Jaroslaw Mirek | Searcharxiv

arXiv2019

On the complexity of fault-tolerant consensus

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.

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arXiv2018

Ordered and Delayed Adversaries and How to Work against Them on a Shared Channel

In this work we define a class of ordered adversaries causing distractions according to some partial order fixed by the adversary before the execution, and study how they affect performance of algorithms. We focus on the Do-All problem of performing t tasks on a shared channel consisting of p crash-prone stations. The channel restricts communication: no message is delivered to the alive stations if more than one station transmits at the same time. The performance measure for the Do-All problem is work: the total number of available processor steps during the whole execution. We address the question of how the ordered adversaries controlling crashes of stations influence work performance of Do-All algorithms. The first presented algorithm solves Do-All with work O(t+p\sqrt{t}\log p) against the Linearly-Ordered adversary, restricted by some pre-defined linear order of crashing stations. Another algorithm runs against the Weakly-Adaptive adversary, restricted by some pre-defined set of f crash-prone stations (it can be seen as restricted by the order being an anti-chain of crashing stations). The work done by this algorithm is O(t+p\sqrt{t}+p\min{p/(p-f),t}\log p). Both results are close to the corresponding lower bounds from [CKL]. We generalize this result to the class of adversaries restricted by a partial order with a maximum anti-chain of size k and complement with the lower bound. We also consider a class of delayed adaptive adversaries, who could see random choices with some delay. We give an algorithm that runs against the 1-RD adversary (seeing random choices of stations with one round delay), achieving close to optimal O(t+p\sqrt{t}\log^2 p) work complexity. This shows that restricting adversary by even 1 round delay results in (almost) optimal work on a shared channel.

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