  We consider several problems relating to strongly-connected directed networks
of identical finite-state processors that work synchronously in discrete time
steps. The conceptually simplest of these is the Wake Up and Report Problem;
this is the problem of having a unique "root" processor send a signal to all
other processors in the network and then enter a special "done" state only when
all other processors have received the signal. The most difficult of the
problems we consider is the classic Firing Squad Synchronization Problem; this
is the much-studied problem of achieving macro-synchronization in a network
given micro-synchronization. We show via a complex algorithmic application of
the "snake" data structure first introduced in Even, Litman, and Winkler [ELW],
that these two problems are asymptotically time-equivalent up to a constant
factor. This result leads immediately to the inclusion of several other related
problems into this new asymptotic time-class.
