Nearly Tight Bounds for Wormhole Routing

We present nearly tight bounds for wormhole routing 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 \Omega\Gammat/1 3 N=(log log N ) 2 ). The best lower bound known earlier was \Omega\Gammas/2 2 N ), which is simply the flit congestion in 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 3 N log log N ), which improves upon the previous best bound of O(log 4 N ). Our method also extends to other networks such as the two-dimensional mesh, where it is nearly optimal. Final..