Randomized and deterministic algorithms for geometric spanners of small diameter

Let S be a set of n points in IR and let t > 1 be a real number. A t-spanner for S is a directed graph having the points of S as its vertices, such that for any pair p and q of points there is a path from p to q of length at most t times the Euclidean distance between p and q. Such a path is called a t-spanner path. The spanner diameter of such a spanner is defined as the smallest integer D such that for any pair p and q of points there is a t-spanner path from p to q containing at most D edges. Randomized and deterministic algorithms are given for constructing t-spanners consisting of O(n) edges and having O(log n) diameter. Also, it is shown how to maintain the randomized t-spanner under random insertions and deletions. Previously, no results were known for spanners with low spanner diameter and for maintaining spanners under insertions and deletions.