The detour and spanning ratio of a graph G embedded in E d measure how well G approximates Euclidean space and the complete Euclidean graph, respectively. In this paper we describe O(n log n) time algorithms for computing the detour and spanning ratio of a planar polygonal path. By generalizing these algorithms, we 18 Discrete Comput Geom (2008) 39: 17-37 obtain O(n log 2 n)-time algorithms for computing the detour or spanning ratio of planar trees and cycles. Finally, we develop subquadratic algorithms for computing the detour and spanning ratio for paths, cycles, and trees embedded in E 3 , and show that computing the detour in E 3 is at least as hard as Hopcroft's problem.

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