Hybrid approaches to attribute reduction based on indiscernibility and discernibility relation

Attribute reduction is one of the key issues in rough set theory. Many heuristic attribute reduction algorithms such as positive-region reduction, information entropy reduction and discernibility matrix reduction have been proposed. However, these methods are usually computationally time-consuming for large data. Moreover, a single attribute significance measure is not good for more attributes with the same greatest value. To overcome these shortcomings, we first introduce a counting sort algorithm with time complexity O(∣C∣ ∣U∣) for dealing with redundant and inconsistent data in a decision table and computing positive regions and core attributes (∣C∣ and ∣U∣ denote the cardinalities of condition attributes and objects set, respectively). Then, hybrid attribute measures are constructed which reflect the significance of an attribute in positive regions and boundary regions. Finally, hybrid approaches to attribute reduction based on indiscernibility and discernibility relation are proposed with time complexity no more than max(O(∣C∣2∣U/C∣), O(∣C∣∣U∣)), in which ∣U/C∣ denotes the cardinality of the equivalence classes set U/C. The experimental results show that these proposed hybrid algorithms are effective and feasible for large data.