Coded distributed computing (CDC) introduced by Li \emph{et al.} can greatly reduce the communication load for MapReduce computing systems. In the general cascaded CDC with $K$ workers, $N$ input files and $Q$ Reduce functions, each input file will be mapped by $r$ workers and each Reduce function will be computed by $s$ workers such that coding techniques can be applied to achieve the maximum multicast gain. The main drawback of most existing CDC schemes is that they require the original data to be split into a large number of input files that grows exponentially with $K$, which can significantly increase the coding complexity and degrade system performance. In this paper, we first use a classic combinatorial structure $t$-design, for any integer $t\geq 2$, to develop a low-complexity and asymptotically optimal CDC with $r=s$. The main advantages of our scheme via $t$-design are two-fold: 1) having much smaller $N$ and $Q$ than the existing schemes under the same parameters $K$, $r$ and $s$; and 2) achieving smaller communication loads compared with the state-of-the-art schemes. Remarkably, unlike the previous schemes that realize on large operation fields, our scheme operates on the minimum binary field $\mathbb{F}_2$. Furthermore, we show that our construction method can incorporate the other combinatorial structures that have a similar property to $t$-design. For instance, we use $t$-GDD to obtain another asymptotically optimal CDC scheme over $\mathbb{F}_2$ that has different parameters from $t$-design. Finally, we show that our construction method can also be used to construct CDC schemes with $r\neq s$ that have small file number and Reduce function number.

翻译：Li等人提出的编码分布式计算（CDC）可大幅降低MapReduce计算系统的通信负载。在含$K$个工作节点、$N$个输入文件和$Q$个Reduce函数的通用级联CDC中，每个输入文件由$r$个工作节点进行映射，每个Reduce函数由$s$个工作节点进行计算，从而可应用编码技术实现最大多播增益。现有大多数CDC方案的主要缺陷在于：原始数据需被拆分为随$K$呈指数增长的巨量输入文件，这将显著增加编码复杂度并降低系统性能。本文首先采用经典组合结构$t$-设计（$t\geq 2$为任意整数），构建了满足$r=s$的低复杂度渐近最优CDC方案。本方案通过$t$-设计具有双重优势：1）在相同参数$K$、$r$、$s$下，所需$N$和$Q$远小于现有方案；2）与当前最优方法相比可实现更低的通信负载。值得关注的是，我们的方案在最小二进制域$\mathbb{F}_2$上运行，而现有方案需基于大操作域实现。进一步研究表明，本文构建方法可兼容其他具有类似$t$-设计特性的组合结构。例如，我们利用$t$-GDD构造了另一个在$\mathbb{F}_2$上具有与$t$-设计不同参数的渐近最优CDC方案。最后，我们证明该方法还可用于构建$r\neq s$且具有较小文件数与Reduce函数数的CDC方案。