On the complexity of the "most general" undirected firing squad synchronization problem

Authors:
Darin Goldstein;Kojiro Kobayashi
Affiliations:
Department of Computer Engineering and Computer Science, California State University, Long Beach;Department of Information Systems Science, Soka University
Venue:
ISAAC'07 Proceedings of the 18th international conference on Algorithms and computation
Year:
2007

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Abstract

We show that if a minimal-time solution exists for a fundamental distributed computation primitive, synchronizing arbitrary undirected networks of finite-state processors, then there must exist an "extraordinarily fast" Õ(D5E) algorithm in the RAM model of computation for exactly determining the diameter D of an arbitrary unweighted undirected graph with E edges. The proof is constructive. At present we know eight variations of the firing squad synchronization problems whose solutions are known but whose minimal-time solutions are not known. Our result essentially completes the program outlined in [3] to show that it is highly unlikely for there to exist minimal-time solutions for these variations.