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 $\Omega(f^{\frac12 - \frac{1}{2k}} n^{1+1/k} +fn)$. 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 $O_k(f^{\frac12 - \frac{1}{2k}} n^{1+1/k} + fn)$, 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 $k$.

中文翻译：

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部分最佳边缘容错扳手

最近的工作已经确定，对于每个正整数$ k $，每个$ n $-节点图的$ O（f ^ {1-1 / k} n ^ {1+ 1 / k}）$边缘可抵抗$ f $边缘或顶点错误。对于顶点错误，此边界很严格。但是，对边缘故障的情况了解得还不够：一般$ k $的最著名下限是$ \ Omega（f ^ {\ frac12-\ frac {1} {2k}} n ^ {1 + 1 / k} + fn）$。我们的主要结果是用改善的上限几乎弥合了这个间隙，从而分离了边缘和顶点断层的情况。对于奇数$ k $，我们的新上限是$ O_k（f ^ {\ frac12-\ frac {1} {2k}} n ^ {1 + 1 / k} + fn）$，这很接近隐藏的$ poly（k）$因素。对于$ k $，我们的新上限是$ O_k（f ^ {1/2} n ^ {1 + 1 / k} + fn）$，这留下了$ poly（k）f ^ {1 / （2k）} $。我们的证明是对容错贪婪算法的分析，该算法需要指数时间，