Exact sampling algorithms for Latin squares and Sudoku matrices via probabilistic divide-and-conquer

TL;DR

This paper introduces exact, uniform random sampling algorithms for Latin squares and Sudoku matrices using probabilistic divide-and-conquer, enhancing sampling efficiency and accuracy.

Contribution

It presents novel PDC-based algorithms, including an 'almost deterministic second half' approach, for exact sampling of Latin squares and Sudoku matrices.

Findings

Algorithms produce exact uniform samples

The 'almost deterministic second half' improves efficiency

Demonstrates practical sampling for complex combinatorial structures

Abstract

We provide several algorithms for the exact, uniform random sampling of Latin squares and Sudoku matrices via probabilistic divide-and-conquer (PDC). Our approach divides the sample space into smaller pieces, samples each separately, and combines them in a manner which yields an exact sample from the target distribution. We demonstrate, in particular, a version of PDC in which one of the pieces is sampled using a brute force approach, which we dub $\textit{almost deterministic second half}$, as it is a generalization to a previous application of PDC for which one of the pieces is uniquely determined given the others.