On the complexity of multivariate blockwise polynomial multiplication

Hoeven, Joris van der; Lecerf, Grégoire

doi:10.1145/2442829.2442861

Cited by 24 publications

(20 citation statements)

References 28 publications

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“…Computer algebra programs including Maple, Mathematica, Sage, and Singular use a sparse representation by default for multivariate polynomials, and there has been considerable recent work on how to efficiently store and compute with sparse polynomials [8,10,19,21,14].…”

Section: Introductionmentioning

confidence: 99%

Output-Sensitive Algorithms for Sumset and Sparse Polynomial Multiplication

Arnold

Roche

2015

Proceedings of the 2015 ACM International Symposium on Symbolic and Algebraic Computation

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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.

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Section: Introductionmentioning

confidence: 99%

Output-Sensitive Algorithms for Sumset and Sparse Polynomial Multiplication

Arnold

Roche

2015

Proceedings of the 2015 ACM International Symposium on Symbolic and Algebraic Computation

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“…Considerable progress has been made toward this open problem, and it seems now nearly within reach. Some authors have looked at special cases, when the support of nonzero coefficients has a certain structure, to reduce to dense multiplication and achieve the desired complexity in those cases [45,79,80].…”

Section: Multiplicationmentioning

confidence: 99%

What Can (and Can't) we Do with Sparse Polynomials?

Roche

2018

Proceedings of the 2018 ACM International Symposium on Symbolic and Algebraic Computation

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Simply put, a sparse polynomial is one whose zero coefficients are not explicitly stored. Such objects are ubiquitous in exact computing, and so naturally we would like to have efficient algorithms to handle them. However, with this compact storage comes new algorithmic challenges, as fast algorithms for dense polynomials may no longer be efficient. In this tutorial we examine the state of the art for sparse polynomial algorithms in three areas: arithmetic, interpolation, and factorization. The aim is to highlight recent progress both in theory and in practice, as well as opportunities for future work. KEYWORDSsparse polynomial, interpolation, arithmetic, factorization ACM Reference Format: Daniel S. Roche. 2018. What Can (and Can't) we Do with Sparse Polynomials?.

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“…These temporary duplicated terms increase largely the pressure on the memory allocator, which is a bottleneck in a parallel context. The FFT or block based methods are mostly suitable for the multiplication of multivariate dense polynomials and not much for very sparse polynomials, even if the barrier between the sparse and dense algorithms becomes weaker [17,16,19].…”

Section: Introductionmentioning

confidence: 99%