论文标题
相对于覆盖维度的零维有限拓扑空间的算法计数
Algorithmic Counting of Zero-Dimensional Finite Topological Spaces With Respect to the Covering Dimension
论文作者
论文摘要
将覆盖尺寸视为拓扑空间维度的概念，我们首先指定{1，...，n} $上的零维t_0空间的Zdim_ {t_0}（n）和零维的数字Zdim（n} $的Zer imenifemensional topical topical topical topical in e {1，n}的均值，n}的均值， {1，...，n}和{1，...，...，n}分区的p（1，...，n}的部分订单的PO（n）。存在两个映射的算法。假设PO可以使用，我们使用ZDIM_ {T_0}（n）的规范，并以一种方式对P进行修改，以便计算ZDIM_ {T_0}（n）而不是P（n）。然后，ZDIM（n）的规范允许将此数字从zdim_ {t_0}（1）计算为ZDIM_ {T_0}（n）和第二种S（n，1）的stirl数字到S（n，n，n）。所得算法已在C中实现，我们还与它们一起提供了实验的结果。为了大大减少计算ZDIM_ {T_0}（N）的运行时间，我们还使用OpenMP库中描述了一种回溯方法及其在C中的并行实现。
Taking the covering dimension dim as notion for the dimension of a topological space, we first specify thenumber zdim_{T_0}(n) of zero-dimensional T_0-spaces on {1,...,n}$ and the number zdim(n) of zero-dimensional arbitrary topological spaces on {1,\ldots,n} by means oftwo mappings po and P that yieldthe number po(n) of partial orders on {1,...,n} and the set P(n) of partitions of {1,...,n}, respectively. Algorithms for both mappings exist. Assuming one for po to be at hand, we use our specification of zdim_{T_0}(n) and modify one for P in such a way that it computes zdim_{T_0}(n) instead of P(n). The specification of zdim(n) then allows to compute this number from zdim_{T_0}(1) to zdim_{T_0}(n) and the Stirling numbers of the second kind S(n,1) to S(n,n). The resulting algorithms have been implemented in C and we also present results of practical experiments with them. To considerably reduce the running times for computing zdim_{T_0}(n), we also describe a backtracking approach and its parallel implementation in C using the OpenMP library.
