Computing the Stretch Factor of Paths, Trees, and Cycles in Weighted Fixed Orientation Metrics

Let G be a connected graph with n vertices embedded in a metric space with metric δ. The stretch factor of G is the maximum over all pairs of distinct vertices u, v ∈ G of the ratio δG(u, v)/δ(u, v), where δG(u, v) is the metric distance in G between u and v. We consider the plane equipped with a weighted fixed orientation metric, i.e. a metric that measures the distance between a pair of points as the length of a shortest path between them using only a given set of σ ≥ 2 weighted fixed orientations. We show how to compute the stretch factor of G in O(σn log n) time when G is a path and in O(σn log n) time when G is a tree or a cycle. For the L1-metric, we generalize the algorithms to d-dimensional space and show that the stretch factor can be computed in O(n log n) time when G is a path and in O(n log n) time when G is a tree or a cycle. All algorithms have O(n) space requirement. Time and space bounds are worst-case bounds.