We study the problem of computing geometric spanners for (additively) weighted point sets. A weighted point set is a set of pairs @math where @math is a point in the plane and @math is a real number. The distance between two points @math and @math is defined as @math . We show that in the case where all @math are positive numbers and @math for all @math (in which case the points can be seen as non-intersecting disks in the plane), a variant of the Yao graph is a @math -spanner that has a linear number of edges. We also show that the Additively Weighted Delaunay graph (the face-dual of the Additively Weighted Voronoi diagram) has constant spanning ratio. The straight line embedding of the Additively Weighted Delaunay graph may not be a plane graph. We show how to compute a plane embedding that also has a constant spanning ratio.
The @cite_18 of a finite set of points @math is a partition of the plane into @math regions such that each region contains exactly those points having the same nearest neighbor in @math . The points in @math are also called . It is well known that the Voronoi diagram of a set of points is the face dual of the Delaunay graph of that set of points @cite_18 , i.e. two points have adjacent Voronoi regions if and only if they share an edge in the Delaunay graph (see Figure ).
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The Voronoi diagram is a famous structure of computational geometry. We show that there is a straightforward equivalent in graph theory which can be efficiently computed. In particular, we give two algorithms for the computation of graph Voronoi diagrams, prove a lower bound on the problem, and identify cases where the algorithms presented are optimal. The space requirement of a graph Voronoi diagram is modest, since it needs no more space than does the graph itself. The investigation of graph Voronoi diagrams is motivated by many applications and problems on networks that can be easily solved with their help. This includes the computation of nearest facilities, all nearest neighbors and closest pairs, some kind of collision free moving, and anticenters and closest points. © 2000 John Wiley & Sons, Inc. We address the problem of replicating a Voronoi diagram @math of a planar point set @math by making proximity queries, which are of three possible (in decreasing order of information content): 1. the exact location of the nearest site(s) in @math ; 2. the distance to and label(s) of the nearest site(s) in @math ; 3. a unique label for every nearest site in @math . We provide algorithms showing how queries of Type 1 and Type 2 allow an exact cloning of @math with @math queries and @math processing time. We also prove that queries of Type 3 can never exactly clone @math , but we show that with @math queries we can construct an @math -approximate cloning of @math . In addition to showing the limits of nearest-neighbor database security, our methods also provide one of the first natural algorithmic applications of retroactive data structures. 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. The Voronoi diagram is an important technique for answering nearest-neighbor queries for spatial databases. In this paper, we study how the Voronoi diagram can be used on uncertain data, which are inherent in scientific and business applications. In particular, we propose the Uncertain-Voronoi Diagram (or UV-diagram in short). Conceptually, the data space is divided into distinct “UV-partitions”, where each UV-partition P is associated with a set S of objects; any point q located in P has the set S as its nearest neighbor with non-zero probabilities. The UV-diagram facilitates queries that inquire objects for having non-zero chances of being the nearest neighbor of a given query point. It also allows analysis of nearest neighbor information, e.g., finding out how many objects are the nearest neighbors in a given area. However, a UV-diagram requires exponential construction and storage costs. To tackle these problems, we devise an alternative representation for UV-partitions, and develop an adaptive index for the UV-diagram. This index can be constructed in polynomial time. We examine how it can be extended to support other related queries. We also perform extensive experiments to validate the effectiveness of our approach.