Sparse Fault-Tolerant BFS Trees

Parter, Merav; Peleg, David

doi:10.1007/978-3-642-40450-4_66

Cited by 31 publications

(95 citation statements)

References 34 publications

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“…Another common variant is faulttolerant spanners or distance preservers, which must (approximately) preserve distances even after some edges "fail." Parter and Peleg [35] obtained matching upper and lower bounds for BFS structures in the presence of one fault, and Parter [34] obtained upper and lower bounds for the two fault case. Interesting fault-tolerant spanners were constructed in [17,33,21,10,15].…”

Section: Other Related Workmentioning

confidence: 95%

Linear Size Distance Preservers

Bodwin

2017

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

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The famous shortest path tree lemma states that, for any node s in a graph G = (V, E), there is a subgraph on O(n) edges that preserves all distances between node pairs in the set {s} × V . A very basic question in distance sketching research, with applications to other problems in the field, is to categorize when else graphs admit sparse subgraphs that preserve distances between a set P of p node pairs, where P has some different structure than {s} × V or possibly no guaranteed structure at all. Trivial lower bounds of a path or a clique show that such a subgraph will need Ω(n + p) edges in the worst case. The question is then to determine when these trivial lower bounds are sharp; that is, when do graphs have linear size distance preservers on O(n + p) edges?In this paper, we make the first new progress on this fundamental question in over ten years. We show:

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Section: Other Related Workmentioning

confidence: 95%

Linear Size Distance Preservers

Bodwin

2017

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

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“…Bilò, Gualà, Leucci, and Proietti [5] showed that any weighted undirected graph has a fault tolerant (1+ )-shortest path subgraph of size O((1/ 2 )n log n). Parter and Peleg [18] showed that any unweighted (un)directed graph has a fault tolerant exact shortest path subgraph with O(n 3/2 ) edges. They also showed a matching lower bound.…”

Section: Related Workmentioning

confidence: 99%

“…Parter and Peleg [18] showed that even in the case of unweighted undirected graphs there are graphs such that the size of their fault tolerant exact shortest path subgraph is at least Ω(n 1.5 ). Bilò, Gualà, Leucci, and Proietti [5] showed that any weighted undirected graph has a fault tolerant (1 + )-shortest path subgraph of size O((1/ 2 )n log n).…”

Section: Introductionmentioning

confidence: 99%

“…Demetrescu, Thorup, Chowdhury and Ramachandran [12] showed that the size of a fault tolerant exact shortest path subgraph in weighted directed/undirected graphs is Ω(m). Parter and Peleg [18] showed that even in the restricted case of unweighted undirected graphs the size of any subgraph for the exact shortest path is at least Ω(n 1.5 ). Therefore, a (1 + )-approximation is the best one can hope for.…”

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confidence: 99%

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Approximate Single Source Fault Tolerant Shortest Path

Baswana

Choudhary

Hussain³

et al. 2018

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

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Let G = (V, E) be an n-vertices m-edges directed graph with edge weights in the range [1, W ] and L = log(W ). Let s ∈ V be a designated source. In this paper we address several variants of the problem of maintaining the (1 + )-approximate shortest path from s to each v ∈ V \ {s} in the presence of a failure of an edge or a vertex. From the graph theory perspective we show that G has a subgraph H with O(nL/ ) edges such that for any x, v ∈ V , the graph H \ x contains a path whose length is a (1 + )-approximation of the length of the shortest path from s to v in G \ x. We show that the size of the subgraph H is optimal (up to logarithmic factors) by proving a lower bound of Ω(nL/ ) edges. Demetrescu, Thorup, Chowdhury and Ramachandran [12] showed that the size of a fault tolerant exact shortest path subgraph in weighted directed/undirected graphs is Ω(m). Parter and Peleg [18] showed that even in the restricted case of unweighted undirected graphs the size of any subgraph for the exact shortest path is at least Ω(n 1.5 ). Therefore, a (1 + )-approximation is the best one can hope for. We consider also the data structure problem and show that there exists an O(nL/ ) size oracle that for any v ∈ V reports a (1 + )-approximate distance of v from s on a failure of any x ∈ V in O(log log 1+ (nW )) time. We show that the size of the oracle is optimal (up to logarithmic factors) by proving a lower bound of Ω(nL/ log n). Finally, we present two distributed algorithms. We present a single source routing scheme that can route on a (1 + )-approximation of the shortest path from a fixed source s to any destination t in the presence of a fault. Each vertex has a label and a routing table of O(L/ ) bits. We present also a labeling scheme that assigns each vertex a label of O(L/ ) bits. For any two vertices x, v ∈ V the labeling scheme outputs a (1 + )-approximation of the distance from s to v in G \ x using only the labels of x and v.

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“…Parter and Peleg [62] considered the problem of computing a subgraph that preserves shortest paths from s sources after a single edge or vertex failure. They proved that Θ(s 1/2 n 3/2 ) edges are necessary and sufficient, for every s. See also [14,17,18,23,31,61,63] for spanners (subgraphs) that preserve approximate distances subject to edge or vertex failures.…”

Section: Introductionmentioning

confidence: 99%

Connectivity Oracles for Graphs Subject to Vertex Failures

Duan¹,

Pettie²

2017

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

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We introduce new data structures for answering connectivity queries in graphs subject to batched vertex failures. Our deterministic structure processes a batch of d ≤ d failed vertices inÕ(d 3 ) time and thereafter answers connectivity queries in O(d) time. It occupies space O(d m log n). We develop a randomized Monte Carlo version of our data structure with update timeÕ(d 2 ), query time O(d), and spaceÕ(m) for any d . This is the first connectivity oracle for general graphs that can efficiently deal with an unbounded number of vertex failures.Our data structures are based on a new decomposition theorem for an undirected graph G = (V, E), which is of independent interest. It states that for any terminal set U ⊆ V we can remove a set B of |U |/(s − 2) vertices such that the remaining graph contains a Steiner forest for U − B with maximum degree s. IntroductionThe dynamic subgraph model [19,21,36,39,37,42,65] is a constrained dynamic graph model. Rather than allow the graph to evolve in completely arbitrary ways (via an unbounded sequence of edge insertions and deletions), there is assumed to be a fixed ideal graph G = (V, E) that can be preprocessed in advance. The ideal graph is susceptible only to the failure of edges/vertices and their subsequent recovery, possibly with a bound d on the number of failures at one time. Queries naturally answer questions about the current failure-free subgraph. This model is useful because it more accurately represents the behavior of many real-world networks: changes to the underlying topology are relatively rare but transient failures very common. More importantly, this model offers the algorithm designer the freedom to explore exotic graph rep-

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Sparse Fault-Tolerant BFS Trees

Cited by 31 publications

References 34 publications

Linear Size Distance Preservers

Linear Size Distance Preservers

Approximate Single Source Fault Tolerant Shortest Path

Connectivity Oracles for Graphs Subject to Vertex Failures

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