The distributed linearly separable computation problem finds extensive applications across domains such as distributed gradient coding, distributed linear transform, real-time rendering, etc. In this paper, we investigate this problem in a fully decentralized scenario, where $\mathsf{N}$ workers collaboratively perform the computation task without a central master. Each worker aims to compute a linearly separable computation that can be manifested as $\mathsf{K}_{\mathrm{c}}$ linear combinations of $\mathsf{K}$ messages, where each message is a function of a distinct dataset. We require that each worker successfully fulfill the task based on the transmissions from any $\mathsf{N}_{\mathrm{r}}$ workers, such that the system can tolerate any $\mathsf{N}-\mathsf{N}_{\mathrm{r}}$ stragglers. We focus on the scenario where the computation cost (the number of uncoded datasets assigned to each worker) is minimum, and aim to minimize the communication cost (the number of symbols the fastest $\mathsf{N}_{\mathrm{r}}$ workers transmit). We propose a novel distributed computing scheme that is optimal under the widely used cyclic data assignment. Interestingly, we demonstrate that the side information at each worker is ineffective in reducing the communication cost when $\mathsf{K}_{\mathrm{c}}\leq {\mathsf{K}}\mathsf{N}_{\mathrm{r}}/{\mathsf{N}}$, while it helps reduce the communication cost as $\mathsf{K}_{\mathrm{c}}$ increases.

翻译：[译摘要] 分布式线性可分计算问题广泛应用于分布式梯度编码、分布式线性变换、实时渲染等领域。本文在全去中心化场景中研究该问题，其中$\mathsf{N}$个工作者在没有中央主节点的情况下协作完成计算任务。每个工作者旨在计算一个可表示为$\mathsf{K}$个消息的$\mathsf{K}_{\mathrm{c}}$个线性组合的线性可分计算，其中每个消息是不同数据集的函数。我们要求每个工作者基于任意$\mathsf{N}_{\mathrm{r}}$个工作者的传输成功完成任务，使系统能容忍任意$\mathsf{N}-\mathsf{N}_{\mathrm{r}}$个掉队者。我们聚焦于计算成本（分配给每个工作者的未编码数据集数量）最小的场景，并致力于最小化通信成本（最快的$\mathsf{N}_{\mathrm{r}}$个工作者传输的符号数）。我们提出了一种新颖的分布式计算方案，该方案在广泛使用的循环数据分配下达到最优。有趣的是，我们证明当$\mathsf{K}_{\mathrm{c}}\leq {\mathsf{K}}\mathsf{N}_{\mathrm{r}}/{\mathsf{N}}$时，每个工作者的边信息在降低通信成本方面无效，而随着$\mathsf{K}_{\mathrm{c}}$增加，边信息有助于降低通信成本。