π/2-Angle Yao Graphs Are Spanners

Bose, Prosenjit; Damian, Mirela; Douïeb, Karim; O’Rourke, Joseph; Seamone, Ben; Smid, Michiel; Wuhrer, Stefanie

doi:10.1007/978-3-642-17514-5_38

Cited by 20 publications

(27 citation statements)

References 4 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…Very little is known for k 6. For instance, it was known that Y 4 -graphs are connected [16] and recently it has been shown that they are 8(29 + 23 √ 2)-spanners 4 [4]. A very recent result [29] states that Y 6 -graphs are 20.4-spanners.…”

Section: Theta-graphsmentioning

confidence: 99%

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

Bonichon

Gavoille

Hanusse

et al. 2010

Lecture Notes in Computer Science

View full text Add to dashboard Cite

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.

show abstract

Section: Theta-graphsmentioning

confidence: 99%

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

Bonichon

Gavoille

Hanusse

et al. 2010

Lecture Notes in Computer Science

View full text Add to dashboard Cite

show abstract

“…Notice that the Yao ∞ 4 (P ) graph is a subgraph of the L ∞ Delaunay graph on P . The Yao ∞ 4 (P ) was shown to be an 8 √ 2-spanner [12].…”

Section: Variants Of Delaunay Graphsmentioning

confidence: 99%

On plane geometric spanners: A survey and open problems

Bose

Smid

2013

Computational Geometry

Self Cite

View full text Add to dashboard Cite

show abstract

“…Due to space restrictions, some of these properties are stated without proofs. The proofs can be found in [1]. The ultimate goal of this section is to show that, if two edges in Y 4 cross, there is a short path between their endpoints (Lemma 8).…”

Section: Y 4 In the L 2 Metricmentioning

confidence: 99%

“…Proof. For the sake of clarity, we only prove here that there is a short path p(a, b) between a and b, and skip the calculations of the actual stretch factor t (which are detailed in the appendix of [1]). We refer to an edge or a path as short if its length is within a constant factor of |ab|.…”

Section: S1mentioning

confidence: 99%

“…Bose et al [2] showed that, for k ≥ 9, Y k is a spanner with stretch factor 1 cos 2π k −sin 2π k . In [1] we improve the stretch factor and show that, in fact, Y k is a spanner for any k ≥ 7. Recently, Molla [5] showed that Y 2 and Y 3 are not spanners, and that Y 4 is a spanner with stretch factor 4(2 + √ 2), for the special case when the nodes in V are in convex position (see also [3]).…”

Section: Introductionmentioning

confidence: 97%

See 1 more Smart Citation

Algorithms and Computation

2010

Lecture Notes in Computer Science

View full text Add to dashboard Cite

Vous avez des questions? Nous pouvons vous aider. Pour communiquer directement avec un auteur, consultez la première page de la revue dans laquelle son article a été publié afin de trouver ses coordonnées. Si vous n'arrivez pas à les repérer, communiquez avec nous à PublicationsArchive-ArchivesPublications@nrc-cnrc.gc.ca. NRC Publications Record / Notice d'Archives des publications de CNRC:http://nparc.cisti-icist.nrc-cnrc.gc.ca/eng/view/object/?id=b5712542-720e-4226-8919-42c3d65ea51e http://nparc.cisti-icist.nrc-cnrc.gc.ca/fra/voir/objet/?id=b5712542-720e-4226-8919-42c3d65ea51e π/2-Angle Yao Graphs are Spanners Abstract. We show that the Yao graph Y4 in the L2 metric is a spanner with stretch factor 8 √ 2(29 + 23 √ 2).

show abstract

π/2-Angle Yao Graphs Are Spanners

Cited by 20 publications

References 4 publications

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

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

On plane geometric spanners: A survey and open problems

Algorithms and Computation

Contact Info

Product

Resources

About