Bounded-Angle Spanning Tree: Modeling Networks with Angular Constraints

We introduce a new structure for a set of points in the plane and an angle α, which is similar in flavor to a bounded-degree MST. We name this structure α-MST. Let P be a set of points in the plane and let 0 < α≤ 2 π be an angle. An α-ST of P is a spanning tree of the complete Euclidean graph induced by P, with the additional property that for each point p∈ P, the smallest angle around p containing all the edges adjacent to p is at most α. An α-MST of P is then an α-ST of P of minimum weight, where the weight of an α-ST is the sum of the lengths of its edges. For α< π/ 3 , an α-ST does not always exist, and, for α≥ π/ 3 , it always exists (Ackerman et al. in Comput Geom Theory Appl 46(3):213–218, 2013; Aichholzer et al. in Comput Geom Theory Appl 46(1):17–28, 2013; Carmi et al. in Comput Geom Theory Appl 44(9):477–485, 2011). In this paper, we study the problem of computing an α-MST for several common values of α. Motivated by wireless networks, we formulate the problem in terms of directional antennas. With each point p∈ P, we associate a wedge Wp of angle α and apex p. The goal is to assign an orientation and a radius r_p to each wedge Wp, such that the resulting graph is connected and its MST is an α-MST (we draw an edge between p and q if p∈Wq, q∈Wp, and | pq| ≤ r_p, r_q). We prove that the problem of computing an α-MST is NP-hard, at least for α= π and α= 2 π/ 3 , and present constant-factor approximation algorithms for α= π/ 2 , 2 π/ 3 , π. One of our major results is a surprising theorem for α= 2 π/ 3 , which, besides being interesting from a geometric point of view, has important applications. For example, the theorem guarantees that given any set P of 3n points in the plane and any partitioning of the points into n triplets, one can orient the wedges of each triplet independently, such that the graph induced by P is connected. We apply the theorem to the antenna conversion problem and to the orientation and power assignment problem.