On some randomized algorithms and their evaluation

TL;DR

This paper explores randomized algorithms for generating Sudoku matrices, focusing on their implementation, evaluation, and relationship with permutation matrices, providing both theoretical insights and practical algorithms.

Contribution

It introduces algorithms for generating random permutation matrices related to Sudoku, with evaluation criteria for probability and computational efficiency.

Findings

Developed an algorithm for generating random  matrices

Evaluated algorithms based on probability and runtime

Analyzed the relationship between permutation matrices and Sudoku matrices

Abstract

The paper considers implementations of some randomized algorithms in connection with obtaining a random $n^2 \times n^2$ Sudoku matrix with programming language C++. For this purpose we describes the set $\Pi_n$ of all $(2n) \times n$ matrices, consisting of elements of the set $\mathbb{Z}_n =\{ 1,2,\ldots ,n\}$, such that every row is a permutation. We emphasize the relationship between these matrices and the $n^2 \times n^2$ Sudoku matrices. An algorithm to obtain random $\Pi_n$ matrices is presented in this paper. Several auxiliary algorithms that are related to the underlying problem have been described. We evaluated all algorithms according to two criteria - probability evaluation, and time for generation of random objects and checking of belonging to a specific set. This evaluations are interesting from both theoretical and practical point of view because they are particularly useful in the analysis of computer programs.