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 \(0< \alpha < 2\pi \) be an angle. An \(\alpha \)-spanning tree (\(\alpha \)-ST) of P is a spanning tree of the complete Euclidean graph over P, with the following property: For each vertex \(p_i \in P\), the (smallest) angle that is spanned by all the edges incident to \(p_i\) is at most \(\alpha \). An \(\alpha \)-minimum spanning tree (\(\alpha \)-MST) is an \(\alpha \)-ST of P of minimum weight, where the weight of an \(\alpha \)-ST is the sum of the lengths of its edges. In this paper, we consider the problem of computing an \(\alpha \)-MST for the important case where \(\alpha = \frac{2\pi }{3}\). 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 \(\frac{16}{3}\), respectively.
To obtain this result, we devise an O(n)-time algorithm that, given any Hamiltonian path \(\varPi \) of P, constructs a \(\frac{2\pi }{3}\)-ST \(\mathcal{T}\) of P, such that \(\mathcal{T}\)’s weight is at most twice that of \(\varPi \) and, moreover, \(\mathcal{T}\) is a 3-hop spanner of \(\varPi \). This latter result is optimal in the sense that for any \({\varepsilon }> 0\) there exists a polygonal path for which every \(\frac{2\pi }{3}\)-ST (of the corresponding set of points) has weight greater than \(2-{\varepsilon }\) times the weight of the path.
M. Katz was supported by grant 1884/16 from the Israel Science Foundation.
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Ashur, S., Katz, M.J. (2021). A 4-Approximation of the \(\frac{2\pi }{3}\)-MST. In: Lubiw, A., Salavatipour, M., He, M. (eds) Algorithms and Data Structures. WADS 2021. Lecture Notes in Computer Science(), vol 12808. Springer, Cham. https://doi.org/10.1007/978-3-030-83508-8_10
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