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217
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 753 (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.
A Delaunay Refinement Algorithm for Quality 2Dimensional Mesh Generation
, 1995
"... We present a simple new algorithm for triangulating polygons and planar straightline graphs. It provides "shape" and "size" guarantees: All triangles have a bounded aspect ratio. The number of triangles is within a constant factor of optimal. Such "quality" triangulatio ..."
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Cited by 242 (0 self)
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We present a simple new algorithm for triangulating polygons and planar straightline graphs. It provides "shape" and "size" guarantees: All triangles have a bounded aspect ratio. The number of triangles is within a constant factor of optimal. Such "quality" triangulations are desirable as meshes for the nite element method, in which the running time generally increases with the number of triangles, and where the convergence and stability may be hurt by very skinny triangles. The technique we use  successive refinement of a Delaunay triangulation  extends a mesh generation technique of Chew by allowing triangles of varying sizes. Compared with previous quadtreebased algorithms for quality mesh generation, the Delaunay refinement approach is much simpler and generally produces meshes with fewer triangles. We also discuss an implementation of the algorithm and evaluate its performance on a variety of inputs.
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 213 (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.
Spectral Partitioning Works: Planar graphs and finite element meshes
 In IEEE Symposium on Foundations of Computer Science
, 1996
"... Spectral partitioning methods use the Fiedler vectorthe eigenvector of the secondsmallest eigenvalue of the Laplacian matrixto find a small separator of a graph. These methods are important components of many scientific numerical algorithms and have been demonstrated by experiment to work extr ..."
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Cited by 199 (10 self)
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Spectral partitioning methods use the Fiedler vectorthe eigenvector of the secondsmallest eigenvalue of the Laplacian matrixto find a small separator of a graph. These methods are important components of many scientific numerical algorithms and have been demonstrated by experiment to work extremely well. In this paper, we show that spectral partitioning methods work well on boundeddegree planar graphs and finite element meshes the classes of graphs to which they are usually applied. While naive spectral bisection does not necessarily work, we prove that spectral partitioning techniques can be used to produce separators whose ratio of vertices removed to edges cut is O( p n) for boundeddegree planar graphs and twodimensional meshes and O i n 1=d j for wellshaped ddimensional meshes. The heart of our analysis is an upper bound on the secondsmallest eigenvalues of the Laplacian matrices of these graphs. 1. Introduction Spectral partitioning has become one of the mos...
Delaunay Refinement Algorithms for Triangular Mesh Generation
 Computational Geometry: Theory and Applications
, 2001
"... Delaunay refinement is a technique for generating unstructured meshes of triangles for use in interpolation, the finite element method, and the finite volume method. In theory and practice, meshes produced by Delaunay refinement satisfy guaranteed bounds on angles, edge lengths, the number of tria ..."
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Cited by 173 (0 self)
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Delaunay refinement is a technique for generating unstructured meshes of triangles for use in interpolation, the finite element method, and the finite volume method. In theory and practice, meshes produced by Delaunay refinement satisfy guaranteed bounds on angles, edge lengths, the number of triangles, and the grading of triangles from small to large sizes. This article presents an intuitive framework for analyzing Delaunay refinement algorithms that unifies the pioneering mesh generation algorithms of L. Paul Chew and Jim Ruppert, improves the algorithms in several minor ways, and most importantly, helps to solve the difficult problem of meshing nonmanifold domains with small angles.
An Optimal Algorithm for Euclidean Shortest Paths in the Plane
 SIAM J. Comput
, 1997
"... We propose an optimaltime algorithm for a classical problem in plane computational geometry: computing a shortest path between two points in the presence of polygonal obstacles. Our algorithm runs in worstcase time O(n log n) and requires O(n log n) space, where n is the total number of vertice ..."
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Cited by 116 (2 self)
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We propose an optimaltime algorithm for a classical problem in plane computational geometry: computing a shortest path between two points in the presence of polygonal obstacles. Our algorithm runs in worstcase time O(n log n) and requires O(n log n) space, where n is the total number of vertices in the obstacle polygons. The algorithm is based on an efficient implementation of wavefront propagation among polygonal obstacles, and it actually computes a planar map encoding shortest paths from a fixed source point to all other points of the plane; the map can be used to answer singlesource shortest path queries in O(logn) time. The time complexity of our algorithm is a significant improvement over all previously published results on the shortest path problem. Finally, we also discuss extensions to more general shortest path problems, involving nonpoint and multiple sources. 1 Introduction 1.1 The Background and Our Result The Euclidean shortest path problem is one of the o...
How Good is Recursive Bisection?
 SIAM J. Sci. Comput
, 1995
"... . The most commonly used pway partitioning method is recursive bisection (RB). It first divides a graph or a mesh into two equal sized pieces, by a "good" bisection algorithm, and then recursively divides the two pieces. Ideally, we would like to use an optimal bisection algorithm. Becaus ..."
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Cited by 109 (5 self)
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. The most commonly used pway partitioning method is recursive bisection (RB). It first divides a graph or a mesh into two equal sized pieces, by a "good" bisection algorithm, and then recursively divides the two pieces. Ideally, we would like to use an optimal bisection algorithm. Because the optimal bisection problem, that partitions a graph into two equal sized subgraphs to minimize the number of edges cut, is NPcomplete, practical RB algorithms use more efficient heuristics in place of an optimal bisection algorithm. Most such heuristics are designed to find the best possible bisection within allowed time. We show that the recursive bisection method, even when an optimal bisection algorithm is assumed, may produce a pway partition that is very far way from the optimal one. Our negative result is complemented by two positive ones: First we show that for some important classes of graphs that occur in practical applications, such as wellshaped finite element and finite difference...
A Replacement for Voronoi Diagrams of Near Linear Size
 In Proc. 42nd Annu. IEEE Sympos. Found. Comput. Sci
, 2001
"... For a set P of n points in R^d, we define a new type of space decomposition. The new diagram provides an εapproximation to the distance function associated with the Voronoi diagram of P, while being of near linear size, for d ≥ 2. This contrasts with the standard Voronoi diagram ..."
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Cited by 102 (9 self)
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For a set P of n points in R^d, we define a new type of space decomposition. The new diagram provides an &epsilon;approximation to the distance function associated with the Voronoi diagram of P, while being of near linear size, for d &ge; 2. This contrasts with the standard Voronoi diagram that has complexity &Omega;(n^&lceil;d/2&rceil;) in the worst case.