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

An execution of a distributed algorithm is a game between the algorithm and an adversary causing distractions to the computation. 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 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 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 partial order of maximum anti-chain of size k and complementary 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.