Random Permutations, Random Sudoku Matrices and Randomized Algorithms

Abstract

Some randomized algorithms, used to obtain a random n 2 × n 2 Sudoku matrix, where n is a natural number, is reviewed in this study. Below is described the set Π n of all (2n) × n matrices, consisting of elements of the set Z n = {1, 2, . . . , n}, such that every row is a permutation. It is proved that such matrices would be particularly useful in developing efficient algorithms in generating Sudoku matrices. An algorithm to obtain random Π n matrices is presented in this paper. The algorithms are evaluated according to two criteria -probability evaluation, and time evaluation. This type of criteria is interesting from both theoretical and practical point of view because they are particularly useful in the analysis of computer programs.

Exact Sampling Algorithms for Latin Squares and Sudoku Matrices via Probabilistic Divide-and-Conquer

“…These algorithms are generally known as rejection algorithms, since the target set of objects S, also referred to as the target region, lies within the complete sample space Ω. Since the algorithm generates a sample uniformly over all elements of Ω, any Sudoku matrix generated by such a hard rejection algorithm is also uniform over S (see, e.g., [32] for a more detailed analysis in a more general setting).…”