Bounded-Angle Spanning Tree: Modeling Networks with Angular Constraints

Abstract

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 ≤ 2π 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. For α < π/3, an α-ST does not always exist, and, for α ≥ π/3, it always exists [1,2,9]. 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 ∧ p of angle α and apex p. The goal is to assign an orientation and a radius r _p to each wedge ∧ p, such that the resulting graph is connected and its MST is an α-MST. (We draw an edge between p and q if p ∈ ∧ q, q ∈ ∧ p, and |pq| ≤ r _p , r _q .) Unsurprisingly, the problem of computing an α-MST is NP-hard, at least for α = π and α = 2π/3. We present constant-factor approximation algorithms for α = π/2, 2π/3, π.

One of our major results is a surprising theorem for α = 2π/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.