Coded distributed computing (CDC), proposed by Li et al., offers significant potential for reducing the communication load in MapReduce computing systems. In the setting of the cascaded CDC that consisting of $K$ nodes, $N$ input files, and $Q$ output functions, the objective is to compute each output function through $s\geq 1$ nodes with a computation load $r\geq 1$, enabling the application of coding techniques during the Shuffle phase to achieve minimum communication load. However, a significant limitation in most existing cascaded CDC schemes is their demand for splitting the original data into an exponentially growing number of input files and requiring an exponentially large number of output functions, which imposes stringent requirements for implementation. In this paper, we focus on the cascaded case of $K/s\in\mathbb{N}$, deliberately designing the strategy of data placement and output functions assignment based on a grouping method, such that a low-complexity Shuffle strategy is achievable. The main advantages of the proposed scheme include: 1) the multicast gains equal to $(r+s-1)(1-1/s)$ and $r+s-1$ which is approximate to $r+s-1$ when $s$ is relatively large, and the communication load is quite approximate to or surprisingly better than the optimal state-of-the-art scheme proposed by Li et al.; 2) the proposed scheme requires significantly less number of input files and output functions; 3) all the operations are implemented over the minimum binary field $\mathbb{F}_2$ in the one-shot fashion. Finally, we derive a new converse bound for the cascaded CDC framework, under the given strategies of data placement and output functions assignment. We demonstrate that the communication load of the proposed scheme is order optimal within a factor of $2$; and is also approximately optimal when $K$ is sufficiently large for a given $r$.

翻译：李等人提出的编码分布式计算（CDC）在降低MapReduce计算系统的通信负载方面展现出巨大潜力。在由$K$个节点、$N$个输入文件和$Q$个输出函数组成的级联CDC场景中，目标是通过$s \geq 1$个节点以计算负载$r \geq 1$计算每个输出函数，从而在Shuffle阶段应用编码技术实现最小通信负载。然而，现有大多数级联CDC方案的一个显著限制是：需将原始数据拆分为指数级增长的输入文件，并要求指数级数量的输出函数，这对实际部署提出了严苛要求。本文针对$K/s \in \mathbb{N}$的级联情形，基于分组方法精心设计数据放置与输出函数分配策略，实现低复杂度Shuffle方案。所提方案的主要优势包括：1）多播增益达到$(r+s-1)(1-1/s)$和$r+s-1$，当$s$较大时趋近于$r+s-1$，通信负载接近甚至显著优于李等人提出的最优现有方案；2）所需输入文件与输出函数数量大幅减少；3）所有操作在最小二进制域$\mathbb{F}_2$上以单次方式实现。最后，本文在给定数据放置与输出函数分配策略下，推导出级联CDC框架的新下界。证明所提方案的通信负载在因子$2$内达到阶最优；且当给定$r$且$K$足够大时接近最优。