A coded distributed computing (CDC) system aims to reduce the communication load in the MapReduce framework. Such a system has $K$ nodes, $N$ input files, and $Q$ Reduce functions. Each input file is mapped by $r$ nodes and each Reduce function is computed by $s$ nodes. The objective is to achieve the maximum multicast gain. There are known CDC schemes that achieve optimal communication load. In some prominent known schemes, however, $N$ and $Q$ grow too fast in terms of $K$, greatly reducing their gains in practical scenarios. To mitigate the situation, some asymptotically optimal cascaded CDC schemes with $r=s$ have been proposed by using symmetric designs. In this paper, we put forward new asymptotically optimal cascaded CDC schemes with $r=s$ by using $1$-designs. Compared with earlier schemes from symmetric designs, ours have much smaller computation loads while keeping the other relevant parameters the same. We also obtain new asymptotically optimal cascaded CDC schemes with more flexible parameters compared with previously best-performing schemes.

翻译：编码分布式计算（CDC）系统旨在降低MapReduce框架中的通信负载。此类系统包含$K$个节点、$N$个输入文件和$Q个Reduce函数$。每个输入文件由$r$个节点映射，每个Reduce函数由$s$个节点计算。其目标是实现最大多播增益。已知存在一些CDC方案能够达到最优通信负载。然而，在部分著名方案中，$N$和$Q$随$K$的增长速度过快，严重降低了它们在实际场景中的增益。为缓解此问题，已有研究者利用对称设计提出了部分满足$r=s$的渐近最优级联CDC方案。本文利用$1$-设计提出了新的满足$r=s$的渐近最优级联CDC方案。与先前基于对称设计的方案相比，本文方案在保持其他相关参数不变的前提下，计算负载显著降低。此外，与先前性能最优方案相比，本文还获得了参数更灵活的渐近最优级联CDC方案。