Approximating the Stretch Factor of Euclidean Graphs

There are several results available in the literature dealing with efficient construction of t-spanners for a given set S of n points in Rd. t-spanners are Euclidean graphs in which distances between vertices in G are at most t times the Euclidean distances between them; in other words, distances in G are “stretched” by a factor of at most t. We consider the interesting dual problem: given a Euclidean graph G whose vertex set corresponds to the set S, compute the stretch factor of G, i.e., the maximum ratio between distances in G and the corresponding Euclidean distances. It can trivially be solved by solving the All-Pairs-Shortest-Path problem. However, if an approximation to the stretch factor is sufficient, then we show it can be efficiently computed by making only O(n) approximate shortest path queries in the graph G. We apply this surprising result to obtain efficient algorithms for approximating the stretch factor of Euclidean graphs such as paths, cycles, trees, planar graphs, and general graphs. The main idea behind the algorithm is to use Callahan and Kosaraju’s well-separated pair decomposition.