Fault-tolerant spanners for general graphs

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

doi:10.1145/1536414.1536475

Cited by 44 publications

(40 citation statements)

References 36 publications

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“…Multipath spanners have some flavors of fault-tolerant spanners, notion introduced in [CLPR10] for general graphs. A subgraph H is an r-fault tolerant s-spanner of G if for any set F of at most r 0 faulty vertices, and for any pair u, v of vertices outside…”

Section: Overviewmentioning

confidence: 99%

“…We observe that many classical spanner constructions (including the greedy one) do not provide boundedhop spanners, although such spanners exist as proved in Section 2.1. Some variant presented in [CLPR10] of the Thorup-Zwick constructions [TZ05] are also bounded-hop (Section 2.2). Combining these specific spanners with the generic construction of fault tolerant spanners of [DK11], we show in Section 2.3 how to obtained a LOCAL distributed algorithm for computing a p-mutlipath spanner of bounded stretch.…”

Section: Overviewmentioning

confidence: 99%

“…The algorithm is a distributed version of the spanner algorithm used in [CLPR10], which is based on the sampling technique of [TZ05]. We make the observation that this algorithm can run in O(k) rounds.…”

Section: Distributed Bounded Hop Spannersmentioning

confidence: 99%

“…The total number of edges is O(kn 1+1/k log 1−1/k n). It is proved in [CLPR10] that for every edge uv of G, there is a cluster C(w) containing u and v. The path of the tree from w to one of the end-point has at most k − 1 edges and cost (k − 1) · ω(uv), and the path from w to the other end-point has at most k edges and cost k · ω(uv). This is therefore a (2k − 1)-hop spanner.…”

Section: Distributed Bounded Hop Spannersmentioning

confidence: 99%

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Node-Disjoint Multipath Spanners and Their Relationship with Fault-Tolerant Spanners

Gavoille

Godfroy

Viennot

2011

Lecture Notes in Computer Science

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Motivated by multipath routing, we introduce a multi-connected variant of spanners. For that purpose we introduce the p-multipath cost between two nodes u and v as the minimum weight of a collection of p internally vertex-disjoint paths between u and v. Given a weighted graph G, a subgraph H is a p-multipath s-spanner if for all u, v, the p-multipath cost between u and v in H is at most s times the p-multipath cost in G. The s factor is called the stretch.Building upon recent results on fault-tolerant spanners, we show how to build p-multipath spanners of constant stretch and ofÕ(n 1+1/k ) edges 1 , for fixed parameters p and k, n being the number of nodes of the graph. Such spanners can be constructed by a distributed algorithm running in O(k) rounds.Additionally, we give an improved construction for the case p = k = 2. Our spanner H has O(n 3/2 ) edges and the p-multipath cost in H between any two node is at most twice the corresponding one in G plus O(W ), W being the maximum edge weight.

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Section: Overviewmentioning

confidence: 99%

Section: Overviewmentioning

confidence: 99%

Section: Distributed Bounded Hop Spannersmentioning

confidence: 99%

Section: Distributed Bounded Hop Spannersmentioning

confidence: 99%

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Node-Disjoint Multipath Spanners and Their Relationship with Fault-Tolerant Spanners

Gavoille

Godfroy

Viennot

2011

Lecture Notes in Computer Science

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“…They provided two constructions of (k, 1 + )-VFTS: one with optimal O(kn) edges, and the other with O(k 2 n) edges and maximum degree O(k 2 ). There has also been research on the trade-off between maximum degree and weight in fault-tolerant Euclidean spanners [13,7], and fault-tolerant spanners for general graphs [6,9].…”

Section: Introductionmentioning

confidence: 99%

Sparse Fault-Tolerant Spanners for Doubling Metrics with Bounded Hop-Diameter or Degree

Chan

2013

Algorithmica

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Abstract. We study fault-tolerant spanners in doubling metrics. A subgraph H for a metric space X is called a k-vertex-fault-tolerant t-spanner ((k, t)-VFTS or simply k-VFTS), if for any subset S ⊆ X with |S| ≤ k, it holds that d H\S (x, y) ≤ t · d(x, y), for any pair of x, y ∈ X \ S. For any doubling metric, we give a basic construction of k-VFTS with stretch arbitrarily close to 1 that has optimal O(kn) edges. In addition, we also consider bounded hop-diameter, which is studied in the context of fault-tolerance for the first time even for Euclidean spanners. We provide a construction of k-VFTS with bounded hop-diameter: for m ≥ 2n, we can reduce the hop-diameter of the above k-VFTS to O(α(m, n)) by adding O(km) edges, where α is a functional inverse of the Ackermann's function. Finally, we construct a fault-tolerant single-sink spanner with bounded maximum degree, and use it to reduce the maximum degree of our basic k-VFTS. As a result, we get a k-VFTS with O(k 2 n) edges and maximum degree O(k 2 ).

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Path-Fault-Tolerant Approximate Shortest-Path Trees

D’Andrea

D’Emidio

Frigioni

et al. 2015

Structural Information and Communication Complexity

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Let G = (V, E) be an n-nodes non-negatively real-weighted undirected graph. In this paper we show how to enrich a single-source shortest-path tree (SPT) of G with a sparse set of auxiliary edges selected from E, in order to create a structure which tolerates effectively a path failure in the SPT. This consists of a simultaneous fault of a set F of at most f adjacent edges along a shortest path emanating from the source, and it is recognized as one of the most frequent disruption in an SPT. We show that, for any integer parameter k ≥ 1, it is possible to provide a very sparse (i.e., of size O(kn · f 1+1/k )) auxiliary structure that carefully approximates (i.e., within a stretch factor of (2k − 1)(2|F | + 1)) the true shortest paths from the source during the lifetime of the failure. Moreover, we show that our construction can be further refined to get a stretch factor of 3 and a size of O(n log n) for the special case f = 2, and that it can be converted into a very efficient approximate-distance sensitivity oracle, that allows to quickly (even in optimal time, if k = 1) reconstruct the shortest paths (w.r.t. our structure) from the source after a path failure, thus permitting to perform promptly the needed rerouting operations. Our structure compares favorably with previous known solutions, as we discuss in the paper, and moreover it is also very effective in practice, as we assess through a large set of experiments.The natural solution is that of modeling the network as a graph (nodes as vertices and links as edges) and building a (fast and compact) structure to be used to transmit the data. In particular, the most common approach of this kind is that of computing a shortest-path tree (SPT), rooted at the desired source node, of such graph.However, the SPT, as any tree-based topology, is prone to unpredictable events that might occur in practice, such as failures of nodes and/or links. Therefore, the use of SPTs might result in a high sensitivity to malfunctioning, which unavoidably causes the undesired effect of disconnecting sets of nodes from the source and thus the interruption of the broadcasting service.Therefore, a general approach to cope with this scenario is to make the SPT fault-tolerant against a given number of simultaneous component failures, by adding to it a set of suitably selected edges from the underlying graph, so that the resulting structure will remain connected w.r.t. the source. In other words, the selected edges can be used to build up alternative paths from the root, each one of them in replacement of a corresponding original shortest path which was affected by the failure. However, if these paths are constrained to be shortest, then it can be easily seen that for a non-negatively real weighted and undirected graph of n nodes and m edges, this may require as much as Θ(m) additional edges, also in the case in which m = Θ(n 2 ). In other words, the set-up costs of the strengthened network may become unaffordable.Thus, a reasonable compromise is that of building sparse and fault-tolerant s...

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Fault-tolerant spanners for general graphs

Cited by 44 publications

References 36 publications

Node-Disjoint Multipath Spanners and Their Relationship with Fault-Tolerant Spanners

Node-Disjoint Multipath Spanners and Their Relationship with Fault-Tolerant Spanners

Sparse Fault-Tolerant Spanners for Doubling Metrics with Bounded Hop-Diameter or Degree

Path-Fault-Tolerant Approximate Shortest-Path Trees

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