2 research outputs found
New and improved spanning ratios for Yao graphs
For a set of points in the plane and a fixed integer k > 0, the Yao graph partitions the space around each point into equiangular cones of angle , and connects each point to a nearest neighbor in each cone. It is known for all Yao graphs, with the sole exception of , whether or not they are geometric spanners. In this paper we close this gap by showing that for odd , the spanning ratio of is at most , which gives the first constant upper bound for , and is an improvement over the previous bound of for odd .We further reduce the upper bound on the spanning ratio for from to \mbox{}, which falls slightly below the lower bound of established for the spanning ratio of (-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 on the spanning ratio of .In addition, we revisit the 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 () whose best spanning ratios have been established by complex arguments. Here we reduce the known spanning ratio of from to , getting closer to the spanning ratio of 2 established for . Finally, we present the first lower bounds on the spanning ratio of Yao graphs with more than six cones, and a construction that shows that the Yao-Yao graph (a bounded-degree variant of the Yao graph) with five cones is not a spanner. </div