Traveling Salesman-Based Curve Reconstruction in Polynomial Time

An instance of the curve reconstruction problem is a finite sample set V of an unknown collection of curves $\gamma$. The task is to connect the points in V in the order in which they lie on $\gamma$. Giesen [Proceedings of the 15th Annual ACM Symposium on Computational Geometry (SCG '99), 1999, pp. 207--216] showed recently that the traveling salesman tour of V solves the reconstruction problem for single closed curves under otherwise weak assumptions on $\gamma$ and V; $\gamma$ must be a single closed curve. We extend his result along several directions: we weaken the assumptions on the sample; we show that traveling salesman-based reconstruction also works for single open curves (with and without specified endpoints) and for collections of closed curves; we give alternative proofs; and we show that in the context of curve reconstruction, the traveling salesman tour can be constructed in polynomial time.

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