Abstract
Let G be an embedded planar graph whose edges may be curves. For two arbitrary points of G, we can compare the length of the shortest path in G connecting them against their Euclidean distance. The maximum of all these ratios is called the geometric dilation of G. Given a finite point set, we would like to know the smallest possible dilation of any graph that contains the given points. In this paper we prove that a dilation of 1.678 is always sufficient, and that π/2 = 1.570... is sometimes necessary in order to accommodate a finite set of points.
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References
Agarwal, P., Klein, R., Knauer, C., Sharir, M.: Computing the detour of polygonal curves. Technical Report B 02-03, FU Berlin (January 2002)
Aichholzer, O., Aurenhammer, F., Icking, C., Klein, R., Langetepe, E., Rote, G.: Generalized self-approaching curves. Discrete Appl. Math. 109, 3–24 (2001)
Bose, P., Gudmundsson, J., Smid, M.: Constructing plane spanners of bounded degree and low weight. In: Möhring, R.H., Raman, R. (eds.) ESA 2002. LNCS, vol. 2461, pp. 234–246. Springer, Heidelberg (2001)
Chen, D.Z., Das, G., Smid, M.: Lower bounds for computing geometric spanners and approximate shortest paths. Discrete Appl. Math. 110, 151–167 (2001)
Ebbers-Baumann, A., Klein, R., Langetepe, E., Lingas, A.: A fast algorithm for approximating the detour of a polygonal chain. In: Meyer auf der Heide, F. (ed.) ESA 2001. LNCS, vol. 2161, pp. 321–332. Springer, Heidelberg (2001)
Eppstein, D.: Spanning trees and spanners. In: Sack, J.-R., Urrutia, J. (eds.) Handbook of Computational Geometry, pp. 425–461. Elsevier, Amsterdam (1999)
Grüne, A.: Umwege in Polygonen. Master’s thesis, Institut für Informatik I, Universit ät Bonn (2002)
Gudmundsson, J., Levcopoulos, C., Narasimhan, G., Smid, M.: Approximate distance oracles revisited. In: Bose, P., Morin, P. (eds.) ISAAC 2002. LNCS, vol. 2518, pp. 357–368. Springer, Heidelberg (2002)
Icking, C., Klein, R., Langetepe, E.: Self-approaching curves. Math. Proc. Camb. Phil. Soc. 125, 441–453 (1999)
Kato, R., Imai, K., Asano, T.: An improved algorithm for the minimum manhattan network problem. In: Bose, P., Morin, P. (eds.) ISAAC 2002. LNCS, vol. 2518, pp. 344–356. Springer, Heidelberg (2002)
Langerman, S., Morin, P., Soss, M.: Computing the maximum detour and spanning ratio of planar chains, trees and cycles. In: Alt, H., Ferreira, A. (eds.) STACS 2002. LNCS, vol. 2285, pp. 250–261. Springer, Heidelberg (2002)
Narasimhan, G., Smid, M.: Approximating the stretch factor of Euclidean graphs. SIAM J. Comput. 30, 978–989 (2000)
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Ebbers-Baumann, A., Grüne, A., Klein, R. (2003). On the Geometric Dilation of Finite Point Sets. In: Ibaraki, T., Katoh, N., Ono, H. (eds) Algorithms and Computation. ISAAC 2003. Lecture Notes in Computer Science, vol 2906. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-540-24587-2_27
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DOI: https://doi.org/10.1007/978-3-540-24587-2_27
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