On the dilation spectrum of paths, cycles, and trees
Computational Geometry: Theory and Applications
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Let G be a plane graph where each edge is a linesegment. We consider the problem of computing the maximum detour ofG, defined as the maximum over all pairs of distinctpoints p and q of G of the ratio betweenthe distance between p and q in G andthe distance |pq|. The fastest known algorithm for thisproblem has θ(n 2) running timewhere n is the number of vertices. We show how to obtainO(n 3/2log3 n)expected running time. We also show that if G has boundedtreewidth, its maximum detour can be computed inO(nlog3 n) expected time.