An Infinite Class of Sparse-Yao Spanners

We show that, for any integer k > 5, the Sparse-Yao graph YY_{6k} (also known as Yao-Yao) is a spanner with stretch factor 11.67. The stretch factor drops down to 4.75 for k > 7.Comment: 17 pages, 12 figure

Pi/2-Angle Yao Graphs are Spanners

We show that the Yao graph Y4 in the L2 metric is a spanner with stretch factor 8(29+23sqrt(2)). Enroute to this, we also show that the Yao graph Y4 in the Linf metric is a planar spanner with stretch factor 8.Comment: 20 pages, 9 figure

On the Stretch Factor of Randomly Embedded Random Graphs

We consider a random graph G(n,p) whose vertex set V has been randomly embedded in the unit square and whose edges are given weight equal to the geometric distance between their end vertices. Then each pair {u,v} of vertices have a distance in the weighted graph, and a Euclidean distance. The stretch factor of the embedded graph is defined as the maximum ratio of these two distances, over all u,v in V. We give upper and lower bounds on the stretch factor (holding asymptotically almost surely), and show that for p not too close to 0 or 1, these bounds are best possible in a certain sense. Our results imply that the stretch factor is bounded with probability tending to 1 if and only if n(1-p) tends to 0, answering a question of O'Rourke.Comment: 12 page

The Stretch Factor of the Delaunay Triangulation Is Less Than 1.998

Let $S$ be a finite set of points in the Euclidean plane. Let $D$ be a Delaunay triangulation of $S$. The {\em stretch factor} (also known as {\em dilation} or {\em spanning ratio}) of $D$ is the maximum ratio, among all points $p$ and $q$ in $S$, of the shortest path distance from $p$ to $q$ in $D$ over the Euclidean distance $||pq||$. Proving a tight bound on the stretch factor of the Delaunay triangulation has been a long standing open problem in computational geometry. In this paper we prove that the stretch factor of the Delaunay triangulation of a set of points in the plane is less than $\rho = 1.998$, improving the previous best upper bound of 2.42 by Keil and Gutwin (1989). Our bound 1.998 is better than the current upper bound of 2.33 for the special case when the point set is in convex position by Cui, Kanj and Xia (2009). This upper bound breaks the barrier 2, which is significant because previously no family of plane graphs was known to have a stretch factor guaranteed to be less than 2 on any set of points.Comment: 41 pages, 16 figures. A preliminary version of this paper appeared in the Proceedings of the 27th Annual Symposium on Computational Geometry (SoCG 2011). This is a revised version of the previous preprint [v1

On the stretch factor of the Theta-4 graph

In this paper we show that the \theta-graph with 4 cones has constant stretch factor, i.e., there is a path between any pair of vertices in this graph whose length is at most a constant times the Euclidean distance between that pair of vertices. This is the last \theta-graph for which it was not known whether its stretch factor was bounded