Distributed Constructions of Dual-Failure Fault-Tolerant Distance Preservers

Fault tolerant distance preservers (spanners) are sparse subgraphs that preserve (approximate) distances between given pairs of vertices under edge or vertex failures. So-far, these structures have been studied thoroughly mainly from a centralized viewpoint. Despite the fact fault tolerant preservers are mainly motivated by the error-prone nature of distributed networks, not much is known on the distributed computational aspects of these structures. In this paper, we present distributed algorithms for constructing fault tolerant distance preservers and +2 additive spanners that are resilient to at most two edge faults. Prior to our work, the only non-trivial constructions known were for the single fault and single source setting by [Ghaffari and Parter SPAA\u2716]. Our key technical contribution is a distributed algorithm for computing distance preservers w.r.t. a subset S of source vertices, resilient to two edge faults. The output structure contains a BFS tree BFS(s,G ? {e?,e?}) for every s ? S and every e?,e? ? G. The distributed construction of this structure is based on a delicate balance between the edge congestion (formed by running multiple BFS trees simultaneously) and the sparsity of the output subgraph. No sublinear-round algorithms for constructing these structures have been known before

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

New Pairwise Spanners

Let G = (V,E) be an undirected unweighted graph on n vertices. A subgraph H of G is called an (all-pairs) purely additive spanner with stretch beta if for every (u,v) in V times V, mathsf{dist}_H(u,v) le mathsf{dist}_G(u,v) + beta. The problem of computing sparse spanners with small stretch beta is well-studied. Here we consider the following relaxation: we are given psubseteq V times V and we seek a sparse subgraph H where mathsf{dist}_H(u,v)le mathsf{dist}_G(u,v) + beta for each (u,v) in p. Such a subgraph is called a pairwise spanner with additive stretch beta and our goal is to construct such subgraphs that are sparser than all-pairs spanners with the same stretch. We show sparse pairwise spanners with additive stretch 4 and with additive stretch 6. We also consider the following special cases: p = S times V and p = S times T, where Ssubseteq V and Tsubseteq V, and show sparser pairwise spanners for these cases

Path-Reporting Distance Oracles with Near-Logarithmic Stretch and Linear Size

Given an $n$-vertex undirected graph $G=(V,E,w)$, and a parameter $k\geq1$, a path-reporting distance oracle (or PRDO) is a data structure of size $S(n,k)$, that given a query $(u,v)\in V^2$, returns an $f(k)$-approximate shortest $u-v$ path $P$ in $G$ within time $q(k)+O(|P|)$. Here $S(n,k)$, $f(k)$ and $q(k)$ are arbitrary functions. A landmark PRDO due to Thorup and Zwick, with an improvement of Wulff-Nilsen, has $S(n,k)=O(k\cdot n^{1+\frac{1}{k}})$, $f(k)=2k-1$ and $q(k)=O(\log k)$. The size of this oracle is $\Omega(n\log n)$ for all $k$. Elkin and Pettie and Neiman and Shabat devised much sparser PRDOs, but their stretch was polynomially larger than the optimal $2k-1$. On the other hand, for non-path-reporting distance oracles, Chechik devised a result with $S(n,k)=O(n^{1+\frac{1}{k}})$, $f(k)=2k-1$ and $q(k)=O(1)$. In this paper we make a dramatic progress in bridging the gap between path-reporting and non-path-reporting distance oracles. We devise a PRDO with size $S(n,k)=O(\lceil\frac{k\log\log n}{\log n}\rceil\cdot n^{1+\frac{1}{k}})$, stretch $f(k)=O(k)$ and query time $q(k)=O(\log\lceil\frac{k\log\log n}{\log n}\rceil)$. We can also have size $O(n^{1+\frac{1}{k}})$, stretch $O(k\cdot\lceil\frac{k\log\log n}{\log n}\rceil)$ and query time $q(k)=O(\log\lceil\frac{k\log\log n}{\log n}\rceil)$. Our results on PRDOs are based on novel constructions of approximate distance preservers, that we devise in this paper. Specifically, we show that for any $\epsilon>0$, any $k=1,2,...$, and any graph $G$ and a collection $\mathcal{P}$ of $p$ vertex pairs, there exists a $(1+\epsilon)$-approximate preserver with $O(\gamma(\epsilon,k)\cdot p+n\log k+n^{1+\frac{1}{k}})$ edges, where $\gamma(\epsilon,k)=(\frac{\log k}{\epsilon})^{O(\log k)}$. These new preservers are significantly sparser than the previous state-of-the-art approximate preservers due to Kogan and Parter.Comment: 61 pages, 3 figure