumed that input sets are well behaved. When an algorithm is implemented with floating-point arithmetic, this assumption can lead to serous errors. We briefly describe a solution to this problem when computing the convex hull in two, three, or four dimensions. The output is a set of “thick” facets that contain all possible exact convex hulls of the input. A variation is effective in five or more dimensions. In this paper we study the problem of flipping edges in triangulations of polygons and point sets. One of the main results is that any triangulation of a set of n points in general position contains at least ( (n-4) 2 ) edges that can be flipped. We also prove that O(n + k 2 ) flips are sufficient to transform any triangulation of an n -gon with k reflex vertices into any other triangulation. We produce examples of n -gons with triangulations T and T' such that to transform T into T' requires Ω(n 2 ) flips. Finally we show that if a set of n points has k convex layers, then any triangulation of the point set can be transformed into any other triangulation using at most O(kn) flips. We describe a simple push-pull optimization (PPO) algorithm for blue-noise sampling by enforcing spatial constraints on given point sets. Constraints can be a minimum distance between samples, a maximum distance between an arbitrary point and the nearest sample, and a maximum deviation of a sample's capacity (area of Voronoi cell) from the mean capacity. All of these constraints are based on the topology emerging from Delaunay triangulation, and they can be combined for improved sampling quality and efficiency. In addition, our algorithm offers flexibility for trading-off between different targets, such as noise and aliasing. We present several applications of the proposed algorithm, including anti-aliasing, stippling, and non-obtuse remeshing. Our experimental results illustrate the efficiency and the robustness of the proposed approach. Moreover, we demonstrate that our remeshing quality is superior to the current state-of-the-art approaches. In this work we consider triangulations of point sets in the Euclidean plane, i.e., maximal straight-line crossing-free graphs on a finite set of points. Given a triangulation of a point set, an edge flip is the operation of removing one edge and adding another one, such that the resulting graph is again a triangulation. Flips are a major way of locally transforming triangular meshes. We show that, given a point set S in the Euclidean plane and two triangulations T"1 and T"2 of S, it is an APX-hard problem to minimize the number of edge flips to transform T"1 to T"2. This paper introduces a randomized variation of the alternating least squares (ALS) algorithm for rank reduction of canonical tensor formats. The aim is to address the potential numerical ill-conditioning of least squares matrices at each ALS iteration. The proposed algorithm, dubbed randomized ALS, mitigates large condition numbers via projections onto random tensors, a technique inspired by well-established randomized projection methods for solving overdetermined least squares problems in a matrix setting. A probabilistic bound on the condition numbers of the randomized ALS matrices is provided, demonstrating reductions relative to their standard counterparts. Additionally, results are provided that guarantee comparable accuracy of the randomized ALS solution at each iteration. The performance of the randomized algorithm is studied with three examples, including manufactured tensors and an elliptic PDE with random inputs. In particular, for the latter, tests illustrate not only improvements in condition... We present a method for reconstructing surfaces from point sets. The main novelty lies in a structure-preserving approach where the input point set is first consolidated by structuring and resampling the planar components, before reconstructing the surface from both the consolidated components and the unstructured points. The final surface is obtained through solving a graph-cut problem formulated on the 3D Delaunay triangulation of the structured point set where the tetrahedra are labeled as inside or outside cells. Structuring facilitates the surface reconstruction as the point set is substantially reduced and the points are enriched with structural meaning related to adjacency between primitives. Our approach departs from the common dichotomy between smooth piecewise-smooth and primitive-based representations by gracefully combining canonical parts from detected primitives and free-form parts of the inferred shape. Our experiments on a variety of inputs illustrate the potential of our approach in terms of robustness, flexibility and efficiency. Given a set of pointsV in the plane, the Euclidean bottleneck matching problem is to match each point with some other point such that the longest Euclidean distance between matched points, resulting from this matching, is minimized. To solve this problem, we definek-relative neighborhood graphs, (kRNG) which are derived from Toussaint's relative neighborhood graphs (RNG). Two points are calledk-relative neighbors if and only if there are less thank points ofV which are closer to both of the two points than the two points are to each other. AkRNG is an undirected graph (V,E r k) whereE r k is the set of pairs of points ofV which arek-relative neighbors. We prove that there exists an optimal solution of the Euclidean bottleneck matching problem which is a subset ofE r 17. We also prove that ¦E r k¦ < 18kn wheren is the number of points in setV. Our algorithm would construct a 17RNG first. This takesO(n2) time. We then use Gabow and Tarjan's bottleneck maximum cardinality matching algorithm for general graphs whose time-complexity isO((n logn)0.5m), wherem is the number of edges in the graph, to solve the bottleneck maximum cardinality matching problem in the 17RNG. This takesO(n1.5 log0.5n) time. The total time-complexity of our algorithm for the Euclidean bottleneck matching problem isO(n2 +n1.5 log0.5n).