Fault tolerant additive and <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si1.gif" overflow="scroll"><mml:mo stretchy="false">(</mml:mo><mml:mi>μ</mml:mi><mml:mo>,</mml:mo><mml:mi>α</mml:mi><mml:mo stretchy="false">)</mml:mo></mml:math>-spanners

Braunschvig, Gilad; Chechik, Shiri; Peleg, David; Sealfon, Adam

doi:10.1016/j.tcs.2015.02.036

Cited by 21 publications

(16 citation statements)

References 12 publications

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“…For f = 0, i.e., the single-fault setting, our +4 spanner has O(n 3/2 ) edges which matches the bound originally attained by Biló, Grandoni, Gualá, Leucci, Proietti [6] and also obtained as corollaries of the previous two papers on subset preservers [8,7]. There are many notable constructions of fault-tolerant additive spanners with other error bounds; see for example [6,28,9,8,10].…”

Section: Applicationssupporting

confidence: 78%

Restorable Shortest Path Tiebreaking for Edge-Faulty Graphs

Bodwin¹,

Parter²

2021

Preprint

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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.

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Section: Applicationssupporting

confidence: 78%

Restorable Shortest Path Tiebreaking for Edge-Faulty Graphs

Bodwin¹,

Parter²

2021

Preprint

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“…Here the aim is to build a compact data structure for a given problem, that is resilient to failure of vertices/edges, and can efficiently report the solution of the problem for any given set of failures. There has been a lot of work in the last two decades on fault tolerant algorithms for connectivity [10,16,21], shortest paths [6,12,15], and spanners [8,11].…”

Section: Introductionmentioning

confidence: 99%

Dynamic DFS in Undirected Graphs: breaking the O(m) barrier

Baswana¹,

Chaudhury²,

Choudhary³

et al. 2015

Proceedings of the Twenty-Seventh Annual ACM-SIAM Symposium on Discrete Algorithms

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Given an undirected graph G = (V, E) on n vertices and m edges, we address the problem of maintaining a DFS tree when the graph is undergoing updates (insertion and deletion of vertices or edges). We present the following results for this problem. Fault tolerant DFS tree:There exists a data structure of sizeÕ(m) 1 such that given any set F of failed vertices or edges, a DFS tree of the graph G \ F can be reported inÕ(n|F|) time. Fully dynamic DFS tree:There exists a fully dynamic algorithm for maintaining a DFS tree that takes worst caseÕ( √ mn) time per update for any arbitrary online sequence of updates. Incremental DFS tree:Given any arbitrary online sequence of edge insertions, we can maintain a DFS tree inÕ(n) worst case time per edge insertion.These are the first o(m) worst case time results for maintaining a DFS tree in a dynamic environment. Moreover, our fully dynamic algorithm provides, in a seamless manner, the first deterministic algorithm with O(1) query time and o(m) worst case update time for the dynamic subgraph connectivity, biconnectivity, and 2-edge connectivity.

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“…Braunschvig, Chechik, Peleg, and Sealfon [15] were the first to introduce fault-tolerance to additive spanners, via the natural extension of Definition 3.…”

Section: Additive Stretchmentioning

confidence: 99%

“…In other words, tolerating one additional fault costs poly(n) in spanner size, and there is no way to tolerate Ω(log n) faults in subquadratic size. Accordingly, constructions of VFT spanners of fixed size have to pay super-constant additive error of type +O(f ) [9,13,15,22].…”

Section: Additive Stretchmentioning

confidence: 99%

Vertex Fault-Tolerant Emulators

Bodwin¹,

Dinitz²,

Nazari³

2021

Preprint

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A k-spanner of a graph G is a sparse subgraph that preserves its shortest path distances up to a multiplicative stretch factor of k, and a k-emulator is similar but not required to be a subgraph of G. A classic theorem by Thorup and Zwick [JACM '05] shows that, despite the extra flexibility available to emulators, the size/stretch tradeoffs for spanners and emulators are equivalent. Our main result is that this equivalence in tradeoffs no longer holds in the commonly-studied setting of graphs with vertex failures. That is: we introduce a natural definition of vertex fault-tolerant emulators, and then we show a three-way tradeoff between size, stretch, and fault-tolerance for these emulators that polynomially surpasses the tradeoff known to be optimal for spanners.We complement our emulator upper bound with a lower bound construction that is essentially tight (within log n factors of the upper bound) when the stretch is 2k − 1 and k is either a fixed odd integer or 2. We also show constructions of fault-tolerant emulators with additive error, demonstrating that these also enjoy significantly improved tradeoffs over those available for fault-tolerant additive spanners.

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Fault tolerant additive and (μ,α)-spanners

Cited by 21 publications

References 12 publications

Restorable Shortest Path Tiebreaking for Edge-Faulty Graphs

Restorable Shortest Path Tiebreaking for Edge-Faulty Graphs

Dynamic DFS in Undirected Graphs: breaking the O(m) barrier

Vertex Fault-Tolerant Emulators

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