Efficient Construction of Spanners in $d$-Dimensions

Abstract: In this paper we consider the problem of efficiently constructing $k$-vertex fault-tolerant geometric $t$-spanners in $\dspace$ (for $k \ge 0$ 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 $O(\weight(MST))$. This resolves an open problem. For $k \ge 1$, we present an efficient method that, given $n$ points in $\dspace$, constructs $k$-vertex fault-tolerant $t$-spanners with the maximum degree bound O(k) and weight bound $O(k^2 \weight(MST))$ 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 $\dspace$ in time $O(n \log n)$.