Nearly Tight Bounds for Wormhole Routing

We present nearly tight bounds f o r 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 fl(10g3 N/(loglog N ) ’ ) . The best lower bound known earlier was Q(log2 N ) , which is simply the frit 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(10g3 N log log N) , which improves upon the previous best bound of O(10g4 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 ofline wormhole routing, where we give optimal algorithms for trees and multidimensional meshes.