One of the fundamental problems in distributed computing is how to efficiently
perform routing in a faulty network in which each link fails with some
probability. This paper investigates how big the failure probability can be,
before the capability to efficiently find a path in the network is lost. Our main
results show tight upper and lower bounds for the failure probability which
permits routing, both for the hypercube and for the d−dimensional mesh. We use
tools from percolation theory to show that in the d−dimensional mesh, once a
giant component appears --- efficient routing is possible. A different behavior is
observed when the hypercube is considered. In the hypercube there is a range of
failure probabilities in which short paths exist with high probability, yet
finding them must involve querying essentially the entire network. Thus the
routing complexity of the hypercube shows an asymptotic phase transition. The
critical probability with respect to routing complexity lies in a different
location then that of the critical probability with respect to connectivity.
Finally we show that an oracle access to links (as opposed to local routing) may
reduce significantly the complexity of the routing problem. We demonstrate this
fact by providing tight upper and lower bounds for the complexity of routing in
the random graph *G _{n,p}*.