Chapter

# Partially Optimal Edge Fault-Tolerant Spanners

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... The technical details of this analysis get surprisingly tricky, and it turns out that we actually cannot consider all k-paths in the algorithm and analysis outlined above, but only a carefully selected subset of them that we call "SALAD" paths. The technical details of these paths are responsible for the exp(k) factors and the even/odd k distinction (a similar even/odd distinction appears in [12], for a similar technical reason). ...
... Hence E[σ ′′ ] ≤ σ d k+1 . Our second inequality needs the following lemma, which was proved implicitly in [12] (it is a generalization of the "intermediate counting lemma" of [12]; we include the proof here for completeness). Lemma 3.9. ...
... Hence E[σ ′′ ] ≤ σ d k+1 . Our second inequality needs the following lemma, which was proved implicitly in [12] (it is a generalization of the "intermediate counting lemma" of [12]; we include the proof here for completeness). Lemma 3.9. ...
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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.
... We note even the (fault-free) Baswana-Sen algorithm provides (2k − 1) spanner with O(kn 1+1/k ) edges. Providing optimal algorithms for optimal EFT-spanners is also a major open problem, in light of the recent work by Bodwin, Dinitz and Robelle [BDR21b]. ...
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We (nearly) settle the time complexity for computing vertex fault-tolerant (VFT) spanners with optimal sparsity (up to polylogarithmic factors). VFT spanners are sparse subgraphs that preserve distance information, up to a small multiplicative stretch, in the presence of vertex failures. These structures were introduced by [Chechik et al., STOC 2009] and have received a lot of attention since then. We provide algorithms for computing nearly optimal $f$-VFT spanners for any $n$-vertex $m$-edge graph, with near optimal running time in several computational models: - A randomized sequential algorithm with a runtime of $\widetilde{O}(m)$ (i.e., independent in the number of faults $f$). The state-of-the-art time bound is $\widetilde{O}(f^{1-1/k}\cdot n^{2+1/k}+f^2 m)$ by [Bodwin, Dinitz and Robelle, SODA 2021]. - A distributed congest algorithm of $\widetilde{O}(1)$ rounds. Improving upon [Dinitz and Robelle, PODC 2020] that obtained FT spanners with near-optimal sparsity in $\widetilde{O}(f^{2})$ rounds. - A PRAM (CRCW) algorithm with $\widetilde{O}(m)$ work and $\widetilde{O}(1)$ depth. Prior bounds implied by [Dinitz and Krauthgamer, PODC 2011] obtained sub-optimal FT spanners using $\widetilde{O}(f^3m)$ work and $\widetilde{O}(f^3)$ depth. An immediate corollary provides the first nearly-optimal PRAM algorithm for computing nearly optimal $\lambda$-\emph{vertex} connectivity certificates using polylogarithmic depth and near-linear work. This improves the state-of-the-art parallel bounds of $\widetilde{O}(1)$ depth and $O(\lambda m)$ work, by [Karger and Motwani, STOC'93].
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This paper concerns eecient broadcast in communication networks, where eeciency is measured with respect to both the total communication cost and the speed of message delivery at all destinations. It is shown that this task can be performed eciently using sparse, light w eight, low stretch spanners. It is then shown how to eciently construct a family of spanners that are near optimal with respect to these three properties. Previous constructions were at least (p n) a way from the lower bounds in at least one of the parameters, and required O(n 2) initialization cost. In comparison, the parameters of our construction as well as its initialization cost are within poly-logarithmic factors from the corresponding lower bounds.
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This paper concerns graph spanners that are resistant to vertex or edge failures. In the failure-free setting it is known how to efficiently construct a (2k-1)-spanner of size O(n 1+1/k ), and this size-stretch trade-off is conjectured to be tight. The notion of fault tolerant spanners was introduced a decade ago in the geometric setting by C. Levcopoulos, G. Narasimhan and M. Smid [in: STOC ’98. Proceedings of the 30th annual ACM symposium on theory of computing, Dallas, TX, USA, May 23-26, 1998. New York, NY: ACM, Association for Computing Machinery. 186–195 (1998; Zbl 1027.68978)]. A subgraph H is an f-vertex fault tolerant k-spanner of the graph G if for any set F⊆V of size at most f and any pair of vertices u,v∈V∖F, the distances in H satisfy δ H∖F (u,v)≤k·δ G∖F (u,v). A fault tolerant geometric spanner with optimal maximum degree and total weight was presented in [A. Czumaj and H. Zhao, Discrete Comput. Geom. 32, No. 2, 207–230 (2004; Zbl 1095.68125)]. This paper also raised as an open problem the question of whether it is possible to obtain a fault tolerant spanner for an arbitrary undirected weighted graph. The current paper answers this question in the affirmative, presenting an f-vertex fault tolerant (2k-1)-spanner of size O(f 2 k f+1 ·n 1+1/k log 1-1/k n). Interestingly, the stretch of the spanner remains unchanged, while the size of the spanner increases only by a factor that depends on the stretch k, on the number of potential faults f, and on logarithmic terms in n. In addition, we consider the simpler setting of f-edge fault tolerant spanners (defined analogously). We present an f-edge fault tolerant (2k-1)-spanner with edge set of size O(f·n 1+1/k ) (only f times larger than standard spanners). For both edge and vertex faults, our results are shown to hold when the given graph G is weighted.
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