  Gradient coding allows a master node to derive the aggregate of the partial
gradients, calculated by some worker nodes over the local data sets, with
minimum communication cost, and in the presence of stragglers. In this paper,
for gradient coding with linear encoding, we characterize the optimum
communication cost for heterogeneous distributed systems with \emph{arbitrary}
data placement, with $s \in \mathbb{N}$ stragglers and $a \in \mathbb{N}$
adversarial nodes. In particular, we show that the optimum communication cost,
normalized by the size of the gradient vectors, is equal to $(r-s-2a)^{-1}$,
where $r \in \mathbb{N}$ is the minimum number that a data partition is
replicated. In other words, the communication cost is determined by the data
partition with the minimum replication, irrespective of the structure of the
placement. The proposed achievable scheme also allows us to target the
computation of a polynomial function of the aggregated gradient matrix. It also
allows us to borrow some ideas from approximation computing and propose an
approximate gradient coding scheme for the cases when the repetition in data
placement is smaller than what is needed to meet the restriction imposed on
communication cost or when the number of stragglers appears to be more than the
presumed value in the system design.
