Generalization of Voronoi Diagrams in the Plane

Lee, D. T.; Drysdale, Robert L. Scot

doi:10.1137/0210006

Cited by 270 publications

(115 citation statements)

References 7 publications

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“…Among various generalizations of Voronoi diagrams, Additively Weighted Voronoi Diagrams have been widely studied (see, for example, [3,24,21]). In such a diagram, the point set is replaced by a set of weighted points S = {(p 1 , w 1 ), .…”

Section: Final Remarksmentioning

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: Final Remarksmentioning

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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“…A fourth independent appearance of the cut locus (or, more exactly, of the ambiguous locus), with a more applied background, under the name of "medial axis", appears in papers by Lee [17], Lee and Drysdale [18], Yap [29], Choi, Choi and Moon [10], and others.…”

Section: Introductionmentioning

confidence: 99%

On the cut locus in Alexandrov spaces and applications to convex surfaces

Zamfirescu¹

2004

Pacific J. Math.

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Alexandrov spaces are a large class of metric spaces that includes Hilbert spaces, Riemannian manifolds and convex surfaces. In the framework of Alexandrov spaces, we examine the ambiguous locus of analysis and the cut locus of differential geometry, proving a general bisecting property, showing how small the ambiguous locus must be, and proving that typically the ambiguous locus and a fortiori the cut locus are dense.

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“…A cell is denoted by Vor(q) and edges of the Voronoi Diagram are straight-line segments (parts of bisectors between pairs of points of Q). The generalized Voronoi Diagram [11,13] of a polygon P , is the generalization of the Voronoi Diagram to the vertices and edges of P . This is a decomposition of the plane into cells such that, in each cell, all points have the same closest vertex/edge.…”

Section: Voronoi Diagramsmentioning

confidence: 99%

Optimization Schemes for Protective Jamming

et al. 2013

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In this paper, we study strategies for allocating and managing friendly jammers, so as to create virtual barriers that would prevent hostile eavesdroppers from tapping sensitive wireless communication. Our scheme precludes the use of any encryption technique. Applications include domains such as (i) protecting the privacy of storage locations where RFID tags are used for item identification, (ii) secure reading of RFID tags embedded in credit cards, (iii) protecting data transmitted through wireless networks, sensor networks, etc. By carefully managing jammers to produce noise, we show how to reduce the SINR of eavesdroppers to below a threshold for successful reception, without jeopardizing network performance.We present algorithms targeted towards optimizing power allocation and number of jammers needed in several settings. Experimental simulations back up our results.

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Generalization of Voronoi Diagrams in the Plane

Cited by 270 publications

References 7 publications

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

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

On the cut locus in Alexandrov spaces and applications to convex surfaces

Optimization Schemes for Protective Jamming

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