Average Stretch Factor: How Low Does It Go?

Abstract

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