Neither the greedy nor the delaunay triangulation of a planar point set approximates the optimal triangulation

Manacher, Glenn K.; Zobrist, A. L.

doi:10.1016/0020-0190(79)90104-2

Cited by 59 publications

(20 citation statements)

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“…The initial population was prepared as randomly generated triangulations. There were two types of testing data: randomly generated points and points sitting on an arc of a circle (N = 10 − 200), which are known to be bad for greedy triangulation (GT) and are used for comparison [36]. Parallel implementation on a four-processor hypercube architecture has also been attempted.…”

Section: Crossover Operatormentioning

confidence: 99%

Multicriteria-optimized triangulations

Kolingerová¹,

Ferko²

2001

The Visual Computer

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Triangulation of a given set of points in a plane is one of the most commonly solved problems in computer graphics and computational geometry. Because they are useful in many applications, triangulations must provide well-shaped triangles. Many criteria have been developed to provide such meshes, namely weight and angular criteria. Each criterion has its pros and cons, some of them are difficult to compute, and sometimes even the polynomial algorithm is not known. By any of the existing deterministic methods, it is not possible to compute a triangulation which satisfies more than one criterion or which contains parts developed according to several criteria. We explain how such a mixture can be generated using genetic optimization.

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Section: Crossover Operatormentioning

confidence: 99%

Multicriteria-optimized triangulations

Kolingerová¹,

Ferko²

2001

The Visual Computer

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“…In the first approach, various workers have attempted to show that certain other standard triangulations approximate the MWT. For instance, it was at one point believed that the Delaunay triangulation actually achieved the minimum weight; however it has since been shown that it can be as far as Ω(n) from the optimum [11,19]. This is pessimal since any triangulation achieves O(n) times the minimum weight [11].…”

Section: Related Workmentioning

confidence: 99%

“…This is pessimal since any triangulation achieves O(n) times the minimum weight [11]. Similarly, the greedy triangulation [10,15] has been proposed as an approximation to the MWT; however the approximation factor can be as bad as Ω( √ n) [13,19]. On the other hand, for convex polygons the greedy triangulation offers an easily computed approximation to the MWT [14,15].…”

Section: Related Workmentioning

confidence: 99%

Approximating the minimum weight steiner triangulation

Eppstein

1994

Discrete Comput Geom

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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).

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“…Shames' thesis [45] discussed the applicability of Delaunay triangulations and Voronoi diagrams to many problems in computational geometry and raised several interesting questions. The relationships among the greedy triangulation, the minimum-weight triangulation, and the Delaunay triangulation are examined in [12], [33] and [35]- [37]. It is shown in [1] that a Delaunay triangulation need not contain a minimum-weight perfect matching.…”

Section: Introductionmentioning

confidence: 99%

Toughness and Delaunay triangulations

Dillencourt

1990

Discrete Comput Geom

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Abstract. We show that nondegenerate Delaunay triangulations satisfy a combinatorial property called 1-toughness. A graph G is 1-tough if for any set P of vertices, c(G -P) < I GI, where c(G -P) is the number of components of the graph obtained by removing P and all attached edges from G, and I GI is the number of vertices in G. This property arises in the study of Hamiltonian graphs: all Hamiltonian graphs are 1-tough, but not conversely. We also show that all Delaunay triangulations T satisfy the following closely related property: for any set P of vertices the number of interior components of T-P is at most IPI -2, where an interior component of T-P is a component that contains no boundary vertex of T. These appear to be the first nontrivial properties of a purely combinatorial nature to be established for Delaunay triangulations. We give examples to show that these bounds are best possible and are independent of one another. We also characterize the conditions under which a degenerate Delaunay triangulation can fail to be 1-tough. This characterization leads to a proof that all graphs that can be realized as polytopes inscribed in a sphere are 1-tough. One consequence of the toughness results is that all Delaunay triangulations and all inscribable graphs have perfect matchings.

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Neither the greedy nor the delaunay triangulation of a planar point set approximates the optimal triangulation

Cited by 59 publications

References 1 publication

Multicriteria-optimized triangulations

Multicriteria-optimized triangulations

Approximating the minimum weight steiner triangulation

Toughness and Delaunay triangulations

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