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A Branch & Bound Algorithm to Derive a Direct Construction for Binary Covering Arrays

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Advances in Artificial Intelligence and Soft Computing (MICAI 2015)

Part of the book series: Lecture Notes in Computer Science ((LNAI,volume 9413))

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Abstract

Covering arrays are used in testing deterministic systems where failures occur as a result of interactions among subsystems. The goal is to reveal if any interaction induces a failure in the system. Application areas include software and hardware testing. A binary covering array CA(N;t,k,2) is an \(N \times k\) array over the alphabet \(\{0,1\}\) with the property that each set of t columns contains all the \(2^t\) possible t-tuples of 0’s and 1’s at least once. In this paper we propose a direct method to construct binary covering arrays using an specific interpretation of binomial coefficients: a binomial coefficient with parameters k and r will be interpreted as the set of all the k-tuples from \(\{0,1\}\) having r ones and \(k-r\) zeroes. For given values of k and t, the direct method uses an explicit formula in terms of both k and t to provide a covering array CA(N;t,k,2) expressed as the juxtaposition of a set of binomial coefficients; this covering array will be of the minimum size that can be obtained by any juxtaposition of binomial coefficients. In order to derive the formula, a Branch & Bound (B&B) algorithm was first developed; the B&B algorithm provided solutions for small values of k and t that allowed the identification of the general pattern of the solutions. Like others previously reported methods, our direct method finds optimal covering arrays for \(k = t+1\) and \(k = t+2\); however, the major achievement is that nine upper bounds were significantly improved by our direct method, plus the fact that the method is able to set an infinite number of new upper bounds for \(t \ge 7\) given that little work has been done to compute binary covering arrays for general values of k and t.

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Notes

  1. 1.

    By identifying the \(t+1\) tClass in the same way as the kClass, i.e. by the number of 1’s in its rows.

  2. 2.

    One kClass is redundant with respect to the other kClass in the current solution if it does not cover at least one tClass not covered by any kClass in the solution.

  3. 3.

    Being \(m=\bigl \lfloor \frac{k-\big (\big \lfloor \frac{t}{2} \big \rfloor \, \text {mod}\,d\big )}{d} \bigr \rfloor \) and \(r=dj+(\lfloor \frac{t}{2} \rfloor \,\text{ mod }\,d)\) with \(d=k-t+1\).

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Acknowledgements

The authors acknowledge General Coordination of Information and Communications Technologies (CGSTIC) at CINVESTAV for providing HPC resources on the Hybrid Cluster Supercomputer “Xiuhcoatl”, that have contributed to the research results reported. The following projects have funded the research reported in this paper: 238469 - CONACyT Métodos Exactos para Construir Covering Arrays Óptimos; 2143 - Cátedras CONACyT.

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Authors and Affiliations

  1. Information Technology Laboratory, CINVESTAV-Tamaulipas, Ciudad Victoria, Tamaulipas, Mexico

    Jose Torres-Jimenez, Idelfonso Izquierdo-Marquez & Aldo Gonzalez-Gomez

  2. HARAMARA TIC-LAB, CICESE-UT3, Andador 10, Ciudad Del Conocimiento, Tepic, Nayarit, Mexico

    Himer Avila-George

  3. Cátedras CONACyT, Avenida Insurgentes Sur 1582, Benito Juárez, Crédito Constructor, 03940, Mexico, D.F., Mexico

    Himer Avila-George

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  1. Jose Torres-Jimenez
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  2. Idelfonso Izquierdo-Marquez
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  3. Aldo Gonzalez-Gomez
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  4. Himer Avila-George
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Corresponding author

Correspondence to Jose Torres-Jimenez .

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Editors and Affiliations

  1. Instituto Politécnico Nacional, Centro de Investigación en Computación, Mexico City, Mexico

    Grigori Sidorov

  2. Facultad de ciencias, Universidad Autónoma Nacional, México, Distrito Federal, Mexico

    Sofía N. Galicia-Haro

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Torres-Jimenez, J., Izquierdo-Marquez, I., Gonzalez-Gomez, A., Avila-George, H. (2015). A Branch & Bound Algorithm to Derive a Direct Construction for Binary Covering Arrays. In: Sidorov, G., Galicia-Haro, S. (eds) Advances in Artificial Intelligence and Soft Computing. MICAI 2015. Lecture Notes in Computer Science(), vol 9413. Springer, Cham. https://doi.org/10.1007/978-3-319-27060-9_13

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  • DOI: https://doi.org/10.1007/978-3-319-27060-9_13

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