Sibson's formula for higher order Voronoi diagrams

Index is a data structure to improve searching and minimizing effort to read the actual data. In spatial data structure, there are various objects can be considered, such as point, regular polygon, and irregular polygon. Irregular polygon is the shape of Voronoi Diagram which is used to divide a region into smaller regions based on its nearest neighbor. Voronoi Diagram has three orders, namely: (1) First Order Voronoi Diagram, (2) Higher Order Voronoi Diagram (HOVD), (3) Highest Order Voronoi Diagram (HSVD). Index uses Minimum Bounding Rectangle (MBR) in HSVD can cause high level of overlapping. The highest level of overlapping found in the HSVD, while the lowest is found in the First Order. VoR-tree (Voronoi R-Tree) is used to minimize overlapping. Previous studies used VoR-tree only to observed First Order Voronoi Diagram. This paper presents an index HVoR-Tree, which uses the VoR-tree structure effectively used in Highest Order Voronoi Diagram (HSVD).

Computational geometry is a mathematical knowlege in the field related to the design and analysis of algorithm to solve geometry problems. Its can be applicated in the fields of mapping, robotics, geometry and so forth. A method can be used is Voronoi diagram. Voronoi diagram is a method of deviding the area to a smaller area based on the principle of the nearest neighboring. This method only used in 1-order voronoi diagram. In voronoi diagram there is a new variation named Highest Order Voronoi Diagram (HSVD). HSVD can be used for all orders voronoi diagram. However, these methods have disadvantage that accessing fragment use linear search. Consequently make data fragment searches to find the region to be slow and takes a long time. Therefore, in this paper will present a index structure that incoperates Highest Order Voronoi Diagrams into Quadtree. Quadtree index used is capable of cutting more than half of the original data. This algorithm makes the search regions faster than before.

The problem of origami design is how to express the intended shape by folding a single sheet of paper. A two-layer origami tessellation (TLOT) is a category of flat origami which makes a polygonal figure that is slightly lifted up from the surrounding paper layers. To construct the crease pattern of a TLOT origami piece, a Voronoi diagram that corresponds to the figure is needed. Although a method to obtain such a Voronoi diagram is known, it is not clear how to evaluate the Voronoi diagram in order to obtain better crease patterns in which there are few too-short creases that are difficult to fold physically. In this paper, we propose a method to evaluate the lengths of the creases from the Voronoi diagram. We also propose a method to interactively construct Voronoi diagrams that create a crease pattern with relatively long creases. We show the effectiveness of the methods by implementing a prototype system on a PC. We confirmed that it is possible to construct appropriate Voronoi diagrams for TLOT.

Computational geometry is concerned with the design and analysis of algorithms for geometrical problems. In addition, other more practically oriented, areas of computer science-such as computer graphics, computer-aided design, robotics, pattern recognition, and operation research-give rise to problems that inherently are geometrical. A method that is commonly used is Voronoi diagram. Voronoi diagram is a diagram that divides plane based on nearest neighbour approach. This method does not require checking objects one by one. In Voronoi diagram there is a new variation named Highest Order Voronoi Diagram. Highest order voronoi diagram can be used directly to identify the region for various type of spatial queries and can be used for all order Voronoi Diagram. From work related there is a method named FLIP. Deficiency from work related is data structure for construction are too high which makes construction of Highest Order Voronoi Diagram is heavy. Therefore, in this paper will present Left with Least-Angle Movement (LAM) method for construction Highest Order Voronoi Diagram that change data structure to scale up data that can be processed.

We consider a generalization of Voronoi diagrams, recently introduced by Barequet et al., in which the distance is measured from a pair of sites to a point. An easy way to define such distance was proposed together with the concept: it can be the sum-of, the product-of, or (the absolute value of) the difference-between Euclidean distances from either site to the respective point. We explore further the last definition, and analyze the complexity of the nearest- and the furthest-neighbor 2-site Voronoi diagrams for points in the plane with Manhattan or Chebyshev underlying metrics, providing extensions to general Minkowsky metrics and, for the nearest-neighbor case, to higher dimensions. In addition, we point out that the observation made earlier in the literature that 2-point site Voronoi diagrams under the sum-of and the product-of Euclidean distances are identical and almost identical to the second order Voronoi diagrams, respectively, holds in a much more general statement.

Let P be a planar n-point set in general position. For k between 1 and n-1, the Voronoi diagram of order k is obtained by subdividing the plane into regions such that points in the same cell have the same set of nearest k neighbors in P. The (nearest point) Voronoi diagram (NVD) and the farthest point Voronoi diagram (FVD) are the particular cases of k=1 and k=n-1, respectively. It is known that the family of all higher-order Voronoi diagrams of order 1 to K for P can be computed in total time O(n K^2 + n log n) using O(K^2(n-K)) space. Also NVD and FVD can be computed in O(n log n) time using O(n) space. For s in {1, ..., n}, an s-workspace algorithm has random access to a read-only array with the sites of P in arbitrary order. Additionally, the algorithm may use O(s) words of Theta(log n) bits each for reading and writing intermediate data. The output can be written only once and cannot be accessed afterwards. We describe a deterministic s-workspace algorithm for computing an NVD and also an FVD for P that runs in O((n^2/s) log s) time. Moreover, we generalize our s-workspace algorithm for computing the family of all higher-order Voronoi diagrams of P up to order K in O(sqrt(s)) in total time O( (n^2 K^6 / s) log^(1+epsilon)(K) (log s / log K)^(O(1)) ) for any fixed epsilon > 0. Previously, for Voronoi diagrams, the only known s-workspace algorithm was to find an NVD for P in expected time O((n^2/s) log s + n log s log^*s). Unlike the previous algorithm, our new method is very simple and does not rely on advanced data structures or random sampling techniques.

Let $P$ be a planar set of $n$ sites in general position. For $k\in\{1,\dots,n-1\}$, the Voronoi diagram of order $k$ for $P$ is obtained by subdividing the plane into cells such that points in the same cell have the same set of nearest $k$ neighbors in $P$. The (nearest site) Voronoi diagram (NVD) and the farthest site Voronoi diagram (FVD) are the particular cases of $k=1$ and $k=n-1$, respectively. For any given $K\in\{1,\dots,n-1\}$, the family of all higher-order Voronoi diagrams of order $k=1,\dots,K$ for $P$ can be computed in total time $O(nK^2+ n\log n)$ using $O(K^2(n-K))$ space [Aggarwal et al., DCG'89; Lee, TC'82]. Moreover, NVD and FVD for $P$ can be computed in $O(n\log n)$ time using $O(n)$ space [Preparata, Shamos, Springer'85]. For $s\in\{1,\dots,n\}$, an $s$-workspace algorithm has random access to a read-only array with the sites of $P$ in arbitrary order. Additionally, the algorithm may use $O(s)$ words, of $\Theta(\log n)$ bits each, for reading and writing intermediate data. The output can be written only once and cannot be accessed or modified afterwards. We describe a deterministic $s$-workspace algorithm for computing NVD and FVD for $P$ that runs in $O((n^2/s)\log s)$ time. Moreover, we generalize our $s$-workspace algorithm so that for any given $K\in O(\sqrt{s})$, we compute the family of all higher-order Voronoi diagrams of order $k=1,\dots,K$ for $P$ in total expected time $O (\frac{n^2 K^5}{s}(\log s+K2^{O(\log^* K)}))$ or in total deterministic time $O(\frac{n^2 K^5}{s}(\log s+K\log K))$. Previously, for Voronoi diagrams, the only known $s$-workspace algorithm runs in expected time $O\bigl((n^2/s)\log s+n\log s\log^* s)$ [Korman et al., WADS'15] and only works for NVD (i.e., $k=1$). Unlike the previous algorithm, our new method is very simple and does not rely on advanced data structures or random sampling techniques.

Abstract Voronoi diagrams [15,16] are based on bisecting curves enjoying simple combinatorial properties, rather than on the geometric notions of sites and circles. They serve as a unifying concept. Once the bisector system of any concrete type of Voronoi diagram is shown to fulfill the AVD properties, structural results and efficient algorithms become available without further effort. For example, the first optimal algorithms for constructing nearest Voronoi diagrams of disjoint convex objects, or of line segments under the Hausdorff metric, have been obtained this way [20].In a concrete order-k Voronoi diagram, all points are placed into the same region that have the same k nearest neighbors among the given sites. This paper is the first to study abstract Voronoi diagrams of arbitrary order k. We prove that their complexity is upper bounded by 2k(n − k). So far, an O(k (n − k)) bound has been shown only for point sites in the Euclidean and L p plane [18,19], and, very recently, for line segments [23]. These proofs made extensive use of the geometry of the sites.Our result on AVDs implies a 2k (n − k) upper bound for a wide range of cases for which only trivial upper complexity bounds were previously known, and a slightly sharper bound for the known cases.Also, our proof shows that the reasons for this bound are combinatorial properties of certain permutation sequences.KeywordsAbstract Voronoi diagramscomputational geometrydistance problemshigher order Voronoi diagramsVoronoi diagrams

This chapter presents a methodology for automated cartographic data input, drawing and editing. This methodology is based on kinematic algorithms for point and line Delaunay triangulation and the Voronoi diagram. It allows one to automate some parts of the manual digitization process and the topological editing of maps that preserve map updates. The manual digitization process is replaced by computer assisted skeletonization using scanned paper maps. We are using the Delaunay triangulation and the Voronoi diagram in order to extract the skeletons that are guaranteed to be topologically correct. The features thus extracted as object centrelines can be stored as vector maps in a Geographic Information System after labelling and editing. This research work can also be used for updates from sources that are either paper copy maps or digital raster images. A prototype application that was developed as part of the research has been presented.

On the complexity of higher order abstract Voronoi diagrams

Present approaches to achieve $k$ -coverage for Wireless Sensor Networks still rely on centralized techniques. In this paper, we devise a distributed method for this problem, namely Distributed VOronoi based Cooperation scheme (DVOC), where nodes cooperate in hole detection and recovery. In previous Voronoi based schemes, each node only monitors its own critical points. Such methods are inefficient for $k$ -coverage because the critical points are far away from their generating nodes in $k$ -order Voronoi diagram, causing high cost for transmission and computing. As a solution, DVOC enables nodes to monitor others’ critical points around themselves by building local Voronoi diagrams (LVDs). Further, DVOC constrains the movement of every node to avoid generating new holes. If a node cannot reach its destination due to the constraint, its hole healing responsibility will fall to other cooperating nodes. The experimental results from the real world testbed demonstrate that DVOC outperforms the previous schemes.

Reinforcement learning is considered an important tool for robotic learning in unknown/uncertain environments. In this article, we suggest that Voronoi space division creates a new Voronoi region which permits an arbitrary point in the plane, say a Voronoi Q-value element (VQE), and constructs a new method for space division using a Voronoi diagram in order to realize multidimensional reinforcement learning. This article shows some results for four-dimensional spaces, and the essential characteristics of VQEs in a continuous state and action are also described. The advantages of learning with a variety of VQEs are enhanced learning speed and reliability for this task.

Decision making support has become essential to conduct an effective management and operational system. It is considered an important part of operational research and management science. There are various decision issues concerning the good governance and management of a supply chain. In this paper, we focus on the conception problem of a supply chain, especially the geographical location implementation of the facilities, also known as multi-facility Weber problem. In fact, business location is considered a strategic decision; it was the focus of various studies in the literature. We propose in this paper an improved approach to solve the multi-facility location decision. The methodology is based on mathamatical modeling using the partition theory and Voronoi diagram in order to define the facilities locations and their assigned demand points (customers). Our objective is to design a network respecting the environmental and economical factors. The obtained results are represented as a diagram defining the subdivisions, the partitions of demand points and the locations of the facilities.

This paper presents a methodology to automate some parts of the manual digitization process. This includes replacing the manual digitization process by computer assisted skeletonization using scanned paper maps. In colour scanned paper maps various features on the map can be distinguished based on their colour. This research work utilizes the Delaunay triangulation and the Voronoi diagram in order to extract the skeletons that are guaranteed to be topologically correct. The features thus extracted as object centrelines can be stored as vector maps in a Geographic Information System after labelling and editing. Map updates are important in any Geographic Information System. Therefore, this research work can also be used for updates from sources that are either paper copy maps or digital raster images. A prototype application that is developed as part of the research has been presented.

In 1983, John Cage used the traditional stone garden, or karesansui at the Zen temple, Ryōan-ji in Kyoto as a model to generate a series of visual and musical works that utilized tracings of a collection of his own rocks. In this article, I analyze the first of the musical works, Ryoanji for oboe, using mixed methods drawn from morphological image analysis and formal concept analysis (FCA). I introduce the aesthetics of the karesansui and then examine the previous work of van Tonder and Lyons regarding the medial axis transform (MAT) of the garden at Ryōan-ji. This leads to the use of the distance transform, local maxima, and Voronoi diagram in order to decompose the two-dimensional image plane of Cage's Ryoanji for oboe. Finally, using the technique of FCA for constructing a number of formal concept lattices, the pitch-class segmentation of Ryoanji for oboe is investigated in regard to the sound gardens and the classes of Voronoi regions found across sound gardens.