A 4-approximation of the 2π3-MST

Stav Ashur, Matthew J. Katz

Abstract: 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<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 pi∈P, the (smallest) angle that is spanned by all the edges incident to pi 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 α=2π3. 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 163, respectively.To obtain this result, we devise an O(n)-time algorithm that, given any Hamiltonian path Π of P, constructs a 2π3-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 ε>0 there exists a polygonal path for which every 2π3-ST (of the corresponding set of points) has weight greater than 2−ε times the weight of the path.