Size-time complexity of Boolean networks for prefix computations

Authors:
G. Bilardi;F. P. Preparata
Affiliations:
Department of Computer Science, Cornell University, Ithaca, NY;Coordinated Science Laboratory, University of Illinois at Urbana-Champaign, Urbana, IL
Venue:
STOC '87 Proceedings of the nineteenth annual ACM symposium on Theory of computing
Year:
1987

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Abstract

The prefix problem consists of computing all the products x0x1…xj (j=0, …, N - 1), given a sequence x = (x0, x1, …, xN - 1) of elements in a semigroup. In this paper we completely characterize the size-time complexity of computing prefixes with boolean networks, which are synchronized interconnections of Boolean gates and one-bit storage devices. This complexity crucially depends upon a property of the underlying semigroup, which we call cycle-freedom (no cycle of length greater than one in the Cayley graph of the semigroup). Denoting by S and T size and computation time, respectively, we have S = &THgr;((N/T) log(N/T)), for non-cycle-free semigroups, and S = &THgr;(N/T), for cycle-free semigroups. In both cases, T ∈ [&OHgr;(logN), O(N)].