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 γ. The task is to connect the points in V in the order in which they lie on γ. 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 γ and V ; γ 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.