ABSTRACT
Consider a plane graph G, drawn with straight lines. For every pair a,b of vertices of G, we compare the shortest-path distance between a and b in G (with Euclidean edge lengths) to their actual distance in the plane. The worst-case ratio of these two values, for all pairs of points, is called the dilation of G. All finite plane graphs of dilation 1 have been classified. They are closely related to the following iterative procedure. For a given point set P ⊆ R2, we connect every pair of points in P by a line segment and then add to P all those points where two such line segments cross. Repeating this process infinitely often, yields a limit point set P∞⊇P. This limit set P∞ is finite if and only if P is contained in the vertex set of a triangulation of dilation 1.The main result of this paper is the following gap theorem: For any finite point set P in the plane for which P∞ is infinite, there exists a threshold λ > 1 such that P is not contained in the vertex set of any finite plane graph of dilation at most λ. As a first ingredient to our proof, we show that such an infinite P∞ must lie dense in a certain region of the plane. In the second, more difficult part, we then construct a concrete point set P0 such that any planar graph that contains this set amongst its vertices must have a dilation larger than 1.0000047.
- P. Agarwal, R. Klein, Ch. Knauer, S. Langerman, P. Morin, M. Sharir, and M. Soss. Computing the Detour and Spanning Ratio of Paths, Trees and Cycles in 2D and 3D. To appear in Discrete Comput. Geom., 2006. Google ScholarDigital Library
- K. Bezdek and J. Pach. A Point Set Everywhere Dense in the Plane. Elemente der Mathematik 40(4): 81--84, 1985.Google Scholar
- A. Dumitrescu, A. Ebbers-Baumann, A. Grüne, R. Klein, and G. Rote, On the Geometric Dilation of Closed Curves, Graphs, and Point Sets. To appear in Computational Geometry: Theory and Applications, 2006. Google ScholarDigital Library
- A. Dumitrescu, A. Ebbers-Baumann, A. Grüne, R. Klein, and G. Rote. On Geometric Dilation and Halving Chords. In F. Dehne, A. López-Ortiz, J.-R. Sack (eds.) Proceedings WADS'05, pp. 244--255, LNCS 3608, Springer-Verlag, 2005. Google ScholarDigital Library
- A. Ebbers-Baumann, R. Klein, E. Langetepe, and A. Lingas. A Fast Algorithm for Approximating the Detour of a Polygonal Chain. Computational Geometry: Theory and Applications, 27:123--134, 2004. Google ScholarDigital Library
- A. Ebbers-Baumann, A. Grüne, and R. Klein. Geometric Dilation of Closed Planar Curves: New Lower Bounds. To appear in Computational Geometry: Theory and Applications, 2006. Google ScholarDigital Library
- A. Ebbers-Baumann, A. Grüne, and R. Klein. On the Geometric Dilation of Finite Point Sets. Algorithmica 44(2), pp. 137--149, 2006. Google ScholarDigital Library
- A. Ebbers-Baumann, A. Grüne, M. Karpinski, R. Klein, Ch. Knauer, and A. Lingas. Embedding Point Sets into Plane Graphs of Small Dilation. In X. Deng und D. Du (eds.) Proceedings ISAAC 2005, pp. 5--16, LNCS 3827, Springer-Verlag, 2005. Google ScholarDigital Library
- D. Eppstein. Spanning trees and spanners. In Handbook of Computational Geometry, J.-R. Sack and J. Urrutia (eds.), pp. 425-461. Elsevier, 1999.Google Scholar
- D. Eppstein. The Geometry Junkyard. http://www.ics.uci.edu/~eppstein/junkyard/dilation-free/.Google Scholar
- D. Eppstein and A. Wortman. Minimum Dilation Stars. Proceedings 21th Annual ACM Symposium on Computational Geomtry, pp. 321--326, 2005. Google ScholarDigital Library
- D. Ismailescu and R. Radoičić. A Dense Planar Point Set from Iterated Line Intersections. Computational Geometry: Theory and Applications, 27(3), pp. 257--267, 2004. Google ScholarDigital Library
- S. Kamali. On the Density of Iterated Segment Intersections. Diploma Thesis. Bonn, 2005.Google Scholar
- J.M. Keil and C.A. Gutwin. The Delaunay Triangulation Closely Approximates the Complete Euclidean Graph. Discrete Comput. Geom. 7, 1992, pp. 13--28. Google ScholarDigital Library
- D. Lorenz. On the Dilation of Finite Point Sets. Diploma Thesis. Bonn, 2005.Google Scholar
- J. Matouvsek. Lectures on Discrete Geometry. Graduate Texts in Mathematics. Springer-Verlag, 2000. Google ScholarDigital Library
- G. Narasimhan and M. Smid. Approximating the Stretch Factor of Euclidean Graphs. SIAM J. Comput. 30(3), 2000, pp. 978--989. Google ScholarDigital Library
- G. Narasimhan and M. Smid. Geometric Spanner Networks. Cambridge University Press, to appear. Google ScholarDigital Library
Index Terms
- The density of iterated crossing points and a gap result for triangulations of finite point sets
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