Single-strip triangulation of manifolds with arbitrary topology

Eppstein, David; Gopi, M.

doi:10.1145/997817.997888

Cited by 11 publications

(25 citation statements)

References 11 publications

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“…By definition, we know that any B 2 -cycle becomes C(3) before becoming C (6). Hence, from Lemma 5 we get that any B 2 -cycle has three attached edges selected by D-branches.…”

Section: Outline Of the Proof Of Propositionmentioning

confidence: 89%

“…. , Q t , until C changes to C (6). Obviously, if one of the configurations (a)-(e), (g), (h) occurs in this sequence we are done as shown above.…”

Section: Proofs Of Lemmata 11 13 and 14mentioning

confidence: 99%

“…Assume first the case C(3) → C(4) that means C once becomes C (4). Let C ′ denote a C(k)-cycle, with 4 ≤ k ≤ 5, used by a branch Q ℓ to perform the next transition of C from C(4) to C (5) or to C (6). Obviously, this means that the C(k) will become dead after the branch is finished.…”

Section: Proofs Of Lemmata 11 13 and 14mentioning

confidence: 99%

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A new upper bound for the traveling salesman problem in cubic graphs

Likiewicz¹,

Schuster²

2014

Journal of Discrete Algorithms

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We provide a new upper bound for traveling salesman problem (TSP) in cubic graphs, i. e. graphs with maximum vertex degree three, and prove that the problem for an n-vertex graph can be solved in O(1.2553 n ) time and in linear space. We show that the exact TSP algorithm of Eppstein, with some minor modifications, yields the stated result. The previous best known upper bound O(1.251 n ) was claimed by Iwama and Nakashima [Proc. COCOON 2007]. Unfortunately, their analysis contains several mistakes that render the proof for the upper bound invalid. ACM Subject Classification◮ Theorem 1. The traveling salesman problem for an n-vertex cubic graph can be solved in O(1.2553 n ) time and in linear space.Our proof techniques are based on ideas used by Eppstein [5] and Iwama and Nakashima [11] and a new, more careful study of worst-case branches in the tree of recursive subdivisions of the problem performed by the algorithm. Thus, our main contribution is more analytical than algorithmic. Nevertheless, we have implemented our algorithm and verified its easy implementability and good performance for graphs up to 114 vertices (for the experimental results see [12]). Related WorkApplying the result by Björklund et al. for an n-vertex graph with maximum degree four one gets that TSP can be solved in O(1.856 n ) time and exponential space. Eppstein [5] showed that the problem can be solved in O(1.890 n ) time but using only polynomial space. Next, Gebauer [9] proposed an algorithm that runs in time O(1.733 n ). This algorithm can also list the Hamiltonian cycles but it uses exponential space. Very recently, Cygan et al. [4] have shown a Monte Carlo algorithm with constant one-sided error probability that solves the Hamiltonian cycle problem in O(1.201 n ) time for cubic graphs and in O(1.588 n ) time for graphs of maximum degree four. Though the technique presented in [4] works well for the Hamiltonian cycle problem, it is not usable for TSP. Outline of Eppstein's AlgorithmEppstein's TSP algorithm [5] (see also Appendix) searches recursively for a minimum weight Hamiltonian cycle H min in a given cubic graph G. The algorithm constructs successively

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“…By definition, we know that any B 2 -cycle becomes C(3) before becoming C (6). Hence, from Lemma 5 we get that any B 2 -cycle has three attached edges selected by D-branches.…”

Section: Outline Of the Proof Of Propositionmentioning

confidence: 89%

“…. , Q t , until C changes to C (6). Obviously, if one of the configurations (a)-(e), (g), (h) occurs in this sequence we are done as shown above.…”

Section: Proofs Of Lemmata 11 13 and 14mentioning

confidence: 99%

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A new upper bound for the traveling salesman problem in cubic graphs

Likiewicz¹,

Schuster²

2014

Journal of Discrete Algorithms

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“…In a recent work, Eppstein and Gopi [19] describe a way to efficiently compute a strip of an arbitrary triangulated surface without boundary. In other words, the surface is cut along some of its edges such that the resulting mesh is a strip.…”

Section: Single-strip Diffusionmentioning

confidence: 99%

Centroidal Voronoi diagrams for isotropic surface remeshing

Alliez¹,

Verdière²,

Devillers³

et al. 2005

Graphical Models

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This paper proposes a new method for isotropic remeshing of triangulated surface meshes. Given a triangulated surface mesh to be resampled and a user-specified density function defined over it, we first distribute the desired number of samples by generalizing error diffusion, commonly used in image halftoning, to work directly on mesh triangles and feature edges. We then use the resulting sampling as an initial configuration for building a weighted centroidal Voronoi diagram in a conformal parameter space, where the specified density function is used for weighting. We finally create the mesh by lifting the corresponding constrained Delaunay triangulation from parameter space. A precise control over the sampling is obtained through a flexible design of the density function, the latter being possibly low-pass filtered to obtain a smoother gradation. We demonstrate the versatility of our approach through various remeshing examples.

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“…in computer graphics in the problem of stripification of triangulated surface models [1,6]. Bounded-degree maximum independent sets had previously been considered [2] but we are unaware of similar work for the traveling salesman problem in bounded degree graphs.…”

Section: Introductionmentioning

confidence: 99%

The Traveling Salesman Problem for Cubic Graphs

Eppstein¹

2007

JGAA

Self Cite

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Abstract. We show how to find a Hamiltonian cycle in a graph of degree at most three with n vertices, in time O(2 n/3 ) ≈ 1.260 n and linear space. Our algorithm can find the minimum weight Hamiltonian cycle (traveling salesman problem), in the same time bound. We can also count or list all Hamiltonian cycles in a degree three graph in time O(2 3n/8 ) ≈ 1.297 n . We also solve the traveling salesman problem in graphs of degree at most four, by randomized and deterministic algorithms with runtime O ((27/4)

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Single-strip triangulation of manifolds with arbitrary topology

Cited by 11 publications

References 11 publications

A new upper bound for the traveling salesman problem in cubic graphs

A new upper bound for the traveling salesman problem in cubic graphs

Centroidal Voronoi diagrams for isotropic surface remeshing

The Traveling Salesman Problem for Cubic Graphs

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