An adaptive global reduction algorithm for wormhole-routed 2D meshes

This paper presents a global reduction algorithm for wormhole-routed 2D meshes. Well-known reduction algorithms that are optimized for short vectors have complexity O(M log N), where N = n × n is the number of nodes, and M the vector length. Algorithms suitable for long vectors have complexity O(√N + M). Previously known asymptotically optimal algorithms with complexity O(log N + M) incur inherent network contention among constituent messages. The proposed algorithm adapts to the given vector length, resulting in complexities O(M log N) for short vectors, O(log N + M) for medium-sized vectors, and O(√N + M) for sufficiently long vectors. The O(√N + M) version is preferred to the O(log N + M) version for long vectors, due to its small coefficient associated with M, the dominating factor for such vectors. The algorithm is contention-free in a synchronous environment. Under asynchronous execution models, depth contention (contention among message-passing steps) may occur. However, simulation studies show that the effect of depth contention on the actual performance is negligible.