# Optimal Vertex Fault Tolerant Spanners (for fixed stretch)

@article{Bodwin2017OptimalVF, title={Optimal Vertex Fault Tolerant Spanners (for fixed stretch)}, author={Gregory Bodwin and Michael Dinitz and Merav Parter and Virginia Vassilevska Williams}, journal={ArXiv}, year={2017}, volume={abs/1710.03164} }

A $k$-spanner of a graph $G$ is a sparse subgraph $H$ whose shortest path distances match those of $G$ up to a multiplicative error $k$. In this paper we study spanners that are resistant to faults. A subgraph $H \subseteq G$ is an $f$ vertex fault tolerant (VFT) $k$-spanner if $H \setminus F$ is a $k$-spanner of $G \setminus F$ for any small set $F$ of $f$ vertices that might "fail." One of the main questions in the area is: what is the minimum size of an $f$ fault tolerant $k$-spanner that…

## 24 Citations

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A unified and simplified approach to derandomize central results in the area of fault-tolerant graph algorithms by presenting efficient deterministic constructions of $(L,f)-RPCs whose covering values almost match the randomized ones, for a wide range of parameters.

### Optimal Vertex Fault-Tolerant Spanners in Polynomial Time

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The first algorithm that produces vertex fault tolerant spanners of optimal size and which runs in polynomial time is given, which reflects an exponential improvement in runtime over [Bodwin-Patel PODC '19], the only previously known algorithm for constructing optimal vertex fault-tolerance spanners.

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

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The first polynomial time algorithm for the f vertex fault tolerant spanner problem is presented, which achieves almost optimal spanner size and is at most a $\log n$ factor away from the upper bound on the worst-case size.

### Vertex Fault-Tolerant Emulators

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A natural definition of vertex fault-tolerant emulators is introduced, and a three-way tradeoff between size, stretch, and fault-Tolerance for these emulators that polynomially surpasses the tradeoff known to be optimal for spanners.

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A natural definition of vertex fault-tolerant emulators is introduced, and a three-way tradeoff between size, stretch, and fault-Tolerance for these emulators that polynomially surpasses the tradeoff known to be optimal for spanners is shown.

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A surprisingly simple algorithm is given which runs in polynomial time and constructs fault-tolerant spanners that are extremely close to optimal (off by only a linear factor in the stretch) by modifying the greedy algorithm to run in poynomial time.

### Nearly optimal vertex fault-tolerant spanners in optimal time: sequential, distributed, and parallel

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The time complexity for computing vertex fault-tolerant (VFT) spanners with optimal sparsity (up to polylogarithmic factors) is settled, with near optimal running time in several computational models.

### Partially Optimal Edge Fault-Tolerant Spanners

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This paper shows that there is a polynomial-time algorithm which creates edge fault tolerant spanners that are larger only by factors of k, and an analysis of the fault-tolerant greedy algorithm, which requires exponential time.

### Approximating Spanners and Directed Steiner Forest

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This work gives an O(n3/5 + ε)-approximation for distance preservers and pairwise spanners and proves Label Cover hardness for approximating additive spanners, even for the cases of additive 1 stretch.

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