Curve Reconstruction, the Traveling Salesman Problem, and Menger’s Theorem on Length

Giesen, Joachim

doi:10.1007/s4540010061

Cited by 15 publications

(14 citation statements)

References 10 publications

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“…Definition 2.1 (see [14]). Let T = { (t 1 , t 2 ) ; t 1 < t 2 , t 1 , t 2 ∈ [0, 1] } and Figure 2.1 shows two semiregular curves.…”

Section: Definitions and Statements Of Resultsmentioning

confidence: 99%

“…Theorem 2.2 (see [14]). For every benign semiregular closed curve γ there exists an > 0 with the following property: If V is a finite sample set of γ so that for every x ∈ γ there is a p ∈ V with pv ≤ , the optimal traveling salesman tour is a polygonal reconstruction of γ.…”

Section: The Curve γ Is Called Left (Right) Regular At γ(T 0 ) With Lmentioning

confidence: 99%

“…Giesen [14] recently obtained the first result for nonsmooth curves. He considered the class of benign semiregular curves.…”

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confidence: 99%

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Traveling Salesman-Based Curve Reconstruction in Polynomial Time

Althaus

Mehlhorn

2001

SIAM J. Comput.

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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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“…Definition 2.1 (see [14]). Let T = { (t 1 , t 2 ) ; t 1 < t 2 , t 1 , t 2 ∈ [0, 1] } and Figure 2.1 shows two semiregular curves.…”

Section: Definitions and Statements Of Resultsmentioning

confidence: 99%

Section: The Curve γ Is Called Left (Right) Regular At γ(T 0 ) With Lmentioning

confidence: 99%

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Traveling Salesman-Based Curve Reconstruction in Polynomial Time

Althaus

Mehlhorn

2001

SIAM J. Comput.

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“…Because of the fundamental role of polygons in geometry, this has made the study of TSP solutions interesting for a wide range of geometric applications. One such context is geometric shape reconstruction, where the objective is to re-compute the original curve from a given set of sample points; see Giesen [12], Althaus and Mehlhorn [1] or Dey, Mehlhorn and Ramos [9] for specific examples. However, this only makes sense when the original shape is known to be simply connected, i.e., bounded by a single closed curve.…”

Section: Cscg] 23 Mar 2016mentioning

confidence: 99%

Computing Nonsimple Polygons of Minimum Perimeter

Fekete

Haas

Hemmer

et al. 2016

Experimental Algorithms

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We provide exact and approximation methods for solving a geometric relaxation of the Traveling Salesman Problem (TSP) that occurs in curve reconstruction: for a given set of vertices in the plane, the problem Minimum Perimeter Polygon (MPP) asks for a (not necessarily simply connected) polygon with shortest possible boundary length. Even though the closely related problem of finding a minimum cycle cover is polynomially solvable by matching techniques, we prove how the topological structure of a polygon leads to NP-hardness of the MPP. On the positive side, we show how to achieve a constant-factor approximation. When trying to solve MPP instances to provable optimality by means of integer programming, an additional difficulty compared to the TSP is the fact that only a subset of subtour constraints is valid, depending not on combinatorics, but on geometry. We overcome this difficulty by establishing and exploiting additional geometric properties. This allows us to reliably solve a wide range of benchmark instances with up to 600 vertices within reasonable time on a standard machine. We also show that using a natural geometry-based sparsification yields results that are on average within 0.5% of the optimum.

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“…[20]), even if efforts [6], [7] were made in correctly re-ascertaining their importance and maintaining their influence, so to say, in other fields, a revival stated only towards the end of the previous century and metric curvatures began to reassume their rightful place in Geometry. This is particularly true as far as Menger's curvature is concerned -it has by now become a rather standard and successful tool in Analysis [24], [21], [31] (but not only -see, for instance [12]). In contrast, Haantjes curvature has received in recent literature far lesser attention, with a very few -but notable -exceptions, e.g.…”

Section: Introductionmentioning

confidence: 99%