A 4-Approximation of the 2π/3-MST
share
Bounded-angle (minimum) spanning trees were first introduced in the context of wireless networks with directional antennas. They are reminiscent of bounded-degree spanning trees, which have received significant attention. Let P = {p_1,…,p_n} be a set of n points in the plane, let Π be the polygonal path (p_1,…,p_n), and let 0 < α < 2π 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 simple 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. In order to obtain this result, we devise a simple O(n)-time algorithm for constructing 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 has weight greater than 2-ε times the weight of the path.
READ FULL TEXT