# A 4-Approximation of the 2π/3-MST.

Stav Ashur, Matthew J. Katz

Research output: Working paper/PreprintPreprint

## 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=2π3. 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 163, respectively.In order to obtain this result, we devise a simple O(n)-time algorithm for constructing 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 in the sense that for any ε>0 there exists a polygonal path for which every 2π3-ST has weight greater than 2−ε times the weight of the path.
Original language English Published - 2020

### Publication series

Name Arxiv preprint

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