  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.
