The iRRAM: Exact Arithmetic in C++
CCA '00 Selected Papers from the 4th International Workshop on Computability and Complexity in Analysis
Turing machines, transition systems, and interaction
Information and Computation
On the expressive power of univariate equations over sets of natural numbers
Information and Computation
Parsing Boolean grammars over a one-letter alphabet using online convolution
Theoretical Computer Science
Relaxing order basis computation
ACM Communications in Computer Algebra
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A Turing machine multiplies binary integers on-line if it receives its inputs, low-order digit first, and produces the jth digit of the product before reading in the (j+1)st digits of the two inputs. We present a general method for converting any off-line multiplication algorithm which forms the product of two n-digit binary numbers in time F(n) into an on-line method which uses time only O(F(n) log n), assuming that F is monotone and satisfies n@?F(n)@?F(2n)/2@?kF(n) for some constant k. Applying this technique to the fast multiplication algorithm of Schonhage and Strassen gives an upper bound of O(n (log n)^2 loglog n) for on-line multiplication of integers. A refinement of the technique yields an optimal method for on-line multiplication by certain sparse integers. Other applications are to the on-line computation of products of polynomials, recognition of palindromes, and multiplication by a constant.