TSP-based curve reconstruction in polynomial time

An instance of the curve reconstruction problem is a finite sample set V of an unknown curve V and the task is to connect the points in V in the order in which they lie on % Giesen [Gie99] showed recently tha t the Traveling Salesman tour of V solves the reconstruction problem under fairly week assumptions on V and V. We extend his result along three dimensions. We weaken the assumptions, we give an al ternate proof, and we show tha t in the context of curve reconstruction the Traveling Salesman tour can be constructed in polynomial time. 1 I n t r o d u c t i o n An instance of the curve reconstruction problem is a finite sample set V of an unknown curve -)' and the task is to construct a graph G on V such that two points in V are connected by an edge of G iff the points are adjacent on % The graph G is called a polygonal reconstruction of 7The curve reconstruction problem and the related surface reconstruction problem received a lot of at tention in the graphics and the computat ional geometry community. We are interested in reconstruction algorithms with guaranteed per/ormanc¢, i.e., a lgori thms which provably solve the reconstruction problem under certain assumptions on ~/ and V. Figure 1 i l lustrates the curve reconstruction problem. If the curve is closed, smooth, and uniformly sampled, several methods are known to work ranging over minimum spanning trees [FG94], a shapes [BB97, EKS83], B-skeletons [KR85], and r-regular shapes [Att97]. A survey of these techniques appears in [Ede98]. The case of non-uniformly sampled closed curves was first t reated successfully by Amenta, Bern and Eppstein [ABE98] and subsequently improved algorithms such as [DK99, Go199] appeared. Open nonuniformly sampled curves were treated in [DMR99]. All papers mentioned so far require that the underlying ~ a x c h partially supported by EG-projects ALCOM-IT and GALIA. tMax-Planck-Institute fiir Inforrnatik, Irn Stadtwald, 66123 Saarbr/icken $Max-Planck-Institute fiir Informatik, Irn Stadtwald, 66123 Saarbriicken . . 1. . ._.