The goal of this work is to show that if a minimal-time solution exists for a fundamental computation primitive, synchronizing networks of finite-state processors (formally known as the firing squad synchronization problems for directed and undirected networks), then there must exist "extraordinarily fast" algorithms for exactly solving the diameter problem in the standard RAM model of computation. These results can therefore be interpreted either as evidence that such minimal-time solutions for synchronization do not exist or as a potentially major algorithmic leap forward in the study of the diameter problem. We conjecture that the former is true. These results essentially complete the program outlined in [K. Kobayashi, Theoret. Comput. Sci., 259 (2001), pp. 129-143] to show that there exist a wide range of natural topologies for which it seems highly unlikely that there exist minimal-time solutions.
We consider several problems relating to stronglyconnected directed networks of identical finite-state processors that work synchronously in discrete time steps. The conceptually simplest of these problems 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 macrosynchronization 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[6] that these two problems in particular are asymptotically timeequivalent up to a constant factor. This result leads immediately to the inclusion of several other related problems into this new asymptotic time-class.
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