Results 1  10
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19
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 180 (8 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" network. ..."
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Cited by 143 (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...
Mesh Generation
 Handbook of Computational Geometry. Elsevier Science
, 2000
"... this article, we emphasize practical issues; an earlier survey by Bern and Eppstein [24] emphasized theoretical results. Although there is inevitably some overlap between these two surveys, we intend them to be complementary. ..."
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Cited by 49 (6 self)
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this article, we emphasize practical issues; an earlier survey by Bern and Eppstein [24] emphasized theoretical results. Although there is inevitably some overlap between these two surveys, we intend them to be complementary.
Edge Insertion for Optimal Triangulations
, 1993
"... The edgeinsertion paradigm improves a triangulation of a finite point set in R² iteratively by adding a new edge, deleting intersecting old edges, and retriangulating the resulting two polygonal regions. After presenting an abstract view of the paradigm, this paper shows that it can be used to o ..."
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Cited by 28 (3 self)
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The edgeinsertion paradigm improves a triangulation of a finite point set in R² iteratively by adding a new edge, deleting intersecting old edges, and retriangulating the resulting two polygonal regions. After presenting an abstract view of the paradigm, this paper shows that it can be used to obtain polynomial time algorithms for several types of optimal triangulations.
Characterizing Proximity Trees
, 1996
"... Complete characterizations are given for those trees that can be drawn as either the relative neighborhood graph, relatively closest graph, gabriel graph or modified gabriel graph of a set of points in the plane. The characterizations give rise to lineartime algorithms for determining whether a tre ..."
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Cited by 18 (8 self)
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Complete characterizations are given for those trees that can be drawn as either the relative neighborhood graph, relatively closest graph, gabriel graph or modified gabriel graph of a set of points in the plane. The characterizations give rise to lineartime algorithms for determining whether a tree has such a drawing; if such a drawing exists one can be constructed in linear time in the real RAM model. The characterization of gabriel graphs settles the conjectures of Matula and Sokal [19].
Approximating the Minimum Weight Steiner Triangulation
, 1994
"... We show that the length of the minimum weight Steiner triangulation (MWST) of a point set can be approximated within a constant factor by a triangulation algorithm based on quadtrees. In O(n log n) time we can compute a triangulation with O(n) new points, and no obtuse triangles, that approximate ..."
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Cited by 14 (0 self)
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We show that the length of the minimum weight Steiner triangulation (MWST) of a point set can be approximated within a constant factor by a triangulation algorithm based on quadtrees. In O(n log n) time we can compute a triangulation with O(n) new points, and no obtuse triangles, that approximates the MWST. We can also approximate the MWST with triangulations having no sharp angles. We generalize some of our results to higher dimensional triangulation problems. No previous polynomial time triangulation algorithm was known to approximate the MWST within a factor better than O(log n). 1 Introduction Optimal triangulation has furnished a number of problems of longstanding interest in computational geometry. These problems have applications to cartography, spatial data analysis, and finite element methods. Optimization criteria for which e#cient algorithms are known include maximizing the minimum angle [20, 24], minimizing the maximum angle [6], minimizing the minimum angle [7], mi...
Approximating the Minimum Weight Triangulation
, 1991
"... We show that the length of the minimum weight Steiner triangulation (MWST) of a point set can be approximated within a constant factor by a triangulation algorithm based on quadtrees. In O(n log n) time we can compute a triangulation with O(n) new points, and no obtuse triangles, that approximat ..."
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Cited by 9 (4 self)
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We show that the length of the minimum weight Steiner triangulation (MWST) of a point set can be approximated within a constant factor by a triangulation algorithm based on quadtrees. In O(n log n) time we can compute a triangulation with O(n) new points, and no obtuse triangles, that approximates the MWST. We can also approximate the MWST with triangulations having no sharp angles. We generalize some of our results to higher dimensional triangulation problems. No previous polynomial time triangulation algorithm was known to approximate the MWST within a factor better than O(log n).
One Strike Against the MinMax Degree Triangulation Problem
 Manuscript, Fachbereich IV, Mathematik und Informatik, Universitat Trier, Postfach 3825, W5500
, 2000
"... In this paper we analyze the computational complexity of the minmax degree triangulation problem. The problem arises in the generation of twodimensional meshes for plane objects. We show that the problem to triangulate a plane geometric graph with degree at most seven is NPcomplete. 1 Introductio ..."
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Cited by 7 (0 self)
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In this paper we analyze the computational complexity of the minmax degree triangulation problem. The problem arises in the generation of twodimensional meshes for plane objects. We show that the problem to triangulate a plane geometric graph with degree at most seven is NPcomplete. 1 Introduction First, we give some denitions. Let V be a set of n points in IR 2 . An edge is a closed line segment connecting two points of V . Let E be a set of edges. Then G = (V; E) is a geometric graph if for every edge ab 2 E, ab \ V = fa; bg. A geometric graph is called plane if for every two edges ab 6= cd in E, either ab \ cd = ; or ab \ cd is an endpoint of both edges. The connected components of IR 2 minus all points in V and on edges of E are the faces of G. If the edges in E are pairwise disjoint, then G is a matching and we have only one unbounded face. If V is xed and E is maximal such that no two edges cross, then G is a geometric triangulation of the convex hull of V . Then, the...