Vertex Fault-Tolerant Emulators

Greg Bodwin; Michael Dinitz; Yasamin Nazari

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arXiv · 2109.08042

Vertex Fault-Tolerant Emulators

Greg Bodwin·Michael Dinitz·Yasamin Nazari

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Abstract

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.

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BibTeX RIS

Greg Bodwin, Michael Dinitz, Yasamin Nazari. 2021-09-16. Vertex Fault-Tolerant Emulators. https://arxiv.org/abs/2109.08042

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