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28
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 216 (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. ..."
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Cited by 154 (2 self)
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We survey results in geometric network design theory, including algorithms for constructing minimum spanning trees and lowdilation graphs.
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 57 (8 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 29 (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 21 (11 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
, 1992
"... 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 16 (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).
A QuasiPolynomial Time Approximation Scheme for Minimum Weight Triangulation
 Proceedings of the 38th ACM Symposium on Theory of Computing
, 2006
"... The Minimum Weight Triangulation problem is to find a triangulation T of minimum length for a given set of points P in the Euclidean plane. It was one of the few longstanding open problems from the famous list of twelve problems with unknown complexity status, published by Garey and Johnson [8] in 1 ..."
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Cited by 10 (1 self)
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The Minimum Weight Triangulation problem is to find a triangulation T of minimum length for a given set of points P in the Euclidean plane. It was one of the few longstanding open problems from the famous list of twelve problems with unknown complexity status, published by Garey and Johnson [8] in 1979. Very recently the problem was shown to be NPhard by Mulzer and Rote. In this paper, we present a quasipolynomial time approximation scheme for Minimum Weight Triangulation.
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).