Average Stretch Factor: How Low Does It Go?

Abstract

In a geometric graph $$G$$G, the stretch factor between two vertices $$u$$u and $$w$$w is the ratio between the Euclidean length of the shortest path from $$u$$u to $$w$$w in $$G$$G and the Euclidean distance between $$u$$u and $$w$$w. The average stretch factor of $$G$$G is the average stretch factor taken over all pairs of vertices in $$G$$G. We show that, for any constant dimension $$d$$d and any set $$V$$V of $$n$$n points in $$\mathbb {R}^d$$Rd, there exists a geometric graph with vertex set $$V$$V that has $$O(n)$$O(n) edges and that has average stretch factor $$1+ o_n(1)$$1+on(1). More precisely, the average stretch factor of this graph is $$1+O\big ((\log n/n)^{1/(2d+1)}\big )$$1+O((logn/n)1/(2d+1)). We complement this upper bound with a lower bound: There exist $$n$$n-point sets in $$\mathbb {R}^2$$R2 for which any graph with $$O(n)$$O(n) edges has average stretch factor $$1+\Omega (1/\sqrt{n})$$1+Ω(1/n). Bounds of this type are not possible for the more commonly studied worst-case stretch factor. In particular, there exist point sets $$V$$V such that any graph with worst-case stretch factor $$1+o_n(1)$$1+on(1) has a superlinear number of edges.