Abstract
In this paper we determine the stretch factor of L 1-Delaunay and L ∞ -Delaunay triangulations, and we show that it is equal to \(\sqrt{4+2\sqrt{2}} \approx 2.61\). Between any two points x,y of such triangulations, we construct a path whose length is no more than \(\sqrt{4+2\sqrt{2}}\) times the Euclidean distance between x and y, and this bound is the best possible. This definitively improves the 25-year old bound of \(\sqrt{10}\) by Chew (SoCG ’86). This is the first time the stretch factor of the L p -Delaunay triangulations, for any real p ≥ 1, is determined exactly.
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References
Bose, P., Carmi, P., Collette, S., Smid, M.: On the Stretch Factor of Convex Delaunay Graphs. In: Hong, S.-H., Nagamochi, H., Fukunaga, T. (eds.) ISAAC 2008. LNCS, vol. 5369, pp. 656–667. Springer, Heidelberg (2008)
Bose, P., Devroye, L., Löffler, M., Snoeyink, J., Verma, V.: Almost all Delaunay triangulations have stretch factor greater than π/2. Comp. Geometry 44, 121–127 (2011)
Bose, P., Fagerberg, R., van Renssen, A., Verdonschot, S.: Competitive routing in the half-θ 6-graph. In: 23rd ACM Symp. on Discrete Algorithms (SODA), pp. 1319–1328 (2012)
Bose, P., Morin, P.: Competitive online routing in geometric graphs. Theoretical Computer Science 324(2-3), 273–288 (2004)
Paul Chew, L.: There is a planar graph almost as good as the complete graph. In: 2nd Annual ACM Symposium on Computational Geometry (SoCG), pp. 169–177 (August 1986)
Paul Chew, L.: There are planar graphs almost as good as the complete graph. Journal of Computer and System Sciences 39(2), 205–219 (1989)
Dobkin, D.P., Friedman, S.J., Supowit, K.J.: Delaunay graphs are almost as good as complete graphs. In: 28th Annual IEEE Symposium on Foundations of Computer Science (FOCS), pp. 20–26. IEEE Computer Society Press (October 1987)
Mark Keil, J., Gutwin, C.A.: Classes of graphs which approximate the complete Euclidean graph. Discrete & Computational Geometry 7(1), 13–28 (1992)
Xia, G.: Improved upper bound on the stretch factor of delaunay triangulations. In: 27th Annual ACM Symposium on Computational Geometry (SoCG), pp. 264–273 (June 2011)
Xia, G., Zhang, L.: Toward the tight bound of the stretch factor of Delaunay triangulations. In: 23rd Canadian Conference on Computational Geometry (CCCG) (August 2011)
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Bonichon, N., Gavoille, C., Hanusse, N., Perković, L. (2012). The Stretch Factor of L 1- and L ∞ -Delaunay Triangulations. In: Epstein, L., Ferragina, P. (eds) Algorithms – ESA 2012. ESA 2012. Lecture Notes in Computer Science, vol 7501. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-33090-2_19
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DOI: https://doi.org/10.1007/978-3-642-33090-2_19
Publisher Name: Springer, Berlin, Heidelberg
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