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 replacement paths around the faulted set. Unfortunately, the complexity of the interaction between replacement paths blows up significantly, even from 1 to 2 faults, and the structure of optimal preservers and spanners is poorly understood. In particular, no nontrivial bounds for preservers and additive spanners are known when the number of faults is bigger than 2. Even the answer to the following innocent question is completely unknown: what is the worstcase size of a preserver for a single pair of nodes in the presence of f edge faults? There are no super-linear lower bounds, nor subquadratic upper bounds for f > 2. In this paper we make substantial progress on this and other fundamental questions: • We present the first truly sub-quadratic size fault-tolerant single-pair preserver in unweighted (possibly directed) graphs: for any n node graph and any fixed number f of faults, Õ(fn2−1/2f ) size suffices. Our result also generalizes to the single-source (all targets) case, and can be used to build new fault-tolerant additive spanners (for all pairs). • The size of the above single-pair preserver grows to O(n2) 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(n2−ε) for ε > 0, then the additive error has to be Ω(εf). This surprisingly matches known upper bounds in the literature. • For weighted graphs, we provide matching upper and lower bounds for the single pair case. Namely, the size of the preserver is Θ(n2) for f ≥ 2 in both directed and undirected graphs, while for f = 1 the size is Θ(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. 1998 ACM Subject Classification G.2.2 Graph Theory – I.1.2 Analysis of algorithms – B.8.1 Reliability, Testing, and Fault-Tolerance ∗ A full version of this paper is available at https://arxiv.org/abs/1703.10293. The first and fourth author were partially supported by NSF Grants CCF-1417238, CCF-1528078 and CCF-1514339, and BSF Grant BSF:2012338. The second author was partially supported by the ERC Starting Grant NEWNET 279352 and the SNSF Grant APPROXNET 200021_159697/1. © Greg Bodwin, Fabrizio Grandoni, Merav Parter, and Virginia Vassilevska Williams; licensed under Creative Commons License CC-BY Leibniz International Proceedings in Informatics Schloss Dagstuhl – Leibniz-Zentrum für Informatik, Dagstuhl Publishing, Germany XX:2 Preserving Distances in Very Faulty Graphs