Abstract
For a set of points in the plane and a fixed integer k > 0, the Yao graph Yk partitions the space around each point into k equiangular cones of angle θ = 2π/k, and connects each point to a nearest neighbor in each cone. It is known for all Yao graphs, with the sole exception of Y5, whether or not they are geometric spanners. In this paper we close this gap by showing that for odd k ≥ 5, the spanning ratio of Yk is at most 1/(1−2sin(3θ/8)), which gives the first constant upper bound for Y5, and is an improvement over the previous bound of 1/(1−2sin(θ/2)) for odd k ≥ 7. We further reduce the upper bound on the spanning ratio for Y5 from 10.9 to 2 + √3 ≈ 3.74, which falls slightly below the lower bound of 3.79 established for the spanning ratio of ⊝5 (⊝-graphs differ from Yao graphs only in the way they select the closest neighbor in each cone). This is the first such separation between a Yao and ⊝-graph with the same number of cones. We also give a lower bound of 2.87 on the spanning ratio of Y5. Finally, we revisit the Y6 graph, which plays a particularly important role as the transition between the graphs (k > 6) for which simple inductive proofs are known, and the graphs (k ≤ 6) whose best spanning ratios have been established by complex arguments. Here we reduce the known spanning ratio of Y6 from 17.6 to 5.8, getting closer to the spanning ratio of 2 established for ⊝6.
Highlights
The complete Euclidean graph defined on a point set S in the plane is the graph with vertex set S and edges connecting each pair of points in S, where each edge xy has as weight the Euclidean distance |xy| between its endpoints x and y
Much effort has gone into the development of various methods for constructing graphs that approximate the complete Euclidean graph
What does it mean to approximate this graph? One standard approach is to construct a spanning subgraph with fewer edges with the additional property that every edge e of the complete Euclidean graph is approximated by a path in the subgraph whose weight is not much more than the weight of e
Summary
The complete Euclidean graph defined on a point set S in the plane is the graph with vertex set S and edges connecting each pair of points in S, where each edge xy has as weight the Euclidean distance |xy| between its endpoints x and y. Applying this result to Y5 yields a spanning ratio of 1/(1 − 2 sin(3π/20)) ≈ 10.868 This is the first known upper bound on the spanning ratio of Y5 and fully settles the question of which Yao graphs are spanners. Instead of applying Lemma 1 for the maximum value of φ (as in the proof of Theorem 2), we apply Lemma 1 only for values φ ≤ θ or ψ ≤ θ, for some threshold angle θ (to be determined later) These cases yield a spanning ratio of t ≥ 1/(1 − 2 sin(θ/2)). The inductive proof of the upper bound on the spanning ratio of Y5 suggests a construction for a lower bound
Sign-in/Register to view highlights & summary Sign-in/Register to access full text options 
Talk to us
Join us for a 30 min session where you can share your feedback and ask us any queries you have
Schedule a callDisclaimer: All third-party content on this website/platform is and will remain the property of their respective owners and is provided on "as is" basis without any warranties, express or implied. Use of third-party content does not indicate any affiliation, sponsorship with or endorsement by them. Any references to third-party content is to identify the corresponding services and shall be considered fair use under The CopyrightLaw.
