Computing the Detour and Spanning Ratio of Paths, Trees, and Cycles in 2D and 3D

Authors:
Pankaj K. Agarwal;Rolf Klein;Christian Knauer;Stefan Langerman;Pat Morin;Micha Sharir;Michael Soss
Affiliations:
Duke University, Department of Computer Science, 27708-0129, Durham, NC, USA;Universität Bonn, Institut für Informatik I, Römerstraße 164, 53117, Bonn, NC, Germany;Freie Universität Berlin, Institut für Informatik, Takustraße 9, 14195, Berlin, NC, Germany;Université Libre de Bruxelles, FNRS, Département d’Informatique, ULB CP212, Boulevard du Triomphe, 1050, Bruxelles, NC, Belgium;Carleton University, School of Computer Science, ULB CP212, 1125 Colonel By Drive, K1S 5B6, Ottawa, ON, Canada;Tel Aviv University, School of Computer Science, ULB CP212, 1125 Colonel By Drive, 69978, Tel Aviv, ON, Israel and New York University, Courant Institute of Mathematical Sciences, New York, NY 100 ...;Goldman Sachs, Foreign Exchange Strategy Division, ULB CP212, 1125 Colonel By Drive, 69978, New York, ON, USA
Venue:
Discrete & Computational Geometry
Year:
2008

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Abstract

The detour and spanning ratio of a graph G embedded in $\mathbb{E}^{d}$ measure how well G approximates Euclidean space and the complete Euclidean graph, respectively. In this paper we describe O(nlog n) time algorithms for computing the detour and spanning ratio of a planar polygonal path. By generalizing these algorithms, we obtain O(nlog 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 $\mathbb{E}^{3}$, and show that computing the detour in $\mathbb{E}^{3}$ is at least as hard as Hopcroft’s problem.