  We consider the problem of massive matrix multiplication, which underlies
many data analytic applications, in a large-scale distributed system comprising
a group of worker nodes. We target the stragglers' delay performance
bottleneck, which is due to the unpredictable latency in waiting for slowest
nodes (or stragglers) to finish their tasks. We propose a novel coding
strategy, named \emph{entangled polynomial code}, for designing the
intermediate computations at the worker nodes in order to minimize the recovery
threshold (i.e., the number of workers that we need to wait for in order to
compute the final output). We demonstrate the optimality of entangled
polynomial code in several cases, and show that it provides orderwise
improvement over the conventional schemes for straggler mitigation.
Furthermore, we characterize the optimal recovery threshold among all linear
coding strategies within a factor of $2$ using \emph{bilinear complexity}, by
developing an improved version of the entangled polynomial code. In particular,
while evaluating bilinear complexity is a well-known challenging problem, we
show that optimal recovery threshold for linear coding strategies can be
approximated within a factor of $2$ of this fundamental quantity. On the other
hand, the improved version of the entangled polynomial code enables further and
orderwise reduction in the recovery threshold, compared to its basic version.
Finally, we show that the techniques developed in this paper can also be
extended to several other problems such as coded convolution and fault-tolerant
computing, leading to tight characterizations.
