Vincent Neiger

Faster Algorithms for Multivariate Interpolation With Multiplicities and Simultaneous Polynomial Approximations

The interpolation step in the Guruswami-Sudan algorithm is a bivariate interpolation problem with multiplicities commonly solved in the literature using either structured linear algebra or basis reduction of polynomial lattices. This problem has been extended to three or more variables; for this generalization, all fast algorithms proposed so far rely on the lattice approach. In this paper, we reduce this multivariate interpolation problem to a problem of simultaneous polynomial approximations, which we solve using fast structured linear algebra. This improves the best known complexity bounds for the interpolation step of the list-decoding of Reed-Solomon codes, Parvaresh-Vardy codes, and folded Reed-Solomon codes. In particular, for Reed-Solomon list-decoding with re-encoding, our approach has complexity O~(ℓω-1m2(n - k)), where ℓ, m, n, and k are the list size, the multiplicity, the number of sample points, and the dimension of the code, and ω is the exponent of linear algebra; this accelerates the previously fastest known algorithm by a factor of ℓ/m.

Fast Computation of Minimal Interpolation Bases in Popov Form for Arbitrary Shifts

We compute minimal bases of solutions for a general interpolation problem, which encompasses Hermite-Pade approximation and constrained multivariate interpolation, and has applications in coding theory and security. This problem asks to find univariate polynomial relations between m vectors of size σ; these relations should have small degree with respect to an input degree shift. For an arbitrary shift, we propose an algorithm for the computation of an interpolation basis in shifted Popov normal form with a cost of O~(mω-1 σ) field operations, where ω is the exponent of matrix multiplication and the notation O~(·) indicates that logarithmic terms are omitted. Earlier works, in the case of Hermite-Pade approximation and in the general interpolation case, compute non-normalized bases. Since for arbitrary shifts such bases may have size Θ(m2 σ), the cost bound O~(mω-1 σ) was feasible only with restrictive assumptions on the shift that ensure small output sizes. The question of handling arbitrary shifts with the same complexity bound was left open. To obtain the target cost for any shift, we strengthen the properties of the output bases, and of those obtained during the course of the algorithm: all the bases are computed in shifted Popov form, whose size is always O(m σ). Then, we design a divide-and-conquer scheme. We recursively reduce the initial interpolation problem to sub-problems with more convenient shifts by first computing information on the degrees of the intermediate bases.

Fast Computation of Shifted Popov Forms of Polynomial Matrices via Systems of Modular Polynomial Equations

We give a Las Vegas algorithm which computes the shifted Popov form of an m x m nonsingular polynomial matrix of degree d in expected ~O(mω d) field operations, where ω is the exponent of matrix multiplication and ~O(·) indicates that logarithmic factors are omitted. This is the first algorithm in ~O(mω d) for shifted row reduction with arbitrary shifts. Using partial linearization, we reduce the problem to the case d ≤ ⌈ σ/m ⌉ where σ is the generic determinant bound, with σ / m bounded from above by both the average row degree and the average column degree of the matrix. The cost above becomes ~O(mω ⌈ σ/m ⌉), improving upon the cost of the fastest previously known algorithm for row reduction, which is deterministic. Our algorithm first builds a system of modular equations whose solution set is the row space of the input matrix, and then finds the basis in shifted Popov form of this set. We give a deterministic algorithm for this second step supporting arbitrary moduli in ~O(mω-1 σ) field operations, where m is the number of unknowns and σ is the sum of the degrees of the moduli. This extends previous results with the same cost bound in the specific cases of order basis computation and M-Pade approximation, in which the moduli are products of known linear factors.

On the Structure of Changes in Dynamic Contact Networks

We present a methodology to investigate the structure of dynamic networks in terms of concentration of changes in the network. We handle dynamic networks as series of graphs on a fixed set of nodes and consider the changes occurring between two consecutive graphs in the series. We apply our methodology to various dynamic contact networks coming from different contexts and we show that changes in these networks exhibit a non-trivial structure: they are not spread all over the network but are instead concentrated around a small fraction of nodes. We compare our observations on real-world networks to three classical dynamic network models and show that they do not capture this key property.

Computing Canonical Bases of Modules of Univariate Relations

We study the computation of canonical bases of sets of univariate relations (p1,...,pm) ∈ K[x]m such that p1 f1 + ⋯ + pm fm = 0; here, the input elements f1,...,fm are from a quotient K[x]n/M, where M is a K[x]-module of rank n given by a basis M ∈ K[x]n x n in Hermite form. We exploit the triangular shape of M to generalize a divide-and-conquer approach which originates from fast minimal approximant basis algorithms. Besides recent techniques for this approach, we rely on high-order lifting to perform fast modular products of polynomial matrices of the form P F mod M. Our algorithm uses O~(mω-1D + nω D/m) operations in K, where D = deg(det(M)) is the K-vector space dimension of K[x]n/M, O~(·) indicates that logarithmic factors are omitted, and ω is the exponent of matrix multiplication. This had previously only been achieved for a diagonal matrix M. Furthermore, our algorithm can be used to compute the shifted Popov form of a nonsingular matrix within the same cost bound, up to logarithmic factors, as the previously fastest known algorithm, which is randomized.

Fast, deterministic computation of the Hermite normal form and determinant of a polynomial matrix

Given a nonsingular

Computing Popov and Hermite Forms of Rectangular Polynomial Matrices

n \times n

Certification of Minimal Approximant Bases

matrix of univariate polynomials over a field

Algorithms for Zero-Dimensional Ideals Using Linear Recurrent Sequences

\mathbb{K}

Two-Point Codes for the Generalized GK Curve

, we give fast and deterministic algorithms to compute its determinant and its Hermite normal form. Our algorithms use