Curve reconstruction, the traveling salesman problem and Menger's theorem on length

We give necessary and sufficient regularity conditions under which the curve reconstruction problem is solved by a traveling salesman tour or path, respectively. For the proof we have to generalize a theorem of Menger [12], [13] on arc length.

[1]  Tamal K. Dey,et al.  A simple provable algorithm for curve reconstruction , 1999, SODA '99.

[2]  K. Menger,et al.  Untersuchungen über allgemeine Metrik. Vierte Untersuchung. Zur Metrik der Kurven , 1930 .

[3]  Kurt Mehlhorn,et al.  TSP-based curve reconstruction in polynomial time , 2000, SODA '00.

[4]  Christopher M. Gold,et al.  Crust and anti-crust: a one-step boundary and skeleton extraction algorithm , 1999, SCG '99.

[5]  David Eppstein,et al.  The Crust and the beta-Skeleton: Combinatorial Curve Reconstruction , 1998, Graph. Model. Image Process..

[6]  K. Falconer.  The geometry of fractal sets , 1985 .

[7]  Yu. G. Reshtenyak.  Some applications of integral geometry to the theory of curves of finite rotation , 1988 .

[8]  Chandrajit L. Bajaj,et al.  Sampling and reconstructing manifolds using alpha-shapes , 1997, CCCG.

[9]  Joachim Giesen,et al.  Curve Reconstruction in Arbitrary Dimension and the Traveling Salesman Problem , 1999, DGCI.

[10]  Kurt Mehlhorn,et al.  Experiments on Curve Reconstruction , 2000 .

[11]  Gerald B. Folland,et al.  Real Analysis: Modern Techniques and Their Applications , 1984 .

[12]  John W. Dawson,et al.  Ergebnisse eines Mathematischen Kolloquiums , 1998 .

[13]  A D Aleksandrov,et al.  Integral curvature of a curve in n-dimensional Euclidean space , 1988 .

[14]  Kurt Mehlhorn,et al.  Curve reconstruction: connecting dots with good reason , 1999, SCG '99.

[15]  K. Falconer.  The geometry of fractal sets: Contents , 1985 .