On the Efficiency of a Local Iterative Algorithm to Compute Delaunay Realizations

Lillis, Kevin M.; Pemmaraju, Sriram V.

doi:10.1007/978-3-540-68552-4_6

Cited by 14 publications

(8 citation statements)

References 22 publications

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“…Based on 3D hyperbolic geometry, Hodgson et al [19] gave a combinatorial characterization of the graphs that are L 2 -Delaunay realizable, leading to a polynomial-time recognition algorithm by the use of integer programming. The algorithm has been simplified later in [18,26]. Other connections between toughness, polyhedra of inscribable type, and L 2 -Delaunay graphs have been developed in [10].…”

Section: Delaunay Realizabilitymentioning

confidence: 99%

Connections between Theta-Graphs, Delaunay Triangulations, and Orthogonal Surfaces

Bonichon

Gavoille

Hanusse

et al. 2010

Lecture Notes in Computer Science

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Abstract. Θ k -graphs are geometric graphs that appear in the context of graph navigation. The shortest-path metric of these graphs is known to approximate the Euclidean complete graph up to a factor depending on the cone number k and the dimension of the space. TD-Delaunay graphs, a.k.a. triangular-distance Delaunay triangulations, introduced by Chew, have been shown to be plane 2-spanners of the 2D Euclidean complete graph, i.e., the distance in the TD-Delaunay graph between any two points is no more than twice the distance in the plane. Orthogonal surfaces are geometric objects defined from independent sets of points of the Euclidean space. Orthogonal surfaces are well studied in combinatorics (orders, integer programming) and in algebra. From orthogonal surfaces, geometric graphs, called geodesic embeddings can be built. In this paper, we introduce a specific subgraph of the Θ6-graph defined in the 2D Euclidean space, namely the half-Θ6-graph, composed of the evencone edges of the Θ6-graph. Our main contribution is to show that these graphs are exactly the TD-Delaunay graphs, and are strongly connected to the geodesic embeddings of orthogonal surfaces of coplanar points in the 3D Euclidean space. Using these new bridges between these three fields, we establish:-Every Θ6-graph is the union of two spanning TD-Delaunay graphs.In particular, Θ6-graphs are 2-spanners of the Euclidean graph, and the bound of 2 on the stretch factor is the best possible. It was not known that Θ6-graphs are t-spanners for some constant t, and Θ7-graphs were only known to be t-spanners for t ≈ 7.562. -Every plane triangulation is TD-Delaunay realizable, i.e., every combinatorial plane graph for which all its interior faces are triangles is the TD-Delaunay graph of some point set in the plane. Such realizability property does not hold for classical Delaunay triangulations.

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Section: Delaunay Realizabilitymentioning

confidence: 99%

Connections between Theta-Graphs, Delaunay Triangulations, and Orthogonal Surfaces

Bonichon

Gavoille

Hanusse

et al. 2010

Lecture Notes in Computer Science

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“…Dhandapani [7] provides an existence proof that two-dimensional Euclidean greedy drawings of triangulations are always possible, but he does not provide a polynomial-time algorithm to find them. Chen et al [4] study methods for producing two-dimensional Euclidean greedy drawings for graphs containing power diagrams, and Lillis and Pemmaraju [17] provide similar methods for graphs containing Delaunay triangulations. It is not clear whether either of these methods runs in polynomial time, however.…”

Section: Prior Related Workmentioning

confidence: 99%

Succinct Greedy Graph Drawing in the Hyperbolic Plane

Eppstein

Goodrich

2009

Graph Drawing

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Abstract. We describe an efficient method for drawing any n-vertex simple graph G in the hyperbolic plane. Our algorithm produces greedy drawings, which support greedy geometric routing, so that a message M between any pair of vertices may be routed geometrically, simply by having each vertex that receives M pass it along to any neighbor that is closer in the hyperbolic metric to the message's eventual destination. More importantly, for networking applications, our algorithm produces succinct drawings, in that each of the vertex positions in one of our embeddings can be represented using O(log n) bits and the calculation of which neighbor to send a message to may be performed efficiently using these representations. These properties are useful, for example, for routing in sensor networks, where storage and bandwidth are limited.

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“…They conjectured that such embeddings are possible in R 2 . This conjecture has drawn a lot of interest [2,3,6,15,22,25,28]. Greedy embeddings in R 2 were first discovered only for graphs containing power diagrams [3], then for graphs containing Delaunay triangulations [25], and then existentially (but not algorithmically) for plane triangulations [6].…”

Section: Virtual-location-based Greedy Routing Via Greedy Drawingmentioning

confidence: 99%

A generalized greedy routing algorithm for 2-connected graphs

Zhang

2010

Theoretical Computer Science

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a b s t r a c tGeometric routing by using virtual locations is an elegant way for solving network routing problems. In its simplest form, greedy routing, a message is simply forwarded to a neighbor that is closer to the destination. One main drawback of this approach is that the coordinates of the virtual locations require Ω(n log n) bits to represent, which makes this scheme infeasible in some applications.The essence of the geometric routing is the following: When an origin vertex u wants to send a message to a destination vertex w, it forwards the message to a neighbor t, solely based on the location information of u, w and all neighbors of u. In the greedy routing scheme, the decision is based on decreasing distance. For this idea to work, however, the decision needs not be based on decreasing distance. As long as the decision is made locally, this scheme will work fine.In this paper, we introduce a version of greedy routing which we call generalized greedy routing algorithm. Instead of relying on decreasing distance, a generalized greedy routing algorithm uses other criteria to determine routing paths, solely based on local information. We present simple generalized greedy routing algorithms based on stcoordinates (consisting of two integers between 0 and n −1), which are derived from an storientation of a 2-connected plane graph. We also generalize this result to arbitrary trees. Both algorithms are natural and simple to be implemented.

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On the Efficiency of a Local Iterative Algorithm to Compute Delaunay Realizations

Cited by 14 publications

References 22 publications

Connections between Theta-Graphs, Delaunay Triangulations, and Orthogonal Surfaces

Connections between Theta-Graphs, Delaunay Triangulations, and Orthogonal Surfaces

Succinct Greedy Graph Drawing in the Hyperbolic Plane

A generalized greedy routing algorithm for 2-connected graphs

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