Π/2-Angle YAO GRAPHS ARE SPANNERS

Bose, Prosenjit; Damian, Mirela; Douïeb, Karim; O’Rourke, Joseph; Seamone, Ben; Smid, Michiel; Wuhrer, Stefanie

doi:10.1142/s0218195912600047

Cited by 30 publications

(17 citation statements)

References 2 publications

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“…For k > 8, Bose et al [6] showed that Y k is a geometric spanner with spanning ratio at most 1/(cos θ − sin θ). This was later strengthened to show that for k > 6, Y k is a 1/(1 − 2 sin(θ/2))-spanner [5]. Damian and Raudonis [8] proved a spanning ratio of 17.64 for Y 6 , which was later improved by Barba et al to 5.8 [4].…”

Section: Introductionmentioning

confidence: 94%

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Continuous Yao graphs

Bakhshesh

Barba

Bose

et al. 2018

Computational Geometry

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In this paper, we introduce a variation of the well-studied Yao graphs. Given a set of points S ⊂ R 2 and an angle 0 < θ 2π, we define the continuous Yao graph cY (θ) with vertex set S and angle θ as follows. For each p, q ∈ S, we add an edge from p to q in cY (θ) if there exists a cone with apex p and aperture θ such that q is a closest point to p inside this cone. We study the spanning ratio of cY (θ) for different values of θ. Using a new algebraic technique, we show that cY (θ) is a spanner when θ 2π/3. We believe that this technique may be of independent interest. We also show that cY (π) is not a spanner, and that cY (θ) may be disconnected for θ > π, but on the other hand is always connected for θ π. Furthermore, we show that cY (θ) is a region-fault-tolerant geometric spanner for convex fault regions when θ < π/3. For half-plane faults, cY (θ) remains connected if θ π. Finally, we show that cY (θ) is not always self-approaching for any value of θ.

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Section: Introductionmentioning

confidence: 94%

“…These graphs have been independently introduced by Flinchbaugh and Jones [11] and Yao [19], and are referred to as Yao graphs in the literature. It has been shown that Yao graphs are good approximations of the complete geometric graph [7,3,6,5,8,10,4].…”

Section: Introductionmentioning

confidence: 99%

Continuous Yao graphs

Bakhshesh

Barba

Bose

et al. 2018

Computational Geometry

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“…Bose et al [13] have recently defined and studied the stretch factor of Yao ∞ 4 , a subgraph of the L ∞ -Delaunay triangulation on a set of points P in the plane. To describe this subgraph, we define a cone to be the region in the plane between two rays that emanate from the same point.…”

Section: Casementioning

confidence: 99%

Tight stretch factors for L1- and L∞-Delaunay triangulations

Gavoille

Hanusse

Perković

2015

Computational Geometry

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In this paper we determine the exact stretch factor of L ∞ -Delaunay triangulations of points in the plane. We do this not only when the distance between the points is defined by the usual L 2 -metric but also when it is defined by the L p -metric, for any p ∈ [1, ∞]. We then apply this result to compute the exact stretch factor of L 1 -Delaunay triangulations when the distance between the points is defined by the L 1 -, L ∞ -, or L 2 -metric. In the important case of the L 2 -metric, we obtain that the stretch factor of L 1 -Delaunay and L ∞ -Delaunay triangulations is exactly 4 + 2 √ 2 ≈ 2.61. This is the first time that the stretch factor of an L p -Delaunay triangulation, for any p ∈ [1, ∞], is determined exactly. We show, in particular, how to construct between any two points a and b of an L 1 -Delaunay or L ∞ -Delaunay triangulation a path whose length is no more than 4 + 2 √ 2 times the Euclidean distance between a and b. This improves the bound of √ 10 by Chew (SoCG '86) [5]. We also describe families of point sets whose L 1 -Delaunay or L ∞ -Delaunay triangulation has a stretch factor that can be made arbitrarily close to 4 + 2 √ 2.

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“…They showed that Yao-graphs with at least 9 cones have spanning ratio at most 1/(cos θ − sin θ). This was later strengthened to show that Yao-graphs with at least 7 cones are 1/ (1 − 2 sin(θ/2))-spanners [13]. Recently, Damian and Raudonis [37] showed that the Y 6 -graph is a 17.64-spanner and Bose et al [14] showed that the Y 4 -graph has spanning ratio at most 663.…”

Section: Constrained Yao-graphsmentioning

confidence: 97%

“…This was later strengthened to show that Yao-graphs with at least 7 cones are 1/ (1 − 2 sin(θ/2))-spanners [13]. Recently, Damian and Raudonis [37] showed that the Y 6 -graph is a 17.64-spanner and Bose et al [14] showed that the Y 4 -graph has spanning ratio at most 663. Barba et al [4] showed that the Y 5 -graph is a 2 + √ 3 -spanner.…”

Section: Constrained Yao-graphsmentioning

confidence: 97%

Theta-Graphs and Other Constrained Spanners

Renssen¹

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We provide improved upper bounds on the spanning ratio of various geometric graphs, one of which being θ-graphs. Given a set of points in the plane, a θ-graph partitions the plane around each vertex into m disjoint cones, each having aperture θ = 2π/m, and adds an edge to the 'closest' vertex in each cone. We provide tight bounds on a large number of these graphs, for different values of m, and improve the upper bounds on the other θ-graphs. We also study the ordered setting, where the θ-graph is built by inserting vertices one at a time and we consider only previously-inserted vertices when determining the 'closest' vertex in each cone. We improve some of the upper bounds in this setting, but our main contribution is that we show that a number of θ-graphs that are spanners in the unordered setting are not spanners in the ordered setting. Our main topic, however, is the constrained setting: We introduce line segment constraints that the edges of the graph are not allowed to cross and show that the upper bounds shown for θ-graphs carry over to constrained θ-graphs. We also construct a bounded-degree plane spanner based on the constrained half-θ 6-graph (the constrained Delaunay graph whose empty convex shape is an equilateral triangle) and we provide a local competitive routing algorithm for the constrained θ 6-graph. Next, we look at constrained Yao-graphs, which are comparable to constrained θgraphs, but use a different distance function, and show that these graphs are spanners. Finally, we look at constrained generalized Delaunay graphs: Delaunay graphs where the empty convex shape is not necessarily a circle, but can be any convex shape. We show that regardless of the convex shape, these graphs are connected, plane spanners. We then proceed to improve the spanning ratio for a subclass of these graphs, where the empty convex shape is an arbitrary rectangle.

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Π/2-Angle YAO GRAPHS ARE SPANNERS

Abstract: We show that the Yao graph Y4 in the L2 metric is a spanner with stretch factor [Formula: see text]. Enroute to this, we also show that the Yao graph [Formula: see text] in the L∞ metric is a plane spanner with stretch factor 8.

Cited by 30 publications

References 2 publications

Continuous Yao graphs

Continuous Yao graphs

Tight stretch factors for L1- and L∞-Delaunay triangulations

Theta-Graphs and Other Constrained Spanners

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