# Partially Optimal Edge Fault-Tolerant Spanners

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.

• 24 publications
• 23 publications
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07/16/2020

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### Fault-Tolerant Spanners against Bounded-Degree Edge Failures: Linearly More Faults, Almost For Free

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### Vertex Fault-Tolerant Emulators

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### Efficient and Simple Algorithms for Fault Tolerant Spanners

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### Epic Fail: Emulators can tolerate polynomially many edge faults for free

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### A Polynomial Time Algorithm for Almost Optimal Vertex Fault Tolerant Spanners

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### 2-Edge-Connectivity and 2-Vertex-Connectivity of an Asynchronous Distributed Network

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