Computing the maximum detour and spanning ratio of planar paths, trees, and cycles

Abstract

The maximum detour and spanning ratio of an embedded graph G are values that 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 maximum detour and spanning ratio of a planar polygonal path. These algorithms solve open problems posed in at least two previous works [5,10]. We also generalize these algorithms to obtain O(n log2 n) time algorithms for computing the maximum detour and spanning ratio of planar trees and cycles.