Improved algorithms for constructing fault-tolerant spanners

Let S be a set of n points in a metric space, and let k be a positive integ er. Algorithms are given that construct k-fault-tolerant spanners for S. If in such a spanner at most k vertices and/or edges are removed, then each p air of points in the remaining graph is still connected by a "short" path. First, an algorithm is given that transforms an arbitrary spanner into a k- fault-tolerant spanner. For the Euclidean metric in Rd, this leads to an O (n log n + c(k) n)-time algorithm that constructs a k-fault-tolerant spanne r of degree O(c(k)), whose total edge length is O(c(k)) times the weight of a minimum spanning tree of S, for some constant c. For constant values of k, this result is optimal, In the second part of the paper, algorithms are presented for the Euclidean metric in Rd. These algorithms construct (i) in O(n log n + k(2)n) time, a k-fault-tolerant spanner with O (k(2)n) edges, and (ii) in O(kn log n) time, such a spanner with O(kn log n) edges.