Efficient Construction of Spanners in $d$-Dimensions

In this paper we consider the problem of efficiently constructing k-vertex fault-tolerant geometric tspanners in R (for k ≥ 0 and t > 1). Vertex fault-tolerant spanners were introduced by Levcopoulus et. al in 1998. For k = 0, we present an O(n log n) method using the algebraic computation tree model to find a t-spanner with degree bound O(1) and weight O(ω(MST )). This resolves an open problem. For k ≥ 1, we present an efficient method that, given n points in R, constructs k-vertex fault-tolerant t-spanners with the maximum degree bound O(k) and weight bound O(kω(MST )) in time O(n log n). Our method achieves the best possible bounds on degree, total edge length, and the time complexity, and solves the open problem of efficient construction of (fault-tolerant) t-spanners in R in time O(n log n).