Not being (super)thin or solid is hard: A study of grid Hamiltonicity

Arkin, Esther M.; Fekete, Sándor P.; Islam, Kamrul; Meijer, Henk; Mitchell, Joseph S. B.; Núòez-Rodríguez, Yurai; Polishchuk, Valentin; Rappaport, David; Xiao, Henry

doi:10.1016/j.comgeo.2008.11.004

Cited by 34 publications

(45 citation statements)

References 32 publications

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“…Following the result of Itai et al [16] about the NP-hardness of the Hamiltonian path problem for rectangular grid graphs, Arkin et al [6] showed the NP-hardness of this problem for some other classes of grid graphs, including hexagonal grid graphs. This immediately implies the NP-hardness of the Euclidean degree-2 BST problem, and its inapproximability in polynomial time by a factor better than √ 3 unless P=NP.…”

Section: Related Work On Bst Ratiosmentioning

confidence: 94%

Euclidean Bottleneck Bounded-Degree Spanning Tree Ratios

Biniaz¹

2020

Proceedings of the Fourteenth Annual ACM-SIAM Symposium on Discrete Algorithms

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Inspired by the seminal works of Khuller et al. (STOC 1994) and Chan (SoCG 2003) we study the bottleneck version of the Euclidean bounded-degree spanning tree problem. A bottleneck spanning tree is a spanning tree whose largest edge-length is minimum, and a bottleneck degree-K spanning tree is a degree-K spanning tree whose largest edge-length is minimum. Let β K be the supremum ratio of the largest edge-length of the bottleneck degree-K spanning tree to the largest edge-length of the bottleneck spanning tree, over all finite point sets in the Euclidean plane. It is known that β 5 = 1, and it is easy to verify that β 2 2, β 3 √ 2, and β 4 > 1.175. It is implied by the Hamiltonicity of the cube of the bottleneck spanning tree that β 2 3. The degree-3 spanning tree algorithm of Ravi et al. (STOC 1993) implies that β 3 2. Andersen and Ras (Networks, 68(4):302-314, 2016) showed that β 4 √ 3. We present the following improved bounds: β 2 √ 7, β 3 √ 3, and β 4 √ 2. As a result, we obtain better approximation algorithms for Euclidean bottleneck degree-3 and degree-4 spanning trees. As parts of our proofs of these bounds we present some structural properties of the Euclidean minimum spanning tree which are of independent interest. * Part of this work has been done while the author was an NSERC postdoctoral fellow at University of Waterloo. 1 The cube of a graph G has the same vertices as G, and has an edge between two distinct vertices if and only if there exists a path, with at most three edges, between them in G.

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Section: Related Work On Bst Ratiosmentioning

confidence: 94%

Euclidean Bottleneck Bounded-Degree Spanning Tree Ratios

Biniaz¹

2020

Proceedings of the Fourteenth Annual ACM-SIAM Symposium on Discrete Algorithms

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“…When compared with the work in [18], the proposed scheme is more robust for dealing with a triangular grid graph. The compared work requires that the planner starts from a boundary cycle of a 2-connected polygonal triangular grid.…”

Section: Coverage Sampling Planner a Coverage Path Planningmentioning

confidence: 99%

“…2(c), then the modification rules of the path segments are shown in the figure. The above-mentioned strategies are referred as the V -and Z-modifications in the work of Arkin et al [18]. The coverage path is updated by iteratively checking and implementing these strategies on all unvisited SLoIs.…”

Section: Coverage Sampling Planner a Coverage Path Planningmentioning

confidence: 99%

Coverage Sampling Planner for UAV-enabled Environmental Exploration and Field Mapping

Teng

Wang

Q.-H.

et al. 2019

2019 IEEE/RSJ International Conference on Intelligent Robots and Systems (IROS)

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Unmanned Aerial Vehicles (UAVs) have been implemented for environmental monitoring by using their capabilities of mobile sensing, autonomous navigation, and remote operation. However, in real-world applications, the limitations of on-board resources (e.g., power supply) of UAVs will constrain the coverage of the monitored area and the number of the acquired samples, which will hinder the performance of field estimation and mapping. Therefore, the issue of constrained resources calls for an efficient sampling planner to schedule UAV-based sensing tasks in environmental monitoring. This paper presents a mission planner of coverage sampling and path planning for a UAV-enabled mobile sensor to effectively explore and map an unknown environment that is modeled as a random field. The proposed planner can generate a coverage path with an optimal coverage density for exploratory sampling, and the associated energy cost is subjected to a power supply constraint. The performance of the developed framework is evaluated and compared with the existing state-of-the-art algorithms, using a real-world dataset that is collected from an environmental monitoring program as well as physical field experiments. The experimental results illustrate the reliability and accuracy of the presented coverage sampling planner in a prior survey for environmental exploration and field mapping.

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“…We start with the formal definitions of OS and shapes in Section 2. As it is NP-hard to decide if a given connected shape of the triangular lattice contains a Hamiltonian path [1], it is also NP-hard to decide if there is an OS that folds into (self-assembles) a given finite shape. We thus explore the folding of upscaled versions of finite shapes.…”

Section: Self-assembly Of Shapes By Folding In Oritatamimentioning

confidence: 99%

“…Figure 3(a) shows an example of a shape which cannot be self-assembled by any OS (at scale 1), as it does not contain any Hamiltonian path. In fact, [1] proves that it is NP-hard to decide if a shape in T has a Hamiltonian path. Note that, if we are given a Hamiltonian path, there is a (hard-coding) OS that "folds" it, by simply using this path as the seed with no transcript.…”

Section: )mentioning

confidence: 99%

Know When to Fold ’Em: Self-assembly of Shapes by Folding in Oritatami

Demaine

Hendricks

Olsen

et al. 2018

Lecture Notes in Computer Science

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These results serve as a foundation for the study of shape-building in this new model of self-assembly, and have the potential to provide better understanding of cotranscriptional folding in biology, as well as improved abilities of experimentalists to design artificial systems that self-assemble via this complex dynamical process.Note that in [13] in the present proceedings, the authors study a slightly different problem: they show that one can design an oritatami transcript that folds an upscaled version of a non-self-intersecting path (instead of a shape). The initial path may come from the triangular grid or from the square grid. The scale of the resulting path is somewhere in between our scales 3 and 4 according to our definition. Note that the cells are only partially covered by their scheme. Combining their result with our theorem 3, their algorithm provides an oritatami transcript partially covering the upscaled version of any shape at scale 6.

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Not being (super)thin or solid is hard: A study of grid Hamiltonicity

Cited by 34 publications

References 32 publications

Euclidean Bottleneck Bounded-Degree Spanning Tree Ratios

Euclidean Bottleneck Bounded-Degree Spanning Tree Ratios

Coverage Sampling Planner for UAV-enabled Environmental Exploration and Field Mapping

Know When to Fold ’Em: Self-assembly of Shapes by Folding in Oritatami

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