Fast Algorithm for Enumerating Diagonal Latin Squares of Small Order

TL;DR

This paper introduces a fast algorithm leveraging symmetry breaking, heuristics, and bit arithmetic to efficiently enumerate diagonal Latin squares of small orders, including symmetric variants.

Contribution

The paper presents a novel algorithm that significantly improves enumeration speed for diagonal Latin squares of small order using advanced computational techniques.

Findings

Enumerated diagonal Latin squares of order up to 9.

Enumerated vertically symmetric diagonal Latin squares of order up to 10.

Demonstrated effectiveness of symmetry breaking and heuristics in enumeration.

Abstract

In this paper we propose an algorithm for enumerating diagonal Latin squares of small order. It relies on specific properties of diagonal Latin squares to employ symmetry breaking techniques, and on several heuristic optimizations and bit arithmetic techniques to make use of computational power of state-of-the-art CPUs. Using this approach we enumerated diagonal Latin squares of order at most 9, and vertically symmetric diagonal Latin squares of order at most 10.