Coded Distributed Computing with Pre-set Assignments of Data and Output Functions

Coded distributed computing can reduce the communication load for distributed computing systems by introducing redundant computation and creating multicasting opportunities. However, the existing schemes require delicate data placement and output function assignment, which is not feasible when distributed nodes fetch data without the orchestration of a master node. In this paper, we consider the general systems where the data placement and output function assignment are arbitrary but pre-set. We propose two coded computing schemes, One-shot Coded Transmission (OSCT) and Few-shot Coded Transmission (FSCT), to reduce the communication load. Both schemes first group the nodes into clusters and divide the transmission of each cluster into multiple rounds, and then design coded transmission in each round to maximize the multicast gain. The key difference between OSCT and FSCT is that the former uses a one-shot transmission where each encoded message can be decoded independently by the intended nodes, while the latter allows each node to jointly decode multiple received symbols to achieve potentially larger multicast gains. Furthermore, based on the lower bound proposed by Yu et al., we derive sufficient conditions for the optimality of OSCT and FSCT, respectively. This not only recovers the existing optimality results but also includes some cases where our schemes are optimal while others are not.