In-depth comparison of the Berlekamp--Massey--Sakata and the Scalar-FGLM algorithms: the adaptive variants

TL;DR

This paper introduces an improved adaptive version of the Berlekamp--Massey--Sakata algorithm, compares it with the adaptive Scalar-FGLM algorithm, and demonstrates that the former reduces computational effort while both are more efficient than their original forms.

Contribution

It develops an adaptive variant of the Berlekamp--Massey--Sakata algorithm and provides a detailed comparison with the adaptive Scalar-FGLM, highlighting efficiency improvements and behavioral differences.

Findings

Adaptive Berlekamp--Massey--Sakata reduces operations in 2D and 3D cases.

Adaptive Scalar-FGLM generally requires fewer queries and operations.

Both adaptive algorithms outperform their original versions.

Abstract

The Berlekamp--Massey--Sakata algorithm and the Scalar-FGLM algorithm both compute the ideal of relations of a multidimensional linear recurrent sequence.Whenever quering a single sequence element is prohibitive, the bottleneck of these algorithms becomes the computation of all the needed sequence terms. As such, having adaptive variants of these algorithms, reducing the number of sequence queries, becomes mandatory.A native adaptive variant of the Scalar-FGLM algorithm was presented by its authors, the so-called Adaptive Scalar-FGLM algorithm.In this paper, our first contribution is to make the Berlekamp--Massey--Sakata algorithm more efficient by making it adaptive to avoid some useless relation test-ings. This variant allows us to divide by four in dimension 2 and by seven in dimension 3 the number of basic operations performed on some sequence family.Then, we compare the two adaptive algorithms. We show that their behaviors differ in a way that it is not possible to tweak one of the algorithms in order to mimic exactly the behavior of the other. We detail precisely the differences and the similarities of both algorithms and conclude that in general the Adaptive Scalar-FGLM algorithm needs fewer queries and performs fewer basic operations than the Adaptive Berlekamp--Massey--Sakata algorithm.We also show that these variants are always more efficient than the original algorithms.