- Toom–Cook multiplication
Toom–Cook, sometimes known as Toom-3, named after
Andrei Toom andStephen Cook , is amultiplication algorithm , a method of multiplying two large integers.Given two large integers, "a" and "b", Toom–Cook splits up "a" and "b" into "k" smaller parts each of length "l", and performs operations on the parts. As "k" grows, one may combine many of the multiplication sub-operations, thus reducing the overall complexity of the algorithm. The multiplication sub-operations can then be computed recursively using Toom–Cook multiplication again, and so on. Although the terms "Toom-3" and "Toom–Cook" are sometimes incorrectly used interchangeably, Toom-3 is only a single instance of the Toom-Cook algorithm, where "k" = 3.
Toom-3 reduces 9 multiplications to 5, and runs in Θ(nlog(5)/log(3)), about Θ(n1.465). In general, Toom-"k" runs in Θ("c"("k") "ne"), where "e" = log(2"k"-1) / log("k"), "ne" is the time spent on sub-multiplications, and "c" is the time spent on additions and multiplication by small constants. [Knuth, p. 296] The
Karatsuba algorithm is a special case of Toom–Cook, where the number is split into two smaller ones. It reduces 4 multiplications to 3 and so operates at Θ("n"log(3)/log(2)), which is about Θ("n"1.585). Ordinary long multiplication is equivalent to Toom-1, with complexity Θ("n"2).Although the exponent "e" can be set arbitrarily close to 1 by increasing "k", the function "c" unfortunately grows very rapidly. [Knuth, p. 296] [Crandall & Pomerance, p. 474] The growth rate for mixed-level Toom-Cook schemes was still an open research problem in 2005. [Crandall & Pomerance, p. 536] An implementation described by
Donald Knuth achieves the time complexity Θ("n" 2√(2 log "n") log "n"). [Knuth, p. 302]Due to its overhead, Toom–Cook is slower than long multiplication with small numbers, and it is therefore typically used for intermediate-size multiplications, before the asymptotically faster
Schönhage-Strassen algorithm (with complexity Θ("n" log "n" log log "n")) becomes practical.Andrei Toom first described this algorithm in 1963, whileStephen Cook improved the algorithm and published it in his PhD thesis in 1966. [http://www.cs.toronto.edu/~sacook/ (thesis)]Details
This section discusses exactly how to perform Toom-"k" for any given value of "k", and is a simplification of a description of Toom–Cook polynomial multiplication described by Marco Bodrato. [Marco Bodrato. Towards Optimal Toom–Cook Multiplication for Univariate and Multivariate Polynomials in Characteristic 2 and 0. In "WAIFI'07 proceedings", volume 4547 of LNCS, pages 116-133. June 21-22, 2007. [http://bodrato.it/papers/#WAIFI2007 author website] ] The algorithm has five main steps: "Splitting", "Evaluation", "Pointwise multiplication", "Interpolation", and "Recomposition".
In a typical large integer implementation, each integer is represented as a sequence of digits in
positional notation , with the base or radix set to some (typically large) value "b"; for this example we use "b" = 10000, so that each digit corresponds to a group of four decimal digits (in a computer implementation, "b" would typically be a power of 2 instead). Say the two integers being multiplied are:As shown, these can also be negative. For large enough numbers, this is the most expensive step, the only step that is not linear in the sizes of "m" and "n".
Interpolation
This is the most complex step, the reverse of the evaluation step: given our "d" points on "r", we need to determine the coefficients of "r". In other words, we want to solve this matrix equation for the vector on the right-hand side:
:
This matrix is constructed the same way as the one in the evaluation step, except that it's "d"×"d". We could solve this equation with a technique like
Gaussian elimination , but this is too expensive. Instead, we use the fact that, provided the evaluation points were chosen suitably, this matrix is invertible, and so::
All that remains is to compute this matrix-vector product. Although the matrix contains fractions, the resulting coefficients will be integers — so this can all be done with integer arithmetic, just additions, subtractions, and multiplication/division by small constants. A difficult design challenge in Toom–Cook is to find an efficient sequence of operations to compute this product; one sequence given by Bodrato for Toom-3 is the following, executed here over the running example:
We now know our product polynomial "r":
:
If we were using different "k""m", "k""n", or evaluation points, the matrix and so our interpolation strategy would change; but it does not depend on the inputs and so can be hard-coded for any given set of parameters.
Recomposition
Finally, we evaluate r(B) to obtain our final answer. This is straightforward since B is a power of "b" and so the multiplications by powers of B are all shifts by a whole number of digits in base "b":
And this is in fact the product of 1234567890123456789012 and 987654321987654321098.
Interpolation matrices for various "k"
Here we give common interpolation matrices for a few different common small values of "k""m" and "k""n".
Toom-1
Toom-1 ("k""m" = "k""n" = 1) requires 1 evaluation point, here chosen to be 0. It degenerates to long multiplication, with an interpolation matrix of the identity matrix:
:
Toom-1.5
Toom-1.5 ("k""m" = 2, "k""n" = 1) requires 2 evaluation points, here chosen to be 0 and ∞. Its interpolation matrix is then the identity matrix:
:
Toom-2
Toom-2 ("k""m" = 2, "k""n" = 2) requires 3 evaluation points, here chosen to be 0, 1, and ∞. It is the same as
Karatsuba multiplication , with an interpolation matrix of::
Toom-2.5
Toom-2.5 ("k""m" = 3, "k""n" = 2) requires 4 evaluation points, here chosen to be 0, 1, -1, and ∞. It then has an interpolation matrix of:
:
Notes
References
* D. Knuth. "
The Art of Computer Programming ", Volume 2. Third Edition, Addison-Wesley, 1997. Section 4.3.3.A: Digital methods, pg.294.
* R. Crandall & C. Pomerance. "Prime Numbers - A Computational Perspective". Second Edition, Springer, 2005. Section 9.5.1: Karatsuba and Toom-Cook methods, pg.433.
* M. Bodrato. " [http://bodrato.it/toom-cook/ Toward Optimal Toom-Cook Multiplication...] ". In WAIFI'07, Springer, 2007.External links
* [http://gmplib.org/manual/Toom-3_002dWay-Multiplication.html Toom-Cook 3-Way Multiplication from GMP Documentations]
Wikimedia Foundation. 2010.
