Average Stretch Factor: How Low Does It Go?

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 set, V , of n points in Rd , there exists a geometric graph with vertex set V , that has O(n) edges, and that has average stretch factor 1 + on(1). More precisely, the average stretch factor of this graph is 1 +O((logn/n)1/(2d+1)). We complement this upper-bound with a lower bound: There exist n point sets in R2 for which any graph with O(n) edges has average stretch factor 1 + Ω(1/ √ n). Bounds of this type are not possible for the more commonly studied worst-case stretch factor. In particular, there exists point sets, V , such that any graph with worst-case stretch factor 1 + on(1) has a superlinear number of edges. ar X iv :1 30 5. 41 70 v1 [ cs .C G ] 1 7 M ay 2 01 3