All-to-All Encode in Synchronous Systems

We define all-to-all encode, a collective communication operation serving as a primitive in decentralized computation and storage systems. Consider a scenario where every processor initially has a data packet and requires a linear combination of all data packets; the linear combinations are distinct from one processor to another, and are specified by a generator matrix of an error correcting code. We use a linear network model, in which processors transmit linear combinations of their data and previously received packets, and adopt a standard synchronous system setting to analyze its communication cost. We provide a universal algorithm which computes any matrix in this model by only varying intermediate coefficients, and prove its optimality. When the generator matrix is of the Vandermonde or Lagrange type, we further optimize the communication efficiency of the proposed algorithm.