关于若干随机化算法及其评估
On some randomized algorithms and their evaluation
AI总结：
本文研究随机数独矩阵生成问题，提出获取随机 $Π_n$ 矩阵的算法及若干辅助算法，并从概率和生成时间两个准则进行评估，为计算机程序分析提供理论与实用依据。
摘要：
本文考虑在C++编程语言下，与获得随机 $n^2 \times n^2$ 数独矩阵相关的若干随机化算法的实现。为此，我们描述了所有 $(2n) \times n$ 矩阵的集合 $Π_n$，这些矩阵由集合 $\mathbb{Z}_n =\{ 1,2,\ldots ,n\}$ 中的元素构成，且每一行都是一个排列。我们强调这些矩阵与 $n^2 \times n^2$ 数独矩阵之间的关系。本文给出了一种获得随机 $Π_n$ 矩阵的算法，并描述了若干与底层问题相关的辅助算法。我们根据两个准则对所有算法进行了评估——概率评估，以及随机对象生成和特定集合归属检查的时间。这些评估从理论和实践的角度来看都很有意义，因为它们在计算机程序分析中特别有用。
英文摘要：
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 $Π_n$ 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 $Π_n$ 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.
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