Fault Tolerant Spanners for General Graphs

Chechik, Shiri; Langberg, Michael; Peleg, David; Roditty, Liam

doi:10.1137/090758039

Cited by 64 publications

(84 citation statements)

References 26 publications

(29 reference statements)

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“…The notion of fault-tolerant spanners for general graphs was initiated by Chechik et al [11] for the case of multiplicative stretch. Specifically, Chechik et al [11] presented algorithms for constructing an f -vertex fault tolerant spanner with multiplicative stretch (2k − 1) and O( f 2 k f +1 · n 1+1/k log 1−1/k n) edges.…”

Section: Introductionmentioning

confidence: 99%

Vertex fault tolerant additive spanners

Parter

2015

Distrib. Comput.

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A 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 fault-tolerant additive spanner, namely, a subgraph H of the network G such that subsequent to the failure of a single vertex, the surviving part of H still contains an additive spanner for (the surviving partthe problem of constructing fault-tolerant additive spanners resilient to the failure of up to f edges has been considered (Braunschvig et al. Proceedings of the WG, pp 206-214, 2012). The problem of handling vertex failures was left open therein. In this paper we develop new techniques for constructing additive FTspanners overcoming the failure of a single vertex in the graph.Our first result is an FT-spanner with additive stretch 2 and O(n 5/3 ) edges. Our second result is an FT-spanner with additive stretch 6 and 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 adapted to failure prone settings. Finally, we also describe two constructions for fault-tolerant multisource additive spanners, aiming to guarantee a bounded additive stretch following a vertex failure, for every pair of vertices in S × V for a given subset of sources S ⊆ V . The additive stretch bounds of our constructions are 4 and 8 (using a different number of edges).

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Section: Introductionmentioning

confidence: 99%

Vertex fault tolerant additive spanners

Parter

2015

Distrib. Comput.

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“…They provided both size and weight bounds for (1+ )-spanners, which were later improved by Lukovski [Luk99] and Czumaj and Zhao [CZ03]. The first result on fault-tolerant spanners for general graphs, by Chechik, Langberg, Peleg, and Roditty [CLPR09], constructs r-fault tolerant (2k−1)-spanners with size O(r 2 k r+1 · n 1+1/k log 1−1/k n), for any integer k ≥ 1. Since it has long been known how to construct (2k − 1)-spanners with size O(n 1+1/k ) (see e.g [ADD + 93]), this means that the extra cost of r-fault tolerance is O(r 2 k r+1 ).…”

Section: Introductionmentioning

confidence: 98%

“…While this is independent of n, it grows rapidly as the number of faults r gets large. We address an important question left open in [CLPR09] of improving this dependence on r from exponential to polynomial.…”

Section: Introductionmentioning

confidence: 99%

Fault-tolerant spanners

Dinitz

Krauthgamer

2011

Proceedings of the 30th Annual ACM SIGACT-SIGOPS Symposium on Principles of Distributed Computing

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A natural requirement for many distributed structures is fault-tolerance: after some failures in the underlying network, whatever remains from the structure should still be effective for whatever remains from the network. In this paper we examine spanners of general graphs that are tolerant to vertex failures, and significantly improve their dependence on the number of faults r for all stretch bounds.For stretch k ≥ 3 we design a simple transformation that converts every k-spanner construction with at most f (n) edges into an r-fault-tolerant k-spanner construction with at most O(r 3 log n) · f (2n/r) edges. Applying this to standard greedy spanner constructions gives r-fault tolerant kspanners withÕ(r 2 n 1+ 2 k+1 ) edges. The previous construction by Chechik, Langberg, Peleg, and Roddity [CLPR09] depends similarly on n but exponentially on r (approximately like k r ).For the case of k = 2 and unit edge-lengths, an O(r log n)approximation is known from recent work of Dinitz and Krauthgamer [DK11], in which several spanner results are obtained using a common approach of rounding a natural flow-based linear programming relaxation. Here we use a different (stronger) LP relaxation and improve the approximation ratio to O(log n), which is, notably, independent of the number of faults r. We further strengthen this bound in terms of the maximum degree by using the Lovász Local Lemma.Finally, we show that most of our constructions are inherently local by designing equivalent distributed algorithms in the LOCAL model of distributed computation.

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“…Algorithms for constructing sparse edge and vertex fault tolerant spanners for arbitrary undirected weighted graphs were presented in [5,8]. Note, however, that the use of costly link reinforcements for attaining fault-tolerance in spanners is less attractive than for FT-BFS structures, since the cost of adding fault-tolerance via backup edges (in the relevant complexity measure) is often low (e.g., merely polylogarithmic in the graph size n), hence the gains expected from using reinforcement are relatively small.…”

Section: Introductionmentioning

confidence: 99%

Fault Tolerant BFS Structures

Parter

Peleg

2015

Proceedings of the 27th ACM Symposium on Parallelism in Algorithms and Architectures

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This paper initiates the study of fault resilient network structures that mix two orthogonal protection mechanisms: (a) backup, namely, augmenting the structure with many (redundant) low-cost but fault-prone components, and (b) reinforcement, namely, acquiring high-cost but fault-resistant components. To study the trade-off between these two mechanisms in a concrete setting, we address the problem of designing a (b, r) fault-tolerant BFS (or (b, r) FT-BFS for short) structure, namely, a subgraph H of the network G consisting of two types of edges: a set E ⊆ E of r(n) fault-resistant reinforcement edges, which are assumed to never fail, and a (larger) set E(H) \ E of b(n) fault-prone backup edges, such that subsequent to the failure of a single fault-prone backup edge e ∈ E \ E , the surviving part of H still contains a BFS spanning tree for (the surviving part of) G, satisfying dist(s, v, H \ {e}) ≤ dist(s, v, G \ {e}) for every v ∈ V and e ∈ E\E . We establish the following tradeoff: For every real ∈ (0, 1], if r(n) =Θ(n 1− ), then b(n) =Θ(n 1+ ) is necessary and sufficient. More specifically, as shown in [14], for = 1, FT-BFS structures (with no reinforced edges) require Θ(n 3/2 ) edges, and this is sufficient. At the other extreme, if = 0, then n − 1 reinforced edges suffice (with no need for backup). Here, we present a polynomial time algorithm that given an undirected graph G = (V, E), a source vertex s and a real ∈ (0, 1], constructs a (b(n), r(n)) FT-BFS with r(n) = O(n 1− ) and b(n) = O(min{1/ · n 1+

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Fault Tolerant Spanners for General Graphs

Cited by 64 publications

References 26 publications

Vertex fault tolerant additive spanners

Vertex fault tolerant additive spanners

Fault-tolerant spanners

Fault Tolerant BFS Structures

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