This work addresses the problem of distributed computation of linearly separable functions, where a master node with access to $K$ datasets, employs $N$ servers to compute $L$ user-requested functions, each defined over the datasets. Servers are instructed to compute subfunctions of the datasets and must communicate computed outputs to the user, who reconstructs the requested outputs. The central challenge is to reduce the per-server computational load and the communication cost from servers to the user, while ensuring recovery for any possible set of $L$ demanded functions. We here establish the fundamental communication-computation tradeoffs for arbitrary $K$ and $L$, through novel task-assignment and communication strategies that, under the linear-encoding and no-subpacketization assumptions, are proven to be either exactly optimal or within a factor of three from the optimum. In contrast to prior approaches that relied on fixed assignments of tasks -- either disjoint or cyclic assignments -- our key innovation is a nullspace-based design that jointly governs task assignment and server transmissions, ensuring exact decodability for all demands, and attaining optimality over all assignment and delivery methods. To prove this optimality, we here uncover a duality between nullspaces and sparse matrix factorizations, enabling us to recast the distributed computing problem as an equivalent factorization task and derive a sharp information-theoretic converse bound. Building on this, we establish an additional converse that, for the first time, links the communication cost to the covering number from the theory of general covering designs.

翻译：本文研究线性可分函数的分布式计算问题，其中主节点可访问$K$个数据集，并利用$N$个服务器计算$L$个用户请求的函数（每个函数定义在这些数据集上）。服务器被要求计算数据集的子函数，并将计算结果传输给用户，用户据此重构所需输出。核心挑战在于降低每个服务器的计算负载及服务器到用户的通信开销，同时确保对任意可能的$L$个需求函数集合均可实现恢复。本文针对任意$K$和$L$建立了基本的通信-计算权衡关系，通过创新的任务分配与通信策略——在线性编码与无子分组假设下——证明其要么严格最优，要么与最优解相差不超过三倍。与以往依赖固定任务分配（不相交分配或循环分配）的方法不同，本研究的核心创新是基于零空间的设计，该设计联合调控任务分配与服务器传输，确保对所有需求实现精确可解码性，并在所有分配与传输方法中达到最优性。为证明此最优性，本文揭示了零空间与稀疏矩阵分解之间的对偶关系，从而将分布式计算问题重构为等价的分解任务，并推导出严格的信息论逆界。在此基础上，我们建立了新的逆界，首次将通信开销与广义覆盖设计理论中的覆盖数联系起来。