Partially Optimal Edge Fault-Tolerant Spanners
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Recent work has established that, for every positive integer k, every n-node graph has a (2k-1)-spanner on O(f^1-1/k n^1+1/k) edges that is resilient to f edge or vertex faults. For vertex faults, this bound is tight. However, the case of edge faults is not as well understood: the best known lower bound for general k is Ω(f^1/2 - 1/2k n^1+1/k +fn). Our main result is to nearly close this gap with an improved upper bound, thus separating the cases of edge and vertex faults. For odd k, our new upper bound is O_k(f^1/2 - 1/2k n^1+1/k + fn), which is tight up to hidden poly(k) factors. For even k, our new upper bound is O_k(f^1/2 n^1+1/k +fn), which leaves a gap of poly(k) f^1/(2k). Our proof is an analysis of the fault-tolerant greedy algorithm, which requires exponential time, but we also show that there is a polynomial-time algorithm which creates edge fault tolerant spanners that are larger only by factors of k.
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