Nearly tight bounds for wormhole routing

We present nearly tight bounds for wormhole muting on Butterfly networks which indicate it is fundamentally different from store-and-forward packet routing. For instance, consider the problem of routing N log N (randomly generated) log N length messages from the inputs to the outputs of an N input Butterfly. We show that with high probability that this must take time at least /spl Omega/(log/sup 3/N/(log log N)/sup 2/). The best lower bound known earlier was /spl Omega/(log/sup 2/ N), which is simply the flit congestion an each link. Thus our lower bound shows that wormhole routing (unlike store-and-forward-routing) is very ineffective in utilizing communication links. We also give a routing algorithm which nearly matches our lower bound. That is, we show that with high probability the time is O(log/sup 3/ N log log N), which improves upon, the previous best bound of O(log/sup 4/ N). Our method also extends to other networks such as the two-dimensional mesh, where it is nearly optimal. Finally, we consider the problem of offline wormhole routing, where we give optimal algorithms for trees and multidimensional meshes.<
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