Improved Purely Additive Fault-Tolerant Spanners

Bilò, Davide; Grandoni, Fabrizio; Gualà, Luciano; Leucci, Stefano; Proietti, Guido

doi:10.1007/978-3-662-48350-3_15

Cited by 24 publications

(27 citation statements)

References 20 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…In general graphs, the construction of sparse fault tolerant subgraphs has received a significant amount of recent attention [43,39]. Construction of purely additive fault tolerant spanners were studied in [16,10,14]. Concerning exact distances from a single (or few) sources, [40] introduced the notion of FT-BFS structures that contain a BFS tree tree from s in G \ {e} for every failing edge e ∈ E(G).…”

Section: Introductionmentioning

confidence: 99%

Optimal Vertex Fault Tolerant Spanners (for fixed stretch)

Bodwin¹,

Dinitz

Parter³

et al. 2018

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

View full text Add to dashboard Cite

A k-spanner of a graph G is a sparse subgraph H whose shortest path distances match those of G up to a multiplicative error k. In this paper we study spanners that are resistant to faults. A subgraph H ⊆ G is an f vertex fault tolerant (VFT) k-spanner if H \ F is a k-spanner of G \ F for any small set F of f vertices that might "fail." One of the main questions in the area is: what is the minimum size of an f fault tolerant k-spanner that holds for all n node graphs (as a function of f , k and n)? In this paper, we settle the question of the optimal size of a VFT spanner, in the setting where the stretch factor k is fixed. Specifically, we prove that every (undirected, possibly weighted) n-node graph G has a (2k − 1)-spanner resilient to f vertex faults with O k (f 1−1/k n 1+1/k ) edges, and this is fully optimal (unless the famous Erdös Girth Conjecture is false). Our lower bound even generalizes to imply that no data structure capable of approximating dist G\F (s, t) similarly can beat the space usage of our spanner in the worst case. To the best of our knowledge, this is the first instance in fault tolerant network design in which introducing fault tolerance to the structure increases the size of the (non-FT) structure by a sublinear factor in f . Another advantage of this result is that our spanners are constructed by a very natural and simple greedy algorithm, which is the obvious extension of the standard * Supported in part by NSF awards 1464239 and 1535887. Supported in part by NSF grants CCF-1417238, CCF-1528078 and CCF-1514339, and BSF grant BSF:2012338. greedy algorithm used to build spanners in the non-faulty setting.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 = 2 (and hence we close the EFT problem for 3-approximations), but it falls to Ω(f 1/2−1/(2k) · n 1+1/k ) for k ≥ 3. We leave it as an open problem to close this gap.

show abstract

Section: Introductionmentioning

confidence: 99%

Optimal Vertex Fault Tolerant Spanners (for fixed stretch)

Bodwin¹,

Dinitz

Parter³

et al. 2018

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

View full text Add to dashboard Cite

show abstract

“…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: 99%

Linear Size Distance Preservers

Bodwin

2017

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

View full text Add to dashboard Cite

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:

show abstract

“…On unweighted graphs, it makes instead sense to study fault-tolerant additive spanners. In particular, Braunshvig et al [9] proposed the following general approach to build an f -EFT additive spanner: Let A be an f -EFT σ-spanner, and let B be an ordinary (1, β)- [7] the corresponding analysis has been refined yielding a better additive bound of 2f (β + σ − 1) + β. Finally, for other results on single edge/vertex failures spanners/oracles on unweighted graphs, we refer the reader to [4,29,7].…”

Section: More Related Work On (Fault-tolerant) Spanners/oraclesmentioning

confidence: 99%

Fault-Tolerant Approximate Shortest-Path Trees

Bilò

Gualà

Leucci

et al. 2014

Algorithms - ESA 2014

Self Cite

View full text Add to dashboard Cite

The resiliency of a network is its ability to remain effectively functioning also when any of its nodes or links fails. However, to reduce operational and set-up costs, a network should be small in size, and this conflicts with the requirement of being resilient. In this paper we address this trade-off for the prominent case of the broadcasting routing scheme, and we build efficient (i.e., sparse and fast) faulttolerant approximate shortest-path trees, for both the edge and vertex single-failure case. In particular, for an n-vertex non-negatively weighted graph, and for any constant ε > 0, we design two structures of size O( n log n ε 2 ) which guarantee (1 + ε)-stretched paths from the selected source also in the presence of an edge/vertex failure. This favorably compares with the currently best known solutions, which are for the edge-failure case of size O(n) and stretch factor 3, and for the vertexfailure case of size O(n log n) and stretch factor 3. Moreover, we also focus on the unweighted case, and we prove that an ordinary spanner can be slightly augmented in order to build efficient fault-tolerant approximate breadth-first-search trees.

show abstract

Improved Purely Additive Fault-Tolerant Spanners

Cited by 24 publications

References 20 publications

Optimal Vertex Fault Tolerant Spanners (for fixed stretch)

Optimal Vertex Fault Tolerant Spanners (for fixed stretch)

Linear Size Distance Preservers

Fault-Tolerant Approximate Shortest-Path Trees

Contact Info

Product

Resources

About