Optimal All-to-all personalized communication with minimum span on Boolean cubes

All-to-all personalized communication is a class of permutations in which each processor sends a unique message to every other processor. We present optimal algorithms for concurrent communication on all channels in Boolean cube networks, both for the case with a single permutation, and the case where multiple permutations shall be performed on the same local data set, but on different sets of processors. For K elements per processor our algorithms give the optimal number of elements transfer, K/2. For a succession of all-to-all personalized communications on disjoint subcubes of β dimensions each, our best algorithm yields K/2 + σ - β element exchanges in sequence, where σ is the total number of processor dimensions in the permutation. An implementation on the Connection Machine of one of the algorithms offers a maximum speed-up of 50% compared to the previously best known algorithm.