Preserving Distances in Very Faulty Graphs

Preservers and additive spanners are sparse (hence cheap to store) subgraphs that preserve the distances between given pairs of nodes exactly or with some small additive error, respectively. Since real-world networks are prone to failures, it makes sense to study fault-tolerant versions of the above structures. This turns out to be a surprisingly difficult task. For every small but arbitrary set of edge or vertex failures, the preservers and spanners need to contain {\em replacement paths} around the faulted set. In this paper we make substantial progress on fault tolerant preservers and additive spanners: (1) We present the first truly sub-quadratic size single-pair preservers in unweighted (possibly directed) graphs for \emph{any} fixed number $f$ of faults. Our result indeed generalizes to the single-source case, and can be used to build new fault-tolerant additive spanners (for all pairs). (2) The size of the above single-pair preservers is $O(n^{2-g(f)})$ for some positive function $g$, and grows to $O(n^2)$ for increasing $f$. We show that this is necessary even in undirected unweighted graphs, and even if you allow for a small additive error: If you aim at size $O(n^{2-\epsilon})$ for $\epsilon>0$, then the additive error has to be $\Omega(\eps f)$. This surprisingly matches known upper bounds in the literature. (3) For weighted graphs, we provide matching upper and lower bounds for the single pair case. Namely, the size of the preserver is $\Theta(n^2)$ for $f\geq 2$ in both directed and undirected graphs, while for $f=1$ the size is $\Theta(n)$ in undirected graphs. For directed graphs, we have a superlinear upper bound and a matching lower bound. Most of our lower bounds extend to the distance oracle setting, where rather than a subgraph we ask for any compact data structure.