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 worst-case 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, O~(fn^{2-1/2^f}) 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(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-eps}) for eps>0, then the additive error has to be Omega(eps f). 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 Theta(n^2) for f > 1 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

Optimal Vertex Fault Tolerant Spanners (for fixed stretch)

A $k$-spanner of a graph $G$ is a sparse subgraph $H$ whose shortest path distances match those of $G$ up to a multiplicative error $k$. In this paper we study spanners that are resistant to faults. A subgraph $H \subseteq G$ is an $f$ vertex fault tolerant (VFT) $k$-spanner if $H \setminus F$ is a $k$-spanner of $G \setminus F$ for any small set $F$ of $f$ vertices that might "fail." One of the main questions in the area is: what is the minimum size of an $f$ fault tolerant $k$-spanner that holds for all $n$ node graphs (as a function of $f$, $k$ and $n$)? This question was first studied in the context of geometric graphs [Levcopoulos et al. STOC '98, Czumaj and Zhao SoCG '03] and has more recently been considered in general undirected graphs [Chechik et al. STOC '09, Dinitz and Krauthgamer PODC '11]. In this paper, we settle the question of the optimal size of a VFT spanner, in the setting where the stretch factor $k$ is fixed. Specifically, we prove that every (undirected, possibly weighted) $n$-node graph $G$ has a $(2k-1)$-spanner resilient to $f$ vertex faults with $O_k(f^{1 - 1/k} n^{1 + 1/k})$ edges, and this is fully optimal (unless the famous Erdos Girth Conjecture is false). Our lower bound even generalizes to imply that no data structure capable of approximating $dist_{G \setminus F}(s, t)$ similarly can beat the space usage of our spanner in the worst case. We also consider the edge fault tolerant (EFT) model, defined analogously with edge failures rather than vertex failures. We show that the same spanner upper bound applies in this setting. Our data structure lower bound extends to the case $k=2$ (and hence we close the EFT problem for $3$-approximations), but it falls to $\Omega(f^{1/2 - 1/(2k)} \cdot n^{1 + 1/k})$ for $k \ge 3$. We leave it as an open problem to close this gap.Comment: To appear in SODA 201

Restorable Shortest Path Tiebreaking for Edge-Faulty Graphs

The restoration lemma by Afek, Bremler-Barr, Kaplan, Cohen, and Merritt [Dist. Comp. '02] proves that, in an undirected unweighted graph, any replacement shortest path avoiding a failing edge can be expressed as the concatenation of two original shortest paths. However, the lemma is tiebreaking-sensitive: if one selects a particular canonical shortest path for each node pair, it is no longer guaranteed that one can build replacement paths by concatenating two selected shortest paths. They left as an open problem whether a method of shortest path tiebreaking with this desirable property is generally possible. We settle this question affirmatively with the first general construction of restorable tiebreaking schemes. We then show applications to various problems in fault-tolerant network design. These include a faster algorithm for subset replacement paths, more efficient fault-tolerant (exact) distance labeling schemes, fault-tolerant subset distance preservers and $+4$ additive spanners with improved sparsity, and fast distributed algorithms that construct these objects. For example, an almost immediate corollary of our restorable tiebreaking scheme is the first nontrivial distributed construction of sparse fault-tolerant distance preservers resilient to three faults