On the Spanning Ratio of Gabriel Graphs and beta-Skeletons

Abstract

The spanning ratio of a graph defined on n points in the Euclidean plane is the maximum ratio over all pairs of data points (u,v) of the minimum graph distance between u and v divided by the Euclidean distance between u and v. A connected graph is said to be an S-spanner if the spanning ratio does not exceed S. For example, for any S there exists a point set whose minimum spanning tree isnot an S-spanner. At the other end of the spectrum, a Delaunay triangulation is guaranteed to be a 2.42-spanner [J. M. Keil and C. A. Gutwin, Discrete Comput. Geom., 7 (1992), pp. 13-28]. For proximity graphs between these two extremes, such as Gabriel graphs [K. R. Gabriel and R. R. Sokal, Systematic Zoology, 18 (1969), pp. 259-278], relative neighborhood graphs [G. T. Toussaint, Pattern Recognition, 12 (1980), pp. 261-268], and $\beta$-skeletons [D. G. Kirkpatrick and J. D. Radke, Comput. Geom., G. T. Toussaint, ed., Elsevier, Amsterdam, 1985, pp. 217-248] with $\beta$ in [0,2] some interesting questions arise. We show that the spanning ratio for Gabriel graphs (which are $\beta$-skeletons with $\beta$ = 1) is $\Theta ( \sqrt{n})$ in the worst case. For all $\beta$-skeletons with $\beta$ in [0,1], we prove that the spanning ratio is at most $O(n^\gamma)$, where $\gamma = (1-\log_2(1+\sqrt{1-\beta^2}))/2$. For all $\beta$-skeletons with $\beta$ in [1,2], we prove that there exist point sets whose spanning ratio is at least $\left( \frac{1}{2} - o(1) \right) \sqrt{n} $. For relative neighborhood graphs [G. T. Toussaint, Pattern Recognition, 12 (1980), pp. 261-268] (skeletons with $\beta$ = 2), we show that there exist point sets where the spanning ratio is $\Omega(n)$. For points drawn independently from the uniform distribution on the unit square, we show that the spanning ratio of the (random) Gabriel graph and all $\beta$-skeletons with $\beta$ in [1,2] tends to $\infty$ in probability as $\sqrt{\log n / \log \log n}$.