Fault Tolerant Additive Spanners

Abstract

Graph spanners are sparse subgraphs that preserve the distances of the original graph, up to some small multiplicative factor or additive term (known as the stretch of the spanner). A number of algorithms are known for constructing sparse spanners with small multiplicative or additive stretch. Recently, the problem of constructing fault-tolerant multiplicative spanners for general graphs was given some algorithms. This paper addresses the analogous problem of constructing fault tolerant additive spanners for general graphs.

We establish the following general result. Given an n-vertex graph G, if H ₁ is an ordinary additive spanner for G with additive stretch α, and H ₂ is a fault tolerant multiplicative spanner for G, resilient against up to f edge failures, with multiplicative stretch μ, then H = H ₁ ∪ H ₂ is an additive fault tolerant spanner of G, resilient against up to f edge failures, with additive stretch $O(\tilde{f}(\alpha+\mu))$ where $\tilde{f}$ is the number of failures that have actually occurred $(\tilde{f}\leq f)$.

This allows us to derive a poly-time algorithm ${\texttt Span}^{f-t}_{add}$ for constructing an additive fault tolerant spanner H of G, relying on the existence of algorithms for constructing fault tolerant multiplicative spanners and (ordinary) additive spanners. In particular, based on some known spanner construction algorithms, we show how to construct for any n-vertex graph G an additive fault tolerant spanner with additive stretch $O(\tilde{f})$ and size O(fn ^4/3).