Approximating average bounded-angle minimum spanning trees

Motivated by the problem of orienting directional antennas in wireless communication networks, we study average bounded-angle minimum spanning trees. Let P be a set of points in the plane and let α be an angle. An α-spanning tree (α-ST) of P is a spanning tree of the complete Euclidean graph induced by P such that all edges incident to each point $p\in P$ lie in a fixed wedge of angle α with apex p. An α-minimum spanning tree (α-MST) of P is an α-ST with minimum total edge length.

An average-α-spanning tree (denoted by $\bar{\alpha }$-ST) is a spanning tree with the relaxed condition that incident edges to all points lie in wedges with average angle α. An average-α-minimum spanning tree ($\bar{\alpha }$-MST) is an $\bar{\alpha }$-ST with minimum total edge length.

Let $A\left(\alpha \right)$ be the smallest ratio of the length of the $\bar{\alpha }$-MST to the length of the standard MST, over all sets of points in the plane. We investigate bounds for $A\left(\alpha \right)$. For $\alpha =\frac{2\pi }{3}$, Biniaz, Bose, Lubiw, and Maheshwari (Algorithmica 2022) showed that $\frac{4}{3}\le A\left(\frac{2\pi }{3}\right)\le \frac{3}{2}$. We improve the upper bound and show that $A\left(\frac{2\pi }{3}\right)\le \frac{13}{9}$. We also study this for $\alpha =\frac{\pi }{2}$ and prove that $\frac{3}{2}\le A\left(\frac{\pi }{2}\right)\le 4$.