36 research outputs found
On the complexity of polynomial reduction
In this paper, we present a new algorithm for reducing a multivariate polynomial with respect to an autoreduced tuple of other polynomials. In a suitable sparse complexity model, it is shown that the execution time is essentially the same (up to a logarithmic factor) as the time needed to verify that the result is correct. This is a first step towards making advantage of fast sparse polynomial arithmetic for the computation of Gröbner bases
On exact division and divisibility testing for sparse polynomials
No polynomial-time algorithm is known to test whether a sparse polynomial G
divides another sparse polynomial . While computing the quotient Q=F quo G
can be done in polynomial time with respect to the sparsities of F, G and Q,
this is not yet sufficient to get a polynomial-time divisibility test in
general. Indeed, the sparsity of the quotient Q can be exponentially larger
than the ones of F and G. In the favorable case where the sparsity #Q of the
quotient is polynomial, the best known algorithm to compute Q has a non-linear
factor #G#Q 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 F, G and Q. 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 S such that QS is sparse and testing divisibility by S can
be done in polynomial time. We identify some structure patterns in the divisor
G for which we can efficiently compute such a polynomial~S
Chunky and Equal-Spaced Polynomial Multiplication
Finding the product of two polynomials is an essential and basic problem in
computer algebra. While most previous results have focused on the worst-case
complexity, we instead employ the technique of adaptive analysis to give an
improvement in many "easy" cases. We present two adaptive measures and methods
for polynomial multiplication, and also show how to effectively combine them to
gain both advantages. One useful feature of these algorithms is that they
essentially provide a gradient between existing "sparse" and "dense" methods.
We prove that these approaches provide significant improvements in many cases
but in the worst case are still comparable to the fastest existing algorithms.Comment: 23 Pages, pdflatex, accepted to Journal of Symbolic Computation (JSC
High Performance Sparse Multivariate Polynomials: Fundamental Data Structures and Algorithms
Polynomials may be represented sparsely in an effort to conserve memory usage and provide a succinct and natural representation. Moreover, polynomials which are themselves sparse – have very few non-zero terms – will have wasted memory and computation time if represented, and operated on, densely. This waste is exacerbated as the number of variables increases. We provide practical implementations of sparse multivariate data structures focused on data locality and cache complexity. We look to develop high-performance algorithms and implementations of fundamental polynomial operations, using these sparse data structures, such as arithmetic (addition, subtraction, multiplication, and division) and interpolation. We revisit a sparse arithmetic scheme introduced by Johnson in 1974, adapting and optimizing these algorithms for modern computer architectures, with our implementations over the integers and rational numbers vastly outperforming the current wide-spread implementations. We develop a new algorithm for sparse pseudo-division based on the sparse polynomial division algorithm, with very encouraging results. Polynomial interpolation is explored through univariate, dense multivariate, and sparse multivariate methods. Arithmetic and interpolation together form a solid high-performance foundation from which many higher-level and more interesting algorithms can be built
Nearly Optimal Sparse Polynomial Multiplication
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.Comment: Accepted to IEEE Transactions on Information Theor
Highly Scalable Multiplication for Distributed Sparse Multivariate Polynomials on Many-core Systems
We present a highly scalable algorithm for multiplying sparse multivariate
polynomials represented in a distributed format. This algo- rithm targets not
only the shared memory multicore computers, but also computers clusters or
specialized hardware attached to a host computer, such as graphics processing
units or many-core coprocessors. The scal- ability on the large number of cores
is ensured by the lacks of synchro- nizations, locks and false-sharing during
the main parallel step.Comment: 15 pages, 5 figure