An algorithm for shifted continued fraction expansions in parallel linear time

The linear complexity profile of a sequence of length n is readily obtained in O(n(2)) steps by the Berlekamp-Massey algorithm (BMA). Piper demands that the linear complexity profiles should be acceptable for every starting point, that is, for all shifted sequences as well. By repetition of the BMA, this can be verified in O(n(3)) steps. This paper describes a transducer that in only C-q.((n+1)(2))F-q operations, where C-2 = 7 and C-q = 6.5 for q greater than or equal to 3, computes the continued fraction expansions of all n shifted sequences (a(1),...,a(n)), (a(2),...,a(n)), to (a(n)). Hence, no additional computational effort is necessary to check Piper's demand. When n transducers are occupied in parallel, the output is obtained in C-q F-q operations per symbol, that is, in parallel linear time, yielding an O(n(2). log n) time-space product. (C) 1999 Elsevier Science B.V. All rights reserved.