Improved bounds on the stretch factor of Y4

Damian, Mirela; Nelavalli, Naresh

doi:10.1016/j.comgeo.2016.12.001

Cited by 4 publications

(3 citation statements)

References 8 publications

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“…• The hinge set centered on a point β ∈ B φ such that β belongs to one or two internal-pairs in Φ: 12 We denote the two internal-pairs by ϕ and φ. 13 See Figure 14a for an illustration. The hinge set centered at β includes: β itself, the child-pair of ϕ closest to β (denote the pair by (ξ 1 , ξ 2 )) and the child-pair of φ closest to β (denoted the pair by (η 1 , η 2 )).…”

Section: Hinge Set Decompositionmentioning

confidence: 99%

“…In this case, β is affiliated to a hinge set centered on other point. 13 If one of the two child-pairs is a leaf-pair, let ϕ = ∅. 14 If α belongs to only one internal-pair of Φ, let ϕ = ∅.…”

Section: Hinge Set Decompositionmentioning

confidence: 99%

“…Bose et al [6] proved that Y 4 is a 663-spanner. Damian and Nelavalli [13] improved this to 54.6 recently. Barba et al [2] showed that Y 5 is a 3.74-spanner.…”

Section: Introductionmentioning

confidence: 95%

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Odd Yao-Yao Graphs are Not Spanners

Jin,

Li,

Zhan

2017

Preprint

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It is a long standing open problem whether Yao-Yao graphs YY k are all spanners [26]. Bauer and Damian [4] showed that all YY 6k for k ≥ 6 are spanners. Li and Zhan [23] generalized their result and proved that all even Yao-Yao graphs YY 2k are spanners (for k ≥ 42). However, their technique cannot be extended to odd Yao-Yao graphs, and whether they are spanners are still elusive. In this paper, we show that, surprisingly, for any integer k ≥ 1, there exist odd Yao-Yao graph YY 2k+1 instances, which are not spanners.

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Section: Hinge Set Decompositionmentioning

confidence: 99%

Section: Hinge Set Decompositionmentioning

confidence: 99%

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Odd Yao-Yao Graphs are Not Spanners

Jin,

Li,

Zhan

2017

Preprint

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Spanning Properties of Theta–Theta-6

Damian

Iacono

Winslow

2020

Graphs and Combinatorics

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We show that, unlike the Yao-Yao graph Y Y 6 , the Theta-Theta graph ΘΘ 6 defined by six cones is a spanner for sets of points in convex position. We also show that, for sets of points in non-convex position, the spanning ratio of ΘΘ 6 is unbounded. IntroductionLet S be a set of n points in the plane and let G = (S, E) be a weighted geometric graph with vertex set S and a set E of (directed or undirected) edges between pairs of points, where the weight of an edge uv ∈ E is equal to the Euclidean distance |uv| between u and v. The length of a path in G is the sum of the weights of its constituent edges. The distance d G (u, v) in G between two points u, v ∈ S is the length of a shortest path in G between u and v. The graph G is called a t-spanner if any two points u, v ∈ S at distance |uv| in the plane are at distance d G (u, v) ≤ t · |uv| in G. The smallest integer t for which this property holds is called the spanning ratio of G.The Yao graph Y k (S) and the Theta graph Θ k (S) are defined for a fixed integer k > 0 as follows. Partition the plane into k equiangular cones by extending k equally-separated rays starting at the origin, with the first ray in the direction of the positive x-axis. Then translate the cones to each point u ∈ S, and connect u to a "nearest" neighbor in each cone. The difference between Yao and Theta graphs is in the way the "nearest" neighbor is defined. For a fixed point u ∈ S and a cone C(u) with apex u, a Yao edge − → uv ∈ C(u) minimizes the Euclidean distance |uv| between u and v, whereas a Theta edge − → uv ∈ C(u) minimizes the projective distance uv from u to v, which is the Euclidean distance between u and the orthogonal projection of v on the bisector of C(u). Ties are arbitrarily broken.Each of the graphs Θ k and Y k has out-degree k, but in-degree n − 1 in the worst case (consider, for example, the case of n − 1 points uniformly distributed on the circumference of a circle centered at the n th point: for any k ≥ 6, the center point has in-degree n − 1). This is a significant drawback in certain wireless networking applications where a wireless node can communicate with only a limited number of neighbors. To reduce the in-degrees, a second filtering step can be applied to the set of incoming edges in each cone. This filtering step eliminates, for each each point u ∈ S and each cone with apex u, all but a "shortest" incoming edge. The result of this filtering step applied on Θ k (Y k ) is the Theta-Theta (Yao-Yao) graph ΘΘ k (Y Y k ). Again, the definition of "shortest" differs for Yao and Theta graphs: a shortest Yao edge − → vu ∈ C(u) minimizes |vu|, and a shortest Theta edge − → vu ∈ C(u) minimizes vu . Again, ties are arbitrarily broken. Yao and Theta graphs (and their Yao-Yao and Theta-Theta sparse variants) have many important applications in wireless networking [1], motion planning [9] and walkthrough animations [15].

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