A 4-approximation of the 2π3-MST

Bounded-angle (minimum) spanning trees were first introduced in the context of wireless networks with directional antennas. They are reminiscent of bounded-degree (minimum) spanning trees, which have received significant attention. Let P be a set of n points in the plane, and let $0<\alpha <2\pi$ be an angle. An α-spanning tree (α-ST) of P is a spanning tree of the complete Euclidean graph over P, with the following property: For each vertex $p_i\in P$, the (smallest) angle that is spanned by all the edges incident to $p_i$ is at most α. An α-minimum spanning tree (α-MST) is an α-ST of P of minimum weight, where the weight of an α-ST is the sum of the lengths of its edges. In this paper, we consider the problem of computing an α-MST for the case where $\alpha =\frac{2\pi }{3}$. We present a 4-approximation algorithm, thus improving upon the previous results of Aschner and Katz and Biniaz et al., who presented algorithms with approximation ratios 6 and $\frac{16}{3}$, respectively.

To obtain this result, we devise an $O\left(n\right)$-time algorithm that, given any Hamiltonian path Π of P, constructs a $\frac{2\pi }{3}$-ST $T$ of P, such that $T$'s weight is at most twice that of Π and, moreover, $T$ is a 3-hop spanner of Π. This latter result is optimal (with respect to $T$'s weight), since for any $\epsilon >0$ there exists a polygonal path for which every $\frac{2\pi }{3}$-ST (of the corresponding set of points) has weight greater than $2-\epsilon$ times the weight of the path.