A data structure for monomial ideals with applications to signature Gr\"obner bases

TL;DR

This paper introduces monomial divisibility diagrams (MDDs), a novel data structure for monomial ideals that enhances efficiency in membership testing and insertion, with practical speed-ups in signature Gr"obner basis computations.

Contribution

The paper presents MDDs, a new canonical tree-based data structure for monomial ideals, and demonstrates their integration into Gr"obner basis algorithms for improved performance.

Findings

MDDs support efficient insertion and membership testing.

Empirical results show MDDs reduce computation time in Gr"obner basis calculations.

MDDs outperform list-based representations in speed and memory usage.

Abstract

We introduce monomial divisibility diagrams (MDDs), a data structure for monomial ideals that supports insertion of new generators and fast membership tests. MDDs stem from a canonical tree representation by maximally sharing equal subtrees, yielding a directed acyclic graph. We establish basic complexity bounds for membership and insertion, and study empirically the size of MDDs. As an application, we integrate MDDs into the signature Gr\"obner basis implementation of the Julia package AlgebraicSolving.jl. Membership tests in monomial ideals are used to detect some reductions to zero, and the use of MDDs leads to substantial speed-ups compared to the existing representation by lists of generators with divmasks.