Bounded-Angle Spanning Tree: Modeling Networks with Angular Constraints

摘要

We introduce a new structure for a set of points in the plane and an angle , which is similar in flavor to a bounded-degree MST. We name this structure -MST. Let P be a set of points in the plane and let be an angle. An -ST of P is a spanning tree of the complete Euclidean graph induced by P, with the additional property that for each point , the smallest angle around p containing all the edges adjacent to p is at most . An -MST of P is then an -ST of P of minimum weight, where the weight of an -ST is the sum of the lengths of its edges. For , an -ST does not always exist, and, for , 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 -MST for several common values of . Motivated by wireless networks, we formulate the problem in terms of directional antennas. With each point , we associate a wedge of angle and apex p. The goal is to assign an orientation and a radius to each wedge , such that the resulting graph is connected and its MST is an -MST (we draw an edge between p and q if , , and ). We prove that the problem of computing an -MST is NP-hard, at least for and , and present constant-factor approximation algorithms for . One of our major results is a surprising theorem for , 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.