An Optimal Algorithm for Computing Angle-Constrained Spanners

Let S be a set of n points in Rd and let t > 1 be a real number. A graph G = (S,E) is called a t-spanner for S, if for any two points p and q in S, the shortest-path distance in G between p and q is at most t|pq|, where |pq| denotes the Euclidean distance between p and q. The graph G is called θ-angle-constrained, if any two distinct edges sharing an endpoint make an angle of at least θ. It is shown that, for any θ with 0 < θ < π/3, a θ-angle-constrained t-spanner can be computed in O(n log n) time, where t depends only on θ. For values of θ approaching 0, we have t = 1 +O(θ).