In this paper, we investigate the problem of multi-user linearly decomposable function computation, where $N$ servers help compute functions for $K$ users, and where each such function can be expressed as a linear combination of $L$ basis subfunctions. The process begins with each server computing some of the subfunctions, then broadcasting a linear combination of its computed outputs to a selected group of users, and finally having each user linearly combine its received data to recover its function. As it has become recently known, this problem can be translated into a matrix decomposition problem $\mathbf{F}=\mathbf{D}\mathbf{E}$, where $\mathbf{F} \in \mathbf{GF}(q)^{K \times L}$ describes the coefficients that define the users' demands, where $\mathbf{E} \in \mathbf{GF}(q)^{N \times L}$ describes which subfunction each server computes and how it combines the computed outputs, and where $\mathbf{D} \in \mathbf{GF}(q)^{K \times N}$ describes which servers each user receives data from and how it combines this data. Our interest here is in reducing the total number of subfunction computations across the servers (cumulative computational cost), as well as the worst-case load which can be a measure of computational delay. Our contribution consists of novel bounds on the two computing costs, where these bounds are linked here to the covering and packing radius of classical codes. One of our findings is that in certain cases, our distributed computing problem -- and by extension our matrix decomposition problem -- is treated optimally when $\mathbf{F}$ is decomposed into a parity check matrix $\mathbf{D}$ of a perfect code, and a matrix $\mathbf{E}$ which has as columns the coset leaders of this same code.

翻译：本文研究了多用户线性可分解函数计算问题，其中$N$台服务器为$K$个用户计算函数，每个函数可表示为$L个基子函数的线性组合。计算过程始于每台服务器计算部分子函数，然后向选定用户群广播其计算输出的线性组合，最后每个用户线性组合接收数据以恢复其函数。如近期所知，该问题可转化为矩阵分解问题$\mathbf{F}=\mathbf{D}\mathbf{E}$，其中$\mathbf{F} \in \mathbf{GF}(q)^{K \times L}$描述用户需求系数，$\mathbf{E} \in \mathbf{GF}(q)^{N \times L}$描述每台服务器计算的子函数及其输出组合方式，$\mathbf{D} \in \mathbf{GF}(q)^{K \times N}$描述每个用户接收数据的服务器及其数据组合方式。本文关注降低服务器间子函数计算总数（累积计算成本）以及作为计算延迟度量的最坏负载。我们的贡献包括两个计算成本的新边界，这些边界与经典码的覆盖半径和填充半径相关联。研究发现，在某些情况下，当$\mathbf{F}$分解为完美码的奇偶校验矩阵$\mathbf{D}$和以该码陪集首为列的矩阵$\mathbf{E}$时，分布式计算问题（进而矩阵分解问题）可得到最优处理。