This work establishes the fundamental limits of the classical problem of multi-user distributed computing of linearly separable functions. In particular, we consider a distributed computing setting involving $L$ users, each requesting a linearly separable function over $K$ basis subfunctions from a master node, who is assisted by $N$ distributed servers. At the core of this problem lies a fundamental tradeoff between communication and computation: each server can compute up to $M$ subfunctions, and each server can communicate linear combinations of their locally computed subfunctions outputs to at most $Δ$ users. The objective is to design a distributed computing scheme that reduces the communication cost (total amount of data from servers to users), and towards this, for any given $K$, $L$, $M$, and $Δ$, we propose a distributed computing scheme that jointly designs the task assignment and transmissions, and shows that the scheme achieves optimal performance in the real field under various conditions using a novel converse. We also characterize the performance of the scheme in the finite field using another converse based on counting arguments.

翻译：本文建立了线性可分函数多用户分布式计算这一经典问题的基本极限。具体而言，我们考虑一个包含 $L$ 个用户的分布式计算场景，每个用户向主节点请求基于 $K$ 个基本子函数的线性可分函数，主节点由 $N$ 个分布式服务器协助。该问题的核心在于通信与计算之间的基本权衡：每个服务器最多可计算 $M$ 个子函数，且每个服务器最多可向 $Δ$ 个用户传输其本地计算子函数输出的线性组合。目标是设计一种能降低通信成本（从服务器到用户的总数据量）的分布式计算方案。为此，针对任意给定的 $K$、$L$、$M$ 和 $Δ$，我们提出了一种联合设计任务分配与传输策略的分布式计算方案，并通过新颖的逆定理证明了该方案在实数域下多种条件下达到最优性能。此外，我们基于计数论证的另一逆定理刻画了该方案在有限域下的性能表现。