A Note on the Performance of Algorithms for Solving Linear Diophantine Equations in the Naturals

TL;DR

This paper compares four algorithms for solving linear Diophantine equations in natural numbers, finding that the graph-based algorithm outperforms others on larger inputs, guiding implementation choices for efficiency.

Contribution

The paper provides a detailed comparison of four algorithms, highlighting the efficiency of the graph-based algorithm for large-scale problems in solving linear Diophantine equations.

Findings

Graph-based algorithm becomes faster than Slopes on larger inputs.

Lexicographic enumeration and completion are slower than the other two algorithms.

Implementation of AC-unification algorithms should prefer the graph-based approach for efficiency.

Abstract

We implement four algorithms for solving linear Diophantine equations in the naturals: a lexicographic enumeration algorithm, a completion procedure, a graph-based algorithm, and the Slopes algorithm. As already known, the lexicographic enumeration algorithm and the completion procedure are slower than the other two algorithms. We compare in more detail the graph-based algorithm and the Slopes algorithm. In contrast to previous comparisons, our work suggests that they are equally fast on small inputs, but the graph-based algorithm gets much faster as the input grows. We conclude that implementations of AC-unification algorithms should use the graph-based algorithm for maximum efficiency.