A 4-Approximation of the 2π/3 -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 < α< 2 π 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∈ 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 important case where α=2π/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 16/3, respectively. To obtain this result, we devise an O(n)-time algorithm that, given any Hamiltonian path Π of P, constructs a 2π/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 in the sense that for any ε> 0 there exists a polygonal path for which every 2π/3 -ST (of the corresponding set of points) has weight greater than 2 - ε times the weight of the path.