Spanning Properties of Theta–Theta-6

Abstract

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].

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

“…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].…”