The expected extremes in a delaunay triangulation

Bern, Marshall; Eppstein, David; Yao, Fei

doi:10.1007/3-540-54233-7_173

Cited by 21 publications

(41 citation statements)

References 11 publications

(9 reference statements)

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“…The proof of this result follows very closely the proof for the Poisson model given in [3,Lemmas 8 and 9]. The only technical difference is that they bound the first probability by 1/n 2 instead of 1/n 4 .…”

Section: Lemma 7 There Exists a Constant C Such Thatsupporting

confidence: 71%

Fast randomized point location without preprocessing in two- and three-dimensional Delaunay triangulations

Mücke

Saias

Zhu

1996

Proceedings of the Twelfth Annual Symposium on Computational Geometry - SCG '96

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This paper studies the point location problem in Delaunay triangulations without preprocessing and additional storage. The proposed procedure finds the query point by simply "walking through" the triangulation, after selecting a "good starting point" by random sampling. The analysis generalizes and extends a recent result for d = 2 dimensions by proving this procedure takes expected time close to O(n 1/(d+1) ) for point location in Delaunay triangulations of n random points in d = 3 dimensions. Empirical results in both two and three dimensions show that this procedure is efficient in practice.

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Section: Lemma 7 There Exists a Constant C Such Thatsupporting

confidence: 71%

Fast randomized point location without preprocessing in two- and three-dimensional Delaunay triangulations

Mücke

Saias

Zhu

1996

Proceedings of the Twelfth Annual Symposium on Computational Geometry - SCG '96

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“…For example, in [3] the expected maximum degree of the random Poisson-Delaunay tessellations is shown to be O( log n log log n ), while in [35] the expected maximum edge length is studied, showing in particular that the expected maximum edge length in the unit disk is Ω( log n n ). We extend this analysis by providing upper bounds on the edge lengths in the unit disk, which depend on the distances of their endpoints from the boundary.…”

Section: Introductionmentioning

confidence: 99%

Geometrically aware communication in random wireless networks

Kozma

Lotker

Sharir

et al. 2004

Proceedings of the Twenty-Third Annual ACM Symposium on Principles of Distributed Computing

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Some of the first routing algorithms for geographically aware wireless networks used the Delaunay triangulation among the network's nodes as the underlying connectivity graph [4]. These solutions were considered impractical, however, because in general the Delaunay triangulation may contain arbitrarily long edges, and because calculating the Delaunay triangulation generally requires a global view of the network. Many other algorithms were then suggested for geometric routing, often assuming random placement of network nodes for analysis or simulation [30,5,31,16]. We show that, when the nodes are uniformly placed in the unit disk, the Delaunay triangulation does not contain long edges, it is easy to compute locally and it is in many ways optimal for geometric routing and flooding.In particular, we prove that, with high probability, the * Work by M.S. has been supported by a grant from the U.S.-Israeli Binational Science Foundation, by a grant from the Israel Science Fund (for a Center of Excellence in Geometric Computing), by NSF Grants CCR-97-32101, CCR-00-98246, and by the Hermann Minkowski-MINERVA Center for Geometry at Tel Aviv University.Permission to make digital or hard copies of all or part of this work for personal or classroom use is granted without fee provided that copies are not made or distributed for profit or commercial advantage and that copies bear this notice and the full citation on the first page. To copy otherwise, to republish, to post on servers or to redistribute to lists, requires prior specific permission and/or a fee. maximal length of an edge in Del(P ), the Delaunay triangulation of a set P of n nodes uniformly placed in the unit disk, is O( 3 log n n ), and that the expected sum of squares of all the edges in Del(P ) is O(1). These geometric results imply that for wireless networks, randomly distributed in a unit disk (1) computing the Delaunay triangulation locally is asymptotically easy; (2) simple "face routing" through the Delaunay triangulation optimizes, up to poly-logarithmic factors, the energy load on the nodes, and (3) flooding the network, an operation quite common in sensor nets, is with high probability optimal up to a constant factor. The last property is particularly important for geocasting [20] because the Delaunay triangulation is known to be a spanner [12].

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“…This paper presents such a local refinement method of the bisection type, which is called n-dimensional since it is applicable to grids of n-simplices, independent of the dimension n. The examination of the new method was initiated with a three-dimensional bisection algorithm of the author published in [21 ], and stimulated by the work by [5]. Among the available generation methods in two, three, or more dimensions, there are methods based on Voronoi's tessellations, as in George and Hecht [15]; methods based on Delaunay triangulations, as in Bern, Eppstein and Ya0 [9], Riedinger et al [25], and Schr6der and Shephard [27]; and methods based on fractal concepts, as in Bova Flaherty [2]. Some techniques guarantee the resulting grid to have certain properties, see, for…”

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confidence: 99%

Local Bisection Refinement for N-Simplicial Grids Generated by Reflection

Maubach¹

1995

SIAM J. Sci. Comput.

262

197

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Abstract. A simple local bisection refinement algorithm for the adaptive refinement of n-simplicial grids is presented. The algorithm requires that the vertices of each simplex be ordered in a special way relative to those in neighboring simplices. It is proven that certain regular simplicial grids on [0, 1]n have this property, and the more general grids to which this method is applicable are discussed. The edges to be bisected are determined by an ordering of the simplex vertices, without local or global computation or communication. Further, the number of congruency classes in a locally refined grid turns out to be bounded above by n, independent of the level of refinement.

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The expected extremes in a delaunay triangulation

Cited by 21 publications

References 11 publications

Fast randomized point location without preprocessing in two- and three-dimensional Delaunay triangulations

Fast randomized point location without preprocessing in two- and three-dimensional Delaunay triangulations

Geometrically aware communication in random wireless networks

Local Bisection Refinement for N-Simplicial Grids Generated by Reflection

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