All-to-All Personalized Communication on Fat-Trees Using Latin Squares

Interconnection networks have a key role in systems with a high number of processors, and are chosen according to their capability of realizing different communication schemes. The all-to-all personalized communication is a relevant commu-nication pattern for many multiprocessor applications, and has been extensively studied in the past for many network topologies. In particular, when using multistage interconnection networks, the method based on the realization of a sequence of permutations forming a Latin square is very advantageous. In particular, a method for realizing the Latin square consisting of the identity permutation and its
$N-1$
rotations,
$\mathcal{R}\mathcal{L}\mathcal{S}$
, has been proposed for the Butterfly-Butterfly network. In this paper, we consider the Fat-Tree, a popular interconnection network used in many distributed systems as well as in Data Centers, and show how to modify and extend the method for the
$\mathcal{R}\mathcal{L}\mathcal{S}$
to work on them, whatever the size of the network and switches considered. We then consider also the Slimmed Fat-Tree, that is a cheaper network in terms of circuitry, and for this reason, it can be conveniently used in very large systems. The proposed method can realize the
$N$
permutations of a Latin Square in pipeline fashion in optimal time, that is
$O(N)$
.