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Entropy 2013, 15(9), 3762-3795; doi:10.3390/e15093762

Improved Time Complexities for Learning Boolean Networks

1,*  and 2,*
1 Faculty of Life Science and Technology, Kunming University of Science and Technology, Kunming 650500, Yunnan, China 2 School of Computer Engineering, Nanyang Technological University, 50 Nanyang Avenue 639798, Singapore
* Authors to whom correspondence should be addressed.
Received: 27 May 2013 / Revised: 2 September 2013 / Accepted: 3 September 2013 / Published: 11 September 2013
Download PDF [1256 KB, 24 February 2015; original version 24 February 2015]


Existing algorithms for learning Boolean networks (BNs) have time complexities of at least O(N · n0:7(k+1)), where n is the number of variables, N is the number of samples and k is the number of inputs in Boolean functions. Some recent studies propose more efficient methods with O(N · n2) time complexities. However, these methods can only be used to learn monotonic BNs, and their performances are not satisfactory when the sample size is small. In this paper, we mathematically prove that OR/AND BNs, where the variables are related with logical OR/AND operations, can be found with the time complexity of O(k·(N+ logn)·n2), if there are enough noiseless training samples randomly generated from a uniform distribution. We also demonstrate that our method can successfully learn most BNs, whose variables are not related with exclusive OR and Boolean equality operations, with the same order of time complexity for learning OR/AND BNs, indicating our method has good efficiency for learning general BNs other than monotonic BNs. When the datasets are noisy, our method can still successfully identify most BNs with the same efficiency. When compared with two existing methods with the same settings, our method achieves a better comprehensive performance than both of them, especially for small training sample sizes. More importantly, our method can be used to learn all BNs. However, of the two methods that are compared, one can only be used to learn monotonic BNs, and the other one has a much worse time complexity than our method. In conclusion, our results demonstrate that Boolean networks can be learned with improved time complexities.
Keywords: Boolean networks; gene regulatory networks; mutual information; entropy; time complexity Boolean networks; gene regulatory networks; mutual information; entropy; time complexity
This is an open access article distributed under the Creative Commons Attribution License (CC BY) which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

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Zheng, Y.; Kwoh, C.K. Improved Time Complexities for Learning Boolean Networks. Entropy 2013, 15, 3762-3795.

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