New and Improved Spanning Ratios for Yao Graphs

Abstract

For a set of points in the plane and a fixed integer k > 0, the Yao graph Y k 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 Y k is at most 1/(1 − 2 sin(3θ/8)), which gives the first constant upper bound for Y5, and is an improvement over the previous bound of 1/(1 − 2 sin(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 Θ5 (Θ-graphs differ from Yao graphs only in the way they select the closest neighbor in each cone). 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 Θ6.

Truly Optimal Euclidean Spanners

“…Is the tradeoff between stretch 1 + and n · O( −d+1 ) edges tight?We remark that the Θ-graph and its variants provide stretch 1 + only for sufficiently small angle Θ. These graphs have also been studied for fixed values of Θ; see [10,6,11,5,40,37,9], and the references therein. The general goal here is to determine the best possible stretch for small values of Θ.…”

Improved bounds on the stretch factor of Y4

“…Graph Stretch Factor [6] Y2, Y3 ∞ [5] Y4 8 √ 2(26 + 23 √ 2) 662.16 [1] Y5 2 + √ 3 3.74 [1] Y6 5.8 [4] Y k , k ≥ 7 (1 + √ 2 − 2 cos θ)/(2 cos θ − 1), where θ = 2π/k [3] Y Our contributions. We show that the stretch factor of Y 4 is at most (11 + 7 √ 2) 4 + 2 √ 2 54.62, which is a significant improvement upon the best previously known upper bound of 662.16 from [5].…”

LAYSTREAM: Composing standard gossip protocols for live video streaming

“…In each sector, n selects the closest node and connects to it with a directed edge. It has been shown that the Yao graph is a spanner for k ≥ 4 [3]. The Yao-Yao graph builds on top of the Yao graph.…”