Vertex Fault-Tolerant Emulators

TL;DR

This paper introduces vertex fault-tolerant emulators that outperform spanners in size, stretch, and fault-tolerance tradeoffs, providing new bounds and constructions for resilient network approximations.

Contribution

It establishes that vertex fault-tolerant emulators have superior size/stretch/fault-tolerance tradeoffs compared to spanners, with tight bounds and new constructions.

Findings

Fault-tolerant emulators surpass spanners in tradeoffs

Tight lower bounds for certain stretch parameters

Efficient constructions with additive error

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.