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Register a new userParallel Integer Polynomial Multiplication
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Added by
Ning Xie
Publication date
2016
fields
Informatics Engineering
and research's language is
English
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We propose a new algorithm for multiplying dense polynomials with integer coefficients in a parallel fashion, targeting multi-core processor architectures. Complexity estimates and experimental comparisons demonstrate the advantages of this new approach.
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In the sparse polynomial multiplication problem, one is asked to multiply two sparse polynomials f and g in time that is proportional to the size of the input plus the size of the output. The polynomials are given via lists of their coefficients F and G, respectively. Cole and Hariharan (STOC 02) have given a nearly optimal algorithm when the coefficients are positive, and Arnold and Roche (ISSAC 15) devised an algorithm running in time proportional to the structural sparsity of the product, i.e. the set supp(F)+supp(G). The latter algorithm is particularly efficient when there not too many cancellations of coefficients in the product. In this work we give a clean, nearly optimal algorithm for the sparse polynomial multiplication problem.
We present a probabilistic algorithm to compute the product of two univariate sparse polynomials over a field with a number of bit operations that is quasi-linear in the size of the input and the output. Our algorithm works for any field of characteristic zero or larger than the degree. We mainly rely on sparse interpolation and on a new algorithm for verifying a sparse product that has also a quasi-linear time complexity. Using Kronecker substitution techniques we extend our result to the multivariate case.
The polynomial multiplication problem has attracted considerable attention since the early days of computer algebra, and several algorithms have been designed to achieve the best possible time complexity. More recently, efforts have been made to improve the space complexity, developing modifi
We present randomized algorithms to compute the sumset (Minkowski sum) of two integer sets, and to multiply two univariate integer polynomials given by sparse representations. Our algorithm for sumset has cost softly linear in the combined size of the inputs and output. This is used as part of our sparse multiplication algorithm, whose cost is softly linear in the combined size of the inputs, output, and the sumset of the supports of the inputs. As a subroutine, we present a new method for computing the coefficients of a sparse polynomial, given a set containing its support. Our multiplication algorithm extends to multivariate Laurent polynomials over finite fields and rational numbers. Our techniques are based on sparse interpolation algorithms and results from analytic number theory.
Matrix multiplication $A^t A$ appears as intermediate operation during the solution of a wide set of problems. In this paper, we propose a new cache-oblivious algorithm for the $A^t A$ multiplication. Our algorithm, A$scriptstyle mathsf{T}$A, calls classical Strassens algorithm as sub-routine, decreasing the computational cost %(expressed in number of performed products) of the conventional $A^t A$ multiplication to $frac{2}{7}n^{log_2 7}$. It works for generic rectangular matrices and exploits the peculiar symmetry of the resulting product matrix for sparing memory. We used the MPI paradigm to implement A$scriptstyle mathsf{T}$A in parallel, and we tested its performances on a small subset of nodes of the Galileo cluster. Experiments highlight good scalability and speed-up, also thanks to minimal number of exchanged messages in the designed communication system. Parallel overhead and inherently sequential time fraction are negligible in the tested configurations.
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