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Fault-Tolerant Spanners: Better and Simpler

Abstract

A natural requirement of many distributed structures is fault-tolerance: after some failures, whatever remains from the structure should still be effective for whatever remains from the network. In this paper we examine spanners of general graphs that are tolerant to vertex failures, and significantly improve their dependence on the number of faults rr, for all stretch bounds. For stretch kβ‰₯3k \geq 3 we design a simple transformation that converts every kk-spanner construction with at most f(n)f(n) edges into an rr-fault-tolerant kk-spanner construction with at most O(r3log⁑n)β‹…f(2n/r)O(r^3 \log n) \cdot f(2n/r) edges. Applying this to standard greedy spanner constructions gives rr-fault tolerant kk-spanners with O~(r2n1+2k+1)\tilde O(r^{2} n^{1+\frac{2}{k+1}}) edges. The previous construction by Chechik, Langberg, Peleg, and Roddity [STOC 2009] depends similarly on nn but exponentially on rr (approximately like krk^r). For the case k=2k=2 and unit-length edges, an O(rlog⁑n)O(r \log n)-approximation algorithm is known from recent work of Dinitz and Krauthgamer [arXiv 2010], where several spanner results are obtained using a common approach of rounding a natural flow-based linear programming relaxation. Here we use a different (stronger) LP relaxation and improve the approximation ratio to O(log⁑n)O(\log n), which is, notably, independent of the number of faults rr. We further strengthen this bound in terms of the maximum degree by using the \Lovasz Local Lemma. Finally, we show that most of our constructions are inherently local by designing equivalent distributed algorithms in the LOCAL model of distributed computation.Comment: 17 page

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