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

## TL;DR

This paper investigates how ordered and delayed adversaries impact the efficiency of algorithms solving the Do-All problem on shared channels, proposing algorithms with near-optimal work bounds under various adversary models.

## Contribution

The paper introduces new algorithms that achieve near-optimal work bounds against ordered and delayed adaptive adversaries, extending understanding of adversarial impact on shared channel algorithms.

## Key findings

- Algorithms achieve near-optimal work bounds against ordered adversaries.
- Restricting adversaries by delay significantly improves algorithm performance.
- Generalization to partial order adversaries with bounded anti-chain size.

## Abstract

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.