Lower bounds for computing geometric spanners and approximate shortest paths

We consider the problems of constructing geometric spanners, possibly containing Steiner points, for a set of n input points in d-dimensional space R-d, and constructing spanners and approximate shortest paths among a collection of polygonal obstacles on the plane. The complexities of these problems are shown to be Omega (n log n) in the algebraic computation tree model. Since O(n log n)-time algorithms are known for solving these problems, our lower bounds are tight up to a constant factor. (C) 2001 Elsevier Science B.V. All rights reserved.