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Voronoi diagrams  a survey of a fundamental geometric data structure
 ACM COMPUTING SURVEYS
, 1991
"... This paper presents a survey of the Voronoi diagram, one of the most fundamental data structures in computational geometry. It demonstrates the importance and usefulness of the Voronoi diagram in a wide variety of fields inside and outside computer science and surveys the history of its development. ..."
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Cited by 699 (5 self)
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This paper presents a survey of the Voronoi diagram, one of the most fundamental data structures in computational geometry. It demonstrates the importance and usefulness of the Voronoi diagram in a wide variety of fields inside and outside computer science and surveys the history of its development. The paper puts particular emphasis on the unified exposition of its mathematical and algorithmic properties. Finally, the paper provides the first comprehensive bibliography on Voronoi diagrams and related structures.
Primitives for the manipulation of general subdivisions and the computations of Voronoi diagrams
 ACM Tmns. Graph
, 1985
"... The following problem is discussed: Given n points in the plane (the sites) and an arbitrary query point 4, find the site that is closest to q. This problem can be solved by constructing the Voronoi diagram of the given sites and then locating the query point in one of its regions. Two algorithms ar ..."
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Cited by 517 (10 self)
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The following problem is discussed: Given n points in the plane (the sites) and an arbitrary query point 4, find the site that is closest to q. This problem can be solved by constructing the Voronoi diagram of the given sites and then locating the query point in one of its regions. Two algorithms are given, one that constructs the Voronoi diagram in O(n log n) time, and another that inserts a new site in O(n) time. Both are based on the use of the Voronoi dual, or Delaunay triangulation, and are simple enough to be of practical value. The simplicity of both algorithms can be attributed to the separation of the geometrical and topological aspects of the problem and to the use of two simple but powerful primitives, a geometric predicate and an operator for manipulating the topology of the diagram. The topology is represented by a new data structure for generalized diagrams, that is, embeddings of graphs in twodimensional manifolds. This structure represents simultaneously an embedding, its dual, and its mirror image. Furthermore, just two operators are sufficient for building and modifying arbitrary diagrams.
The quadtree and related hierarchical data structures
 ACM Computing Surveys
, 1984
"... A tutorial survey is presented of the quadtree and related hierarchical data structures. They are based on the principle of recursive decomposition. The emphasis is on the representation of data used in applications in image processing, computer graphics, geographic information systems, and robotics ..."
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Cited by 503 (11 self)
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A tutorial survey is presented of the quadtree and related hierarchical data structures. They are based on the principle of recursive decomposition. The emphasis is on the representation of data used in applications in image processing, computer graphics, geographic information systems, and robotics. There is a greater emphasis on region data (i.e., twodimensional shapes) and to a lesser extent on point, curvilinear, and threedimensional data. A number of operations in which such data structures find use are examined in greater detail.
Fast Computation of Generalized Voronoi Diagrams Using Graphics Hardware
, 1999
"... We present a new approach for computing generalized 2D and 3D Voronoi diagrams using interpolationbased polygon rasterization hardware. We compute a discrete Voronoi diagram by rendering a three dimensional distance mesh for each Voronoi site. The polygonal mesh is a boundederror approximation of ..."
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Cited by 227 (26 self)
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We present a new approach for computing generalized 2D and 3D Voronoi diagrams using interpolationbased polygon rasterization hardware. We compute a discrete Voronoi diagram by rendering a three dimensional distance mesh for each Voronoi site. The polygonal mesh is a boundederror approximation of a (possibly) nonlinear function of the distance between a site and a 2D planar grid of sample points. For each sample point, we compute the closest site and the distance to that site using polygon scanconversion and the Zbuffer depth comparison. We construct distance meshes for points, line segments, polygons, polyhedra, curves, and curved surfaces in 2D and 3D. We generalize to weighted and farthestsite Voronoi diagrams, and present efficient techniques for computing the Voronoi boundaries, Voronoi neighbors, and the Delaunay triangulation of points. We also show how to adaptively refine the solution through a simple windowing operation. The algorithm has been implemented on SGI workstations and PCs using OpenGL, and applied to complex datasets. We demonstrate the application of our algorithm to fast motion planning in static and dynamic environments, selection in complex userinterfaces, and creation of dynamic mosaic effects.
On constructing minimum spanning trees in kdimensional space and related problems
 SIAM Journal on Computing
, 1982
"... Abstract. The problem of finding a minimum spanning tree connecting n points in a kdimensional space is discussed under three common distance metrics: Euclidean, rectilinear, and L. By employing a subroutine that solves the post office problem, we show that, for fixed k _> 3, such a minimum spa ..."
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Cited by 222 (0 self)
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Abstract. The problem of finding a minimum spanning tree connecting n points in a kdimensional space is discussed under three common distance metrics: Euclidean, rectilinear, and L. By employing a subroutine that solves the post office problem, we show that, for fixed k _> 3, such a minimum spanning tree can be found in time O(n2a<k)(1og n)la<k)), where a(k) = 2+1). The bound can be improved to O((n log n) 1"8) for points in 3dimensional Euclidean space. We also obtain o(n 2) algorithms for finding a farthest pair in a set of n points and for other related problems. Key words, algorithm, minimum spanning tree, nearest neighbor, post office problem 1. Introduction. Given
Mesh Generation And Optimal Triangulation
, 1992
"... We survey the computational geometry relevant to finite element mesh generation. We especially focus on optimal triangulations of geometric domains in two and threedimensions. An optimal triangulation is a partition of the domain into triangles or tetrahedra, that is best according to some cri ..."
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Cited by 206 (7 self)
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We survey the computational geometry relevant to finite element mesh generation. We especially focus on optimal triangulations of geometric domains in two and threedimensions. An optimal triangulation is a partition of the domain into triangles or tetrahedra, that is best according to some criterion that measures the size, shape, or number of triangles. We discuss algorithms both for the optimization of triangulations on a fixed set of vertices and for the placement of new vertices (Steiner points). We briefly survey the heuristic algorithms used in some practical mesh generators.
Spanning Trees and Spanners
, 1996
"... We survey results in geometric network design theory, including algorithms for constructing minimum spanning trees and lowdilation graphs. 1 Introduction This survey covers topics in geometric network design theory. The problem is easy to state: connect a collection of sites by a "good" ..."
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Cited by 146 (2 self)
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We survey results in geometric network design theory, including algorithms for constructing minimum spanning trees and lowdilation graphs. 1 Introduction This survey covers topics in geometric network design theory. The problem is easy to state: connect a collection of sites by a "good" network. For instance, one may wish to connect components of a VLSI circuit by networks of wires, in a way that uses little surface area on the chip, draws little power, and propagates signals quickly. Similar problems come up in other applications such as telecommunications, road network design, and medical imaging [1]. One network design problem, the Traveling Salesman problem, is sufficiently important to have whole books devoted to it [79]. Problems involving some form of geometric minimum or maximum spanning tree also arise in the solution of other geometric problems such as clustering [12], mesh generation [56], and robot motion planning [93]. One can vary the network design problem in many w...
Efficient exact arithmetic for computational geometry
 In Ninth Annual Symposium on Computational Geometry
, 1993
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Optimal expectedtime algorithms for closest point problems
 ACM Transactions of Mathematical Software
, 1980
"... Geometric closest potnt problems deal with the proxLmity relationships in kdimensional point sets. Examples of closest point problems include building minimum spanning trees, nearest neighbor searching, and triangulation constructmn Shamos and Hoey [17] have shown how the Voronoi dtagram can be use ..."
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Cited by 93 (0 self)
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Geometric closest potnt problems deal with the proxLmity relationships in kdimensional point sets. Examples of closest point problems include building minimum spanning trees, nearest neighbor searching, and triangulation constructmn Shamos and Hoey [17] have shown how the Voronoi dtagram can be used to solve a number of planar closest point problems in optimal worst case tune. In this paper we extend thmr work by giving optimal expected.trine algorithms for solving a number of closest point problems in kspace, including nearest neighbor searching, finding all nearest neighbors, and computing planar minimum spanning trees. In addition to establishing theoretical bounds, the algorithms in this paper can be implemented to solve practical problems very efficiently. Key Words and Phrases ' computational geometry, closest point problems, minunum spanning trees, nearest neighbor searching, optimal algorithms, probabfllstm analysis of algorithms, Voronoi diagrams CR Categories: 3.74, 5 25, 5.31, 5.32 1.