  By integrating edge computing with parallel computing, distributed edge
computing (DEC) makes use of distributed devices in edge networks to perform
computing in parallel, which can substantially reduce service delays. In this
paper, we explore DEC that exploits distributed edge devices connected by a
wireless network to perform a computation task offloaded from an end device. In
particular, we study the fundamental problem of minimizing the delay of
executing a distributed algorithm of the computation task. We first establish
some structural properties of the optimal communication scheduling policy.
Then, given these properties, we characterize the optimal computation
allocation policy, which can be found by an efficient algorithm. Next, based on
the optimal computation allocation, we characterize the optimal scheduling
order of communications for some special cases, and develop an efficient
algorithm with a finite approximation ratio to find it for the general case.
Last, based on the optimal computation allocation and communication scheduling,
we further show that the optimal selection of devices can be found efficiently
for some special cases. Our results provide some useful insights for the
optimal computation-communication co-design. We evaluate the performance of the
theoretical findings using simulations.
