AN OPTIMAL ALGORITHM FOR COMPUTING ANGLE-CONSTRAINED SPANNERS

Let S be a set of n points in R-d 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 vertical bar p q vertical bar, where vertical bar p q vertical bar denotes the Euclidean distance between p and q. The graph G is called theta- angle-constrained, if any two distinct edges sharing an endpoint make an angle of at least theta. It is shown that, for any theta with 0 < theta < pi/3, a theta-angle- constrained t-spanner can be computed in O (n log n) time, where t depends only on theta. For values of theta approaching 0, we have t = 1 + O (theta).