16 research outputs found
Parallel sparse interpolation using small primes
To interpolate a supersparse polynomial with integer coefficients, two
alternative approaches are the Prony-based "big prime" technique, which acts
over a single large finite field, or the more recently-proposed "small primes"
technique, which reduces the unknown sparse polynomial to many low-degree dense
polynomials. While the latter technique has not yet reached the same
theoretical efficiency as Prony-based methods, it has an obvious potential for
parallelization. We present a heuristic "small primes" interpolation algorithm
and report on a low-level C implementation using FLINT and MPI.Comment: Accepted to PASCO 201
A local decision test for sparse polynomials
An â„“-sparse (multivariate) polynomial is a polynomial containing at most â„“-monomials in its explicit description. We assume that a polynomial is implicitly represented as a black-box: on an input query from the domain, the black-box replies with the evaluation of the polynomial at that input. We provide an efficient, randomized algorithm, that can decide whether a polynomial [MathML] given as a black-box is â„“-sparse or not, provided that q is large compared to the polynomial's total degree. The algorithm makes only queries, which is independent of the domain size. The running time of our algorithm (in the bit-complexity model) is , where d is an upper bound on the degree of each variable. Existing interpolation algorithms for polynomials in the same model run in time . We provide a similar test for polynomials with integer coefficients
Revisit Sparse Polynomial Interpolation based on Randomized Kronecker Substitution
In this paper, a new reduction based interpolation algorithm for black-box
multivariate polynomials over finite fields is given. The method is based on
two main ingredients. A new Monte Carlo method is given to reduce black-box
multivariate polynomial interpolation to black-box univariate polynomial
interpolation over any ring. The reduction algorithm leads to multivariate
interpolation algorithms with better or the same complexities most cases when
combining with various univariate interpolation algorithms. We also propose a
modified univariate Ben-or and Tiwarri algorithm over the finite field, which
has better total complexity than the Lagrange interpolation algorithm.
Combining our reduction method and the modified univariate Ben-or and Tiwarri
algorithm, we give a Monte Carlo multivariate interpolation algorithm, which
has better total complexity in most cases for sparse interpolation of black-box
polynomial over finite fields
Interpolation of Shifted-Lacunary Polynomials
Given a "black box" function to evaluate an unknown rational polynomial f in
Q[x] at points modulo a prime p, we exhibit algorithms to compute the
representation of the polynomial in the sparsest shifted power basis. That is,
we determine the sparsity t, the shift s (a rational), the exponents 0 <= e1 <
e2 < ... < et, and the coefficients c1,...,ct in Q\{0} such that f(x) =
c1(x-s)^e1+c2(x-s)^e2+...+ct(x-s)^et. The computed sparsity t is absolutely
minimal over any shifted power basis. The novelty of our algorithm is that the
complexity is polynomial in the (sparse) representation size, and in particular
is logarithmic in deg(f). Our method combines previous celebrated results on
sparse interpolation and computing sparsest shifts, and provides a way to
handle polynomials with extremely high degree which are, in some sense, sparse
in information.Comment: 22 pages, to appear in Computational Complexit
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
Detecting lacunary perfect powers and computing their roots
We consider solutions to the equation f = h^r for polynomials f and h and
integer r > 1. Given a polynomial f in the lacunary (also called sparse or
super-sparse) representation, we first show how to determine if f can be
written as h^r and, if so, to find such an r. This is a Monte Carlo randomized
algorithm whose cost is polynomial in the number of non-zero terms of f and in
log(deg f), i.e., polynomial in the size of the lacunary representation, and it
works over GF(q)[x] (for large characteristic) as well as Q[x]. We also give
two deterministic algorithms to compute the perfect root h given f and r. The
first is output-sensitive (based on the sparsity of h) and works only over
Q[x]. A sparsity-sensitive Newton iteration forms the basis for the second
approach to computing h, which is extremely efficient and works over both
GF(q)[x] (for large characteristic) and Q[x], but depends on a number-theoretic
conjecture. Work of Erdos, Schinzel, Zannier, and others suggests that both of
these algorithms are unconditionally polynomial-time in the lacunary size of
the input polynomial f. Finally, we demonstrate the efficiency of the
randomized detection algorithm and the latter perfect root computation
algorithm with an implementation in the C++ library NTL.Comment: to appear in Journal of Symbolic Computation (JSC), 201