Õptimal Vertex Fault-Tolerant Spanners in Õptimal Time: Sequential, Distributed and Parallel
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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 O(m) (i.e., independent in the number of faults f). The state-of-the-art time bound is O(f^1-1/k· n^2+1/k+f^2 m) by [Bodwin, Dinitz and Robelle, SODA 2021]. - A distributed congest algorithm of O(1) rounds. Improving upon [Dinitz and Robelle, PODC 2020] that obtained FT spanners with near-optimal sparsity in O(f^2) rounds. - A PRAM (CRCW) algorithm with O(m) work and O(1) depth. Prior bounds implied by [Dinitz and Krauthgamer, PODC 2011] obtained sub-optimal FT spanners using O(f^3m) work and O(f^3) depth. An immediate corollary provides the first nearly-optimal PRAM algorithm for computing nearly optimal λ-vertex connectivity certificates using polylogarithmic depth and near-linear work. This improves the state-of-the-art parallel bounds of O(1) depth and O(λ m) work, by [Karger and Motwani, STOC'93].
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