On Resilient Graph Spanners

Ausiello, Giorgio; Franciosa, Paolo Giulio; Italiano, Giuseppe F.; Ribichini, Andrea

doi:10.1007/s00453-015-0006-x

Cited by 7 publications

(12 citation statements)

References 27 publications

(36 reference statements)

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“…Another setting which is very close in spirit to fault-tolerant spanners is the recent work on fault-tolerant approximate shortest-path trees, both for unweighted [19] and for weighted [5,7] graphs. In [3] it was introduced the resembling concept of resilient spanners, i.e., spanners that approximately preserve the relative increase of distances due to an edge failure.…”

Section: Related Workmentioning

confidence: 99%

“…H−e (s, t) ≤ d A (s, cnt(z)) + d H−e (cnt(z), cnt(z ′ )) + d A (cnt(z ′ ), t) ≤ d G−e (s, z) + β − 1 + d G−e (z, z ′ ) + 2 + d G−e (z ′ , t) + β − d G−e (s, t) + 2β.Finally, when e = (cnt(z ′ ), z ′ ), we have:d H−e (s, t) ≤ d A (s, cnt(z)) + d H−e (cnt(z), cnt(z ′ )) + d H−e (cnt(z ′ ), z ′ ) + d A (z ′ , t) ≤ d G−e (s, z) + β − 1 + d G−e (z, z ′ ) + 2 + 1 + d G−e (z ′ , t) + β ≤ d G (s, t) + 2β + 2.This concludes the proof.⊓ ⊔This result can immediately be applied to the 6-additive spanner of size O(n ) in[4], which is clustering-based and uses O(n ) clusters. Using the 5multiplicative EFT spanner M of size O(n 4/3 ) from[10], we obtain: There exists a polynomial time algorithm to compute a 14-additive EFT spanner of size O(n 43 ).…”

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confidence: 99%

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Improved Purely Additive Fault-Tolerant Spanners

Bilò

Grandoni

Gualà

et al. 2015

Algorithms - ESA 2015

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Abstract. Let G be an unweighted n-node undirected graph. A β-additive spanner of G is a spanning subgraph H of G such that distances in H are stretched at most by an additive term β w.r.t. the corresponding distances in G. A natural research goal related with spanners is that of designing sparse spanners with low stretch. In this paper, we focus on fault-tolerant additive spanners, namely additive spanners which are able to preserve their additive stretch even when one edge fails. We are able to improve all known such spanners, in terms of either sparsity or stretch. In particular, we consider the sparsest known spanners with stretch 6, 28, and 38, and reduce the stretch to 4, 10, and 14, respectively (while keeping the same sparsity). Our results are based on two different constructions. On one hand, we show how to augment (by adding a small number of edges) a faulttolerant additive sourcewise spanner (that approximately preserves distances only from a given set of source nodes) into one such spanner that preserves all pairwise distances. On the other hand, we show how to augment some known fault-tolerant additive spanners, based on clustering techniques. This way we decrease the additive stretch without any asymptotic increase in their size. We also obtain improved fault-tolerant additive spanners for the case of one vertex failure, and for the case of f edge failures.

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Section: Related Workmentioning

confidence: 99%

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confidence: 99%

Improved Purely Additive Fault-Tolerant Spanners

Bilò

Grandoni

Gualà

et al. 2015

Algorithms - ESA 2015

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“…For a comparison, the sparsest known (2k − 1)-multiplicative ordinary spanner has size O(n 1+1/k ) [1], and this is believed to be asymptotically tight due to the long-standing girth conjecture of Erdős [18]. Finally, we mention that in [3] it was introduced the resembling concept of resilient spanners, i.e., spanners such that whenever any edge in G fails, then the relative distance increases in the spanner are very close to those in G, and it was shown how to build a resilient spanner by augmenting an ordinary spanner.…”

Section: Other Related Resultsmentioning

confidence: 99%

Fault-Tolerant Approximate Shortest-Path Trees

et al. 2017

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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.

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“…For a comparison, the sparsest known (2k − 1)-multiplicative ordinary spanner has size O(n 1+1/k ) [2], and this is believed to be asymptotically tight due to the girth conjecture of Erdős [21]. Then, in [3] it was introduced the resembling concept of 1-EFT resilient spanners, i.e., spanners such that whenever any edge in G fails, then the relative distance increases in the spanner are very close to those in G.…”

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

confidence: 99%