This article represents the computational model for spacial addresation of the sensors in the dynamically changing real-time internet of things system. The model bases on the Voronoi diagrams as a basic data structure. Problem - the correct data addresation without time delays in real-time processing and database indexation in distributed storages. Goal - to develop the real-time processing model of the object location identification in the Voronoi diagram data structure. Relevance - the research and development on the contemporary issues on the convergence (the N limit up to which the model presents the algorithm convergence), time-delay, correct indexation in the database transactions and adressation throughout wireless radio frequencies in the distribiuted sensor systems. Solution proposes the Voronoi diagram computational geometry data structure and the sweeping curve algorithm. Methods represents the following steps: simulation of the dynamically changing agent system using set of points that are based on a contemporary paths of the public transport routes, bykes, vehicles circulations; 3D map rendering on the SITG Canton of Geneva map; Voronoi diagrams calculation with the Fortune's algorithm.
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。
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…
The Voronoi Diagram of Rotating Rays with applications to Floodlight Illumination
Università Roma Tre（罗马第三大学）；Università della Svizzera italiana（瑞士意大利语联邦理工学院）；University of Novi Sad（诺维萨德大学）；Universitat Politècnica de Catalunya（加泰罗尼亚理工大学）
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 $\tilde O(nb^2)$ time (footnote: The $\tilde O$ notation hides polylogarithmic factors.) so that one can compute any additively weighted Voronoi diagram for these sites in $\tilde O(b)$ time. We use this construction to compute the diameter of a directed planar graph with real arc lengths in $\tilde{O}(n^{5/3})$ time. This improves the recent breakthrough res…
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 $S$ such that (i) the points prefer sites closer to them and sites prefer points closer to them, and (ii) each site has a quota or "appetite" indicating the area of the set of points that can be matched to it. Thus, a stable-matching Voronoi diagram is a solution to the well-known post office problem with the added (realistic) constraint that each post office has …
Given a set $S$ of $m$ point sites in a simple polygon $P$ of $n$ vertices, we consider the problem of computing the geodesic farthest-point Voronoi diagram for $S$ in $P$. It is known that the problem has an $Ω(n+m\log m)$ time lower bound. Previously, a randomized algorithm was proposed [Barba, SoCG 2019] that can solve the problem in $O(n+m\log m)$ expected time. The previous best deterministic algorithms solve the problem in $O(n\log \log n+ m\log m)$ time [Oh, Barba, and Ahn, SoCG 2016] or in $O(n+m\log m+m\log^2 n)$ time [Oh and Ahn, SoCG 2017]. In this paper, we present a deterministic …
We consider preprocessing a set $S$ of $n$ points in convex position in the plane into a data structure supporting queries of the following form: given a point $q$ and a directed line $\ell$ in the plane, report the point of $S$ that is farthest from (or, alternatively, nearest to) the point $q$ among all points to the left of line $\ell$. We present two data structures for this problem. The first data structure uses $O(n^{1+\varepsilon})$ space and preprocessing time, and answers queries in $O(2^{1/\varepsilon} \log n)$ time, for any $0 < \varepsilon < 1$. The second data structure uses $O(n …
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…
In this paper, we propose a novel space partitioning strategy for implicit hierarchy visualization such that the new plot not only has a tidy layout similar to the treemap, but also is flexible to data changes similar to the Voronoi treemap. To achieve this, we define a new distance function and neighborhood relationship between sites so that space will be divided by axis-aligned segments. Then a sweepline+skyline based heuristic algorithm is proposed to allocate the partitioned spaces to form an orthogonal Voronoi diagram with orthogonal rectangles. To the best of our knowledge, it is the fir…
We describe a new data structure for dynamic nearest neighbor queries in the plane with respect to a general family of distance functions. These include $L_p$-norms and additively weighted Euclidean distances. Our data structure supports general (convex, pairwise disjoint) sites that have constant description complexity (e.g., points, line segments, disks, etc.). Our structure uses $O(n \log^3 n)$ storage, and requires polylogarithmic update and query time, improving an earlier data structure of Agarwal, Efrat and Sharir that required $O(n^\varepsilon)$ time for an update and $O(\log n)$ time …
We introduce the inverse Voronoi diagram problem in graphs: given a graph $G$ with positive edge-lengths and a collection $\mathbb{U}$ of subsets of vertices of $V(G)$, decide whether $\mathbb{U}$ is a Voronoi diagram in $G$ with respect to the shortest-path metric. We show that the problem is NP-hard, even for planar graphs where all the edges have unit length. We also study the parameterized complexity of the problem and show that the problem is W[1]-hard when parameterized by the number of Voronoi cells or by the pathwidth of the graph. For trees we show that the problem can be solved in $O…
