An execution of a distributed algorithm is often seen as a game between the algorithm and a conceptual adversary causing specific distractions to the computation. In this work we define a class of ordered adaptive adversaries, which cause distractions—in particular crashes—online according to some partial order of the participating stations, which is fixed by the adversary before the execution. We distinguish: Linearly-Ordered adversary, restricted by some pre-defined linear order of (potentially) crashing stations; Anti-Chain-Ordered adversary, previously known as the Weakly-Adaptive adversary, which is restricted by some pre-defined set of crash-prone stations (it can be seen as an ordered adversary with the order being an anti-chain, i.e., a collection of incomparable elements, consisting of these stations); k-Thick-Ordered adversary restricted by partial orders of stations with a maximum anti-chain of size k. We initiate a study of how they affect performance of algorithms. For this purpose, we focus on the well-known Do-All problem of performing t tasks by p synchronous crash-prone stations communicating on a shared channel. The channel restricts communication by the fact that no message is delivered to the operational stations if more than one station transmits at the same time. The question addressed in this work is how the ordered adversaries controlling crashes of stations influence work performance, defined as the total number of available processor steps during the whole execution and introduced by Kanellakis and Shvartsman (Distrib Comput 5(4):201–217, 1992) in the context of Write-All algorithms. The first presented algorithm solves the Do-All problem with work $${\mathcal {O}}(t+p \sqrt{t}\log p)$$O(t+ptlogp) against the Linearly-Ordered adversary. Surprisingly, the upper bound on performance of this algorithm does not depend on the number of crashes f and is close to the absolute lower bound $$\varOmega (t+p\sqrt{t})$$Ω(t+pt) proved in Chlebus et al. (Distrib Comput 18(6):435–451, 2006). Another algorithm is developed against the Weakly-Adaptive adversary. Work done by this algorithm is $$\mathcal {O}(t + p\sqrt{t} + p\min \left\{ p/(p-f),t\right\} \log p ),$$O(t+pt+pminp/(p-f),tlogp), which is close to the lower bound $$\varOmega (t + p\sqrt{t} + p\min \left\{ p/(p-f),t\right\} )$$Ω(t+pt+pminp/(p-f),t) proved in [11] and answers the open questions posed there. We generalize this result to the class of k-Thick-Ordered adversaries, in which case the work of the algorithm is bounded by $$\mathcal {O}(t + p\sqrt{t} + p\min \left\{ p/(p-f),k,t\right\} \log p ).$$O(t+pt+pminp/(p-f),k,tlogp). We complement this result by proving the almost matching lower bound $$\begin{aligned} \varOmega (t + p\sqrt{t} + p\min \left\{ p/(p-f),k,t\right\} ). \end{aligned}$$Ω(t+pt+pminp/(p-f),k,t).Independently from the results for the ordered adversaries, we consider a class of delayed adaptive adversaries, which could see random choices with some delay. We present an algorithm that works efficiently against the 1-RD adversary, which could see random choices of stations with one round delay, achieving close to optimal $${\mathcal {O}}(t+p \sqrt{t}\log ^{2} p)$$O(t+ptlog2p) work complexity. This shows that restricting the adversary by not allowing it to react on random decisions immediately makes it significantly weaker, in the sense that there is an algorithm achieving (almost) optimal work performance.

中文翻译：

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有序和延迟的对手以及如何在共享渠道上对抗他们

分布式算法的执行通常被视为算法与概念对手之间的博弈，导致特定的计算分心。在这项工作中，我们定义了一类有序的自适应对手，它们会根据参与站的某些部分顺序在网上引起干扰——尤其是崩溃，这是由对手在执行前固定的。我们区分： 线性排序的对手，受（潜在）碰撞站的一些预定义线性顺序的限制；Anti-Chain-Ordered 敌手，以前称为 Weakly-Adaptive 敌手，它受到一些预定义的一组容易崩溃的站点的限制（它可以被看作是一个有序的敌手，其顺序是一个反链，即，由这些站组成的无与伦比的元素的集合）；k-Thick-Ordered 敌手受最大反链大小为 k 的站点的偏序限制。我们开始研究它们如何影响算法的性能。为此，我们关注众所周知的 Do-All 问题，即通过在共享信道上通信的 p 个同步易崩溃站执行 t 个任务。如果一个以上的站点同时发送消息，则该信道通过以下事实限制通信：没有消息传递到操作站点。这项工作中解决的问题是控制站崩溃的有序对手如何影响工作性能，定义为整个执行期间可用处理器步骤的总数，由 Kanellakis 和 Shvartsman 引入（Distrib Comput 5(4):201–217， 1992）在全写算法的背景下。第一个提出的算法解决了 Do-All 问题，工作 $${\mathcal {O}}(t+p \sqrt{t}\log p)$$O(t+ptlogp) 对抗线性排序的对手。令人惊讶的是，该算法的性能上限不依赖于崩溃次数 f 并且接近绝对下限 $$\varOmega (t+p\sqrt{t})$$Ω(t+pt) 证明在 Chlebus 等人中。(Distribu Comput 18(6):435–451, 2006)。另一种算法是针对弱自适应对手开发的。这个算法做的功是 $$\mathcal {O}(t + p\sqrt{t} + p\min \left\{ p/(pf),t\right\} \log p ),$$O( t+pt+pminp/(pf),tlogp)，接近下界$$\varOmega(t + p\sqrt{t} + p\min \left\{ p/(pf),t\right \} )$$Ω(t+pt+pminp/(pf),t) 在 [11] 中得到证明并回答那里提出的开放性问题。我们将这个结果推广到 k-Thick-Ordered 对手类，在这种情况下，算法的工作受限于 $$\mathcal {O}(t + p\sqrt{t} + p\min \left\{ p/(pf),k,t\right\} \log p ).$$O(t+pt+pminp/(pf),k,tlogp)。我们通过证明几乎匹配的下界 $$\begin{aligned} \varOmega (t + p\sqrt{t} + p\min \left\{ p/(pf),k,t\right\} ）。\end{aligned}$$Ω(t+pt+pminp/(pf),k,t).独立于有序对手的结果，我们考虑一类延迟自适应对手，它可以看到一些延迟的随机选择. 我们提出了一种有效对抗 1-RD 对手的算法，该算法可以看到具有一轮延迟的站点的随机选择，实现接近最优的 $${\mathcal {O}}(t+p \sqrt{t}\log ^{2} p)$$O(t+ptlog2p) 工作复杂度。