  This paper studies the distributed linearly separable computation problem,
which is a generalization of many existing distributed computing problems such
as distributed gradient descent and distributed linear transform. In this
problem, a master asks $N$ distributed workers to compute a linearly separable
function of $K$ datasets, which is a set of $K_c$ linear combinations of $K$
messages (each message is a function of one dataset). We assign some datasets
to each worker, which then computes the corresponding messages and returns some
function of these messages, such that from the answers of any $N_r$ out of $N$
workers the master can recover the task function. In the literature, the
specific case where $K_c = 1$ or where the computation cost is minimum has been
considered. In this paper, we focus on the general case (i.e., general $K_c$
and general computation cost) and aim to find the minimum communication cost.
We first propose a novel converse bound on the communication cost under the
constraint of the popular cyclic assignment (widely considered in the
literature), which assigns the datasets to the workers in a cyclic way.
Motivated by the observation that existing strategies for distributed computing
fall short of achieving the converse bound, we propose a novel distributed
computing scheme for some system parameters. The proposed computing scheme is
optimal for any assignment when $K_c$ is large and is optimal under cyclic
assignment when the numbers of workers and datasets are equal or $K_c$ is
small. In addition, it is order optimal within a factor of 2 under cyclic
assignment for the remaining cases.
