On the Spanning and Routing Ratio of Theta-Four

Bose, Prosenjit; Carufel, Jean-Lou De; Hill, Darryl; Smid, Michiel

doi:10.1137/1.9781611975482.144

Cited by 13 publications

(10 citation statements)

References 24 publications

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“…Is the tradeoff between stretch 1 + and n · O( −d+1 ) edges tight?We remark that the Θ-graph and its variants provide stretch 1 + only for sufficiently small angle Θ. These graphs have also been studied for fixed values of Θ; see [10,6,11,5,40,37,9], and the references therein. The general goal here is to determine the best possible stretch for small values of Θ.…”

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confidence: 99%

Truly Optimal Euclidean Spanners

Solomon

2019

2019 IEEE 60th Annual Symposium on Foundations of Computer Science (FOCS)

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Euclidean spanners are important geometric structures, having found numerous applications over the years. Cornerstone results in this area from the late 80s and early 90s state that for any ddimensional n-point Euclidean space, there exists a (1 + )-spanner with n · O( −d+1 ) edges and lightness (normalized weight) O( −2d ). 1 Surprisingly, the fundamental question of whether or not these dependencies on and d for small d can be improved has remained elusive, even for d = 2. This question naturally arises in any application of Euclidean spanners where precision is a necessity (thus is tiny). In the most extreme case is inverse polynomial in n, and then one could potentially improve the size and lightness bounds by factors that are polynomial in n.The state-of-the-art bounds n · O( −d+1 ) and O( −2d ) on the size and lightness of spanners are realized by the greedy spanner. In 2016, Filtser and Solomon [25] proved that, in low dimensional spaces, the greedy spanner is "near-optimal"; informally, their result states that the greedy spanner for dimension d is just as sparse and light as any other spanner but for dimension larger by a constant factor. Hence the question of whether the greedy spanner is truly optimal remained open to date.The contribution of this paper is two-fold.1 Introduction Background and motivationSparse spanners. Let P be a set of n points in R d , d ≥ 2, and consider the complete weighted graph G P = (P, P 2 ) induced by P , where the weight of any edge (x, y) ∈ P 2 is the Euclidean distance |xy| between its endpoints. Let H = (P, E) be a spanning subgraph of G P , with E ⊆ P 2 , where, as in G P , the weight function is given by the Euclidean distances. For any t ≥ 1, H is called a t-spanner for P if for every x, y ∈ P , the distance d G (x, y) between x and y in G is at most t|xy|; the parameter t is called the stretch of the spanner and the most basic goal is to get it down to 1 + , for arbitrarily small > 0, 1 The lightness of a spanner is the ratio of its weight and the MST weight. without using too many edges. Euclidean spanners were introduced in the pioneering SoCG'86 paper of Chew [16], who showed that O(n) edges can be achieved with stretch √ 10, and later improved the stretch bound to 2 [17]. The first Euclidean spanners with stretch 1 + , for an arbitrarily small > 0, were presented independently in the seminal works of Clarkson [18] (FOCS'87) and Keil [38] (see also [39]), which introduced the Θ-graph in R 2 and R 3 , and soon afterwards was generalized for any R d in [45,2]. The Θ-graph is a natural variant of the Yao graph, introduced by Yao [51] in 1982, where, roughly speaking, the space R d around each point p ∈ P is partitioned into cones of angle Θ each, and then edges are added between each point p ∈ P and its closest points in each of the cones centered around it. The Θ-graph is defined similarly, where, instead of connecting p to its closest point in each cone, we connect it to a point whose orthogonal projection to some fixed ray contained in the cone is closest to p....

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confidence: 99%

Truly Optimal Euclidean Spanners

Solomon

2019

2019 IEEE 60th Annual Symposium on Foundations of Computer Science (FOCS)

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“…The Θ 6 -graphs, and the more general Θ k -graphs, which are defined in terms of k cones, have some properties that are relevant in a number of application areas. In particular, they are sparse-Θ k (P ) has at most k|P | edges [27]-and they are spanners-the ratio (known as the spanning ratio) of the length of the shortest path between any two vertices in Θ k , k ≥ 4, to the Euclidean distance between the vertices is at most a constant [15,17,18,25]. Because of these properties, Θ k -graphs have applications in many areas including wireless networking [4,16], motion planning [19], real-time animation [24], and approximating complete Euclidean graphs [18,26].…”

Section: Introductionmentioning

confidence: 99%

“…Among Θ k -graphs, Θ 6 has some nice properties that make it suitable for communications in wireless sensor networks. In particular, k = 6 is the smallest integer for which: (i) Θ k has spanning ratio 2 [14,15,17]; (ii) the so-called ΘΘ k -graph, which is a subgraph of Θ k where each vertex has only one incoming edge per cone, is a spanner [20]; and (iii) so-called half-Θ k -graphs, which is another subgraph of Θ k , admit a deterministic local competitive routing strategy [16].…”

Section: Introductionmentioning

confidence: 99%

Maximum Matchings and Minimum Blocking Sets in $$\varTheta _6$$ -Graphs

Biedl

Biniaz

Irvine

et al. 2019

Graph-Theoretic Concepts in Computer Science

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Θ6-Graphs graphs are important geometric graphs that have many applications especially in wireless sensor networks. They are equivalent to Delaunay graphs where empty equilateral triangles take the place of empty circles. We investigate lower bounds on the size of maximum matchings in these graphs. The best known lower bound is n/3, where n is the number of vertices of the graph. Babu et al. (2014) conjectured that any Θ6-graph has a (near-)perfect matching (as is true for standard Delaunay graphs). Although this conjecture remains open, we improve the lower bound to (3n − 8)/7.We also relate the size of maximum matchings in Θ6-graphs to the minimum size of a blocking set. Every edge of a Θ6-graph on point set P corresponds to an empty triangle that contains the endpoints of the edge but no other point of P . A blocking set has at least one point in each such triangle. We prove that the size of a maximum matching is at least β(n)/2 where β(n) is the minimum, over all Θ6-graphs with n vertices, of the minimum size of a blocking set. In the other direction, lower bounds on matchings can be used to prove bounds on β, allowing us to show that β(n) ≥ 3n/4 − 2.

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“…Molla [20] showed that Y 2 and Y 3 may not be spanners, and her examples can be used to show that Θ 2 and Θ 3 are not spanners either. On the other hand, it has been shown that, for any k ≥ 4, Y k and Θ k are spanners: Y 4 is a 54.6-spanner [13] and Θ 4 is a 17-spanner [4]; Y 5 is a 3.74-spanner [2] and Θ 5 is a 9.96-spanner [6]; Y 6 is a 5.8-spanner [2] and Θ 6 is a 2-spanner [3]; for k ≥ 7, the spanning ratio of Y k is 1+ √ 2−2 cos(2π/k) 2 cos(2π/k)−1 [5] and the spanning ratio of Θ k is 1 1−2 sin(π/k) [22]; improved bounds on the spanning ratio of Y k for odd k ≥ 5, and for Θ k for even k ≥ 6, also exist [7].…”

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Spanning Properties of Theta–Theta-6

Damian

Iacono

Winslow

2020

Graphs and Combinatorics

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We show that, unlike the Yao-Yao graph Y Y 6 , the Theta-Theta graph ΘΘ 6 defined by six cones is a spanner for sets of points in convex position. We also show that, for sets of points in non-convex position, the spanning ratio of ΘΘ 6 is unbounded. IntroductionLet S be a set of n points in the plane and let G = (S, E) be a weighted geometric graph with vertex set S and a set E of (directed or undirected) edges between pairs of points, where the weight of an edge uv ∈ E is equal to the Euclidean distance |uv| between u and v. The length of a path in G is the sum of the weights of its constituent edges. The distance d G (u, v) in G between two points u, v ∈ S is the length of a shortest path in G between u and v. The graph G is called a t-spanner if any two points u, v ∈ S at distance |uv| in the plane are at distance d G (u, v) ≤ t · |uv| in G. The smallest integer t for which this property holds is called the spanning ratio of G.The Yao graph Y k (S) and the Theta graph Θ k (S) are defined for a fixed integer k > 0 as follows. Partition the plane into k equiangular cones by extending k equally-separated rays starting at the origin, with the first ray in the direction of the positive x-axis. Then translate the cones to each point u ∈ S, and connect u to a "nearest" neighbor in each cone. The difference between Yao and Theta graphs is in the way the "nearest" neighbor is defined. For a fixed point u ∈ S and a cone C(u) with apex u, a Yao edge − → uv ∈ C(u) minimizes the Euclidean distance |uv| between u and v, whereas a Theta edge − → uv ∈ C(u) minimizes the projective distance uv from u to v, which is the Euclidean distance between u and the orthogonal projection of v on the bisector of C(u). Ties are arbitrarily broken.Each of the graphs Θ k and Y k has out-degree k, but in-degree n − 1 in the worst case (consider, for example, the case of n − 1 points uniformly distributed on the circumference of a circle centered at the n th point: for any k ≥ 6, the center point has in-degree n − 1). This is a significant drawback in certain wireless networking applications where a wireless node can communicate with only a limited number of neighbors. To reduce the in-degrees, a second filtering step can be applied to the set of incoming edges in each cone. This filtering step eliminates, for each each point u ∈ S and each cone with apex u, all but a "shortest" incoming edge. The result of this filtering step applied on Θ k (Y k ) is the Theta-Theta (Yao-Yao) graph ΘΘ k (Y Y k ). Again, the definition of "shortest" differs for Yao and Theta graphs: a shortest Yao edge − → vu ∈ C(u) minimizes |vu|, and a shortest Theta edge − → vu ∈ C(u) minimizes vu . Again, ties are arbitrarily broken. Yao and Theta graphs (and their Yao-Yao and Theta-Theta sparse variants) have many important applications in wireless networking [1], motion planning [9] and walkthrough animations [15].

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On the Spanning and Routing Ratio of Theta-Four

Cited by 13 publications

References 24 publications

Truly Optimal Euclidean Spanners

Truly Optimal Euclidean Spanners

Maximum Matchings and Minimum Blocking Sets in $$\varTheta _6$$ -Graphs

Spanning Properties of Theta–Theta-6

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