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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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