arXiv 2102.072212021-02-14cs.DC

拥塞Clique中的近优调度

Near-Optimal Scheduling in the Congested Clique

Keren Censor-Hillel, Yannic Maus, Volodymyr Polosukhin

备注：: To appear in SIROCCO 2021

摘要：

本文提出了三种近似最优的算法调度$\mathsf{CLIQUE}$模型中的$t$作业。首先，我们提出了一个确定性的调度算法，运行在$O(\mathsf{GlobalCongestion} + \mathsf{dilation})$轮的工作，是足够有效的内存。$\mathsf{dilation}$是任何给定作业的最大轮复杂度，而$\mathsf{GlobalCongestion}$是所有作业中的消息总数除以$\mathsf{CLIQUE}$模型的$n^2$的每轮带宽。两者都是任何调度算法的固有下限。然后，我们提出了一种随机调度算法，该算法在$O(\mathsf{GlobalCongestion} + \mathsf{dilation}\cdot\log{n}+t)$轮中运行$t$作业，并且仅要求输入和输出不超过每个节点的$O(n\log n)$位，例如，几乎所有的图形问题。最后，我们从$\mathsf{CLIQUE}$模型调整\emph{random-delay-based}调度算法[Ghaffari，PODC'15]，并获得在$O(t / n + \mathsf{LocalCongestion} + \mathsf{dilation}\cdot\log{n})$轮中调度任何$t$作业的算法，其中$\mathsf{LocalCongestion}$与$\mathsf{CLIQUE}$的单个节点处的拥塞有关。我们将此算法与以前的方法进行比较，并显示其优点。我们调度的一组工作的飞行，没有先验知识的参数或通信模式的工作。根据固有的下限，我们所有的算法都是接近最优的。我们通过分析最先进的MIS协议[Ghaffari，Gouleakis，Konrad，Mitrovic和Rubinfeld，PODC'18]的消息复杂性来验证我们算法的能力，并且我们表明我们可以在$O(t + \log\logΔ\log{n})$轮中解决MIS的$t$实例，即在$O(1)$分摊时间内，对于$t\geq \log\logΔ\log{n}$。

英文摘要：

This paper provides three nearly-optimal algorithms for scheduling $t$ jobs in the $\mathsf{CLIQUE}$ model. First, we present a deterministic scheduling algorithm that runs in $O(\mathsf{GlobalCongestion} + \mathsf{dilation})$ rounds for jobs that are sufficiently efficient in terms of their memory. The $\mathsf{dilation}$ is the maximum round complexity of any of the given jobs, and the $\mathsf{GlobalCongestion}$ is the total number of messages in all jobs divided by the per-round bandwidth of $n^2$ of the $\mathsf{CLIQUE}$ model. Both are inherent lower bounds for any scheduling algorithm. Then, we present a randomized scheduling algorithm which runs $t$ jobs in $O(\mathsf{GlobalCongestion} + \mathsf{dilation}\cdot\log{n}+t)$ rounds and only requires that inputs and outputs do not exceed $O(n\log n)$ bits per node, which is met by, e.g., almost all graph problems. Lastly, we adjust the \emph{random-delay-based} scheduling algorithm [Ghaffari, PODC'15] from the $\mathsf{CLIQUE}$ model and obtain an algorithm that schedules any $t$ jobs in $O(t / n + \mathsf{LocalCongestion} + \mathsf{dilation}\cdot\log{n})$ rounds, where the $\mathsf{LocalCongestion}$ relates to the congestion at a single node of the $\mathsf{CLIQUE}$. We compare this algorithm to the previous approaches and show their benefit. We schedule the set of jobs on-the-fly, without a priori knowledge of its parameters or the communication patterns of the jobs. In light of the inherent lower bounds, all of our algorithms are nearly-optimal. We exemplify the power of our algorithms by analyzing the message complexity of the state-of-the-art MIS protocol [Ghaffari, Gouleakis, Konrad, Mitrovic and Rubinfeld, PODC'18], and we show that we can solve $t$ instances of MIS in $O(t + \log\logΔ\log{n})$ rounds, that is, in $O(1)$ amortized time, for $t\geq \log\logΔ\log{n}$.

机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。

摘要：

英文摘要：

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