A $k$-spanner of a graph $G$ is a sparse subgraph that preserves its shortest path distances up to a multiplicative stretch factor of $k$, and a $k$-emulator is similar but not required to be a subgraph of $G$. A classic theorem by Thorup and Zwick [JACM '05] shows that, despite the extra flexibility available to emulators, the size/stretch tradeoffs for spanners and emulators are equivalent. Our main result is that this equivalence in tradeoffs no longer holds in the commonly-studied setting of graphs with vertex failures. That is: we introduce a natural definition of vertex fault-tolerant emulators, and then we show a three-way tradeoff between size, stretch, and fault-tolerance for these emulators that polynomially surpasses the tradeoff known to be optimal for spanners. We complement our emulator upper bound with a lower bound construction that is essentially tight (within $\log n$ factors of the upper bound) when the stretch is $2k-1$ and $k$ is either a fixed odd integer or $2$. We also show constructions of fault-tolerant emulators with additive error, demonstrating that these also enjoy significantly improved tradeoffs over those available for fault-tolerant additive spanners.

中文翻译：

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Vertex 容错仿真器

图 $G$ 的 $k$-spanner 是一个稀疏子图，在乘法拉伸因子 $k$ 内保持其最短路径距离，$k$-模拟器类似但不需要是$G$。Thorup 和 Zwick 的经典定理 [JACM '05] 表明，尽管模拟器具有额外的灵活性，但扳手和模拟器的大小/拉伸权衡是等效的。我们的主要结果是，这种权衡的等价性不再适用于普遍研究的具有顶点故障的图设置。也就是说：我们引入了顶点容错仿真器的自然定义，然后我们展示了这些仿真器的大小、拉伸和容错之间的三向权衡，这些权衡在多项式上超过了已知对扳手最佳的权衡。当拉伸为 $2k-1$ 且 $k$ 是一个固定的奇数整数或 $2$ 时，我们用一个本质上紧的下界构造来补充我们的模拟器上限（在上限的 $\log n$ 个因子内） . 我们还展示了具有附加错误的容错仿真器的构造，证明与容错附加扳手相比，这些也具有显着改善的权衡。