Supporting multiple partial computations efficiently at each of the workers is a keystone in distributed coded computing in order to speed up computations and to fully exploit the resources of heterogeneous workers in terms of communication, storage, or computation capabilities. Multivariate polynomial coding schemes have recently been shown to deliver faster results for distributed matrix-matrix multiplication compared to conventional univariate polynomial coding schemes by supporting multiple partial coded computations at each worker at reduced communication costs. In this work, we extend multivariate coding schemes to also support arbitrary matrix partitions. Generalized matrix partitions have been proved useful to trade-off between computation speed and communication costs in distributed (univariate) coded computing. We first formulate the computation latency-communication trade-off in terms of the computation complexity and communication overheads required by coded computing approaches as compared to a single server uncoded computing system. Then, we propose two novel multivariate coded computing schemes supporting arbitrary matrix partitions. The proposed schemes are shown to improve the studied trade-off as compared to univariate schemes.

翻译：在分布式编码计算中，高效支持每个工作节点执行多个部分计算是加速计算过程、充分利用异构工作节点在通信、存储或计算能力方面资源的关键。与传统的单变量多项式编码方案相比，多元多项式编码方案通过在每个工作节点支持多个部分编码计算并降低通信成本，已被证明能为分布式矩阵乘法提供更快的计算结果。本研究将多元编码方案扩展至支持任意矩阵划分。广义矩阵划分已被证明有助于在分布式（单变量）编码计算中权衡计算速度与通信成本。我们首先从计算复杂度和通信开销的角度，将计算延迟与通信的权衡关系形式化，并与单服务器非编码计算系统进行比较。随后，我们提出了两种支持任意矩阵划分的新型多元编码计算方案。研究表明，与单变量方案相比，所提方案能改善所研究的权衡关系。