Traveling Salesman-Based Curve Reconstruction in Polynomial Time

Abstract

Abstract. 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.Key words. traveling salesman, polynomial time, curve reconstruction AMS subject classifications. 68Q25, 05C85PII. S0097539700366115 1.Introduction. An instance of the curve reconstruction problem is a finite sample set V of an unknown collection of curves γ. The task is to construct a graph G on V so that two points in V are connected by an edge of G iff the points are adjacent on γ. The curve reconstruction problem and the related surface reconstruction problem have received a lot of attention in the graphics and the computational geometry community. We are interested in reconstruction algorithms with guaranteed performance, i.e., algorithms which provably solve the reconstruction problem under certain assumptions on γ and V . Figure 1.1 illustrates the curve reconstruction problem.Many curve reconstruction algorithms have been proposed in the past; we restrict our discussion to algorithms that provably solve the reconstruction problem for a certain class of curves and under certain assumptions on the sample set. The algorithms differ with respect to the following aspects:• whether a collection of curves or just a single curve can be handled;• whether (collections of) open and closed curves can be handled or only (collections of) closed curves; • whether the sampling must be uniform or not. Uniform sampling with density requires that the sample set V contains at least one point from every curve segment of length . In nonuniform sampling, the sampling frequency may depend on local properties of the curve, e.g., can be lower in parts of low curvature;• whether nonsmooth curves can be handled or not. A smooth curve has a

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