All-to-all personalized communication on multistage interconnection networks

In parallel/distributed computing systems, the all-to-all personalized communication (or complete exchange) is required in numerous applications of parallel processing. In this paper, we consider this problem for log N stage Multistage Interconnection Networks (MINs). It is proved that the set of admissible permutations for a MIN can be partitioned in Latin Squares. Since routing permutations belonging to a Latin Square provides the all-to-all personalized communication, a method to realize the complete exchange with time complexity O(N), that is optimal, can be derived. This method, compared with other ones in literature, does not necessitate of neither pre-computation nor memory allocation to record the Latin Square, because an explicit construction of it is not required; furthermore it is applicable to any log N stage multistage networks since it is independent of the topology