A 4-Approximation of the $\\frac{2\\pi }{3}$-MST

Abstract

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 = {p1, . . . , pn} be a set of n points in the plane, let Π be the polygonal path (p1, . . . , pn), 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 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. 2012 ACM Subject Classification Theory of computation → Computational geometry; Theory of computation → Design and analysis of algorithms