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

The detour and spanning ratio of a graph embedded in measure how well approximates Euclidean space and the complete Euclidean graph, respectively. In this paper we describe time algorithms for computing the detour and spanning ratio of a planar polygonal path. By generalizing these algorithms, we obtain -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 ! , and show that computing the detour in "! is at least as hard as Hopcroft’s problem. # This research was partly funded by CRM, FCAR, MITACS, and NSERC. P.A. was supported by NSF under grants CCR-00-86013 EIA-99-72879, EIA-01-31905, and CCR-02-04118, by ARO grants W911NF-04-1-0278 and DAAD1903-1-0352, and by a grant from the U.S.-Israeli Binational Science Foundation. R.K. was supported by DFG grant Kl 655/14-1. M.S. was supported by NSF Grants CCR-97-32101 and CCR-00-98246, by a grant from the U.S.-Israeli Binational Science Foundation (jointly with P.A.), by a grant from the Israeli Academy of Sciences for a Center of Excellence in Geometric Computing at Tel Aviv University, and by the Hermann Minkowski–MINERVA Center for Geometry at Tel Aviv University. $ Some of these results have appeared in a preliminary form in [2, 20]. % Department of Computer Science, Duke University, Durham, NC 27708-0129, U.S.A., [email protected]. & Institut für Informatik I, Universität Bonn, Römerstraße 164, D-53117 Bonn, Germany, [email protected]. ' Institut für Informatik, Freie Universität Berlin, Takustraße 9, D-14195 Berlin, Germany, [email protected]. ( Chercheur qualifié du FNRS, Département d’Informatique, Université Libre de Bruxelles, ULB CP212, boulevard du Triomphe, 1050 Bruxelles, [email protected] # # School of Computer Science, Carleton University, 1125 Colonel By Drive, Ottawa, Ontario K1S 5B6, Canada, [email protected]. $)$ School of Computer Science, Tel Aviv University, Tel Aviv 69978, Israel; and Courant Institute of Mathematical Sciences, New York University, New York, NY 10012, USA. [email protected]. %)% Foreign Exchange Strategy Division, Goldman Sachs, New York, [email protected]