  This paper formulates a distributed computation problem, where a master asks
$N$ distributed workers to compute a linearly separable function. The task
function can be expressed as $K_c$ linear combinations of $K$ messages, where
each message is a function of one dataset. Our objective is to find the optimal
tradeoff between the computation cost (number of uncoded datasets assigned to
each worker) and the communication cost (number of symbols the master must
download), such that from the answers of any $N_r$ out of $N$ workers the
master can recover the task function with high probability, where the
coefficients of the $K_c$ linear combinations are uniformly i.i.d. over some
large enough finite field. The formulated problem can be seen as a generalized
version of some existing problems, such as distributed gradient coding and
distributed linear transform.
  In this paper, we consider the specific case where the computation cost is
minimum, and propose novel achievability schemes and converse bounds for the
optimal communication cost. Achievability and converse bounds coincide for some
system parameters; when they do not match, we prove that the achievable
distributed computing scheme is optimal under the constraint of a widely used
`cyclic assignment' scheme on the datasets. Our results also show that when $K
= N$, with the same communication cost as the optimal distributed gradient
coding scheme proposed by Tandon et al. from which the master recovers one
linear combination of $K$ messages, our proposed scheme can let the master
recover any additional $N_r - 1$ independent linear combinations of messages
with high probability.
