Output-Sensitive Algorithms for Sumset and Sparse Polynomial Multiplication

Arnold, Andrew; Roche, Daniel S.

doi:10.1145/2755996.2756653

Cited by 26 publications

(57 citation statements)

References 27 publications

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“…We assume the polynomials have the worst possible bitsize, that is O(n1n2D lg(D) + nDτ ). The cost of each multiplication is OB(n1n2D 2 lg(D) + nD 2 τ ), using a probabilistic [3,Thm. 7.1] or a worst case [28,Cor.…”

Section: Square Systems Of Dimensionmentioning

confidence: 99%

On the Bit Complexity of Solving Bilinear Polynomial Systems

Emiris

Mantzaflaris

Tsigaridas

2016

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

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We bound the Boolean complexity of computing isolating hyperboxes for all complex roots of systems of bilinear polynomials. The resultant of such systems admits a family of determinantal Sylvester-type formulas, which we make explicit by means of homological complexes. The computation of the determinant of the resultant matrix is a bottleneck for the overall complexity. We exploit the quasi-Toeplitz structure to reduce the problem to efficient matrix-vector products, corresponding to multivariate polynomial multiplication. For zero-dimensional systems, we arrive at a primitive element and a rational univariate representation of the roots. The overall bit complexity of our probabilistic algorithm is OB(n 4 D 4 + n 2 D 4 τ ), where n is the number of variables, D equals the bilinear Bézout bound, and τ is the maximum coefficient bitsize. Finally, a careful infinitesimal symbolic perturbation of the system allows us to treat degenerate and positive dimensional systems, thus making our algorithms and complexity analysis applicable to the general case.√ 3 3 ; −1, 2 ∓ √ 3) plus the root (x0, x1; y0, y1, y2) = (0, 1; 0, −1, 1) at infinity.

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Section: Square Systems Of Dimensionmentioning

confidence: 99%

On the Bit Complexity of Solving Bilinear Polynomial Systems

Emiris

Mantzaflaris

Tsigaridas

2016

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

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“…The problem is trivial for addition and subtraction. For multiplication, though many algorithms have been proposed [5,8,15,16,21,24,26,29], none of them was quasi-optimal in the general case. Only recently, we proposed a quasi-optimal algorithm for the multiplication of sparse polynomials over finite fields of large characteristic or over the integers [12].…”

Section: Introductionmentioning

confidence: 99%

On Exact Division and Divisibility Testing for Sparse Polynomials

Giorgi

Grenet

Cray

2021

Proceedings of the 2021 International Symposium on Symbolic and Algebraic Computation

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No polynomial-time algorithm is known to test whether a sparse polynomial divides another sparse polynomial . While computing the quotient = quo can be done in polynomial time with respect to the sparsities of , and , this is not yet sufficient to get a polynomial-time divisibility test in general. Indeed, the sparsity of the quotient can be exponentially larger than the ones of and . In the favorable case where the sparsity # of the quotient is polynomial, the best known algorithm to compute has a non-linear factor # # in the complexity, which is not optimal.In this work, we are interested in the two aspects of this problem. First, we propose a new randomized algorithm that computes the quotient of two sparse polynomials when the division is exact. Its complexity is quasi-linear in the sparsities of , and . Our approach relies on sparse interpolation and it works over any finite field or the ring of integers. Then, as a step toward faster divisibility testing, we provide a new polynomial-time algorithm when the divisor has a specific shape. More precisely, we reduce the problem to finding a polynomial such that is sparse and testing divisibility by can be done in polynomial time. We identify some structure patterns in the divisor for which we can efficiently compute such a polynomial . CCS CONCEPTS• Computing methodologies → Algebraic algorithms; • Theory of computation → Design and analysis of algorithms; • Mathematics of computing → Probabilistic algorithms.

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“…When the support of the product is known or structured, work in [Roc08, Roc11, VDHL12, VDHL13] indicates how to perform the multiplication fast. Using techniques from spare interpolation, Aarnold and Roche [AR15] have given an algorithm that runs in time that is nearly linear in the "structural sparsity" of the product, i.e. the sumset of the supports of the two polynomials.…”

Section: Introductionmentioning

confidence: 99%

Nearly Optimal Sparse Polynomial Multiplication

Nakos

2020

IEEE Trans. Inform. Theory

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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.

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Output-Sensitive Algorithms for Sumset and Sparse Polynomial Multiplication

Cited by 26 publications

References 27 publications

On the Bit Complexity of Solving Bilinear Polynomial Systems

On the Bit Complexity of Solving Bilinear Polynomial Systems

On Exact Division and Divisibility Testing for Sparse Polynomials

Nearly Optimal Sparse Polynomial Multiplication

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