577 research outputs found

    Distributed Constructions of Dual-Failure Fault-Tolerant Distance Preservers

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    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

    Sparse Fault-Tolerant BFS Trees

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    This paper addresses the problem of designing a sparse {\em fault-tolerant} BFS tree, or {\em FT-BFS tree} for short, namely, a sparse subgraph TT of the given network GG such that subsequent to the failure of a single edge or vertex, the surviving part TT' of TT still contains a BFS spanning tree for (the surviving part of) GG. Our main results are as follows. We present an algorithm that for every nn-vertex graph GG and source node ss constructs a (single edge failure) FT-BFS tree rooted at ss with O(n \cdot \min\{\Depth(s), \sqrt{n}\}) edges, where \Depth(s) is the depth of the BFS tree rooted at ss. This result is complemented by a matching lower bound, showing that there exist nn-vertex graphs with a source node ss for which any edge (or vertex) FT-BFS tree rooted at ss has Ω(n3/2)\Omega(n^{3/2}) edges. We then consider {\em fault-tolerant multi-source BFS trees}, or {\em FT-MBFS trees} for short, aiming to provide (following a failure) a BFS tree rooted at each source sSs\in S for some subset of sources SVS\subseteq V. Again, tight bounds are provided, showing that there exists a poly-time algorithm that for every nn-vertex graph and source set SVS \subseteq V of size σ\sigma constructs a (single failure) FT-MBFS tree T(S)T^*(S) from each source siSs_i \in S, with O(σn3/2)O(\sqrt{\sigma} \cdot n^{3/2}) edges, and on the other hand there exist nn-vertex graphs with source sets SVS \subseteq V of cardinality σ\sigma, on which any FT-MBFS tree from SS has Ω(σn3/2)\Omega(\sqrt{\sigma}\cdot n^{3/2}) edges. Finally, we propose an O(logn)O(\log n) approximation algorithm for constructing FT-BFS and FT-MBFS structures. The latter is complemented by a hardness result stating that there exists no Ω(logn)\Omega(\log n) approximation algorithm for these problems under standard complexity assumptions

    Multiple-Edge-Fault-Tolerant Approximate Shortest-Path Trees

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    Let GG be an nn-node and mm-edge positively real-weighted undirected graph. For any given integer f1f \ge 1, we study the problem of designing a sparse \emph{f-edge-fault-tolerant} (ff-EFT) σ\sigma{\em -approximate single-source shortest-path tree} (σ\sigma-ASPT), namely a subgraph of GG having as few edges as possible and which, following the failure of a set FF of at most ff edges in GG, contains paths from a fixed source that are stretched at most by a factor of σ\sigma. To this respect, we provide an algorithm that efficiently computes an ff-EFT (2F+1)(2|F|+1)-ASPT of size O(fn)O(f n). Our structure improves on a previous related construction designed for \emph{unweighted} graphs, having the same size but guaranteeing a larger stretch factor of 3(f+1)3(f+1), plus an additive term of (f+1)logn(f+1) \log n. Then, we show how to convert our structure into an efficient ff-EFT \emph{single-source distance oracle} (SSDO), that can be built in O~(fm)\widetilde{O}(f m) time, has size O(fnlog2n)O(fn \log^2 n), and is able to report, after the failure of the edge set FF, in O(F2log2n)O(|F|^2 \log^2 n) time a (2F+1)(2|F|+1)-approximate distance from the source to any node, and a corresponding approximate path in the same amount of time plus the path's size. Such an oracle is obtained by handling another fundamental problem, namely that of updating a \emph{minimum spanning forest} (MSF) of GG after that a \emph{batch} of kk simultaneous edge modifications (i.e., edge insertions, deletions and weight changes) is performed. For this problem, we build in O(mlog3n)O(m \log^3 n) time a \emph{sensitivity} oracle of size O(mlog2n)O(m \log^2 n), that reports in O(k2log2n)O(k^2 \log^2 n) time the (at most 2k2k) edges either exiting from or entering into the MSF. [...]Comment: 16 pages, 4 figure

    Vertex Fault Tolerant Additive Spanners

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    A {\em fault-tolerant} structure for a network is required to continue functioning following the failure of some of the network's edges or vertices. In this paper, we address the problem of designing a {\em fault-tolerant} additive spanner, namely, a subgraph HH of the network GG such that subsequent to the failure of a single vertex, the surviving part of HH still contains an \emph{additive} spanner for (the surviving part of) GG, satisfying dist(s,t,H{v})dist(s,t,G{v})+βdist(s,t,H\setminus \{v\}) \leq dist(s,t,G\setminus \{v\})+\beta for every s,t,vVs,t,v \in V. Recently, the problem of constructing fault-tolerant additive spanners resilient to the failure of up to ff \emph{edges} has been considered by Braunschvig et. al. The problem of handling \emph{vertex} failures was left open therein. In this paper we develop new techniques for constructing additive FT-spanners overcoming the failure of a single vertex in the graph. Our first result is an FT-spanner with additive stretch 22 and O~(n5/3)\widetilde{O}(n^{5/3}) edges. Our second result is an FT-spanner with additive stretch 66 and O~(n3/2)\widetilde{O}(n^{3/2}) edges. The construction algorithm consists of two main components: (a) constructing an FT-clustering graph and (b) applying a modified path-buying procedure suitably adopted to failure prone settings. Finally, we also describe two constructions for {\em fault-tolerant multi-source additive spanners}, aiming to guarantee a bounded additive stretch following a vertex failure, for every pair of vertices in S×VS \times V for a given subset of sources SVS\subseteq V. The additive stretch bounds of our constructions are 4 and 8 (using a different number of edges)

    Optimal Vertex Fault Tolerant Spanners (for fixed stretch)

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    A kk-spanner of a graph GG is a sparse subgraph HH whose shortest path distances match those of GG up to a multiplicative error kk. In this paper we study spanners that are resistant to faults. A subgraph HGH \subseteq G is an ff vertex fault tolerant (VFT) kk-spanner if HFH \setminus F is a kk-spanner of GFG \setminus F for any small set FF of ff vertices that might "fail." One of the main questions in the area is: what is the minimum size of an ff fault tolerant kk-spanner that holds for all nn node graphs (as a function of ff, kk and nn)? 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 kk is fixed. Specifically, we prove that every (undirected, possibly weighted) nn-node graph GG has a (2k1)(2k-1)-spanner resilient to ff vertex faults with Ok(f11/kn1+1/k)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 distGF(s,t)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=2k=2 (and hence we close the EFT problem for 33-approximations), but it falls to Ω(f1/21/(2k)n1+1/k)\Omega(f^{1/2 - 1/(2k)} \cdot n^{1 + 1/k}) for k3k \ge 3. We leave it as an open problem to close this gap.Comment: To appear in SODA 201

    Dual Failure Resilient BFS Structure

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    We study {\em breadth-first search (BFS)} spanning trees, and address the problem of designing a sparse {\em fault-tolerant} BFS structure, or {\em FT-BFS } for short, resilient to the failure of up to two edges in the given undirected unweighted graph GG, i.e., a sparse subgraph HH of GG such that subsequent to the failure of up to two edges, the surviving part HH' of HH still contains a BFS spanning tree for (the surviving part of) GG. FT-BFS structures, as well as the related notion of replacement paths, have been studied so far for the restricted case of a single failure. It has been noted widely that when concerning shortest-paths in a variety of contexts, there is a sharp qualitative difference between a single failure and two or more failures. Our main results are as follows. We present an algorithm that for every nn-vertex unweighted undirected graph GG and source node ss constructs a (two edge failure) FT-BFS structure rooted at ss with O(n5/3)O(n^{5/3}) edges. To provide a useful theory of shortest paths avoiding 2 edges failures, we take a principled approach to classifying the arrangement these paths. We believe that the structural analysis provided in this paper may decrease the barrier for understanding the general case of f2f\geq 2 faults and pave the way to the future design of ff-fault resilient structures for f2f \geq 2. We also provide a matching lower bound, which in fact holds for the general case of f1f \geq 1 and multiple sources SVS \subseteq V. It shows that for every f1f\geq 1, and integer 1σn1 \leq \sigma \leq n, there exist nn-vertex graphs with a source set SVS \subseteq V of cardinality σ\sigma for which any FT-BFS structure rooted at each sSs \in S, resilient to up to ff-edge faults has Ω(σ1/(f+1)n21/(f+1))\Omega(\sigma^{1/(f+1)} \cdot n^{2-1/(f+1)}) edges
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