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Rough sets theory is not good at discovering knowledge from digraphs which is a kind of relational data. In order to solve this problem, we introduce binary relations derived from simple digraphs and propose a new concept of k -step R -related set in the framework of generalized rough sets theory. In addition, we first investigate the relationships between generalized rough sets theory and graph theory on the basis of mutual representation between binary relations and digraphs. The relationships established in this work make it possible to use generalized rough sets theory to find strongly connected components of simple directed graphs, which previously can be solved only by graph algorithms. An algorithm is correspondingly developed based on the above works, especially k -step R -related set. A series of experiments are carried out to test the proposed algorithm. The results show that our algorithm provides comparable performance to the classical Tarjan algorithm. In addition, the proposed algorithm can be implemented in parallel. And its parallel performance is comparable to existing state-of-the-art parallel algorithms. [ABSTRACT FROM AUTHOR]

Abstract As an extension of Pawlak's rough sets, rough fuzzy sets is proposed to deal with the target concept which is typically fuzzy or uncertain. It is worthwhile to introduce cost-sensitive learning into granular computing, as the granularity with the optimal cost can be selected. In the terms of decision making, test cost and decision cost are the most popular cost types. In the sequential three-way decisions (S3WD) models, the granularity is sensitive to the test cost. Meanwhile, the accuracy is sensitive to the decision cost. Selecting an optimal cost-sensitive granularity for problem solving is helpful for achieving optimal results at the lowest total cost in S3WD. However, it is difficult to evaluate test cost precisely and objectively in real-life applications, and existing works only focus on searching for the total cost as the objective function. In this paper, we firstly present a sequential three-way decisions model with rough fuzzy sets (S3WDRFS). Then, for S3WDRFS and its three regions, the changing rules of their decision cost are revealed in a hierarchical granular structure. By considering user requirements, we propose an optimization mechanism to achieve the optimal cost-sensitive granularity selection based on S3WDRFS model. Finally, the experimental results demonstrate that exemplary optimal granularities can be obtained for high quality decision-making under certain constraints. [ABSTRACT FROM AUTHOR]