Delaunay graphs are almost as good as complete graphs

Dobkin, David P.; Friedman, Steven J.; Supowit, Kenneth J.

doi:10.1109/sfcs.1987.18

Cited by 76 publications

(68 citation statements)

References 3 publications

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“…The DT of a planar point set has small "dilation", that is, the distance between any two sites in the DT graph is at most a constant times the Euclidean distance. 2 Secondly, the DT of a planar point set maximizes the minimum angle among all triangulations. 3,4 Sharp angles in a mesh adversely affect numerical stability and convergence time of finite-element computations.…”

Section: Introductionmentioning

confidence: 99%

The expected extremes in a delaunay triangulation

Bern

Eppstein

Yao

1991

Lecture Notes in Computer Science

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We give an expected-case analysis of Delaunay triangulations. To avoid edge effects we consider a unit-intensity Poisson process in Euclidean d-space, and then limit attention to the portion of the triangulation within a cube of side n 1/d . For d equal to two, we calculate the expected maximum edge length, the expected minimum and maximum angles, and the average aspect ratio of a triangle. We also show that in any fixed dimension the expected maximum vertex degree is Θ(log n/ log log n). Altogether our results provide some measure of the suitability of the Delaunay triangulation for certain applications, such as interpolation and mesh generation.

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Section: Introductionmentioning

confidence: 99%

The expected extremes in a delaunay triangulation

Bern

Eppstein

Yao

1991

Lecture Notes in Computer Science

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“…Let P be a set of points and let p, q ∈ P. Without loss of generality we assume that p and q are on the x-axis such that x(p) < x(q). According to Dobkin et al [12] we can construct an x-monotone path in the Delaunay graph D(P) of P as follows. Let V (P) denote the Voronoi diagram of P and let p 1 ,... , p k be the ordered points corresponding to the Voronoi cells that are traversed when following the line from p to q.…”

Section: Lemma 3 Given a (Complete) Geometric Graph G = (P E) The mentioning

confidence: 99%

Generalizing Geometric Graphs

et al. 2012

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Abstract. Network visualization is essential for understanding the data obtained from huge real-world networks such as flight-networks, the AS-network or social networks. Although we can compute layouts for these networks reasonably fast, even the most recent display media are not capable of displaying these layouts in an adequate way. Moreover, the human viewer may be overwhelmed by the displayed level of detail. The increasing amount of data therefore requires techniques aiming at a sensible reduction of the visual complexity of huge layouts.We consider the problem of computing a generalization of a given layout reducing the complexity of the drawing to an amount that can be displayed without clutter and handled by a human viewer. We take a first step at formulating graph generalization within a mathematical model and we consider the resulting problems from an algorithmic point of view. Although these problems are NP-hard in general, we provide efficient approximation algorithms as well as efficient and effective heuristics. At the end of the paper we showcase some sample generalizations.

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“…In [2] an algorithm is described which constructs a spanner in k-dimensions with a constant maximum degree (that is, not dependent on n), however the degree is exponentially dependent on k and hence quite large for higher dimensions. In 2 dimensions, a degree-7 spanner is reported in [6], and a degree-5 spanner in [13]. It is known that the lower bound for the degree is 3 because, given n set of points arranged in a grid, a Hamiltonian path or circuit cannot be a spanner with a constant stretch factor.…”

Section: B For K < 3 Its Weight Is O(1) Wt(mst) and For K > 3 Its mentioning

confidence: 99%

Constructing degree-3 spanners with other sparseness properties

Das

Heffernan

1993

Algorithms and Computation

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Let V be any set of n points in k-dimensional Euclidean space. A subgraph of the complete Euclidean graph is a t-spanner if for any ~ and ~ in V, the length of the shortest path from u to v in the spanner is at most t times d(~, ~). We show that for any 5 > 1, there exists a polynomial-time constructihle tspanner (where ~ is a constant that depends only on 5 and k) with the following properties. Its maximum degree is 3, it has at most n 9 6 edges, and its total edge weight is comparable to the minimum spanning tree of V (for/~ < 3 its weight is O(1). wt(MgT), and for k > 3 its weight is O(log n). wt(MST)).

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Delaunay graphs are almost as good as complete graphs

Cited by 76 publications

References 3 publications

The expected extremes in a delaunay triangulation

The expected extremes in a delaunay triangulation

Generalizing Geometric Graphs

Constructing degree-3 spanners with other sparseness properties

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