On plane geometric spanners: A survey and open problems

Bose, Prosenjit; Smid, Michiel

doi:10.1016/j.comgeo.2013.04.002

Cited by 73 publications

(60 citation statements)

References 53 publications

(69 reference statements)

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“…The stretch factor (also known as the spanning ratio or the dilation) is a well studied parameter in discrete geometry, see for instance the book [6] or the recent survey [2]. An important problem in this context is the following.…”

Section: Introductionmentioning

confidence: 99%

On the Stretch Factor of Randomly Embedded Random Graphs

Mehrabian

Wormald

2012

Discrete Comput Geom

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We consider a random graph G(n, p) whose vertex set V has been randomly embedded in the unit square and whose edges are given weight equal to the geometric distance between their end vertices. Then each pair {u, v} of vertices have a distance in the weighted graph, and a Euclidean distance. The stretch factor of the embedded graph is defined as the maximum ratio of these two distances, over all {u, v} ⊆ V . We give upper and lower bounds on the stretch factor (holding asymptotically almost surely), and show that for p not too close to 0 or 1, these bounds are best possible in a certain sense. Our results imply that the stretch factor is bounded with probability tending to 1 if and only if n(1 − p) tends to 0, answering a question of O'Rourke.

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Section: Introductionmentioning

confidence: 99%

On the Stretch Factor of Randomly Embedded Random Graphs

Mehrabian

Wormald

2012

Discrete Comput Geom

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“…Either s 11 ∈ π(s 6 , s 18 ) or s 0 ∈ π(s 6 , s 18 ). If s 11 ∈ π(s 6 , s 18 ), then δ(s 6 , s 18 ) ≥ |ρ (6,11,18)|/|s 6 s 18 | ≥ f (5, 7) = 1.4514 . .…”

Section: A New Lower Bound On the Dilation Of Plane Spannersunclassified

“…Similarly, let the degree k dilation of P , denoted by δ 0 (P, k), be the minimum stretch factor of a plane geometric graph of degree at most k on vertex set P . Clearly, δ 0 (P, k) ≥ δ 0 (P ) holds for any k. Furthermore, δ 0 (P, j) ≥ δ 0 (P, k) holds for any j < k. (Note that the term dilation has been also used with different meanings in the literature, see for instance [11,30]. )…”

Section: Introductionmentioning

confidence: 99%

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Lower Bounds on the Dilation of Plane Spanners

Dumitrescu

Ghosh

2016

Int. J. Comput. Geom. Appl.

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Abstract(I) We exhibit a set of 23 points in the plane that has dilation at least 1.4308, improving the previous best lower bound of 1.4161 for the worst-case dilation of plane spanners.(II) For every n ≥ 13, there exists an n-element point set S such that the degree 3 dilation of S equals 1 + √ 3 = 2.7321 . . . in the domain of plane geometric spanners. In the same domain, we show that for every n ≥ 6, there exists a an n-element point set S such that the degree 4 dilation of S equals 1 + (5 − √ 5)/2 = 2.1755 . . . The previous best lower bound of 1.4161 holds for any degree.(III) For every n ≥ 6, there exists an n-element point set S such that the stretch factor of the greedy triangulation of S is at least 2.0268.

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“…[2]. Considering our use cases, we focus here on results with such planar considerations; see [4] for a survey.…”

Section: Related Workmentioning

confidence: 99%

Partitioning Polygons via Graph Augmentation

Haunert

Meulemans

2016

Geographic Information Science

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This is the accepted version of the paper.This version of the publication may differ from the final published version. wouter.meulemans@city.ac.uk Permanent repository linkAbstract. We study graph augmentation under the dilation criterion. In our case, we consider a plane geometric graph G = (V, E) and a set C of edges. We aim to add to G a minimal number of nonintersecting edges from C to bound the ratio between the graph-based distance and the Euclidean distance for all pairs of vertices described by C. Motivated by the problem of decomposing a polygon into natural subregions, we present an optimal linear-time algorithm for the case that P is a simple polygon and C models an internal triangulation of P . The algorithm admits some straightforward extensions. Most importantly, in pseudopolynomial time, it can approximate a solution of minimum total length or, if C is weighted, compute a solution of minimum total weight. We show that minimizing the total length or the total weight is weakly NP-hard. Finally, we show how our algorithm can be used for two well-known problems in GIS: generating variable-scale maps and area aggregation.

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On plane geometric spanners: A survey and open problems

Cited by 73 publications

References 53 publications

On the Stretch Factor of Randomly Embedded Random Graphs

On the Stretch Factor of Randomly Embedded Random Graphs

Lower Bounds on the Dilation of Plane Spanners

Partitioning Polygons via Graph Augmentation

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