# On the complexity of fault-tolerant consensus

## TL;DR

This paper investigates the complexity of achieving fault-tolerant consensus in distributed systems under various constrained adversarial crash models, proposing efficient algorithms and establishing tight bounds.

## Contribution

It introduces new consensus algorithms tailored for constrained adversaries, analyzes their message complexity, and extends lower bounds to cover weaker adversary models, advancing the understanding of fault-tolerant consensus.

## Key findings

- Proposed time-efficient consensus algorithms for constrained adversaries.
- Improved message complexity bounds for these algorithms.
- Extended lower bounds to weaker adversary models, including partial order constraints.

## Abstract

The paper studies the problem of reaching agreement in a distributed message-passing system prone to crash failures. Crashes are generated by \constrained\ adversaries - a \wadapt\ adversary, who has to fix in advance the set of $f$ crash-prone processes, or a \chainadapt\ adversary, who orders all the processes into $k$ disjoint chains and has to follow this pattern when crashing them. Apart from these constraints, both of them may crash processes in an adaptive way at any time. While commonly used \sadapt\ adversaries model attacks and \noadapt\ ones -- pre-defined faults, the constrained adversaries model more realistic scenarios when there are fault-prone dependent processes, e.g., in hierarchical or dependable software/hardware systems. We propose time-efficient consensus algorithms against such adversaries and also show how to improve the message complexity of proposed solutions. Finally, we show how to reach consensus against a \kthick\ adversary, limited by an arbitrary partial order \dk{with a maximal anti-chain of length $k$}. We complement our algorithmic results with (almost) tight lower bounds, and extend the one for \wadapt\ adversaries to hold also for (syntactically) weaker \noadapt\ adversaries. Together with the consensus algorithm against \wadapt\ adversaries (which automatically translates to \noadapt\ adversaries), these results extend the state-of-the-art of the popular class of \noadapt\ adversaries, in particular the result of Chor, Meritt and Shmoys~\cite{CMS}, and prove general separation between \sadapt\ and the constrained adversaries (including \noadapt) analyzed by Bar-Joseph and Ben-Or~\cite{BB} and others.